**Advances in Boundary Value Problems for Fractional Differential Equations**

Editor

**Rodica Luca**

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*Editor* Rodica Luca Department of Mathematics "Gheorghe Asachi" Technical University of Iasi Iasi Romania

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## **Contents**


#### **Daliang Zhao and Yongyang Liu**


## **About the Editor**

#### **Rodica Luca**

Rodica Luca is a Full Professor of Mathematics at the "Gheorghe Asachi" Technical University of Iasi, Romania. She received her PhD in Mathematics from the "Alexandru Ioan Cuza" University of Iasi in 1996 and her Habilitation Certificate in Mathematics from the School of Advanced Studies of the Romanian Academy in Bucharest in 2017. She has published five monographs and more than 180 papers. Her research interests are boundary value problems for nonlinear ordinary differential equations, finite difference equations, fractional differential equations and systems, and initial-boundary value problems for nonlinear hyperbolic systems.

## *Editorial* **Advances in Boundary Value Problems for Fractional Differential Equations**

**Rodica Luca**

Department of Mathematics, Gh. Asachi Technical University, 700506 Iasi, Romania; rluca@math.tuiasi.ro

Fractional-order differential and integral operators and fractional differential equations have extensive applications in the mathematical modelling of real-world phenomena which occur in scientific and engineering disciplines such as physics, chemistry, biophysics, biology, medical sciences, financial economics, ecology, bioengineering, control theory, signal and image processing, aerodynamics, transport dynamics, thermodynamics, viscoelasticity, hydrology, statistical mechanics, electromagnetics, astrophysics, cosmology, and rheology. Fractional differential equations are also regarded as a better tool for the description of hereditary properties of various materials and processes than the corresponding integer-order differential equations. The Special Issue "Advances in Boundary Value Problems for Fractional Differential Equations" covers aspects of the recent developments in the theory and applications of fractional differential equations, inclusions, inequalities, and systems of fractional differential equations with Riemann–Liouville derivatives, Caputo derivatives, or other generalized fractional derivatives subject to various boundary conditions. In the papers published in this Special Issue, the authors study the existence, uniqueness, multiplicity, and nonexistence of classical or mild solutions, the approximation of solutions, and the approximate controllability of mild solutions for diverse models. I will present these papers in the following, grouped according to their subject.

#### **1. Equations and Systems of Equations with Sequential Fractional Derivatives**

In paper [1], the authors investigate the differential equation

$$D^{\sigma\_{\text{tr}}} \mathbf{z}(t) = \mathcal{A} \mathbf{z}(t) + \mathbf{f}(t), \ t \in (0, T]. \tag{1}$$

with the initial conditions

$$\mathbf{D}^{\sigma\_k} \mathbf{z}(0) = \mathbf{z}\_{k\prime} \ k = 0, 1, \ldots, n - 1,\tag{2}$$

where the operator A : *D*<sup>A</sup> ⊂Z→Z is linear and closed with its domain *D*<sup>A</sup> (a dense set), <sup>Z</sup> is a Banach space, f : [0, *<sup>T</sup>*] → Z is a given function, and *<sup>D</sup>σ<sup>k</sup>* , *<sup>k</sup>* <sup>=</sup> 0, 1, ... , *<sup>n</sup>* are the Dzhrbashyan–Nersesyan fractional derivatives. For the set of numbers {*αk*}*<sup>n</sup>* <sup>0</sup> , with *<sup>α</sup><sup>k</sup>* <sup>∈</sup> (0, 1], *<sup>k</sup>* <sup>=</sup> 0, 1, ... , *<sup>n</sup>*, they introduced the numbers *<sup>σ</sup><sup>k</sup>* <sup>=</sup> <sup>∑</sup>*<sup>k</sup> <sup>j</sup>*=<sup>0</sup> *α<sup>j</sup>* − 1, *k* = 0, 1, . . . , *n*, with the condition *σ<sup>n</sup>* > 0. The fractional derivatives *Dσ<sup>k</sup>* , *k* = 0, 1, ... , *n* are given by *Dσ*<sup>0</sup> z(*t*) = *Dα*0−<sup>1</sup> *<sup>t</sup>* <sup>z</sup>(*t*), *<sup>D</sup>σk*z(*t*) = *<sup>D</sup>αk*−<sup>1</sup> *<sup>t</sup> <sup>D</sup>αk*−<sup>1</sup> *<sup>t</sup> <sup>D</sup>αk*−<sup>2</sup> *<sup>t</sup>* ... *<sup>D</sup>α*<sup>0</sup> *<sup>t</sup>* z(*t*), for *k* = 1, 2, ... , *n*, where *Dβ <sup>t</sup>* is the Riemann–Liouville integral for *β* ≤ 0 and the Riemann–Liouville derivative for *β* > 0. The Dzhrbashyan–Nersesyan fractional derivative *Dσ<sup>n</sup>* is a generalization of the Riemann–Liouville and Caputo fractional derivatives. The authors prove firstly the existence and uniqueness of the *k*-resolving families of operators (for *k* = 0, ... , *n* − 1) for the homogeneous equation *<sup>D</sup>σ<sup>n</sup> <sup>z</sup>*(*t*) = <sup>A</sup>*z*(*t*), and then they give a criterion for the existence and uniqueness of analytic *k*-resolving families, namely A belongs to a class of operators denoted by A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0). Different properties of the resolving families are also studied, and a perturbation theorem for operators from A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0) is presented. Then, the authors prove the existence and uniqueness of a solution for problem (1),(2), where f is continuous in the graph norm of A or it is a Hölderian function. As an application,

**Citation:** Luca, R. Advances in Boundary Value Problems for Fractional Differential Equations. *Fractal Fract.* **2023**, *7*, 406. https://doi.org/10.3390/ fractalfract7050406

Received: 14 May 2023 Accepted: 15 May 2023 Published: 17 May 2023

**Copyright:** © 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

they show the existence of a unique solution for an initial boundary value problem to a fractional linearized model of viscoelastic Oldroyd fluid dynamics.

Paper [2] deals with a nonlinear coupled system of sequential fractional differential equations

$$\begin{cases} \left(^c \mathbf{D}^{q+1} + ^c \mathbf{D}^q \right) \mathbf{x}(t) = \mathbf{f}(t, \mathbf{x}(t), \mathbf{y}(t)), & t \in [0, 1], \\ \left(^c \mathbf{D}^{p+1} + ^c \mathbf{D}^p \right) \mathbf{y}(t) = \mathbf{g}(t, \mathbf{x}(t), \mathbf{y}(t)), & t \in [0, 1], \end{cases} \tag{3}$$

supplemented with the coupled multipoint and Riemann–Stieltjes integral boundary conditions

$$\begin{cases} \mathbf{x}(0) = \mathbf{0}, \; \mathbf{x}'(0) = \mathbf{0}, \; \mathbf{x}'(1) = \mathbf{0},\\ \mathbf{x}(1) = k \int\_0^\rho \mathbf{y}(s) \, d\mathcal{A}(s) + \sum\_{i=1}^{n-2} \alpha\_i \mathbf{y}(\sigma\_i) + k\_1 \int\_\nu^1 \mathbf{y}(s) \, d\mathcal{A}(s),\\ \mathbf{y}(0) = \mathbf{0}, \; \mathbf{y}'(0) = \mathbf{0}, \; \mathbf{y}'(1) = \mathbf{0},\\ \mathbf{y}(1) = h \int\_0^\rho \mathbf{x}(s) \, d\mathcal{A}(s) + \sum\_{i=1}^{n-2} \beta\_i \mathbf{x}(\sigma\_i) + h\_1 \int\_\nu^1 \mathbf{x}(s) \, d\mathcal{A}(s), \end{cases} \tag{4}$$

where *<sup>p</sup>*, *<sup>q</sup>* <sup>∈</sup> (2, 3], *<sup>c</sup> <sup>D</sup><sup>κ</sup>* denotes the Caputo fractional derivative of order *<sup>κ</sup>* ∈ {*q*, *<sup>p</sup>*}, <sup>0</sup> <sup>&</sup>lt; *<sup>ρ</sup>* <sup>&</sup>lt; *<sup>σ</sup><sup>i</sup>* <sup>&</sup>lt; *<sup>ν</sup>* <sup>&</sup>lt; 1, f, g : [0, 1] <sup>×</sup> <sup>R</sup> <sup>×</sup> <sup>R</sup> <sup>→</sup> <sup>R</sup> are continuous functions, *<sup>k</sup>*, *<sup>h</sup>*, *<sup>k</sup>*1, *<sup>h</sup>*1, *<sup>α</sup>i*, *<sup>β</sup><sup>i</sup>* <sup>∈</sup> <sup>R</sup>, for *i* = 1, 2, ... , *n* − 2, and A is a function of bounded variation. The word sequential is used in the sense that the operator *<sup>c</sup> Dq*+<sup>1</sup> +*cDq* can be written as the composition of operators *c D<sup>q</sup>* and *D* + *I*, where *D* is the usual differential operator and *I* is the identity operator. Under some assumptions of the data of the problem, the authors prove the existence and uniqueness of solutions for problem (3),(4) by applying the Leray–Schauder alternative and the Banach contraction mapping principle.

#### **2. Resonance Problems for Caputo Fractional Differential Equations**

Paper [3] is concerned with the nonlinear boundary value problem for a fractional differential equation of variable order at resonance

$$\begin{cases} \ ^c \mathbf{D}\_{0+}^{\mathbf{u}(t)} \mathbf{x}(t) = \mathbf{g}(t, \mathbf{x}(t)), \ t \in [0, T],\\ \mathbf{x}(0) = \mathbf{x}(T), \end{cases} \tag{5}$$

where *<sup>c</sup> D*u(*t*) <sup>0</sup><sup>+</sup> is the Caputo derivative of variable order u(*t*) with u : [0, *T*] → (0, 1] and g : [0, *<sup>T</sup>*] <sup>×</sup> <sup>R</sup> <sup>→</sup> <sup>R</sup> is a continuous function. This problem is at resonance, that is, the corresponding linear homogeneous boundary value problem has non-trivial solutions. The authors transform firstly problem (5) to an equivalent standard boundary value problem at resonance with a fractional derivative of constant order by using some generalized intervals and piece-wise constant functions. Then, by applying Mawhin's continuation theorem, they demonstrate the existence of at least one solution to (5).

In paper [4], the authors study the fractional differential equation in space R*<sup>n</sup>*

$$\mathbf{f}^{\complement} \mathbf{D}\_{0+}^{\mu} \mathbf{u}(t) = \mathbf{f}(t, \mathbf{u}(t), ^{\complement} \mathbf{D}\_{0+}^{\mu - 1} \mathbf{u}), \ t \in (0, 1), \tag{6}$$

subject to the boundary conditions

$$\mathbf{u}(0) = \mathcal{B}\mathbf{u}(\xi), \ \mathbf{u}(1) = \mathcal{C}\mathbf{u}(\eta), \tag{7}$$

where *<sup>c</sup> Dk* <sup>0</sup><sup>+</sup> denotes the Caputo fractional derivative of order *k* ∈ {*α*, *α* − 1}, *ξ*, *η* ∈ (0, 1), *<sup>α</sup>* <sup>∈</sup> (1, 2], f : [0, 1] <sup>×</sup> <sup>R</sup>2*<sup>n</sup>* <sup>→</sup> <sup>R</sup>*<sup>n</sup>* satisfies Carathéodory conditions, and <sup>B</sup>, <sup>C</sup> are *<sup>n</sup>*-order nonzero square matrices. They prove the existence of solutions of problem (6),(7) by using Mawhin coincidence degree theory.

#### **3. Approximations of Solutions for Caputo Fractional Differential Equations**

Paper [5] is devoted to the Caputo fractional differential equation with variable coefficients

$$D\_x^\lambda \mathbf{u}(\mathbf{x}) + c\_1(\mathbf{x})\mathbf{u}'(\mathbf{x}) + c\_0(\mathbf{x})\mathbf{u}(\mathbf{x}) = \mathbf{g}(\mathbf{x}), \ 0 < \mathbf{x} < 1,\tag{8}$$

with the boundary conditions

$$p\_0 \mathbf{u}(0) - q\_0 \mathbf{u}'(0) = b\_{0\prime} \ p\_1 \mathbf{u}(1) + q\_1 \mathbf{u}'(1) = b\_{1\prime} \tag{9}$$

where *<sup>λ</sup>* <sup>∈</sup> (1, 2], *<sup>D</sup><sup>λ</sup> <sup>x</sup>* is the Caputo fractional derivative of order *λ*, *c*1, *c*0, and g are continuous functions, *p*0, *p*1, *q*0, *q*<sup>1</sup> ≥ 0, and *p*<sup>0</sup> *p*<sup>1</sup> + *p*0*q*<sup>1</sup> + *q*<sup>0</sup> *p*<sup>1</sup> = 0. By using the shifted Chebyshev polynomials of the first kind and the collocation method, the authors present approximate solutions to problem (8),(9).

#### **4. Systems of Fractional Differential Equations with** *p***-Laplacian Operators**

In paper [6], the authors investigate the system of fractional differential equations with *r*1-Laplacian and *r*2-Laplacian operators

$$\begin{cases} D\_{0+}^{\gamma\_1} \left( \boldsymbol{\varrho} \boldsymbol{r}\_1 (D\_{0+}^{\delta\_1} \mathbf{u}(t)) \right) = \mathbf{f} \left( t, \mathbf{u}(t), \mathbf{v}(t), I\_{0+}^{\sigma\_1} \mathbf{u}(t), I\_{0+}^{\sigma\_2} \mathbf{v}(t) \right), & t \in (0, 1), \\\ D\_{0+}^{\gamma\_2} \left( \boldsymbol{\varrho} \boldsymbol{r}\_2 (D\_{0+}^{\delta\_2} \mathbf{v}(t)) \right) = \mathbf{g} \left( t, \mathbf{u}(t), \mathbf{v}(t), I\_{0+}^{\xi\_1} \mathbf{u}(t), I\_{0+}^{\xi\_2} \mathbf{v}(t) \right), & t \in (0, 1), \end{cases} \tag{10}$$

supplemented with the uncoupled nonlocal boundary conditions

$$\begin{cases} \begin{aligned} \mathbf{u}^{(i)}(0) = 0, \; i = 0, \dots, p-2, \; D\_{0+}^{\delta\_1} \mathbf{u}(0) = 0, \\ \displaystyle\rho\_{\mathbf{r}\_1}(D\_{0+}^{\delta\_1} \mathbf{u}(1)) = \int\_0^1 \rho\_{\mathbf{r}\_1}(D\_{0+}^{\delta\_1} \mathbf{u}(\tau)) \, d\mathcal{H}\_0(\tau), \; D\_{0+}^{\delta\_0} \mathbf{u}(1) = \sum\_{k=1}^n \int\_0^1 D\_{0+}^{\delta\_k} \mathbf{u}(\tau) \, d\mathcal{H}\_k(\tau), \\ \mathbf{v}^{(j)}(0) = 0, \; j = 0, \dots, q-2, \; D\_{0+}^{\delta\_2} \mathbf{v}(0) = 0, \\ \displaystyle\rho\_{\mathbf{r}\_2}(D\_{0+}^{\delta\_2} \mathbf{v}(1)) = \int\_0^1 \rho\_{\mathbf{r}\_2}(D\_{0+}^{\delta\_2} \mathbf{v}(\tau)) \, d\mathcal{K}\_0(\tau), \; D\_{0+}^{\delta\_0} \mathbf{v}(1) = \sum\_{k=1}^m \int\_0^1 D\_{0+}^{\delta\_k} \mathbf{v}(\tau) \, d\mathcal{K}\_k(\tau), \end{aligned} \tag{11}$$

where *<sup>γ</sup>*1, *<sup>γ</sup>*<sup>2</sup> <sup>∈</sup> (1, 2], *<sup>p</sup>*, *<sup>q</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>p</sup>*, *<sup>q</sup>* <sup>≥</sup> 3, *<sup>δ</sup>*<sup>1</sup> <sup>∈</sup> (*<sup>p</sup>* <sup>−</sup> 1, *<sup>p</sup>*], *<sup>δ</sup>*<sup>2</sup> <sup>∈</sup> (*<sup>q</sup>* <sup>−</sup> 1, *<sup>q</sup>*], *<sup>n</sup>*, *<sup>m</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>σ</sup>*1, *<sup>ς</sup>*1, *<sup>σ</sup>*2, *<sup>ς</sup>*<sup>2</sup> <sup>&</sup>gt; 0, *<sup>α</sup><sup>i</sup>* <sup>∈</sup> <sup>R</sup>, *<sup>i</sup>* <sup>=</sup> 0, ... , *<sup>n</sup>*, 0 <sup>≤</sup> *<sup>α</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>α</sup>*<sup>2</sup> <sup>&</sup>lt; ... <sup>&</sup>lt; *<sup>α</sup><sup>n</sup>* <sup>≤</sup> *<sup>α</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>δ</sup>*<sup>1</sup> <sup>−</sup> 1, *<sup>α</sup>*<sup>0</sup> <sup>≥</sup> 1, *<sup>β</sup><sup>j</sup>* <sup>∈</sup> <sup>R</sup>, *<sup>j</sup>* <sup>=</sup> 0, ... , *<sup>m</sup>*, 0 <sup>≤</sup> *<sup>β</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>β</sup>*<sup>2</sup> <sup>&</sup>lt; ... <sup>&</sup>lt; *<sup>β</sup><sup>m</sup>* <sup>≤</sup> *<sup>β</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>δ</sup>*<sup>2</sup> <sup>−</sup> 1, *<sup>β</sup>*<sup>0</sup> <sup>≥</sup> 1, *ϕrk* (*τ*) = |*τ*| *rk*−2*τ*, *rk* <sup>&</sup>gt; 1, *<sup>k</sup>* <sup>=</sup> 1, 2, the functions f, g : (0, 1) <sup>×</sup> <sup>R</sup><sup>4</sup> <sup>+</sup> <sup>→</sup> <sup>R</sup><sup>+</sup> are continuous, singular at *<sup>t</sup>* = 0 and/or *<sup>t</sup>* = 1, (R<sup>+</sup> = [0, <sup>∞</sup>)), *<sup>I</sup>*<sup>κ</sup> <sup>0</sup><sup>+</sup> is the Riemann–Liouville fractional integral of order <sup>κ</sup> (for <sup>κ</sup> <sup>=</sup> *<sup>σ</sup>*1, *<sup>ς</sup>*1, *<sup>σ</sup>*2, *<sup>ς</sup>*2), *<sup>D</sup>*<sup>κ</sup> <sup>0</sup><sup>+</sup> is the Riemann–Liouville fractional derivative of order κ (for κ = *γ*1, *γ*2, *δ*1, *δ*2, *α*0, ... , *αn*, *β*0, ... , *βm*), and the integrals from the boundary conditions (11) are Riemann–Stieltjes integrals with <sup>H</sup>*<sup>i</sup>* : [0, 1] <sup>→</sup> <sup>R</sup>, *<sup>i</sup>* <sup>=</sup> 0, ... , *<sup>n</sup>* and <sup>K</sup>*<sup>j</sup>* : [0, 1] <sup>→</sup> <sup>R</sup>, *<sup>j</sup>* <sup>=</sup> 0, ... , *<sup>m</sup>* functions of bounded variation. By using the Guo– Krasnoselskii fixed point theorem of cone expansion and norm-type compression, they prove the existence and multiplicity of positive solutions for problem (10),(11).

Paper [7] is focused on the system of fractional differential equations (10) subject to the nonlocal coupled boundary conditions

$$\begin{cases} \begin{aligned} \mathbf{u}^{(i)}(0) = 0, \; i = 0, \dots, p-2, \; D^{\delta\_1}\_{0+} \mathbf{u}(0) = 0, \\ \displaystyle \boldsymbol{\varrho}\_{\gamma\_1}(D^{\delta\_1}\_{0+} \mathbf{u}(1)) = \int\_0^1 \boldsymbol{\varrho}\_{\gamma\_1}(D^{\delta\_1}\_{0+} \mathbf{u}(\tau)) \, d\mathcal{H}\_0(\tau), \; D^{\delta\_0}\_{0+} \mathbf{u}(1) = \sum\_{i=1}^n \int\_0^1 D^{\delta\_i}\_{0+} \mathbf{v}(\tau) \, d\mathcal{H}\_i(\tau), \\ \mathbf{v}^{(j)}(0) = 0, \; j = 0, \dots, q-2, \; D^{\delta\_2}\_{0+} \mathbf{v}(0) = 0, \\ \displaystyle \boldsymbol{\varrho}\_{\gamma\_2}(D^{\delta\_2}\_{0+} \mathbf{v}(1)) = \int\_0^1 \boldsymbol{\varrho}\_{\gamma\_2}(D^{\delta\_2}\_{0+} \mathbf{v}(\tau)) \, d\mathcal{K}\_0(\tau), \; D^{\delta\_0}\_{0+} \mathbf{v}(1) = \sum\_{j=1}^m \int\_0^1 D^{\delta\_j}\_{0+} \mathbf{u}(\tau) \, d\mathcal{K}\_j(\tau), \end{aligned} \tag{12}$$

where *<sup>α</sup><sup>i</sup>* <sup>∈</sup> <sup>R</sup>, *<sup>i</sup>* <sup>=</sup> 0, ... , *<sup>n</sup>*, 0 <sup>≤</sup> *<sup>α</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>α</sup>*<sup>2</sup> <sup>&</sup>lt; ... <sup>&</sup>lt; *<sup>α</sup><sup>n</sup>* <sup>≤</sup> *<sup>β</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>δ</sup>*<sup>2</sup> <sup>−</sup> 1, *<sup>β</sup>*<sup>0</sup> <sup>≥</sup> 1, *<sup>β</sup><sup>j</sup>* <sup>∈</sup> <sup>R</sup>, *j* = 0, ... , *m*, 0 ≤ *β*<sup>1</sup> < *β*<sup>2</sup> < ... < *β<sup>m</sup>* ≤ *α*<sup>0</sup> < *δ*<sup>1</sup> − 1, *α*<sup>0</sup> ≥ 1. The authors present existence

and multiplicity results for the positive solutions of problem (10),(12) by applying the Guo–Krasnoselskii fixed point theorem.

Paper [8] deals with a system of fractional differential equations with 1-Laplacian and 2-Laplacian operators

$$\begin{cases} D\_{0+}^{\gamma\_1} (\varrho\_{\ell1} (D\_{0+}^{\delta\_1} u(t))) + a(t) f(v(t)) = 0, & t \in (0, 1), \\\ D\_{0+}^{\gamma\_2} (\varrho\_{\ell2} (D\_{0+}^{\delta\_2} v(t))) + b(t) g(u(t)) = 0, & t \in (0, 1), \end{cases} \tag{13}$$

with the coupled nonlocal boundary conditions

$$\begin{cases} \begin{aligned} u^{(j)}(0) = 0, \; i = 0, \dots, p-2; \; D\_{0+}^{\delta\_1} u(0) = 0, \; D\_{0+}^{\kappa\_0} u(1) = \sum\_{i=1}^{n} \int\_0^1 D\_{0+}^{\kappa\_i} v(\tau) \, d\mathcal{H}\_i(\tau) + c\_0, \\\ v^{(j)}(0) = 0, \; j = 0, \dots, q-2; \; D\_{0+}^{\delta\_2} v(0) = 0, \; D\_{0+}^{\delta\_0} v(1) = \sum\_{j=1}^{m} \int\_0^1 D\_{0+}^{\delta\_j} u(\tau) \, d\mathcal{K}\_j(\tau) + d\_0. \end{aligned} \end{cases} \tag{14}$$

where *<sup>γ</sup>*1, *<sup>γ</sup>*<sup>2</sup> <sup>∈</sup> (0, 1], *<sup>p</sup>*, *<sup>q</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>p</sup>*, *<sup>q</sup>* <sup>≥</sup> 3, *<sup>δ</sup>*<sup>1</sup> <sup>∈</sup> (*<sup>p</sup>* <sup>−</sup> 1, *<sup>p</sup>*], *<sup>δ</sup>*<sup>2</sup> <sup>∈</sup> (*<sup>q</sup>* <sup>−</sup> 1, *<sup>q</sup>*], *<sup>n</sup>*, *<sup>m</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>α</sup><sup>i</sup>* <sup>∈</sup> <sup>R</sup> for all *<sup>i</sup>* <sup>=</sup> 0, 1, ... , *<sup>n</sup>*, 0 <sup>≤</sup> *<sup>α</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>α</sup>*<sup>2</sup> <sup>&</sup>lt; ... <sup>&</sup>lt; *<sup>α</sup><sup>n</sup>* <sup>≤</sup> *<sup>β</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>δ</sup>*<sup>2</sup> <sup>−</sup> 1, *<sup>β</sup>*<sup>0</sup> <sup>≥</sup> 1, *<sup>β</sup><sup>j</sup>* <sup>∈</sup> <sup>R</sup> for all *j* = 0, 1, ... , *m*, 0 ≤ *β*<sup>1</sup> < *β*<sup>2</sup> < ... < *β<sup>m</sup>* ≤ *α*<sup>0</sup> < *δ*<sup>1</sup> − 1, *α*<sup>0</sup> ≥ 1, the functions *<sup>f</sup>* , *<sup>g</sup>* : <sup>R</sup><sup>+</sup> <sup>→</sup> <sup>R</sup><sup>+</sup> and *<sup>a</sup>*, *<sup>b</sup>* : [0, 1] <sup>→</sup> <sup>R</sup><sup>+</sup> are continuous, *<sup>c</sup>*<sup>0</sup> and *<sup>d</sup>*<sup>0</sup> are positive parameters, 1, <sup>2</sup> > 1, *ϕ<sup>i</sup>* (*ζ*) = |*ζ*| *i*−2*ζ*, *<sup>i</sup>* <sup>=</sup> 1, 2, the functions <sup>H</sup>*j*, *<sup>j</sup>* <sup>=</sup> 1, ... , *<sup>n</sup>* and <sup>K</sup>*i*, *<sup>i</sup>* <sup>=</sup> 1, ... , *<sup>m</sup>* have bounded variation, and *D<sup>κ</sup>* <sup>0</sup><sup>+</sup> denotes the Riemann–Liouville derivative of order *κ* (for *κ* = *γ*1, *γ*2, *δ*1, *δ*2, *α<sup>i</sup>* for *i* = 0, 1, ... , *n*, *β<sup>j</sup>* for *j* = 0, 1, ... , *m*). The authors give sufficient conditions for the functions *f* and *g*, and intervals for the parameters *c*<sup>0</sup> and *d*<sup>0</sup> such that problem (13),(14) have at least one positive solution or they have no positive solutions. They apply the Schauder fixed point theorem in the proof of the main existence result.

In paper [9], the authors study a system of nonlinear Fredholm fractional integrodifferential equations with *p*-Laplacian operator

$$\begin{cases} \ \_ \mathrm{l} \, \_ \mathrm{D}\_{\mathrm{T}}^{\gamma j} (\mathbf{k}\_{\mathrm{j}}(t) \boldsymbol{\phi}\_{\mathrm{p}} (\mathbf{\zeta} \mathbf{D}\_{\mathrm{t}}^{\gamma j} \mathbf{z}\_{j}(t))) + \mathbf{l}\_{\mathrm{j}}(t) \boldsymbol{\phi}\_{\mathrm{p}} (\mathbf{z}\_{j}(t)) \\ \quad = \mathrm{l} \, \_ \mathrm{z}\_{\mathrm{j}} (t, \mathbf{z}\_{1}(t), \dots, \mathbf{z}\_{\mathrm{m}}(t)) + \int\_{0}^{T} \mathbf{g}\_{\mathrm{j}} (t, s) \boldsymbol{\phi}\_{\mathrm{p}} (\mathbf{z}\_{j}(s)) \, ds, \; t \in [0, T], \; j = 1, 2, \dots, m, \\\ \mathbf{z}\_{\mathrm{j}}(t) = \int\_{0}^{T} \mathbf{g}\_{\mathrm{j}} (t, s) \boldsymbol{\phi}\_{\mathrm{p}} (\mathbf{z}\_{\mathrm{j}}(s)) \, ds, \; t \in [0, T], \; j = 1, 2, \dots, m. \end{cases} \tag{15}$$

supplemented with the Sturm–Liouville boundary conditions

$$\begin{cases} c\_{j}\mathbf{k}\_{\dot{\jmath}}(0)\boldsymbol{\phi}\_{p}(\mathbf{z}\_{\dot{\jmath}}(0)) - c\_{j}^{\prime}D\_{T}^{\gamma\_{j}-1}(\mathbf{k}\_{\dot{\jmath}}(0)\boldsymbol{\phi}\_{p}(^{\prime}D\_{t}^{\gamma\_{j}}\mathbf{z}\_{\dot{\jmath}}(0))) = 0, & j = 1,2,\ldots,m, \\\ d\_{\dot{\jmath}}\mathbf{k}\_{\dot{\jmath}}(T)\boldsymbol{\phi}\_{p}(\mathbf{z}\_{\dot{\jmath}}(T)) + d\_{\dot{\jmath}^{\prime}}^{\prime}D\_{T}^{\gamma\_{j}-1}(\mathbf{k}\_{\dot{\jmath}}(T)\boldsymbol{\phi}\_{p}(^{\prime}D\_{t}^{\gamma\_{j}}\mathbf{z}\_{\dot{\jmath}}(T))) = 0, & j = 1,2,\ldots,m, \end{cases} \tag{16}$$

where *<sup>λ</sup>* is a positive parameter, k*i*, l*<sup>i</sup>* <sup>∈</sup> *<sup>L</sup>*∞[0, *<sup>T</sup>*] with ess inf[0,*T*]k*i*(*t*) <sup>&</sup>gt; 0 and ess inf[0,*T*] l*i*(*t*) ≥ 0, *ci*, *di*, *c i* , *d i* , *i* = 1, 2, ... , *m*, are positive constants, *p* ∈ (1, ∞), *φp*(*s*) = |*s*| *<sup>p</sup>*−2*s*, (*<sup>s</sup>* <sup>=</sup> <sup>0</sup>), *<sup>φ</sup>p*(0) = 0, the functions f : [0, *<sup>T</sup>*] <sup>×</sup> <sup>R</sup>*<sup>m</sup>* <sup>→</sup> <sup>R</sup> and g*<sup>i</sup>* : [0, *<sup>T</sup>*] <sup>×</sup> [0, *<sup>T</sup>*] <sup>→</sup> <sup>R</sup>, *i* = 1, ... , *m* satisfy some conditions, and *<sup>c</sup>* 0*Dγ<sup>j</sup> <sup>t</sup>* and *tDγ<sup>j</sup> <sup>T</sup>* denote the left Caputo fractional derivative and the right Riemann–Liouville fractional derivative of order *γj*, respectively. By using the critical point theory, they prove the existence of infinitely many solutions of problem (15),(16).

#### **5. Approximate Controllability for Fractional Differential Equations in Banach Spaces**

Paper [10] is concerned with the fractional evolution equation of Sobolev type in the Hilbert space *X*, with a control and a nonlocal condition

$$\begin{cases} \ ^L D\_t^\mathbf{u}(\mathcal{E}\mathbf{x}(t)) = \mathcal{A}\mathbf{x}(t) + \mathbf{f}(t, \mathbf{x}(t)) + \mathcal{B}\mathbf{u}(t), \ t \in (0, b],\\\ \ ^{1-a}\_t(\mathcal{E}\mathbf{x}(t))|\_{t=0} + \mathbf{g}(\mathbf{x}) = \mathbf{x}\_0. \end{cases} \tag{17}$$

where *α* ∈ (0, 1), A : *D*(A) ⊂ *X* → *X* and E : *D*(E) ⊂ *X* → *X* are linear operators, B : *U* → *X* is a linear bounded operator, *U* is another Hilbert space, the control function <sup>u</sup> <sup>∈</sup> *<sup>L</sup>p*([0, *<sup>b</sup>*], *<sup>U</sup>*) for *<sup>p</sup><sup>α</sup>* <sup>&</sup>gt; 1, *<sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>X</sup>*, the functions f and g satisfy some assumptions, *<sup>I</sup>* 1−*α t* is the Riemann–Liouville fractional integral operator of order 1 <sup>−</sup> *<sup>α</sup>*, and *<sup>L</sup> D<sup>α</sup> <sup>t</sup>* denotes the Riemann–Liouville fractional derivative of order *α*. By using the Schauder fixed point theorem and operator semigroup theory, the authors prove firstly the existence of mild solutions for problem (17) without the compactness of the operator semigroup. Then, they show that if the corresponding linear problem is approximately controllable on [0, *b*], then problem (17) is also approximately controllable on [0, *b*]. An example with an initial boundary value problem for a partial differential equation with Riemann–Liouville fractional derivatives is finally presented.

Paper [11] is devoted to the fractional differential evolution equation in the Banach space *X* with a finite delay and a control

$$^cD^\beta \mathbf{x}(t) = \mathcal{A}\mathbf{x}(t) + \mathbf{f}(t, \mathbf{x}\_l) + \mathcal{B}\mathbf{u}(t), \ t \in [0, a],\tag{18}$$

subject to the initial date

$$\mathbf{x}(t) = \phi(t), \ t \in [-b, 0], \tag{19}$$

or to the nonlocal condition with a parameter

$$
\lambda \mathbf{x}(t) + \lambda \mathbf{g}\_t(\mathbf{x}) = \phi(t), \ t \in [-b, 0], \tag{20}
$$

where A : D ⊂ *X* → *X* is a closed linear unbounded operator on *X*, where its domain D is a dense set; u is the control function; <sup>B</sup> : *<sup>L</sup>*2([0, *<sup>a</sup>*]; *<sup>U</sup>*) <sup>→</sup> *<sup>L</sup>*2([0, *<sup>a</sup>*]; <sup>D</sup>) is a linear bounded operator, where *<sup>U</sup>* is another Banach space; *<sup>φ</sup>* <sup>∈</sup> *<sup>L</sup>*1([−*b*, 0]; *<sup>X</sup>*), x*<sup>t</sup>* denotes the history of the state function defined by x*t*(*θ*) = {x(*t* + *θ*), if *t* + *θ* ≥ 0; *φ*(*t* + *θ*), if *t* + *θ* ≤ 0} for *θ* ∈ [−*b*, 0]; *λ* is a parameter; g*<sup>t</sup>* : *C*([−*b*, *a*]; *X*) → *X* is a given function satisfying some assumptions; and *<sup>c</sup> <sup>D</sup><sup>β</sup>* is the Caputo fractional derivative of order *<sup>β</sup>*, with *<sup>β</sup>* <sup>∈</sup> (1/2, 1]. Under the assumption that A is the infinitesimal generator of a differentiable resolvent operator, the authors prove the existence and uniqueness of mild solutions for problems (18),(19) and (18),(20) by utilizing the Banach contraction mapping principle. Then, based on the iterative method, they give sufficient conditions for the approximate controllability of (18),(19) and (18),(20). As an application, an example of a Caputo fractional partial differential equation with delay in the space *X* = *L*2([0, *π*]) is finally addressed.

#### **6. Fractional Differential Inclusions and Inequalities**

In paper [12], the authors investigate the neutral impulsive semi-linear fractional differential inclusion with delay and initial date

$$\begin{cases} \ ^cD\_{0,t}^a[\mathbf{x}(t) - \mathbf{h}(t, \varkappa(t)\mathbf{x})] \in \mathcal{A}\mathbf{x}(t) + \mathcal{F}(t, \varkappa(t)\mathbf{x}), \text{ a.e. } t \in [0, b] \\\ I\_i \mathbf{x}(t\_i^-) = \mathbf{x}(t\_i^-) - \mathbf{x}(t\_i^+), \ i = 1, \dots, m\_\prime \\\ \mathbf{x}(t) = \boldsymbol{\psi}(t), \ t \in [-r, 0], \end{cases} \tag{21}$$

where *α* ∈ (0, 1), 0 = *t*<sup>0</sup> < *t*<sup>1</sup> < ... < *tm* < *tm*+<sup>1</sup> = *b*, *r* > 0, the operator A is the infinitesimal generator of the non-compact semigroup T = {*Y*(*t*), *t* ≥ 0} on the Banach space *<sup>E</sup>*, and <sup>F</sup> : [0, *<sup>b</sup>*] <sup>×</sup> <sup>Θ</sup> <sup>→</sup> <sup>2</sup>*<sup>E</sup>* \ {*φ*} is a multifunction. Here, h : [0, *<sup>b</sup>*] <sup>×</sup> <sup>Θ</sup> <sup>→</sup> *<sup>E</sup>*, *Ii* : *<sup>E</sup>* <sup>→</sup> *<sup>E</sup>*, *<sup>i</sup>* <sup>=</sup> 1, ... , *<sup>m</sup>*, *<sup>ψ</sup>* <sup>∈</sup> <sup>Θ</sup>, and for every *<sup>t</sup>* <sup>∈</sup> [0, *<sup>b</sup>*], the function <sup>κ</sup>(*t*) : H → <sup>Θ</sup> is defined by (κ(*t*)x)(*θ*) = <sup>x</sup>(*<sup>t</sup>* <sup>+</sup> *<sup>θ</sup>*) for *<sup>θ</sup>* <sup>∈</sup> [−*r*, 0]. *<sup>c</sup> D<sup>α</sup>* 0,*<sup>t</sup>* denotes the Caputo fractional derivative of order *α* and the spaces Θ and H are defined in the paper. They show that the set of mild solutions to problem (21) is nonempty, compact, and an *Rδ*-set in a complete metric space *H*.

Paper [13] is focused on the Hilfer fractional neutral integro-differential inclusion with initial date

$$\begin{cases} \mathbf{D}\_{0+}^{k\varepsilon}[\mathbf{y}(t) - \mathcal{N}(t, \mathbf{y}(t))] \in \mathcal{A}\mathbf{y}(t) + \mathcal{G}\left(t, \mathbf{y}(t), \int\_{0}^{t} \mathbf{e}(t, \mathbf{s}, \mathbf{y}(s)) \, ds\right), & t \in (0, d],\\\ I\_{0+}^{(1-k)(1-\varepsilon)}\mathbf{y}(0) = y\_{0\prime} \end{cases} \tag{22}$$

where *Dk*, <sup>0</sup><sup>+</sup> denotes the Hilfer fractional derivative of order *k* and type , with *k* ∈ (0, 1) and ∈ [0, 1], *I* (1−*k*)(1−) <sup>0</sup><sup>+</sup> is the Riemann–Liouville fractional integral of order (1 − *k*)(1 − ), and A is an almost sectorial operator of the analytic semigroup {T (*t*), *t* ≥ 0} on the Banach space *<sup>Y</sup>*. Here, <sup>G</sup> : [0, *<sup>d</sup>*] <sup>×</sup> *<sup>Y</sup>* <sup>×</sup> *<sup>Y</sup>* <sup>→</sup> <sup>2</sup>*<sup>Y</sup>* \ {*φ*} is a nonempty, bounded, closed, convex multivalued map and N : [0, *d*] × *Y* → *Y* and e : [0, *d*] × [0, *d*] × *Y* → *Y* are appropriate functions. By using the Martelli fixed point theorem, the authors prove the existence of mild solutions to problem (22).

In paper [14], the authors study the damped wave inequality

$$\frac{\partial^2 \mathbf{u}}{\partial t^2} - \frac{\partial^2 \mathbf{u}}{\partial x^2} + \frac{\partial \mathbf{u}}{\partial t} \ge \mathbf{x}^\sigma |\mathbf{u}|^p, \ t > 0, \ \mathbf{x} \in (0, L), \tag{23}$$

subject to initial boundary conditions

$$\begin{cases} \left(\mathbf{u}(t,0), \mathbf{u}(t,L)\right) = \left(\mathbf{f}(t), \mathbf{g}(t)\right), \ t > 0, \\\left(\mathbf{u}(0,\mathbf{x}), \frac{\partial \mathbf{u}}{\partial t}(0,\mathbf{x})\right) = \left(u\_0(\mathbf{x}), u\_1(\mathbf{x})\right), \ \mathbf{x} \in (0,L), \end{cases} \tag{24}$$

where *<sup>L</sup>* <sup>&</sup>gt; 0, *<sup>σ</sup>* <sup>∈</sup> <sup>R</sup>, *<sup>p</sup>* <sup>&</sup>gt; 1, f <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *loc*([0, ∞)), g(*t*) = *Cgt <sup>γ</sup>* with *Cg* <sup>≥</sup> 0 and *<sup>γ</sup>* <sup>&</sup>gt; <sup>−</sup>1, and *<sup>u</sup>*0, *<sup>u</sup>*<sup>1</sup> <sup>∈</sup> *<sup>L</sup>*1([0, *<sup>L</sup>*]). They also investigate the time-fractional damped wave inequality

$$
\frac{\partial^{\alpha}\mathbf{u}}{\partial t^{\alpha}} - \frac{\partial^{2}\mathbf{u}}{\partial \mathbf{x}^{2}} + \frac{\partial^{\beta}\mathbf{u}}{\partial t^{\beta}} \ge \mathbf{x}^{\sigma}|\mathbf{u}|^{p}, \ t > 0, \ \mathbf{x} \in (0, L), \tag{25}
$$

supplemented with the initial boundary conditions in (24), where *α* ∈ (1, 2), *β* ∈ (0, 1), and *<sup>∂</sup><sup>κ</sup> <sup>∂</sup>t<sup>κ</sup>* is the time Caputo fractional derivative of order *κ*, for *κ* ∈ {*α*, *β*}. By using the test function method, the authors give sufficient conditions depending on the above data under which problems (23),(24) and (23),(25) admit no global weak solutions.

#### **7. Fractional** *q***-Difference Equations and Systems**

Paper [15] deals with the fractional *q*-difference equation in a Banach space *E*, with nonlinear integral conditions

$$\begin{cases} \left(\mathbf{^C D\_q^\mathbf{y}}\mathbf{y}\right)(t) = \mathbf{f}(t, \mathbf{y}(t)), \text{ a.e. } t \in [0, T],\\ \mathbf{y}(0) - \mathbf{y}'(0) = \int\_0^T \mathbf{g}(s, \mathbf{y}(s)) \, ds, \\\ \mathbf{y}(T) + \mathbf{y}'(T) = \int\_0^T \mathbf{h}(s, \mathbf{y}(s)) \, ds, \end{cases} \tag{26}$$

where *<sup>T</sup>* <sup>&</sup>gt; 0, *<sup>q</sup>* <sup>∈</sup> (0, 1), *<sup>c</sup> D<sup>α</sup> <sup>q</sup>* denotes the Caputo fractional *q*-derivative of order *α*, with *α* ∈ (1, 2], and f, g, h : [0, *T*] × *E* → *E* are given functions satisfying some assumptions. By using the measures of noncompactness technique and the Mönch fixed point theorem, the authors prove the existence of solutions to problem (26).

Paper [16] is concerned with the system of nonlinear fractional *q*-difference equations

$$\begin{cases} \left(D\_q^{\mathbf{u}} \mathbf{u}\right)(t) + \mathcal{P}(t, \mathbf{u}(t), \mathbf{v}(t), I\_q^{\omega\_1} \mathbf{u}(t), I\_q^{\delta\_1} \mathbf{v}(t)) = \mathbf{0}, & t \in (0, 1), \\\left(D\_q^{\beta} \mathbf{v}\right)(t) + \mathcal{Q}(t, \mathbf{u}(t), \mathbf{v}(t), I\_q^{\omega\_2} \mathbf{u}(t), I\_q^{\delta\_2} \mathbf{v}(t)) = \mathbf{0}, & t \in (0, 1), \end{cases} \tag{27}$$

subject to the coupled nonlocal boundary conditions

$$\begin{cases} \mathbf{u}(0) = D\_{\boldsymbol{q}} \mathbf{u}(0) = \dots = D\_{\boldsymbol{q}}^{n-2} \mathbf{u}(0) = 0, \ D\_{\boldsymbol{q}}^{\tilde{\zeta}\_0} \mathbf{u}(1) = \int\_0^1 D\_{\boldsymbol{q}}^{\tilde{\zeta}} \mathbf{v}(t) \, d\_{\boldsymbol{q}} \mathcal{H}(t), \\\ \mathbf{v}(0) = D\_{\boldsymbol{q}} \mathbf{v}(0) = \dots = D\_{\boldsymbol{q}}^{m-2} \mathbf{v}(0) = 0, \ D\_{\boldsymbol{q}}^{\tilde{\zeta}\_0} \mathbf{v}(1) = \int\_0^1 D\_{\boldsymbol{q}}^{\tilde{\zeta}} \mathbf{u}(t) \, d\_{\boldsymbol{q}} \mathcal{K}(t), \end{cases} \tag{28}$$

where *<sup>q</sup>* <sup>∈</sup> (0, 1), *<sup>α</sup>*, *<sup>β</sup>* <sup>∈</sup> <sup>R</sup>, *<sup>α</sup>* <sup>∈</sup> (*<sup>n</sup>* <sup>−</sup> 1, *<sup>n</sup>*], *<sup>β</sup>* <sup>∈</sup> (*<sup>m</sup>* <sup>−</sup> 1, *<sup>m</sup>*], *<sup>n</sup>*, *<sup>m</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>n</sup>*, *<sup>m</sup>* <sup>≥</sup> 2, *ω*1, *δ*1, *ω*2, *δ*<sup>2</sup> > 0, *ζ* ∈ [0, *β* − 1), *ξ* ∈ [0, *α* − 1), *ζ*<sup>0</sup> ∈ [0, *α* − 1), *ξ*<sup>0</sup> ∈ [0, *β* − 1). Here, *Dκ <sup>q</sup>* denotes the Riemann–Liouville *<sup>q</sup>*-derivative of order *<sup>κ</sup>* for *<sup>κ</sup>* ∈ {*α*, *<sup>β</sup>*, *<sup>ζ</sup>*0, *<sup>ζ</sup>*, *<sup>ξ</sup>*0, *<sup>ξ</sup>*}, *<sup>I</sup><sup>k</sup> <sup>q</sup>* is the Riemann–Liouville *q*-integral of order *k* for *k* ∈ {*ω*1, *δ*1, *ω*2, *δ*2}, P and Q are nonlinear functions, and the integrals from conditions (28) are Riemann–Stieltjes integrals with H, K functions of bounded variation. By applying varied fixed point theorems, the authors obtain existence and uniqueness results for the solutions of problem (27),(28).

Finally, I would like to thank all the authors for submitting papers to this Special Issue, and hope that their results will be useful to other researchers working in the field of fractional differential equations.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


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## *Article* **Analytic Resolving Families for Equations with the Dzhrbashyan–Nersesyan Fractional Derivative**

**Vladimir E. Fedorov 1,\* , Marina V. Plekhanova 1,2 and Elizaveta M. Izhberdeeva <sup>1</sup>**


**Abstract:** In this paper, a criterion for generating an analytic family of operators, which resolves a linear equation solved with respect to the Dzhrbashyan–Nersesyan fractional derivative, via a linear closed operator is obtained. The properties of the resolving families are investigated and applied to prove the existence of a unique solution for the corresponding initial value problem of the inhomogeneous equation with the Dzhrbashyan–Nersesyan fractional derivative. A solution is presented explicitly using resolving families of operators. A theorem on perturbations of operators from the found class of generators of resolving families is proved. The obtained results are used for a study of an initial-boundary value problem to a model of the viscoelastic Oldroyd fluid dynamics. Thus, the Dzhrbashyan–Nersesyan initial value problem is investigated in the essentially infinitedimensional case. The use of the proved abstract results to study initial-boundary value problems for a system of partial differential equations is demonstrated.

**Keywords:** fractional Dzhrbashyan–Nersesyan derivative; differential equation with fractional derivatives; resolving family of operators; perturbation theorem; initial value problem; initial-boundary value problem; viscoelastic Oldroyd fluid

**MSC:** 34G10; 35R11; 34A08

#### **1. Introduction**

Consider the differential equation

$$D^{\sigma\_{\Pi}}z(t) = Az(t) + f(t), \quad t \in (0, T], \tag{1}$$

where *A* is a linear closed operator, which has a dense domain *DA* in a Banach space Z, *<sup>T</sup>* <sup>&</sup>gt; 0, *<sup>f</sup>* : [0, *<sup>T</sup>*] → Z is a given function. Let *<sup>D</sup><sup>β</sup> <sup>t</sup>* be the Riemann–Liouville integral for *β* ≤ 0 and the Riemann–Liouville derivative for *β* > 0. Here *Dσ<sup>n</sup> z* := *Dαn*−<sup>1</sup> *<sup>t</sup> <sup>D</sup>αn*−<sup>1</sup> *<sup>t</sup> <sup>D</sup>αn*−<sup>2</sup> *<sup>t</sup>* ... *<sup>D</sup>α*<sup>0</sup> *<sup>t</sup> z*(*t*), where *α<sup>k</sup>* ∈ (0, 1], is the Dzhrbashyan–Nersesyan fractional derivative [1]. Note that this derivative includes as partial cases the Gerasimov–Caputo (*α<sup>k</sup>* = 1, *k* = 0, 1, ... , *n* − 1, *α<sup>n</sup>* = *α* − *n* + 1) and the Riemann–Liouville (*α*<sup>0</sup> = *α* − *n* + 1, *α<sup>k</sup>* = 1, *k* = 1, 2, ... , *n*) fractional derivatives of an order *α* from (*n* − 1, *n*].

In recent decades, fractional-order equations have been actively used in modeling various complex systems and processes in physics, chemistry, social sciences, and humanities [2–6]. We note recent works [7–12], combining theoretical studies in various fields of fractional integro-differential calculus and their use in real-world modeling problems, particularly when modeling biological processes in virology, which is especially important at present. Readers should also note the works [13,14], which consider some applied problems with the Dzhrbashyan–Nersesyan fractional derivative.

**Citation:** Fedorov, V.E.; Plekhanova, M.V.; Izhberdeeva, E.M. Analytic Resolving Families for Equations with the Dzhrbashyan–Nersesyan Fractional Derivative. *Fractal Fract.* **2022**, *6*, 541. https://doi.org/ 10.3390/fractalfract6100541

Academic Editor: Carlo Cattani

Received: 2 August 2022 Accepted: 22 September 2022 Published: 25 September 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

The initial value problem

$$D^{\varphi\_k}z(0) = z\_{k\prime} \quad k = 0, 1, \ldots, n-1,\tag{2}$$

with

$$D^{\sigma\_0}z(t) := D\_t^{a\_0-1}z(t), \quad D^{\sigma\_k}z(t) := D\_t^{a\_k-1}D\_t^{a\_{k-1}}D\_t^{a\_{k-2}} \dots D\_t^{a\_0}z(t), \quad k = 1, 2, \dots, n, \quad t \ge 0$$

for Equation (1) in the scalar case (<sup>Z</sup> <sup>=</sup> <sup>R</sup>, *<sup>A</sup>* <sup>∈</sup> <sup>R</sup>) is studied by M.M. Dzrbashyan, A.B. Nersesyan in [1]. The unique solvability theorem for such a problem with <sup>Z</sup> <sup>=</sup> <sup>R</sup>*<sup>n</sup>* and a matrix *A* was obtained in [15]. Various equations with partial derivatives of Dzhrbashyan and Nersesyan were studied in papers [16–21]. Problem (1), (2) with a linear continuous operator *A* ∈ L(Z) in an arbitrary Banach space Z was researched in [22] considering the methods used to resolve families of operators; see [23].

The results obtained in this work generalize the corresponding results of the theory of analytic semigroups of operators solving first-order equations in Banach spaces [24,25]. We also note the works in which the theory of analytical resolving families is constructed for evolutionary integral equations [26], equations with a Gerasimov–Caputo [27] or Riemann– Liouville [28] derivative, fractional multi-term linear differential equations in Banach spaces [29], and equations with various distributed fractional derivatives [30–34].

After the Introduction and Preliminaries, in the second section of the present work, the notion of a *k*-resolving family for homogeneous Equation (1), i.e., with *f* ≡ 0, *k* = 0, 1, . . . , *n* − 1, is introduced. In the third section, it is shown that the existence of *k*-resolving families, *k* = 1, 2, ... , *n* − 1, follows from the existence of a zero-resolving family. In the fourth section, a criterion of the existence of a zero-resolving family of operators to the homogeneous Equation (1) is found in terms of conditions for a linear closed operator *A*. The class of operators which satisfy these conditions is denoted as A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0). Various properties of the resolving families are investigated, and a perturbation theorem for operators from A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0) is proved in the fifth section. For problem (1), (2) with a function *<sup>f</sup>* , which is continuous in the graph norm of *A* or Hölderian, the existence of a unique solution is obtained in the sixth section. In the last section, this result is used to prove the theorem on a unique solution existence for an initial-boundary value problem to a fractional linearized model of the viscoelastic Oldroyd fluid dynamics.

The theoretical significance of the obtained results lies in the fact that they give a correct statement of an initial problem and conditions for its unique solvability for equations with the Dzhrbashyan–Nersesian fractional derivative and with an unbounded linear operator at the unknown function. The unboundedness of the operator in the equation makes it possible to reduce initial-boundary value problems to various equations and systems of partial differential equations in problems of this type.

#### **2. Preliminaries**

Let <sup>Z</sup> be a Banach space. For the function *<sup>z</sup>* : <sup>R</sup><sup>+</sup> → Z, the Riemann–Liouville fractional integral of an order *β* > 0 has the form

$$J\_t^\beta z(t) := \int\_0^t \frac{(t-s)^{\beta-1}}{\Gamma(\beta)} z(s)ds, \quad t > 0.$$

For the function *z*, the Riemann–Liouville fractional derivative of an order *α* ∈ (*m* − 1, *m*], where *<sup>m</sup>* <sup>∈</sup> <sup>N</sup> is defined as *<sup>D</sup><sup>α</sup> <sup>t</sup> z*(*t*) := *D<sup>m</sup> t J <sup>m</sup>*−*<sup>α</sup> <sup>t</sup> <sup>z</sup>*(*t*), *<sup>D</sup><sup>m</sup> <sup>t</sup>* :<sup>=</sup> *<sup>d</sup><sup>m</sup> dt<sup>m</sup>* . Further, we use the notation *D*−*<sup>α</sup> <sup>t</sup>* := *<sup>J</sup><sup>α</sup> <sup>t</sup>* for *α* > 0; *D*<sup>0</sup> *<sup>t</sup>* = *J*<sup>0</sup> *<sup>t</sup>* is the identical operator.

Let {*αk*}*<sup>n</sup>* <sup>0</sup> be a set of numbers *<sup>α</sup><sup>k</sup>* <sup>∈</sup> (0, 1], *<sup>k</sup>* <sup>=</sup> 0, 1, ... , *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. We will use the denotations *<sup>σ</sup><sup>k</sup>* :<sup>=</sup> *<sup>k</sup>* ∑ *j*=0 *α<sup>j</sup>* − 1, *k* = 0, 1, . . . , *n*, hence *σ<sup>k</sup>* ∈ (−1, *k* − 1]. Further, we will assume that *σ<sup>n</sup>* > 0. Define the Dzhrbashyan–Nersesyan fractional derivatives, which correspond to the sequence {*αk*}*<sup>n</sup>* <sup>0</sup> , by relations

$$D^{\sigma\_0}z(t) := D\_t^{\alpha\_0 - 1}z(t),\tag{3}$$

$$D^{\sigma\_k}z(t) := D\_t^{a\_k - 1}D\_t^{a\_{k-1}}D\_t^{a\_{k-2}}\dots D\_t^{a\_0}z(t), \quad k = 1, 2, \dots, n. \tag{4}$$

**Example 1.** *Take α* ∈ (*n* − 1, *n*]*, α*<sup>0</sup> = *α* − *n* + 1 ∈ (0, 1]*, α<sup>k</sup>* = 1*, k* = 1, 2, ... , *n, then <sup>D</sup>σ*<sup>0</sup> *<sup>z</sup>*(*t*) := *<sup>D</sup>α*−*<sup>n</sup> <sup>t</sup> <sup>z</sup>*(*t*) := *<sup>J</sup> <sup>n</sup>*−*<sup>α</sup> <sup>t</sup> <sup>z</sup>*(*t*)*, <sup>D</sup>σ<sup>k</sup> <sup>z</sup>*(*t*) := *<sup>D</sup>k*−<sup>1</sup> *<sup>t</sup> <sup>D</sup>α*−*n*+<sup>1</sup> *<sup>t</sup> <sup>z</sup>*(*t*) = *<sup>D</sup><sup>k</sup> t J <sup>n</sup>*−*<sup>α</sup> <sup>t</sup> <sup>z</sup>*(*t*) := *<sup>D</sup>k*−*n*+*<sup>α</sup> <sup>t</sup> <sup>z</sup>*(*t*)*, <sup>k</sup>* <sup>=</sup> 1, 2, ... , *n, are the Riemann–Liouville fractional derivatives. In particular, Dσ<sup>n</sup> z*(*t*) = *D<sup>n</sup> t J <sup>n</sup>*−*<sup>α</sup> <sup>t</sup> <sup>z</sup>*(*t*) := *<sup>D</sup><sup>α</sup> <sup>t</sup> z*(*t*)*.*

**Example 2.** *If <sup>α</sup>* <sup>∈</sup> (*<sup>n</sup>* <sup>−</sup> 1, *<sup>n</sup>*]*, <sup>α</sup><sup>k</sup>* <sup>=</sup> <sup>1</sup>*, <sup>k</sup>* <sup>=</sup> 0, 1, ... , *<sup>n</sup>* <sup>−</sup> <sup>1</sup>*, <sup>α</sup><sup>n</sup>* <sup>=</sup> *<sup>α</sup>* <sup>−</sup> *<sup>n</sup>* <sup>+</sup> <sup>1</sup>*, then <sup>D</sup>σ<sup>k</sup> <sup>z</sup>*(*t*) :<sup>=</sup> *Dk <sup>t</sup> <sup>z</sup>*(*t*)*, <sup>k</sup>* <sup>=</sup> 0, 1, ... , *<sup>n</sup>* <sup>−</sup> <sup>1</sup>*, <sup>D</sup>σ<sup>n</sup> <sup>z</sup>*(*t*) :<sup>=</sup> *<sup>D</sup>α*−*<sup>n</sup> <sup>t</sup> <sup>D</sup><sup>n</sup> <sup>t</sup> z*(*t*) := *J <sup>n</sup>*−*<sup>α</sup> <sup>t</sup> <sup>D</sup><sup>n</sup> <sup>t</sup> z*(*t*) :=*CD<sup>α</sup> <sup>t</sup> is the Gerasimov– Caputo fractional derivative.*

**Example 3.** *In [23], it is shown that the compositions of the Gerasimov–Caputo and the Riemann– Liouville fractional derivatives D<sup>α</sup> <sup>t</sup> <sup>D</sup><sup>β</sup> <sup>t</sup> , <sup>D</sup><sup>α</sup> t CD<sup>β</sup> <sup>t</sup> , CD<sup>α</sup> <sup>t</sup> <sup>D</sup><sup>β</sup> <sup>t</sup> , CD<sup>α</sup> t CD<sup>β</sup> <sup>t</sup> may be presented as Dzhrbashyan– Nersesyan fractional derivatives Dσ<sup>n</sup> for some sequences* {*σ*0, *<sup>σ</sup>*1,..., *<sup>σ</sup>n*}*.*

Let *<sup>α</sup>* <sup>∈</sup> (*<sup>m</sup>* <sup>−</sup> 1, *<sup>m</sup>*], *<sup>m</sup>* <sup>∈</sup> <sup>N</sup>. Then, for a function *<sup>z</sup>* : <sup>R</sup><sup>+</sup> → Z, we use *<sup>z</sup>* to denote the Laplace transform, and for too-large expressions for *z* as Lap[*z*]. In [22], it is proved that

$$
\widehat{D^{\sigma\_n}z}(\lambda) = \lambda^{\sigma\_n}\widehat{z}(\lambda) - \sum\_{k=0}^{n-1} \lambda^{\sigma\_n - \sigma\_k - 1} D^{\sigma\_k}z(0). \tag{5}
$$

L(Z) denotes the Banach space of all linear continuous operators on a Banach space Z; C*l*(Z) denotes the set of all linear closed operators, which are densely defined in Z and act into Z. For an operator *A* ∈ C*l*(Z), its domain *DA* is endowed by the norm · *DA* := · <sup>Z</sup> + *A* · <sup>Z</sup> , which is a Banach space due to the closedness of *A*.

Consider the initial value problem

$$D^{\mathcal{O}\_k}z(0) = z\_k \quad k = 0, 1, \ldots, n-1. \tag{6}$$

to the linear homogeneous equation

$$D^{\sigma\_{\overline{n}}}z(t) = Az(t), \quad t > 0,\tag{7}$$

where *<sup>A</sup>* ∈ C*l*(Z), *<sup>D</sup>σ<sup>n</sup>* is the Dzhrbashyan–Nersesyan fractional derivative, associated with a set of real numbers {*αk*}*<sup>n</sup>* <sup>0</sup> , 0 <sup>&</sup>lt; *<sup>α</sup><sup>k</sup>* <sup>≤</sup> 1, *<sup>k</sup>* <sup>=</sup> 0, 1, . . . , *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>, by (3), (4), *<sup>σ</sup><sup>n</sup>* <sup>&</sup>gt; 0.

A solution to problem (6), (7) is a function *<sup>z</sup>* <sup>∈</sup> *<sup>C</sup>*(R+; *DA*), such that *<sup>D</sup>σ<sup>k</sup> <sup>t</sup> <sup>z</sup>* <sup>∈</sup> *<sup>C</sup>*(R+; <sup>Z</sup>), *<sup>k</sup>* <sup>=</sup> 0, 1, ... , *<sup>n</sup>* <sup>−</sup> 1, *<sup>D</sup>σ<sup>n</sup> <sup>t</sup> <sup>z</sup>* <sup>∈</sup> *<sup>C</sup>*(R+; <sup>Z</sup>), (7) holds for all *<sup>t</sup>* <sup>∈</sup> <sup>R</sup><sup>+</sup> and conditions (6) are valid. Hereafter, <sup>R</sup><sup>+</sup> :<sup>=</sup> <sup>R</sup><sup>+</sup> ∪ {0}.

Denote *<sup>S</sup>θ*,*<sup>a</sup>* :<sup>=</sup> {*<sup>λ</sup>* <sup>∈</sup> <sup>C</sup> : <sup>|</sup> arg(*<sup>λ</sup>* <sup>−</sup> *<sup>a</sup>*)<sup>|</sup> <sup>&</sup>lt; *<sup>θ</sup>*, *<sup>λ</sup>* <sup>=</sup> *<sup>a</sup>*}, *<sup>θ</sup>* <sup>∈</sup> [*π*/2, *<sup>π</sup>*], *<sup>a</sup>* <sup>∈</sup> <sup>R</sup>, <sup>Σ</sup>*<sup>ψ</sup>* :<sup>=</sup> {*<sup>t</sup>* <sup>∈</sup> <sup>C</sup> : | arg *t*| < *ψ*, *t* = 0} for *ψ* ∈ (0, *π*/2] and formulate an assertion that is important for further considerations.

**Theorem 1** ([34])**.** *Let <sup>θ</sup>*<sup>0</sup> <sup>∈</sup> (*π*/2, *<sup>π</sup>*]*, <sup>a</sup>* <sup>∈</sup> <sup>R</sup>*, <sup>β</sup>* <sup>∈</sup> [0, 1)*,* <sup>X</sup> *be a Banach space, <sup>H</sup>* : (*a*, <sup>∞</sup>) → X *. Then, the next statements are equivalent.*

(i) *There exists an analytic function <sup>F</sup>* : <sup>Σ</sup>*θ*0−*π*/2 → X *. For every <sup>θ</sup>* ∈ (*π*/2, *<sup>θ</sup>*0)*, there exists such a C*(*θ*) > 0 *that the inequality F*(*t*) <sup>X</sup> ≤ *<sup>C</sup>*(*θ*)|*t*| <sup>−</sup>*βea*Re *<sup>t</sup> is satisfied for all <sup>t</sup>* <sup>∈</sup> <sup>Σ</sup>*θ*−*<sup>π</sup>*/2; *for <sup>λ</sup>* <sup>&</sup>gt; *<sup>a</sup> <sup>F</sup>*(*λ*) = *<sup>H</sup>*(*λ*)*.*

(ii) *H is analytically extendable on Sθ*0,*a*; *for every θ* ∈ (*π*/2, *θ*0) *there exists K*(*θ*) > 0*, such that for all λ* ∈ *Sθ*,*<sup>a</sup>*

$$\|H(\lambda)\|\_{\mathcal{X}} \le \frac{K(\theta)}{|\lambda - a|^{1 - \beta}}.$$

#### **3.** *k***-Resolving Families of Operators**

**Definition 1.** *A set of linear bounded operators* {*Sl*(*t*) ∈ L(Z) : *t* > 0} *is called k-resolving family, k* ∈ {0, 1, . . . , *n* − 1}*, for Equation* (7)*, if it satisfies the next conditions:*

(i) *Sk*(*t*) *is a strongly continuous family at t* > 0*;*

(ii) *Sk*(*t*)[*DA*] ⊂ *DA, for all x* ∈ *DA, t* > 0 *Sk*(*t*)*Ax* = *ASk*(*t*)*x;*

(iii) *For every zk* <sup>∈</sup> *DA Sk*(*t*)*zk is a solution of initial value problem <sup>D</sup>σ<sup>k</sup> <sup>z</sup>*(0) = *zk, <sup>D</sup>σ<sup>l</sup> <sup>z</sup>*(0) = <sup>0</sup>*, l* ∈ {0, . . . , *<sup>n</sup>* <sup>−</sup> <sup>1</sup>}\{*k*} *to Equation* (7)*.*

Let *<sup>ρ</sup>*(*A*) :<sup>=</sup> {*<sup>λ</sup>* <sup>∈</sup> <sup>C</sup> : *<sup>R</sup>λ*(*A*) := (*λ<sup>I</sup>* <sup>−</sup> *<sup>A</sup>*)−<sup>1</sup> ∈ L(Z)} be the resolvent set of operator *A*.

**Proposition 1.** *Let α<sup>l</sup>* ∈ (0, 1]*, l* = 0, 1, ... , *n, σ<sup>n</sup>* > 0*. For k* ∈ {0, 1, ... , *m* − 1} *there exists a k-resolving family of operators* {*Sk*(*t*) ∈ L(Z) : *t* > 0} *for Equation* (7)*, such that at some K* > 0*, <sup>a</sup>* <sup>∈</sup> <sup>R</sup>*, <sup>β</sup>* <sup>∈</sup> [0, 1) *Sk*(*t*) <sup>L</sup>(Z) <sup>≤</sup> *Keatt* <sup>−</sup>*<sup>β</sup> for all t* <sup>&</sup>gt; <sup>0</sup>*. Then, <sup>λ</sup>σ<sup>n</sup>* <sup>∈</sup> *<sup>ρ</sup>*(*A*) *for* Re*<sup>λ</sup>* <sup>&</sup>gt; *a,*

$$
\widehat{S}\_k(\lambda) = \lambda^{\sigma\_\mathbb{N} - \sigma\_k - 1} R\_{\lambda^{\mathfrak{N}}}(A) \tag{8}
$$

*and a k-resolving family of operators for Equation* (7) *is unique.*

**Proof.** Due to identity (5) and Definition 1 for arbitrary *zk* <sup>∈</sup> *DA*, Re*<sup>λ</sup>* <sup>&</sup>gt; *<sup>a</sup> <sup>λ</sup>σ<sup>n</sup> <sup>S</sup>k*(*λ*)*zk* <sup>−</sup> *<sup>λ</sup>σn*−*σk*−<sup>1</sup>*zk* <sup>=</sup> *ASk*(*λ*)*zk* <sup>=</sup> *<sup>S</sup>k*(*λ*)*Azk*. Therefore, the operator *<sup>λ</sup>σ<sup>n</sup> <sup>I</sup>* <sup>−</sup> *<sup>A</sup>* : *DA* → Z is invertible and equality (8) holds. Since *<sup>S</sup>k*(*λ*) ∈ L(Z) for Re*<sup>λ</sup>* <sup>&</sup>gt; *<sup>a</sup>*, we have *<sup>λ</sup>σ<sup>n</sup>* <sup>∈</sup> *<sup>ρ</sup>*(*A*). Due to equality (8) from the uniqueness of the inverse Laplace transform, we see the uniqueness of a *k*-resolving family for Equation (7).

**Proposition 2.** *Let α<sup>k</sup>* ∈ (0, 1]*, k* = 0, 1, ... , *n, σ<sup>n</sup>* > 0*. There exists a* 0*-resolving family* {*S*0(*t*) ∈ L(Z) : *<sup>t</sup>* <sup>&</sup>gt; <sup>0</sup>} *for* (7)*, such that at some <sup>K</sup>* <sup>&</sup>gt; <sup>0</sup>*, <sup>a</sup>* <sup>∈</sup> <sup>R</sup> *S*0(*t*) <sup>L</sup>(Z) <sup>≤</sup> *Keatt σ*0 *for all t* > 0*. Then, for every k* = 0, 1, ... , *n* − 1*, there exists a unique k-resolving family* {*Sk*(*t*) ∈ L(Z) : *t* > 0}*. Moreover, Sk*(*t*) ≡ *J σk*−*σ*<sup>0</sup> *<sup>t</sup> S*0(*t*) *and Sk*(*t*) <sup>L</sup>(Z) <sup>≤</sup> *<sup>K</sup>*1*eatt <sup>σ</sup><sup>k</sup> at some K*<sup>1</sup> > 0 *for all t* > 0*, k* = 1, 2, . . . , *n* − 1*.*

**Proof.** Since every *<sup>z</sup>*<sup>0</sup> <sup>∈</sup> *DA* \ {0} *<sup>J</sup>*1−*α*0*S*0(*t*)*z*<sup>0</sup> has a nonzero limit *<sup>z</sup>*<sup>0</sup> as *<sup>t</sup>* <sup>→</sup> <sup>0</sup>+, due to ([29], Lemma 1) *S*0(*t*)*z*<sup>0</sup> = *t <sup>α</sup>*0−1*z*0/Γ(*α*0) + *o*(*t <sup>α</sup>*0−1) as *<sup>t</sup>* <sup>→</sup> <sup>0</sup>+. Therefore, for every *z*<sup>0</sup> ∈ Z, *T* > 0 *S*0(*t*)*z*<sup>0</sup> ∈ *L*1(0, *T*; Z) and there are Riemann–Liouville fractional integrals for this function.

Define for *k* = 1, 2, ... , *n* − 1 the families {*Sk*(*t*) := *J σk*−*σ*<sup>0</sup> *<sup>t</sup> S*0(*t*) ∈ L(Z) : *t* > 0}. By this construction, it satisfies condition (i) in the Definition 1. For *x* ∈ *DA*, *t* > 0

$$J\_t^{\sigma\_k - \sigma\_0} S\_0(t) A \ge \int\_0^t \frac{(t - s)^{\sigma\_k - \sigma\_0 - 1}}{\Gamma(\sigma\_k - \sigma\_0)} S\_0(s) A \ge ds = A J\_t^{\sigma\_k - \sigma\_0} S\_0(t) \ge \frac{1}{2}$$

since {*S*0(*t*) ∈ L(Z) : *t* > 0} satisfies condition (ii) in Definition 1 and the operator *A* is closed. So, condition (ii) holds for {*Sk*(*t*) ∈ L(Z) : *t* > 0}, where *k* = 1, 2, . . . , *n* − 1.

Further, we have

$$\|\|S\_k(t)\|\|\_{\mathcal{L}(\mathcal{Z})} \le K \int\_0^t \frac{(t-s)^{\sigma\_k - \sigma\_0 - 1}}{\Gamma(\sigma\_k - \sigma\_0)} s^{\sigma\_0} e^{as} ds \le \frac{K e^{at} t^{\sigma\_k} \Gamma(\sigma\_0 + 1)}{\Gamma(\sigma\_k + 1)} = K\_1 e^{at} t^{\sigma\_k}, \quad t > 0.$$

For *zk* <sup>∈</sup> *DA*, multiply the equality *<sup>λ</sup>σ<sup>n</sup> <sup>S</sup>*0(*λ*)*zk* <sup>−</sup> *<sup>λ</sup>σn*−*σ*0−<sup>1</sup>*zk* <sup>=</sup> *AS*0(*λ*)*zk*, which follows from point (iii) of Definition 1 for *k* = 0 after the Laplace transform action, by *λσ*0−*σ<sup>k</sup>* and obtain the equality *λσ<sup>n</sup> J σk*−*σ*<sup>0</sup> *<sup>t</sup> <sup>S</sup>*0(*λ*)*zk* <sup>−</sup> *<sup>λ</sup>σn*−*σk*−<sup>1</sup>*zk* <sup>=</sup> *AJ σk*−*σ*<sup>0</sup> *<sup>t</sup> <sup>S</sup>*0(*λ*)*zk*, i.e., *<sup>λ</sup>σ<sup>n</sup> <sup>S</sup>k*(*λ*)*zk* <sup>−</sup> *<sup>λ</sup>σn*−*σk*−<sup>1</sup>*zk* <sup>=</sup> *ASk*(*λ*)*zk*, which means that {*Sk*(*t*) ∈ L(Z) : *<sup>t</sup>* <sup>&</sup>gt; <sup>0</sup>} is a *<sup>k</sup>*-resolving family for Equation (7) due to the uniqueness of the inverse Laplace transform. Hence equality (8) is valid and a *k*-resolving family of Equation (7) is unique by Proposition 1.

**Remark 1.** *The parameter σ*<sup>0</sup> *in the formulation of Proposition 2 defines the power singularity of the family* {*S*0(*t*) ∈ L(Z) : *t* > 0} *at zero. At the beginning of the proof of Proposition 2, it was shown that we have two possibilities only: the singularity at zero has a power of σ*<sup>0</sup> := *α*<sup>0</sup> − 1 < 0*, or a singularity is absent in the case α*<sup>0</sup> = 1*. Due to Proposition 2, the k-resolving family* {*Sk*(*t*) ∈ L(Z) : *t* > 0} *has the singularity of the power σ<sup>k</sup>* < 0*, or it is absent at zero, if σ<sup>k</sup>* ≥ 0*.*

**Theorem 2.** *Let α<sup>l</sup>* ∈ (0, 1]*, l* = 0, 1, ... , *n, σ<sup>n</sup>* > 0*, there exist a k-resolving family of operators* {*Sk*(*t*) ∈ L(Z) : *t* > 0} *of* (7) *for some k* ∈ {0, 1, ... , *n* − 1}*, such that Sk*(*t*) <sup>L</sup>(Z) <sup>≤</sup> *Keatt σk at some <sup>K</sup>* <sup>&</sup>gt; <sup>0</sup>*, <sup>a</sup>* <sup>∈</sup> <sup>R</sup> *for all <sup>t</sup>* <sup>&</sup>gt; <sup>0</sup>*. Then, there exists a limit* lim *t*→0+ *D<sup>σ</sup>kSk*(*t*) = *I in the norm of the space* L(Z)*, if and only if A* ∈ L(Z)*.*

**Proof.** Note that *D <sup>σ</sup>kSk* <sup>=</sup> *<sup>λ</sup>σkS<sup>k</sup>* <sup>=</sup> *<sup>λ</sup>σn*−1(*λσ<sup>n</sup> <sup>I</sup>* <sup>−</sup> *<sup>A</sup>*)−<sup>1</sup> due to (5), Definition 1 andProposition 1. Hence for *zk* ∈ *DA*, *b* > *a*

$$(D^{\sigma\_k} S\_k(t) z\_k = \int\_{b-i\infty}^{b+i\infty} \lambda^{\sigma\_{\text{fl}}-1} R\_{\lambda^{\text{Pn}}}(A) e^{\lambda t} z\_k d\lambda = z\_k + \int\_{b-i\infty}^{b+i\infty} \lambda^{-1} R\_{\lambda^{\text{Sn}}}(A) e^{\lambda t} A z\_k d\lambda. \tag{9}$$

Since, for large enough |*λ*|

$$\|\lambda^{-1} R\_{\lambda^{\sigma\_0}}(A)\|\_{\mathcal{L}(\mathcal{Z})} \le \frac{\mathsf{C}\_1}{|\lambda|^{\sigma\_n - \sigma\_0 + \kappa\_0}} = \frac{\mathsf{C}\_1}{|\lambda|^{\sigma\_0 + 1}},$$

we have *<sup>D</sup><sup>σ</sup>kSk*(*t*)*zk* <sup>Z</sup> <sup>≤</sup> *<sup>K</sup>*1*ebt*. For Re*λ* > *b*

$$\int\_0^\infty e^{-\lambda t} (D^{\sigma\_k} S\_k(t) - I) dt = \lambda^{\sigma\_n - 1} R\_{\lambda^{\sigma\_n}}(A) - \lambda^{-1} I.$$

Assume that *η*(*t*) := *<sup>D</sup><sup>σ</sup>kSk*(*t*) <sup>−</sup> *<sup>I</sup>* <sup>L</sup>(Z) is a continuous function on [0, 1] and *<sup>η</sup>*(0) = 0. For arbitrary *ε* > 0, take *δ* > 0, such that for all *t* ∈ [0, *δ*] *η*(*t*) ≤ *ε*; therefore, due to the inequality *<sup>η</sup>*(*t*) <sup>≤</sup> *<sup>K</sup>*1*ebt* <sup>+</sup> 1 for *<sup>t</sup>* <sup>≥</sup> 0, we have

$$\left\|\lambda^{\sigma\_{\mathbb{H}}-1}R\_{\lambda^{\sigma\_{\mathbb{H}}}}(A) - \lambda^{-1}I\right\|\_{\mathcal{L}(\mathcal{Z})} \leq \int\_0^\delta e^{-\lambda t}\eta(t)dt + \int\_\delta^\infty e^{-\lambda t}\eta(t)dt \leq \frac{\varepsilon}{\lambda} + o\left(\frac{1}{\lambda}\right)$$

as Re*<sup>λ</sup>* → +∞. Hence, for large enough Re*<sup>λ</sup>* > <sup>0</sup> *λσn*−1*Rλσ<sup>n</sup>* (*A*) <sup>−</sup> *<sup>λ</sup>*−<sup>1</sup> *<sup>I</sup>* <sup>L</sup>(Z) <sup>&</sup>lt; 1. Consequently, *Rλσ<sup>n</sup>* (*A*) is a continuously invertible operator, so *A* ∈ L(Z).

Let *A* ∈ L(Z), *R* > *A* 1/*σn* L(Z) , <sup>Γ</sup>1,*<sup>R</sup>* :<sup>=</sup> {*Rei<sup>ϕ</sup>* : *<sup>ϕ</sup>* <sup>∈</sup> (−*π*, *<sup>π</sup>*)}, <sup>Γ</sup>2,*<sup>R</sup>* :<sup>=</sup> {*rei<sup>π</sup>* : *<sup>r</sup>* <sup>∈</sup> [*R*, <sup>∞</sup>)}, <sup>Γ</sup>3,*<sup>R</sup>* :<sup>=</sup> {*re*−*i<sup>π</sup>* : *<sup>r</sup>* <sup>∈</sup> [*R*, <sup>∞</sup>)}, <sup>Γ</sup>*<sup>R</sup>* :<sup>=</sup> <sup>Γ</sup>1,*<sup>R</sup>* <sup>∪</sup> <sup>Γ</sup>2,*<sup>R</sup>* <sup>∪</sup> <sup>Γ</sup>3,*R*. Due to equality (9), we obtain for *t* > 0

$$D^{\sigma\_k} S\_k(t) = I + \frac{1}{2\pi i} \int\_{\Gamma\_R} \lambda^{-1} R\_{\lambda^{\sigma\_k}}(A) A e^{\lambda t} d\lambda = I + \frac{1}{2\pi i} \int\_{\Gamma\_R} \frac{1}{\lambda} \sum\_{l=1}^{\infty} \frac{A^l e^{\lambda t} d\lambda}{\lambda^{l\sigma\_n}}.$$

Take *R* = 1/*t* for small *t* > 0; then,

$$\| |D^{\sigma\_k} S\_k(t) - I| \|\_{\mathcal{L}(\mathcal{Z})} \le \mathbb{C}\_1 \sum\_{k=1}^3 \sum\_{l=1}^\infty \int\_{\Gamma\_{k,\mathbb{R}}} \frac{| |A| |\_{\mathcal{L}(\mathcal{Z})}^l | d\lambda |}{|\lambda|^{l\sigma\_n + 1}} \le \frac{\mathbb{C}\_2 t^{\sigma\_n} \| |A| \|\_{\mathcal{L}(\mathcal{Z})}}{1 - t^{\sigma\_n} \| |A| \|\_{\mathcal{L}(\mathcal{Z})}} \to 0$$

as *t* → 0+.

**Remark 2.** *An analogous result of Theorem 2 is well-known for resolving semigroups of operators for first-order equations (see, e.g., [35]). On resolving families of operators for equations, which are solved with respect to a Gerasimov–Caputo derivative, a similar theorem was obtained in work [27].*

#### **4. Generation of Analytic** *k***-Resolving Families**

Let *k* ∈ {0, 1, ... , *n* − 1}. A *k*-resolving family of operators is called *analytic*, if at some *ψ*<sup>0</sup> ∈ (0, *π*/2] it has an analytic continuation to Σ*ψ*<sup>0</sup> . An analytic *k*-resolving family of operators {*Sk*(*t*) ∈ L(Z) : *<sup>t</sup>* <sup>&</sup>gt; <sup>0</sup>} has a type (*ψ*0, *<sup>a</sup>*0, *<sup>β</sup>*) at some *<sup>ψ</sup>*<sup>0</sup> <sup>∈</sup> (0, *<sup>π</sup>*/2], *<sup>a</sup>*<sup>0</sup> <sup>∈</sup> <sup>R</sup>, *β* ≥ 0, if, for arbitrary *ψ* ∈ (0, *ψ*0), *a* > *a*0, there exists *C*(*ψ*, *a*), such that the inequality *Sk*(*t*) <sup>L</sup>(Z) <sup>≤</sup> *<sup>C</sup>*(*ψ*, *<sup>a</sup>*)*ea*Re *<sup>t</sup>* |*t*| <sup>−</sup>*<sup>β</sup>* is satisfied for all *<sup>t</sup>* <sup>∈</sup> <sup>Σ</sup>*ψ*.

**Remark 3.** *From Proposition 2 and Remark 1 it follows that for a k-resolving family of operators* {*Sk*(*t*) ∈ L(Z) : *t* > 0}*, we may have β* = −*σk, or β* = 0*.*

**Definition 2.** *An operator <sup>A</sup>* ∈ C*l*(Z) *belongs to the class* A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0)*, <sup>θ</sup>*<sup>0</sup> ∈ (*π*/2, *<sup>π</sup>*)*, <sup>a</sup>*<sup>0</sup> ≥ <sup>0</sup>*, α<sup>k</sup>* ∈ (0, 1]*, k* = 0, 1, . . . , *n, σ<sup>n</sup>* > 0*, if:*

(i) *For all <sup>λ</sup>* <sup>∈</sup> *<sup>S</sup>θ*0,*a*<sup>0</sup> *we have <sup>λ</sup>σ<sup>n</sup>* <sup>∈</sup> *<sup>ρ</sup>*(*A*);

(ii) *For arbitrary θ* ∈ (*π*/2, *θ*0)*, a* > *a*0*, there exists a constant K*(*θ*, *a*) > 0*, such that for every λ* ∈ *Sθ*,*<sup>a</sup>*

$$\|\|R\_{\lambda^{\eta\_0}}(A)\|\|\_{\mathcal{L}(\mathcal{Z})} \le \frac{K(\theta, a)}{|\lambda - a|^{\alpha\_0} |\lambda|^{\sigma\_n - \sigma\_0 - 1}}.$$

If *<sup>A</sup>* ∈ A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0), the operators

$$Z\_k(t) = \frac{1}{2\pi i} \int\_{\gamma} \lambda^{\sigma\_n - \sigma\_k - 1} R\_{\lambda^{\sigma\_n}}(A) e^{\lambda t} d\lambda, \quad t > 0, \ k = 0, 1, \dots, n - 1, \dots$$

are defined, where <sup>Γ</sup> :<sup>=</sup> <sup>Γ</sup><sup>+</sup> <sup>∪</sup> <sup>Γ</sup><sup>−</sup> <sup>∪</sup> <sup>Γ</sup>0, <sup>Γ</sup><sup>±</sup> :<sup>=</sup> {*<sup>λ</sup>* <sup>∈</sup> <sup>C</sup> : *<sup>λ</sup>* <sup>=</sup> *<sup>a</sup>* <sup>+</sup> *re*±*iθ*, *<sup>r</sup>* <sup>∈</sup> (*δ*, <sup>∞</sup>)}, <sup>Γ</sup><sup>0</sup> :<sup>=</sup> {*<sup>λ</sup>* <sup>∈</sup> <sup>C</sup> : *<sup>λ</sup>* <sup>=</sup> *<sup>a</sup>* <sup>+</sup> *<sup>δ</sup>eiϕ*, *<sup>ϕ</sup>* <sup>∈</sup> (−*θ*, *<sup>θ</sup>*)}, *<sup>θ</sup>* <sup>∈</sup> (*π*/2, *<sup>θ</sup>*0), *<sup>a</sup>* <sup>&</sup>gt; *<sup>a</sup>*0, *<sup>δ</sup>* <sup>&</sup>gt; 0.

**Theorem 3.** *Let α<sup>k</sup>* ∈ (0, 1]*, k* = 0, 1, . . . , *n, α*<sup>0</sup> + *α<sup>n</sup>* > 0*, θ*<sup>0</sup> ∈ (*π*/2, *π*]*, a*<sup>0</sup> ≥ 0*.*

(i) *If there exists an analytic* 0*-resolving family of operators of the type* (*θ*<sup>0</sup> − *π*/2, *a*0, −*σ*0) *for* (7)*, then A* ∈ A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0).

(ii) *If <sup>A</sup>* ∈ A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0)*, then for every <sup>k</sup>* = 0, 1, ... , *<sup>n</sup>* − <sup>1</sup> *there exists a unique analytic k-resolving family of operators* {*Sk*(*t*) ∈ L(Z) : *t* > 0} *of the type* (*θ*<sup>0</sup> − *π*/2, *a*0, max{−*σk*, 0}) *for* (7)*. Moreover, for t* > 0*, k* = 0, 1, . . . , *n* − 1 *Sk*(*t*) ≡ *Zk*(*t*) ≡ *J σk*−*σ*<sup>0</sup> *<sup>t</sup> Z*0(*t*)*.*

**Proof.** Choose *R* > *δ*,

$$\Gamma\_R := \bigcup\_{k=1}^4 \Gamma\_{k, \mathbb{R}} \quad \Gamma\_{1, \mathbb{R}} := \Gamma\_0 \quad \Gamma\_{2, \mathbb{R}} := \{ \lambda \in \mathbb{C} : \lambda = a + \operatorname{Re}^{i\varphi}, \ \varphi \in (-\theta, \theta) \}\_{\lambda}$$

<sup>Γ</sup>3,*<sup>R</sup>* :<sup>=</sup> {*<sup>λ</sup>* <sup>∈</sup> <sup>C</sup> : *<sup>λ</sup>* <sup>=</sup> *<sup>a</sup>* <sup>+</sup> *reiθ*, *<sup>r</sup>* <sup>∈</sup> [*δ*, *<sup>R</sup>*]}, <sup>Γ</sup>4,*<sup>R</sup>* :<sup>=</sup> {*<sup>λ</sup>* <sup>∈</sup> <sup>C</sup> : *<sup>λ</sup>* <sup>=</sup> *<sup>a</sup>* <sup>+</sup> *re*−*iθ*, *<sup>r</sup>* <sup>∈</sup> [*δ*, *<sup>R</sup>*]}, Γ*<sup>R</sup>* is the positively oriented closed loop,

$$\Gamma\_{5,\mathbb{R}} := \{ \lambda \in \mathbb{C} : \lambda = a + re^{i\theta}, r \in [\mathbb{R}, \infty) \}, \quad \Gamma\_{6,\mathbb{R}} := \{ \lambda \in \mathbb{C} : \lambda = a + re^{-i\theta}, r \in [\mathbb{R}, \infty) \},$$

then Γ = Γ5,*<sup>R</sup>* ∪ Γ6,*<sup>R</sup>* ∪ Γ*<sup>R</sup>* \ Γ2,*R*.

If *<sup>A</sup>* ∈ A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0), then following Theorem 1 with X = L(Z), the operators family {*Z*0(*t*) ∈ L(Z) : *t* > 0} is analytic of the type (*θ*<sup>0</sup> − *π*/2, *a*0, −*σ*0), it implies point (i) of Definition 1, and point (ii) of this definition is evidently fulfilled.

For any *θ* ∈ (*π*/2, *θ*0), *a* > *a*0, we have such a *K*(*θ*, *a*) > 0, that for every *λ* ∈ *Sθ*,*<sup>a</sup>*

$$\left\| \lambda^{\sigma\_{\mathfrak{n}} - \sigma\_{\mathfrak{k}} - 1} R\_{\lambda^{\sigma\_{\mathfrak{n}}}}(A) \right\|\_{\mathcal{L}(\mathcal{Z})} \leq C \left\| \lambda^{\sigma\_{\mathfrak{n}} - \sigma\_{\mathfrak{0}} - 1} R\_{\lambda^{\sigma\_{\mathfrak{n}}}}(A) \right\|\_{\mathcal{L}(\mathcal{Z})} \leq \frac{C K(\theta, a)}{|\lambda - a|^{a\_0}}.$$

So, for *<sup>k</sup>* <sup>=</sup> 0, 1, ... , *<sup>n</sup>* <sup>−</sup> 1, Re*<sup>λ</sup>* <sup>&</sup>gt; *<sup>a</sup>*<sup>0</sup> there exists the Laplace transforms *<sup>Z</sup>k*(*λ*) = *λσn*−*σk*−1*Rλσ<sup>n</sup>* (*A*), *J β <sup>t</sup> Zk*(*λ*) = *<sup>λ</sup>σn*−*σk*−1−*βRλσ<sup>n</sup>* (*A*), *<sup>β</sup>* > 0, therefore, *Zk*(*t*) = *<sup>J</sup> σk*−*σ*<sup>0</sup> *<sup>t</sup> Z*0(*t*). For *z*<sup>0</sup> ∈ *DA*

$$D\_t^{\alpha\_0 - 1} Z\_0(t) z\_0 = \frac{1}{2\pi i} \int\_{\Gamma} \lambda^{\sigma\_n - 1} R\_{\lambda^{\sigma\_n}}(A) e^{\lambda t} z\_0 d\lambda = 0$$

$$\mathcal{I} = \frac{1}{2\pi i} \int\_{\Gamma} \frac{e^{\lambda t}}{\lambda} z\_0 d\lambda + \frac{1}{2\pi i} \int\_{\Gamma} \lambda^{-1} \mathcal{R}\_{\lambda^{\varepsilon\_0}}(A) e^{\lambda t} A z\_0 d\lambda = z\_0 + \frac{1}{2\pi i} \int\_{\Gamma} \lambda^{-1} \mathcal{R}\_{\lambda^{\varepsilon\_0}}(A) e^{\lambda t} A z\_0 d\lambda.$$
 
$$\text{If } t \in [0, 1], \lambda \in \Gamma \text{ } \{\mu \in \mathbb{C} : |\mu| \le 2a\}, \text{ then}$$

$$\left\|\lambda^{-1}R\_{\lambda^{\sigma\_{\rm II}}}(A)e^{\lambda t}Az\_{0}\right\|\_{\mathcal{Z}} \leq \frac{e^{a+\delta}K(\theta,a)\|Az\_{0}\|\_{\mathcal{Z}}}{|\lambda-a|^{a\_{0}}|\lambda|^{\sigma\_{\rm II}-\sigma\_{0}}} \leq \frac{C\_{1}}{|\lambda|^{\sigma\_{\rm II}+1}}.$$

Hence,

$$\frac{1}{2\pi i} \int\_{\Gamma} \lambda^{-1} R\_{\lambda^{\varepsilon\_{\Pi}}}(A) e^{\lambda t} A z\_0 d\lambda = $$
 
$$ = \lim\_{R \to \infty} \frac{1}{2\pi i} \left( \int\_R - \int\_{\Gamma\_{2,R}} + \int\_{\Gamma\_{5,R}} + \int\_{\Gamma\_{6,R}} \right) \lambda^{-1} R\_{\lambda^{\varepsilon\_{\Pi}}}(A) e^{\lambda t} A z\_0 d\lambda = 0, $$

since by the Cauchy theorem

$$\left\| \int\_{\Gamma\_R} \lambda^{-1} R\_{\lambda^{\oplus n}}(A) e^{\lambda t} A z\_0 d\lambda = 0, \quad \left\| \int\_{\Gamma\_{s,R}} \lambda^{-1} R\_{\lambda^{\oplus n}}(A) e^{\lambda t} A z\_0 d\lambda \right\|\_{\mathcal{Z}} \le \frac{C\_2}{R^{\sigma\_n}} \to 0$$

as *R* → ∞ for *s* = 2, 5, 6.

At the same time, due equality (5)

$$\text{Lap}[D^{\sigma\_1} Z\_0(\cdot) z\_0](\lambda) = \lambda^{\sigma\_0 - \sigma\_0 - 1 + \sigma\_1} R\_{\lambda^{\sigma\_0}}(A) z\_0 - \lambda^{\sigma\_1 - \sigma\_0 - 1} z\_0 = \lambda^{a\_1 - 1} R\_{\lambda^{\sigma\_0}}(A) A z\_0.$$

for *<sup>λ</sup>* <sup>∈</sup> <sup>Γ</sup> \ {*<sup>μ</sup>* <sup>∈</sup> <sup>C</sup> : <sup>|</sup>*μ*| ≤ <sup>2</sup>*a*}

$$\|\lambda^{a\_1 - 1} R\_{\lambda^{\sigma\_\Pi}}(A) A z\_0\|\_{\mathcal{Z}} \le \frac{\mathsf{C}\_3}{|\lambda|^{\sigma\_\Pi - \sigma\_0 + a\_0 - a\_1}} = \frac{\mathsf{C}\_3}{|\lambda|^{a\_0 + a\_2 + a\_3 + \dots + a\_n}}\lambda$$

*<sup>α</sup>*<sup>0</sup> <sup>+</sup> *<sup>α</sup>*<sup>2</sup> <sup>+</sup> *<sup>α</sup>*<sup>3</sup> <sup>+</sup> ··· <sup>+</sup> *<sup>α</sup><sup>n</sup>* <sup>&</sup>gt; *<sup>α</sup>*<sup>0</sup> <sup>+</sup> *<sup>α</sup><sup>n</sup>* <sup>&</sup>gt; 1, hence *<sup>D</sup>σ*1*Z*0(0)*z*<sup>0</sup> <sup>=</sup> 0. Further, for every *k* = 2, 3, . . . , *n* − 1

$$\operatorname{Lap}[D^{\sigma\_k} Z\_0(\cdot) z\_0](\lambda) = \lambda^{\sigma\_k - \sigma\_0 - 1 + \sigma\_k} R\_{\lambda^{\sigma\_k}}(A) z\_0 - \lambda^{\sigma\_k - \sigma\_0 - 1} z\_0 = \lambda^{\sigma\_k - \sigma\_0 - 1} R\_{\lambda^{\sigma\_k}}(A) A z\_0.$$

for *<sup>λ</sup>* <sup>∈</sup> <sup>Γ</sup> \ {*<sup>μ</sup>* <sup>∈</sup> <sup>C</sup> : <sup>|</sup>*μ*| ≤ <sup>2</sup>*a*} *<sup>λ</sup>σk*−*σ*0−1*Rλσ<sup>n</sup>* (*A*)*Az*<sup>0</sup> <sup>Z</sup> <sup>≤</sup> *<sup>C</sup>*<sup>3</sup> <sup>|</sup>*λ*|*σn*−*σk*+*α*<sup>0</sup> <sup>=</sup> *<sup>C</sup>*<sup>3</sup> |*λ*| *<sup>α</sup>*0+*αk*+1+*αk*+2+···+*α<sup>n</sup>* , thus, *DσkZ*0(0)*z*<sup>0</sup> = 0. Finally,

$$\operatorname{Lap}[D^{\sigma\_{\overline{n}}}Z\_0(\cdot)z\_0](\lambda) = \lambda^{\sigma\_{\overline{n}}-\sigma\_0-1+\sigma\_{\overline{n}}}R\_{\lambda^{\sigma\_{\overline{n}}}}(A)z\_0 - \lambda^{\sigma\_{\overline{n}}-\sigma\_0-1}z\_0 = A\lambda^{\sigma\_{\overline{n}}-\sigma\_0-1}R\_{\lambda^{\sigma\_{\overline{n}}}}(A)z\_0.$$

Acting on the inverse Laplace transform, we get the equality *DσnZ*0(*t*)*z*<sup>0</sup> = *AZ*0(*t*)*z*0, so {*Z*0(*t*) ∈ L(Z) : *t* > 0} is a zero-resolving family of operators for Equation (7). Then, by Proposition 2 for every *k* = 1, 2, ... , *n* − 1, there exists a *k*-resolving family of operators, which coincide with operators *J σk*−*σ*<sup>0</sup> *<sup>t</sup> Z*0(*t*) = *Zk*(*t*). Every such family is analytic with the type (*θ*<sup>0</sup> − *π*/2, *a*0, max{−*σk*, 0}); see the proof of Proposition 2 and Remark 3.

If there exists a zero-resolving family with the type (*θ*<sup>0</sup> − *π*/2, *a*0, −*σ*0), equality (8) at *<sup>k</sup>* = 0 and Theorem 1 with X = L(Z) implies that *<sup>A</sup>* ∈ A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0).

**Remark 4.** *Note that σ<sup>n</sup>* > 0*, if α*<sup>0</sup> + *α<sup>n</sup>* > 1*.*

**Remark 5.** *An analogous for Theorem 3 result on the first-order equations is called the Solomyak– Yosida theorem on generation of analytic semigroups of operators [24,25]. Previously, similar results were obtained for evolutionary integral equations [26], differential equations with a Gerasimov– Caputo fractional derivative [27], with a Riemann–Liouville derivative [28], for multi-term linear fractional differential equations in Banach spaces [29], and equations with distributed fractional derivatives [30,31,33,34].*

**Corollary 1.** *Let <sup>A</sup>* ∈ A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0)*, <sup>α</sup><sup>k</sup>* ∈ (0, 1]*, <sup>k</sup>* = 0, 1, ... , *n, <sup>α</sup>*<sup>0</sup> + *<sup>α</sup><sup>n</sup>* > <sup>0</sup>*, <sup>θ</sup>*<sup>0</sup> ∈ (*π*/2, *<sup>π</sup>*]*, a*<sup>0</sup> ≥ 0*. Then, for any z*0, *z*1, ... , *zn*−<sup>1</sup> ∈ *DA problem* (6)*,* (7) *has a unique solution, and it has the form*

$$z(t) = \sum\_{k=0}^{n-1} Z\_k(t) z\_k.$$

*The solution is analytic in* <sup>Σ</sup>*θ*0−*π*/2*.*

**Proof.** After Theorem 3, we need to prove the uniqueness of a solution only. If problem (6), (7) has two solutions *y*1, *y*2, then the difference *y* = *y*<sup>1</sup> − *y*<sup>2</sup> is a solution of (7) with the initial conditions *<sup>D</sup>σky*(0) = 0, *<sup>k</sup>* <sup>=</sup> 0, 1, ... , *<sup>n</sup>* <sup>−</sup> 1. Redefine *<sup>y</sup>* on (*T*, <sup>∞</sup>) for any *<sup>T</sup>* <sup>&</sup>gt; 0 as a zero function. The got function *yT* satisfies equality (7) at *t* > 0 without the point *T*. Using the Laplace transform obtained from Equation (7) and zero initial conditions, the equality *<sup>λ</sup>σ<sup>n</sup> <sup>y</sup>T*(*λ*) = *AyT*(*λ*). Since *<sup>A</sup>* ∈ A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0), we have *<sup>y</sup>T*(*λ*) <sup>≡</sup> 0 for *<sup>λ</sup>* <sup>∈</sup> *<sup>S</sup>θ*0,*a*<sup>0</sup> . Therefore, *yT* <sup>≡</sup> 0 for arbitrary *<sup>T</sup>* <sup>&</sup>gt; 0, hence *<sup>y</sup>* <sup>≡</sup> 0 on <sup>R</sup><sup>+</sup> and a solution of problem (6), (7) is unique.

**Remark 6.** *For A* ∈ L(Z) *the k-resolving operators of Equation* (7) *have the form (see [22])*

$$Z\_k(t) = t^{\sigma\_k} E\_{\sigma\_n \sigma\_k + 1}(t^{\sigma\_n} A), \quad t \in \mathbb{S}\_{\pi, 0}, \quad k = 0, 1, \ldots, n - 1.$$

*Here, according to Eβ*,*<sup>γ</sup> the Mittag–Leffler function is denoted. Indeed, decomposing the resolvent Rσ<sup>n</sup>* (*A*) *in the series for large enough* |*λ*| *and using the Hankel integral, we obtain these equalities.*

**Theorem 4.** *Let <sup>A</sup>* ∈ A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0), *<sup>α</sup><sup>k</sup>* ∈ (0, 1]*, <sup>k</sup>* = 0, 1, ... , *n, <sup>α</sup>*<sup>0</sup> + *<sup>α</sup><sup>n</sup>* > <sup>0</sup>*, <sup>σ</sup><sup>n</sup>* ≥ <sup>2</sup>*, <sup>θ</sup>*<sup>0</sup> ∈ (*π*/2, *π*]*, a*<sup>0</sup> ≥ 0*. Then A* ∈ L(Z)*.*

**Proof.** For some *<sup>ν</sup>*<sup>0</sup> <sup>∈</sup> <sup>C</sup>, such that <sup>|</sup>*ν*0| ≥ *<sup>R</sup>σ<sup>n</sup>* , take *<sup>λ</sup>*<sup>0</sup> <sup>=</sup> *<sup>ν</sup>*1/*σ<sup>n</sup>* <sup>0</sup> , hence |*λ*0| ≥ *R*, arg *λ*<sup>0</sup> = arg *ν*0/*σ<sup>n</sup>* ∈ [−*π*/2, *π*/2], since *σ<sup>n</sup>* ≥ 2. Then, *λ*<sup>0</sup> ∈ *Sθ*0,*a*<sup>0</sup> for sufficiently large *R* > 0. Therefore, {*<sup>ν</sup>* <sup>∈</sup> <sup>C</sup> : <sup>|</sup>*ν*| ≥ *<sup>R</sup>σ<sup>n</sup>* } ⊂ [*Sθ*0,*a*<sup>0</sup> ] *<sup>σ</sup><sup>n</sup>* <sup>⊂</sup> *<sup>ρ</sup>*(*A*), since *<sup>A</sup>* ∈ A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0). Here, we use the principal branch of the power function.

So, for <sup>|</sup>*ν*| ≥ *<sup>R</sup>σ<sup>n</sup>* , where *<sup>ν</sup>* <sup>=</sup> *<sup>λ</sup>σ<sup>n</sup>* ,

$$||\nu R\_{\boldsymbol{\nu}}(A)||\_{\mathcal{L}(\mathcal{Z})} \le \frac{K(\theta, a)|\lambda|^{\sigma\_0+1}}{|\lambda - a|^{\alpha\_0}} \le C$$

and by Lemma 5.2 [36] the operator *A* is bounded.

**Remark 7.** *For strongly continuous resolving families of the equation with a Gerasimov–Caputo derivative, such a result was proved in [27].*

#### **5. Inhomogeneous Equation**

Let *f* ∈ *C*([0, *T*]; Z). Consider the equation

$$D^{\sigma\_0}z(t) = Az(t) + f(t), \quad t \in (0, T]. \tag{10}$$

A solution of the initial value problem

$$D^{\varphi\_k}z(0) = z\_{k\prime} \quad k = 0, 1, \ldots, n-1,\tag{11}$$

to Equation (10) is a function *<sup>z</sup>* <sup>∈</sup> *<sup>C</sup>*((0, *<sup>T</sup>*]; *DA*), such that *<sup>D</sup>σ<sup>k</sup> <sup>z</sup>* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*]; <sup>Z</sup>), *<sup>k</sup>* <sup>=</sup> 0, 1, ... , *<sup>n</sup>* <sup>−</sup> 1, *<sup>D</sup>σ<sup>n</sup> <sup>z</sup>* <sup>∈</sup> *<sup>C</sup>*((0, *<sup>T</sup>*]; <sup>Z</sup>), for all *<sup>t</sup>* <sup>∈</sup> (0, *<sup>T</sup>*] equality (10) is fulfilled and conditions (11) are valid.

Denote

$$Z(t) = \frac{1}{2\pi i} \int\_{\Gamma} \mathcal{R}\_{\lambda^{\boxtimes \mathfrak{n}}}(A) e^{\lambda t} d\lambda, \quad \mathcal{Y}\_{\beta}(t) = \frac{1}{2\pi i} \int\_{\Gamma} \lambda^{\beta} \mathcal{R}\_{\lambda^{\boxtimes \mathfrak{n}}}(A) e^{\lambda t} d\lambda, \quad \beta \in \mathbb{R}.$$

**Lemma 1.** *Let <sup>A</sup>* ∈ A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0)*, <sup>α</sup><sup>k</sup>* ∈ (0, 1]*, <sup>k</sup>* = 0, 1, ... , *n, <sup>α</sup>*<sup>0</sup> + *<sup>α</sup><sup>n</sup>* > <sup>0</sup>*, <sup>θ</sup>*<sup>0</sup> ∈ (*π*/2, *<sup>π</sup>*]*, a*<sup>0</sup> ≥ 0*, f* ∈ *C*([0, *T*]; *DA*)*. Then,*

$$z\_f(t) = \int\_0^t Z(t-s)f(s)ds\tag{12}$$

*is a unique solution for the initial value problem*

$$D^{\sigma\_k}z(0) = 0, \quad k = 0, 1, \ldots, n - 1,\tag{13}$$

*to* (10)*.*

**Proof.** Since *<sup>A</sup>* ∈ A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0), for sufficiently large |*λ*| *Rλσ<sup>n</sup>* (*A*) <sup>L</sup>(Z) ≤ *C*|*λ*| <sup>−</sup>*σ<sup>n</sup>* , hence for Re*<sup>λ</sup>* <sup>&</sup>gt; *<sup>a</sup>*<sup>0</sup> *<sup>Z</sup>*(*λ*) = *<sup>R</sup>λσ<sup>n</sup>* (*A*), *<sup>D</sup> σ*0*Z*(*λ*) = *<sup>λ</sup>σ*0*Rλσ<sup>n</sup>* (*A*), *Z*(*t*) <sup>L</sup>(Z) <sup>≤</sup> *Ctσn*<sup>−</sup>1, *<sup>D</sup>σ*0*Z*(*t*) <sup>L</sup>(Z) <sup>≤</sup> *Ctσn*−*σ*0−<sup>1</sup> for *<sup>t</sup>* <sup>∈</sup> (0, *<sup>T</sup>*]. Analogously, *Yβ*(*t*) <sup>L</sup>(Z) <sup>≤</sup> *Ctσn*−*β*−<sup>1</sup> for *<sup>t</sup>* <sup>∈</sup> (0, *<sup>T</sup>*], *<sup>β</sup>* <sup>∈</sup> <sup>R</sup>.

Further,

$$\| |D^{\sigma\_0} z\_f(t)| \|\_{Z} = \left\| \int\_0^t \chi\_{\sigma\_0}(t - s) f(s) ds \right\|\_{Z} \le \mathbb{C} \max\_{s \in [0, T]} \| f(s) \|\_{Z} t^{\sigma\_0 - \sigma\_0} \sqrt{\rho}$$

hence *<sup>D</sup>σ*<sup>0</sup> *<sup>z</sup> <sup>f</sup>*(0) = 0. Define *<sup>f</sup>* by zero outside the segment [0, *<sup>T</sup>*]; then, *<sup>z</sup> <sup>f</sup>* <sup>=</sup> *<sup>Z</sup>* <sup>∗</sup> *<sup>f</sup>* , *<sup>z</sup> <sup>f</sup>*(*λ*) = *<sup>Z</sup>*(*λ*)*<sup>f</sup>* (*λ*) = *<sup>R</sup>λσ<sup>n</sup>* (*A*)*<sup>f</sup>* (*λ*), *<sup>D</sup> σ*<sup>1</sup> *<sup>z</sup> <sup>f</sup>*(*λ*) = *<sup>λ</sup>σ*1*Rλσ<sup>n</sup>* (*A*)*<sup>f</sup>* (*λ*),

$$\|D^{\sigma\_1}z\_f(t)\|\_{\mathcal{Z}} = \left\|\int\_0^t \mathbf{Y}\_{\sigma\_1}(t-s)f(s)ds\right\|\_{\mathcal{Z}} \le \mathbb{C} \max\_{s\in[0,T]} \|f(s)\|\_{\mathcal{Z}} t^{\sigma\_n-\sigma\_1}, \quad t\in(0,T], \mathcal{Z}$$

*Dσ*<sup>1</sup> *z <sup>f</sup>*(0) = 0. Repeating the analogous reasoning sequentially, we get *k* = 2, 3, ... , *n* − 1 *D σ<sup>k</sup> <sup>z</sup> <sup>f</sup>*(*λ*) = *<sup>λ</sup>σkRλσ<sup>n</sup>* (*A*)*<sup>f</sup>* (*λ*), *<sup>D</sup>σ<sup>k</sup> <sup>z</sup> <sup>f</sup>*(*t*) <sup>L</sup>(Z) ≤ *C* max *s*∈[0,*T*] *f*(*s*) Z *t σn*−*σ<sup>k</sup>*

for *<sup>t</sup>* <sup>∈</sup> (0, *<sup>T</sup>*], *<sup>D</sup>σ<sup>k</sup> <sup>z</sup> <sup>f</sup>*(0) = 0, *<sup>D</sup> σ<sup>n</sup> <sup>z</sup> <sup>f</sup>*(*λ*) = *<sup>λ</sup>σnRλσ<sup>n</sup>* (*A*)*<sup>f</sup>* (*λ*). Since *f* ∈ *C*([0, *T*]; *DA*), we have

$$
\widehat{A}\widehat{z}\_f(\lambda) = \widehat{z}\_{Af}(\lambda) = AR\_{\lambda^{\sigma\_\Omega}}(A)\widehat{f}(\lambda) = \lambda^{\sigma\_\Omega}R\_{\lambda^{\sigma\_\Omega}}(A)\widehat{f}(\lambda) - \widehat{f}(\lambda),
$$

so, *Az <sup>f</sup>*(*t*) = *<sup>D</sup>σ<sup>n</sup> <sup>z</sup> <sup>f</sup>*(*t*) <sup>−</sup> *<sup>f</sup>*(*t*) for all *<sup>t</sup>* <sup>&</sup>gt; 0. Thus, the function *<sup>z</sup> <sup>f</sup>* satisfies equality (10). The proof of a solution's uniqueness is the same as for the homogeneous equation.

Let *<sup>C</sup>γ*([0, *<sup>T</sup>*]; <sup>Z</sup>) for some *<sup>γ</sup>* <sup>∈</sup> (0, 1] be the set of all functions *<sup>f</sup>* : [0, *<sup>T</sup>*] → Z, satisfying the Hölder condition:

$$\exists \mathcal{C} > 0 \quad \forall s, t \in [0, T] \quad ||f(s) - f(t)||\_{\mathcal{Z}} \le \mathcal{C}|s - t|^\gamma.$$

**Lemma 2.** *Let <sup>A</sup>* ∈ A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0)*, <sup>α</sup><sup>k</sup>* ∈ (0, 1]*, <sup>k</sup>* = 0, 1, ... , *n, <sup>α</sup>*<sup>0</sup> + *<sup>α</sup><sup>n</sup>* > <sup>0</sup>*, <sup>θ</sup>*<sup>0</sup> ∈ (*π*/2, *<sup>π</sup>*]*, <sup>a</sup>*<sup>0</sup> <sup>≥</sup> <sup>0</sup>*, <sup>f</sup>* <sup>∈</sup> *<sup>C</sup>γ*([0, *<sup>T</sup>*]; <sup>Z</sup>)*, <sup>γ</sup>* <sup>∈</sup> (0, 1]*. Then, problem* (10)*,* (13) *has a unique solution; it has form* (12)*.*

**Proof.** Since *A* is closed,

$$AZ(t) = \frac{1}{2\pi i} \int\_{\Gamma} AR\_{\lambda^{\mathbb{F}\_{\mathbb{R}}}}(A)e^{\lambda t} d\lambda = \frac{1}{2\pi i} \int\_{\Gamma} \lambda^{\sigma\_{\mathbb{R}}} R\_{\lambda^{\mathbb{F}\_{\mathbb{R}}}}(A)e^{\lambda t} d\lambda = \mathcal{Y}\_{\sigma\_{\mathbb{R}}}(t), \quad t > 0,$$

therefore, im*Z*(*t*) ⊂ *DA*, as *t* → 0+ *AZ*(*t*) <sup>L</sup>(Z) = *<sup>O</sup>*(*<sup>t</sup>* <sup>−</sup>1) (see the previous proof). Therefore, for all *t*,*s* ∈ (0, *T*]

$$||AZ(t-s)(f(s)-f(t))||\_{\mathcal{Z}} \le \mathcal{C}|t-s|^{\gamma-1}.$$

Then

$$\int\_0^t A Z(t-s) f(s) ds = \int\_0^t A Z(t-s) (f(s) - f(t)) ds + \int\_0^t Y\_{\sigma\_n}(t-s) f(t) ds,$$

$$\int\_0^t Y\_{\sigma\_n}(t-s) f(t) ds = -\int\_0^t D\_s^1 Y\_{\sigma\_n - 1}(t-s) f(t) ds = (Y\_{\sigma\_n - 1}(t) - Y\_{\sigma\_n - 1}(0)) f(t).$$

Note that for any *x* ∈ *DA*

$$\mathcal{Y}\_{\sigma\_n - 1}(t)\mathfrak{x} = \mathfrak{x} + \frac{1}{2\pi i} \int\_{\Gamma} \lambda^{-1} \mathcal{R}\_{\lambda^{\sigma\_n}}(A) e^{\lambda t} A \mathfrak{x} d\lambda \to \mathfrak{x}, \quad t \to 0+,$$

since for large enough |*λ*| *<sup>λ</sup>*−1*Rλσ<sup>n</sup>* (*A*)*Ax* <sup>Z</sup> ≤ *C Ax* <sup>Z</sup> |*λ*| <sup>−</sup>*σn*<sup>−</sup>1. At the same time, for sufficiently large |*λ*| *<sup>λ</sup>σn*−1*Rλσ<sup>n</sup>* (*A*) <sup>L</sup>(Z) ≤ *C*|*λ*| <sup>−</sup>1; therefore, the family {*Yσn*−1(*t*) ∈ L(Z) : *t* > 0} is bounded uniformly. Since *DA* is dense in Z, for every *x* ∈ Z lim *t*→0+ *Yσn*−1(*t*)*x* = *x*.

Thus,

$$\begin{aligned} \left\| \int\_0^t A Z(t-s) f(s) ds \right\|\_{\mathcal{Z}} &\leq \mathsf{C}\_1 t^{\gamma} + \left\| Y\_{\sigma\_n - 1}(t) - Y\_{\sigma\_n - 1}(0) \right\|\_{\mathcal{L}(\mathcal{Z})} \left\| f(t) - f(0) \right\|\_{\mathcal{Z}} + \epsilon \\ &\quad + \left\| (Y\_{\sigma\_n - 1}(t) - Y\_{\sigma\_n - 1}(0)) f(0) \right\|\_{\mathcal{Z}} \leq \\ &\leq \mathsf{C}\_1 t^{\gamma} + \mathsf{C}\_2 \left\| f(t) - f(0) \right\|\_{\mathcal{Z}} + \left\| (Y\_{\sigma\_n - 1}(t) - Y\_{\sigma\_n - 1}(0)) f(0) \right\|\_{\mathcal{Z}} \to 0 \end{aligned}$$

as *t* → 0+. Therefore, *z <sup>f</sup>*(*t*) ∈ *DA*, *z <sup>f</sup>* ∈ *C*([0, *T*]; *DA*). Other arguing is the same as in the proof of the previous lemma.

Corollary 1, Lemma 1 and Lemma 2 imply the following result.

**Theorem 5.** *Let <sup>A</sup>* ∈ A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0)*, <sup>α</sup><sup>k</sup>* ∈ (0, 1]*, <sup>k</sup>* = 0, 1, ... , *n, <sup>α</sup>*<sup>0</sup> + *<sup>α</sup><sup>n</sup>* > <sup>0</sup>*, <sup>θ</sup>*<sup>0</sup> ∈ (*π*/2, *<sup>π</sup>*]*, <sup>a</sup>*<sup>0</sup> <sup>≥</sup> <sup>0</sup>*, <sup>γ</sup>* <sup>∈</sup> (0, 1]*, <sup>f</sup>* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*]; *DA*) <sup>∪</sup> *<sup>C</sup>γ*([0, *<sup>T</sup>*]; <sup>Z</sup>)*. Then problem* (10)*,* (11) *has a unique solution, it has the form*

$$z(t) = \sum\_{k=0}^{n-1} Z\_k(t)z\_k + \int\_0^t Z(t-s)f(s)ds.$$

#### **6. Perturbation Theorem**

**Theorem 6.** *Let <sup>A</sup>* ∈ A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0)*, <sup>α</sup><sup>k</sup>* ∈ (0, 1]*, <sup>k</sup>* = 0, 1, ... , *n, <sup>α</sup>*<sup>0</sup> + *<sup>α</sup><sup>n</sup>* > <sup>0</sup>*, <sup>θ</sup>*<sup>0</sup> ∈ (*π*/2, *<sup>π</sup>*]*, a*<sup>0</sup> ≥ 0*, B* ∈ C*l*(Z)*, for some β*, *γ* ≥ 0

$$\|\|B\mathbf{x}\|\|\_{\mathcal{Z}} \le \beta \|\|A\mathbf{x}\|\|\_{\mathcal{Z}} + \gamma \|\|\mathbf{x}\|\|\_{\mathcal{Z}'} \quad \mathbf{x} \in D\_A \subset D\_{B'} \tag{14}$$

*there exists q* ∈ (0, 1)*, such that β*(1 + *K*(*θ*, *a*)) < *q for all θ* ∈ (*π*/2, *θ*0)*, a* > *a*0*. Then, <sup>A</sup>* + *<sup>B</sup>* ∈ A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*1) *for sufficiently large a*<sup>1</sup> > *<sup>a</sup>*0*.*

**Proof.** Choose *<sup>l</sup>* <sup>&</sup>gt; sin−<sup>1</sup> *<sup>θ</sup>*0, *<sup>λ</sup>* <sup>∈</sup> *<sup>S</sup>θ*,*la* <sup>⊂</sup> *<sup>S</sup>θ*,*<sup>a</sup>* for some *<sup>θ</sup>* <sup>∈</sup> (*π*/2, *<sup>θ</sup>*0), *<sup>a</sup>* <sup>&</sup>gt; *<sup>a</sup>*0, then from (14), it follows that

$$\begin{aligned} \|\beta R\_{\lambda^{\sigma\_0}}(A)\|\_{\mathcal{L}(\mathcal{Z})} &\leq \beta \|\langle AR\_{\lambda^{\sigma\_0}}(A)\rangle\|\_{\mathcal{L}(\mathcal{Z})} + \gamma \|R\_{\lambda^{\sigma\_0}}(A)\|\_{\mathcal{L}(\mathcal{Z})} \leq \beta \\ &\leq \beta \left(1 + \frac{|\lambda|^{\sigma\_0+1} K\_A(\theta, a)}{|\lambda - a|^{a\_0}}\right) + \frac{\gamma K\_A(\theta, a)}{|\lambda - a|^{a\_0} |\lambda|^{\sigma\_0 - \sigma\_0 - 1}}. \end{aligned}$$

where *KA*(*θ*, *a*) is the constant from Definition 2. Note that the value

$$\frac{|\lambda|^{\alpha\_0}}{|\lambda - a|^{\alpha\_0}} \le \frac{1}{\left(1 - \frac{a}{|\lambda|}\right)^{\alpha\_0}} \le \frac{1}{\left(1 - \frac{1}{l\sin\theta\_0}\right)^{\alpha\_0}}$$

is close to one, for a sufficiently large number *l*

$$\frac{|\lambda|^{\alpha\_0}}{|\lambda - a|^{\alpha\_0} |\lambda|^{\sigma\_n}} \le \frac{1}{\left(1 - \frac{1}{l \sin \theta\_0}\right)^{\alpha\_0} (la\_0 \sin \theta\_0)^{\sigma\_n}}$$

is close to zero. So, for such a *l*, we have

$$\|\|BR\_{\lambda^{\varrho\_{\mathbb{Z}}}}(A)\|\|\_{\mathcal{L}(\mathcal{Z})} \le \beta \left(1 + \frac{K\_A(\theta, a)}{\left(1 - \frac{1}{I \sin \theta\_0}\right)^{a\_0}}\right) + \frac{\gamma K\_A(\theta, a)}{\left(1 - \frac{1}{I \sin \theta\_0}\right)^{a\_0} (la\_0 \sin \theta\_0)^{\sigma\_{\mathbb{Z}}}} \le \frac{\gamma}{\beta}$$

$$\le \beta (1 + K(\theta, a)) + \varepsilon \le q < 1.$$

Therefore,

$$R\_{\lambda^{\sigma\_{\mathbb{H}}}}(A+B) = R\_{\lambda^{\sigma\_{\mathbb{H}}}}(A)(I - BR\_{\lambda^{\sigma\_{\mathbb{H}}}}(A))^{-1} = R\_{\lambda^{\sigma\_{\mathbb{H}}}}(A) \sum\_{k=0}^{\infty} [BR\_{\lambda^{\sigma\_{\mathbb{H}}}}(A)]^k.$$

$$\frac{|\lambda - la|}{|\lambda - a|} = \left| 1 - \frac{(l-1)a}{\lambda - a} \right| \le 1 + \frac{(l-1)a}{|\lambda - a|} < 1 + \frac{1}{\sin \theta\_0}.$$

$$\|\|R\_{\lambda^{\mathfrak{D}\_{1}}}(A+B)\|\|\_{\mathcal{L}(\mathcal{Z})} \leq \frac{K\_{A}(\theta,a)}{(1-q)|\lambda-a|^{a\_{0}}|\lambda|^{\varpi\_{n}-\varpi\_{0}-1}} \leq \frac{K\_{A}(\theta,a)\left(1+\frac{1}{\sin\theta\_{0}}\right)^{a\_{0}}}{(1-q)|\lambda-ia|^{a\_{0}}|\lambda|^{\varpi\_{n}-\varpi\_{0}-1}}.$$
 
$$\text{So, } A+B \in \mathcal{A}\_{\{a\_{k}\}}(\theta\_{0},a\_{1}) \text{ with } a\_{1} = la\_{0}, \text{ for all } \theta \in (\pi/2,\theta\_{0}), a > a\_{1}$$
 
$$\dots \text{ } \dots \text{ } \dots$$

$$K\_{A+B}(\theta\_\prime a) = \frac{K\_A(\theta\_\prime a/l)}{1-q} \left(1 + \frac{1}{\sin \theta\_0} \right)^{\kappa\_0}.$$

**Remark 8.** *For every B* ∈ L(Z) *condition,* (14) *is satisfied with β* = 0*, γ* = *B* <sup>L</sup>(Z)*.*

**Remark 9.** *Theorem 6 generalizes the similar theorem for generators of analytic semigroups of operators [37]. Note that there are also analogous results for generators of resolving families for equations with distributed fractional derivatives in [30].*

#### **7. Application to a Model of a Viscoelastic Oldroyd Fluid**

Let *<sup>α</sup><sup>k</sup>* <sup>∈</sup> (0, 1], *<sup>k</sup>* <sup>=</sup> 0, 1, ... , *<sup>n</sup>*, *<sup>α</sup>*<sup>0</sup> <sup>+</sup> *<sup>α</sup><sup>n</sup>* <sup>&</sup>gt; 1, *<sup>σ</sup><sup>n</sup>* <sup>∈</sup> (0, 2), <sup>Ω</sup> <sup>⊂</sup> <sup>R</sup>*<sup>d</sup>* be a bounded region, which has a smooth boundary *∂*Ω. We consider a fractional linearized model of the viscoelastic Oldroyd fluid dynamics with the order *N* = 1 (see [38])

$$D^{\mathcal{P}\_k}v(\mathbf{x},0) = v\_k(\mathbf{x}), \; D^{\mathcal{P}\_k}w(\mathbf{x},0) = w\_k(\mathbf{x}), \; \mathbf{x} \in \Omega, \; k = 0, 1, \ldots, n - 1,\tag{15}$$

$$w(\mathbf{x},t) = 0, \quad w(\mathbf{x},t) = 0, \quad (\mathbf{x},t) \in \partial\Omega \times (0,T]. \tag{16}$$

$$D^{\sigma\_n}v = \mu \Delta v + \Delta w - \nabla p + \mathcal{g}\_{\prime} \quad (\ge, t) \in \Omega \times (0, T], \tag{17}$$

$$D^{\sigma\_n} w = bv + cw + h, \quad (\ge, t) \in \Omega \times (0, T], \tag{18}$$

$$
\nabla \cdot \boldsymbol{v} = 0, \quad \nabla \cdot \boldsymbol{w} = 0, \quad (\boldsymbol{x}, t) \in \Omega \times (0, T]. \tag{19}
$$

Here, *T* > 0, *Dσ<sup>k</sup>* , *k* = 0, 1, ... , *n*, are Dzhrbashyan–Nersesyan fractional derivatives with respect to time *t*, *x* = (*x*1, *x*2, ... , *xd*) are spatial variables, *v* = (*v*1, *v*2, ... , *vd*) is the fluid velocity vector, *w* = (*w*1, *w*2, ... , *wd*) is a function of memory for the velocity, which is defined by a Volterra integral with respect to *t* for *v*, ∇*p* = (*px*<sup>1</sup> , *px*<sup>2</sup> , ... , *pxd* ) is the pressure gradient of the fluid, Δ is the Laplace operator with respect to all the spatial variables, Δ*v* = (Δ*v*1, Δ*v*2, ... , Δ*vd*), Δ*w* = (Δ*w*1, Δ*w*2, ... , Δ*wd*), ∇ · *v* = *v*1*x*<sup>1</sup> + *v*2*x*<sup>2</sup> + ··· + *vdxd* , ∇ · *<sup>w</sup>* <sup>=</sup> *<sup>w</sup>*1*x*<sup>1</sup> <sup>+</sup> *<sup>w</sup>*2*x*<sup>2</sup> <sup>+</sup> ··· <sup>+</sup> *wdxd* . The constants *<sup>μ</sup>*, *<sup>b</sup>*, *<sup>c</sup>* <sup>∈</sup> <sup>R</sup> and the functions *<sup>g</sup>*, *<sup>h</sup>* : <sup>Ω</sup> <sup>×</sup> [0, *<sup>T</sup>*] <sup>→</sup> <sup>R</sup>*<sup>d</sup>* are given.

Take L<sup>2</sup> := (*L*2(Ω))*d*, H<sup>1</sup> := (*W*<sup>1</sup> <sup>2</sup> (Ω))*d*, <sup>H</sup><sup>2</sup> := (*W*<sup>2</sup> <sup>2</sup> (Ω))*d*. The closure of <sup>L</sup> :<sup>=</sup> {*<sup>u</sup>* <sup>∈</sup> (*C*<sup>∞</sup> <sup>0</sup> (Ω))*<sup>d</sup>* : ∇ · *<sup>u</sup>* <sup>=</sup> <sup>0</sup>} in the norm of <sup>L</sup><sup>2</sup> will be denoted by <sup>H</sup>*σ*, and in the norm of the space H<sup>1</sup> by H<sup>1</sup> *<sup>σ</sup>*. We also denote H<sup>2</sup> *<sup>σ</sup>* := H<sup>1</sup> *<sup>σ</sup>* <sup>∩</sup> <sup>H</sup>2, <sup>H</sup>*<sup>π</sup>* is the orthogonal complement for <sup>H</sup>*<sup>σ</sup>* in the Hilbert space <sup>L</sup>2, <sup>Σ</sup> : <sup>L</sup><sup>2</sup> <sup>→</sup> <sup>H</sup>*σ*, <sup>Π</sup> :<sup>=</sup> *<sup>I</sup>* <sup>−</sup> <sup>Σ</sup> : <sup>L</sup><sup>2</sup> <sup>→</sup> <sup>H</sup>*<sup>π</sup>* are the projectors.

The operator *B* = ΣΔ, extended to a closed operator in the space H*<sup>σ</sup>* with the domain H2 *σ*, has a real, negative, discrete spectrum with finite multiplicities of eigenvalues, condensed at −∞ only [39]. Denote by {*λk*} eigenvalues of *B*, numbered in non-increasing order, taking into account their multiplicities. Then, {*ϕk*} will be used to denote the orthonormal system of eigenfunctions, which forms a basis in H*<sup>σ</sup>* [39].

In order for Equation (19) to be fulfilled, take <sup>Z</sup> <sup>=</sup> <sup>H</sup>*<sup>σ</sup>* <sup>×</sup> <sup>H</sup>*<sup>σ</sup>* and define in <sup>Z</sup> an operator

$$A = \left(\begin{array}{c} \mu B \\ bI \end{array}\begin{array}{c} B \\ cI \end{array}\right) \in \mathcal{Cl}(\mathcal{Z}), \quad D\_A = \mathbb{H}\_{\sigma}^2 \times \mathbb{H}\_{\sigma}^2. \tag{20}$$

**Theorem 7.** *Let <sup>α</sup><sup>k</sup>* <sup>∈</sup> (0, 1]*, <sup>k</sup>* <sup>=</sup> 0, 1, ... , *n, <sup>σ</sup><sup>n</sup>* <sup>∈</sup> [1, 2)*, <sup>μ</sup>* <sup>&</sup>gt; <sup>0</sup>*, <sup>b</sup>*, *<sup>c</sup>* <sup>∈</sup> <sup>R</sup>*,* <sup>Z</sup> <sup>=</sup> <sup>H</sup>*<sup>σ</sup>* <sup>×</sup> <sup>H</sup>*σ, the operator A be defined by* (20)*. Then, for some <sup>θ</sup>*<sup>0</sup> ∈ (*π*/2, *<sup>π</sup>*)*, a*<sup>0</sup> > <sup>0</sup> *<sup>A</sup>* ∈ A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0)*.*

**Proof.** Let *θ*<sup>0</sup> ∈ (*π*/2, *π*), *θ* ∈ (*π*/2, *θ*0), *a*<sup>0</sup> > 0, *a* > *a*0, then for *λ* ∈ *Sθ*,*<sup>a</sup>*

$$\frac{|\lambda - a|}{|\lambda|} \le 1 + \frac{a}{|\lambda|} \le 1 + \frac{1}{\sin \theta\_0}.$$

so,

$$\frac{1}{|\lambda|^{\sigma\_n}} = \frac{|\lambda - a|^{a\_0}}{|\lambda|^{a\_0}} \frac{1}{|\lambda - a|^{a\_0} |\lambda|^{\sigma\_n - a\_0}} \le \frac{\left(1 + \frac{1}{\sin \theta\_0}\right)^{a\_0}}{|\lambda - a|^{a\_0} |\lambda|^{\sigma\_n - \sigma\_0 - 1}}$$

and instead of estimates of the form *Rλσ<sup>n</sup>* (*A*) <sup>L</sup>(Z) <sup>≤</sup> *<sup>K</sup>* |*λ*−*a*| *<sup>α</sup>*<sup>0</sup> <sup>|</sup>*λ*<sup>|</sup> *<sup>σ</sup>n*−*σ*0−<sup>1</sup> , it will be enough to get inequalities *Rλσ<sup>n</sup>* (*A*) <sup>L</sup>(Z) <sup>≤</sup> *<sup>K</sup>* <sup>|</sup>*λ*|*σ<sup>n</sup>* .

Take *<sup>θ</sup>*<sup>0</sup> <sup>∈</sup> (*π*/2, *<sup>π</sup>*/*σn*), *<sup>a</sup>*<sup>0</sup> = (*l*|*c*|)1/*σ<sup>n</sup>* , where *<sup>l</sup>* <sup>&</sup>gt; 1 is sufficiently large, then for *λ* ∈ *Sθ*0,*a*<sup>0</sup>

$$
\lambda^{\sigma\_{\Pi}}I - A = \sum\_{k=1}^{\infty} \begin{pmatrix} \lambda^{\sigma\_{\Pi}} - \mu \lambda\_{k} & -\lambda\_{k} \\ -b & \lambda^{\sigma\_{\Pi}} - c \end{pmatrix} \langle \cdot, \rho\_{k} \rangle \rho\_{k\prime}
$$

$$
(\lambda^{\sigma\_{\Pi}}I - A)^{-1} = \sum\_{k=1}^{\infty} \begin{pmatrix} \frac{\lambda^{\sigma\_{\Pi}} - c}{(\lambda^{\sigma\_{\Pi}} - c)(\lambda^{\sigma\_{\Pi}} - \mu \lambda\_{k}) - b\lambda\_{k}} & \frac{\lambda\_{k}}{(\lambda^{\sigma\_{\Pi}} - c)(\lambda^{\sigma\_{\Pi}} - \mu \lambda\_{k}) - b\lambda\_{k}} \\ \frac{b}{(\lambda^{\sigma\_{\Pi}} - c)(\lambda^{\sigma\_{\Pi}} - \mu \lambda\_{k})} & \frac{\lambda^{\sigma\_{\Pi}} - \mu \lambda\_{k}}{(\lambda^{\sigma\_{\Pi}} - c)(\lambda^{\sigma\_{\Pi}} - \mu \lambda\_{k}) - b\lambda\_{k}} \end{pmatrix} \langle \cdot, \rho\_{k} \rangle \rho\_{k\prime}.
$$

Since *<sup>λ</sup>σ<sup>n</sup>* <sup>∈</sup> *<sup>S</sup>θ*0*σn*,*l*|*c*<sup>|</sup> for *<sup>λ</sup>* <sup>∈</sup> *<sup>S</sup>θ*0,*a*<sup>0</sup> , we have <sup>|</sup>*λσ<sup>n</sup>* <sup>−</sup> *<sup>c</sup>*| ≥ (*<sup>l</sup>* <sup>−</sup> <sup>1</sup>)|*c*<sup>|</sup> sin(*<sup>π</sup>* <sup>−</sup> *<sup>θ</sup>*0*σn*), for sufficiently large *<sup>l</sup>*, the value <sup>|</sup>*b*(*λσ<sup>n</sup>* <sup>−</sup> *<sup>c</sup>*)−1<sup>|</sup> is small enough and

$$\arg\left(\mu + \frac{b}{\lambda^{\sigma\_n} - c}\right) < \frac{1}{2}(\pi - \theta\_0 \sigma\_n).$$

Fix *<sup>l</sup>*, *<sup>a</sup>*<sup>0</sup> = (*l*|*c*|)1/*σ<sup>n</sup>* ; then, for *<sup>λ</sup>* <sup>∈</sup> *<sup>S</sup>θ*0,*a*<sup>0</sup> , we have *<sup>λ</sup>σ<sup>n</sup>* <sup>∈</sup> *<sup>S</sup>θ*0*σn*,*l*|*c*<sup>|</sup> <sup>⊂</sup> *<sup>S</sup>θ*0*σn*,0 and

$$\left|\frac{\lambda^{\sigma\_{n}}-c}{(\lambda^{\sigma\_{n}}-c)(\lambda^{\sigma\_{n}}-\mu\lambda\_{k})-b\lambda\_{k}}\right| = \frac{1}{\left|\lambda^{\sigma\_{n}}-\lambda\_{k}\left(\mu+\frac{b}{\lambda^{\sigma\_{n}}-c}\right)\right|} \leq \frac{1}{|\lambda|^{\sigma\_{n}}\sin\frac{\pi-\theta\_{0}\sigma\_{n}}{2}},$$

$$\left|\frac{\lambda\_{k}}{(\lambda^{\sigma\_{n}}-c)(\lambda^{\sigma\_{n}}-\mu\lambda\_{k})-b\lambda\_{k}}\right| = \frac{1}{\left|(\lambda^{\sigma\_{n}}-c)\left(\frac{\lambda^{\sigma\_{n}}}{\lambda\_{k}}-\mu\right)-b\right|} \leq$$

$$\leq \frac{1}{|\lambda|^{\sigma\_{n}}\sin(\pi-\theta\_{0}\sigma\_{n})\inf\_{k\in\mathbb{N},\lambda\in\mathcal{S}\_{0,\mathrm{a},0}}\left|\frac{\lambda^{\sigma\_{n}}}{\lambda\_{k}}-\mu\right|-b} \leq$$

$$\leq \frac{2}{|\lambda|^{\sigma\_{n}}\sin(\pi-\theta\_{0}\sigma\_{n})\inf\_{k\in\mathbb{N},\lambda\in\mathcal{S}\_{0,\mathrm{a},0}}\left|\frac{\lambda^{\sigma\_{n}}}{\lambda\_{k}}-\mu\right|} \!/\!/ $$

if we take *l*, such that

$$\begin{aligned} |b| &< \frac{l|\varepsilon|}{2} \sin^{\sigma\_{\mathfrak{n}}} \theta\_0 \sin(\pi - \theta\_0 \sigma\_{\mathfrak{n}}) \inf\_{k \in \mathbb{N}, \lambda \in S\_{\theta\_0, \mathfrak{a}\_0}} \left| \frac{\lambda^{\sigma\_{\mathfrak{n}}}}{\lambda\_k} - \mu \right| \le \varepsilon \\ &\le \frac{|\lambda|^{\sigma\_{\mathfrak{n}}}}{2} \sin(\pi - \theta\_0 \sigma\_{\mathfrak{n}}) \inf\_{k \in \mathbb{N}, \lambda \in S\_{\theta\_0, \mathfrak{a}\_0}} \left| \frac{\lambda^{\sigma\_{\mathfrak{n}}}}{\lambda\_k} - \mu \right|. \end{aligned}$$

Further, for large *<sup>k</sup>* <sup>∈</sup> <sup>N</sup>

$$\left| \frac{b}{(\lambda^{\sigma\_{\mathfrak{n}}} - c)(\lambda^{\sigma\_{\mathfrak{n}}} - \mu \lambda\_k) - b\lambda\_k} \right| \le 1$$

$$\begin{split} \leq & \left| \left( \lambda^{\sigma\_{\overline{n}}} - \frac{c + \mu \lambda\_{k} + \sqrt{\frac{(c - \mu \lambda\_{k})^{2} - 4b\lambda\_{k}}{2}}}{2} \right) \left( \lambda^{\sigma\_{\overline{n}}} - \frac{c + \mu \lambda\_{k} - \sqrt{\frac{(c - \mu \lambda\_{k})^{2} - 4b\lambda\_{k}}{2}}}{2} \right) \right|^{\frac{1}{2}} \\ \leq & \frac{|b|}{|\lambda|^{2\sigma\_{\overline{n}}} \sin^{2}(\pi - \theta\_{0}\sigma\_{\overline{n}})} \leq \frac{|b|(l|c|)^{-1} \sin^{-\sigma\_{\overline{n}}} \theta\_{0}}{|\lambda|^{\sigma\_{\overline{n}}} \sin^{2}(\pi - \theta\_{0}\sigma\_{\overline{n}})}, \\ \left| \frac{\lambda^{\sigma\_{\overline{n}}} - \mu \lambda\_{k}}{(\lambda^{\sigma\_{\overline{n}}} - c)(\lambda^{\sigma\_{\overline{n}}} - \mu \lambda\_{k}) - b\lambda\_{k}} \right| \leq & \frac{1}{|\lambda^{\sigma\_{\overline{n}}} - c - \frac{b\lambda\_{k}}{\lambda^{\sigma\_{\overline{n}}} - \mu \lambda\_{k}}|} \leq \frac{2}{|\lambda|^{\sigma\_{\overline{n}}}} \end{split}$$

for sufficiently large *l*, since

$$\sup\_{k \in \mathbb{N}, \lambda \in S\_{\mathfrak{a}\_0, \theta\_0}} \left| c + \frac{b\lambda\_k}{\lambda^{\sigma\_\aleph} - \mu \lambda\_k} \right| < \infty.$$

Thus, *<sup>A</sup>* ∈ A{*<sup>α</sup>k*}(*θ*0, *<sup>a</sup>*0) with *<sup>θ</sup>*<sup>0</sup> <sup>∈</sup> (*π*/2, *<sup>π</sup>*/*σn*), *<sup>a</sup>*<sup>0</sup> = (*l*|*c*|)1/*σ<sup>n</sup>* with a chosen sufficiently large *l* > 1.

**Theorem 8.** *Let <sup>α</sup><sup>k</sup>* <sup>∈</sup> (0, 1]*, <sup>k</sup>* <sup>=</sup> 0, 1, ... , *n, <sup>α</sup>*<sup>0</sup> <sup>+</sup> *<sup>α</sup><sup>n</sup>* <sup>&</sup>gt; <sup>1</sup>*, <sup>σ</sup><sup>n</sup>* <sup>∈</sup> [1, 2)*,* <sup>Σ</sup>*g*, *<sup>h</sup>* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*]; <sup>H</sup><sup>2</sup> *<sup>σ</sup>*) ∪ *<sup>C</sup>γ*([0, *<sup>T</sup>*]; <sup>H</sup>*σ*), *<sup>γ</sup>* <sup>∈</sup> (0, 1]*. Then, problem* (15)*–*(19) *has a unique solution.*

**Proof.** Problem (15)–(19) is represented as abstract problem (10), (11) due to the above choice of Z and *A*. Since we find the vector functions *v*(·, *t*) and *w*(·, *t*) with the values in <sup>H</sup>*<sup>σ</sup>* for every *<sup>t</sup>* <sup>∈</sup> (0, *<sup>T</sup>*], instead of Equation (17), we consider its projection on <sup>H</sup>*<sup>σ</sup>*

$$D^{\sigma\_{\mathfrak{n}}}v = \mu Bv + Bw + \Sigma \mathfrak{g}\_{\prime} \quad (\mathfrak{x}, t) \in \Omega \times (0, T]\_{\prime}$$

In this case, the projection of Equation (18) on H*<sup>σ</sup>* has the form

$$D^{\sigma\_{\Omega}}w = bv + cw + \Sigma h, \quad (\mathfrak{x}, t) \in \Omega \times (0, T]\_{\Lambda}$$

hence, Π*h* ≡ 0. Theorem 7 and Theorem 5 imply the required statement.

**Remark 10.** *If we found v*(*x*, *t*) *and w*(*x*, *t*)*, we obtain the pressure gradient using the formula* ∇*p*(·, *t*) = *μ*ΠΔ*v*(·, *t*) + ΠΔ*w*(·, *t*) + Π *f*(·, *t*) *from the projection of Equation* (17) *on the subspace* H*π.*

#### **8. Conclusions**

On the one hand, the results obtained will become the basis for the study of various classes of semilinear and quasilinear equations with the Dzhrbashyan–Nersesyan derivative. It is supposed to consider cases when the nonlinearity in the equation is continuous in the norm of the graph of the operator *A* and when it is Hölderian. In addition, there are plans to investigate similar equations with a degenerate linear operator at the Dzhrbashyan– Nersesyan derivative, linear, semi-linear and quasilinear. On the other hand, abstract results will be used to study various initial-boundary value problems for partial differential equations and their systems encountered in applications.

**Author Contributions:** Conceptualization, V.E.F. and M.V.P.; methodology, V.E.F.; software, E.M.I.; validation, E.M.I. and M.V.P.; formal analysis, E.M.I. and M.V.P.; investigation, E.M.I. and V.E.F.; resources, E.M.I.; data curation, E.M.I.; writing—original draft preparation, E.M.I. and M.V.P.; writing review and editing, V.E.F. and M.V.P.; visualization, E.M.I.; supervision, V.E.F. and M.V.P.; project administration, V.E.F.; funding acquisition, V.E.F. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Russian Science Foundation, grant number 22-21-20095.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


## *Article* **Existence Results for Coupled Nonlinear Sequential Fractional Differential Equations with Coupled Riemann–Stieltjes Integro-Multipoint Boundary Conditions**

**Ymnah Alruwaily 1, Bashir Ahmad 2,\* , Sotiris K. Ntouyas <sup>3</sup> and Ahmed S. M. Alzaidi 4,\***

	- P.O. Box 11099, Taif 21944, Saudi Arabia

**Abstract:** This paper is concerned with the existence of solutions for a fully coupled Riemann– Stieltjes, integro-multipoint, boundary value problem of Caputo-type sequential fractional differential equations. The given system is studied with the aid of the Leray–Schauder alternative and contraction mapping principle. A numerical example illustrating the abstract results is also presented.

**Keywords:** sequential fractional differential equations; Caputo fractional derivative; Riemann– Stieltjes integro-multipoint boundary conditions; existence and uniqueness; fixed point

#### **1. Introduction**

Coupled systems of fractional-order differential equations appear in the mathematical models of several real-world problems. Examples include chaos and fractional dynamics [1], bio-engineering [2], ecology [3], financial economics [4], etc. The topic of fractional differential systems, complemented by different kinds of boundary conditions, has been one a popular and important area of scientific investigation. Many researchers have contributed to the development of this subject by publishing numerous articles, Special Issues, etc. The modern methods of functional analysis areof great support in achieving existence and uniqueness results for these problems [5,6]. For some recent works on fractional or sequential fractional differential equations with nonlocal integral boundary conditions, we refer the reader to a series of papers [7–13].

In the article of [14], the authors investigated the solvability of an initial value problem involving a sequential fractional differential equation by means of fixed-point theorems in partially ordered sets. In [15], the existence and uniqueness results for a periodic boundary value problem of nonlinear sequential fractional differential equations were obtained by the method of upper and lower solutions, together with the monotone iterative technique.

Now, we briefly describe some recent works on sequential fractional-order coupled systems equipped with coupled boundary conditions. A fully coupled two-parameter system of sequential fractional integro-differential equations with nonlocal integro-multipoint boundary conditions was studied in [16]. The authors discussed the existence and uniqueness of solutions for a system of Hilfer–Hadamard sequential fractional differential equations with two-point boundary conditions in [17]. The sequential hybrid inclusion boundary value problem with three-point integro-derivative boundary conditions was investigated by using the analytic methods relying on *α*-*ψ*-contractive mappings, endpoints, and the fixed points of the product operators in [18]. The authors studied the existence and uniqueness

**Citation:** Alruwaily, Y.; Ahmad, B.; Ntouyas, S.K.; Alzaidi, A.S.M. Existence Results for Coupled Nonlinear Sequential Fractional Differential Equations with Coupled Riemann–Stieltjes Integro-Multipoint Boundary Conditions. *Fractal Fract.* **2022**, *6*, 123. https://doi.org/ 10.3390/fractalfract6020123

Academic Editor: Maria Rosaria Lancia

Received: 8 January 2022 Accepted: 16 February 2022 Published: 20 February 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

of solutions for an initial value problem of coupled sequential fractional differential equations in [19]. The existence results for a nonlocal coupled system of sequential fractional differential equations involving *ψ*-Hilfer fractional derivatives were presented in [20].

The objective of the present work is to develop the existence theory for a new class of nonlinear coupled systems of sequential fractional differential equations supplemented with coupled, non-conjugate, Riemann–Stieltjes, integro-multipoint boundary conditions. In precise terms, we investigate the following system:

$$\begin{cases} (^cD^{q+1} + ^cD^q) \mathcal{X}(t) = \mathfrak{f}(t, \mathcal{X}(t), \mathcal{Y}(t)), & 2 < q \le 3, \ t \in [0, 1], \\ (^cD^{p+1} + ^cD^p) \mathcal{Y}(t) = \mathfrak{g}(t, \mathcal{X}(t), \mathcal{Y}(t)), & 2 < p \le 3, \ t \in [0, 1], \end{cases} \tag{1}$$

subject to the coupled boundary conditions:

$$\begin{cases} \begin{aligned} \mathcal{X}(0) = 0, \mathcal{X}'(0) = 0, \mathcal{X}'(1) = 0, \mathcal{X}(1) = k \int\_0^\rho \mathcal{Y}(s) dA(s) + \sum\_{l=1}^{n-2} a\_l \mathcal{Y}(\sigma\_l) + k\_1 \int\_\nu^1 \mathcal{Y}(s) dA(s), \\ \mathcal{Y}(0) = 0, \mathcal{Y}'(0) = 0, \mathcal{Y}'(1) = 0, \mathcal{Y}(1) = h \int\_0^\rho \mathcal{X}(s) dA(s) + \sum\_{l=1}^{n-2} \beta\_l \mathcal{X}(\sigma\_l) + h\_1 \int\_\nu^1 \mathcal{X}(s) dA(s), \end{aligned} \end{cases} \tag{2}$$

where *cD<sup>ξ</sup>* denotes the Caputo fractional derivative of order *<sup>ξ</sup>* ∈ {*q*, *<sup>p</sup>*}, 0 <sup>&</sup>lt; *<sup>ρ</sup>* <sup>&</sup>lt; *<sup>σ</sup><sup>i</sup>* <sup>&</sup>lt; *<sup>ν</sup>* <sup>&</sup>lt; 1, <sup>f</sup>, <sup>g</sup> : [0, 1] <sup>×</sup> <sup>R</sup> <sup>×</sup> <sup>R</sup> <sup>→</sup> <sup>R</sup> are given continuous functions, *<sup>k</sup>*, *<sup>k</sup>*1, *<sup>h</sup>*, *<sup>h</sup>*1, *<sup>α</sup>i*, *<sup>β</sup><sup>i</sup>* <sup>∈</sup> <sup>R</sup>, i=1,2, ··· , *n* − 2 and *A* is a function of bounded variation.

Riemann–Stieltjes boundary conditions are quite general, since they include multipoint and integral boundary conditions as special cases [21]. The Riemann–Stieltjes integral is a generalization of the Riemann integral due to the Dutch astronomer T. J. Stieltjes and has potential applications in probability theory [22]. In addition, the Riemann–Stieltjes integral of the random variable with respect to its distribution function interprets the expected value of random variable [23]. Moreover, the boundary conditions (2) have useful applications in diffraction-free and self-healing optoelectronic devices. For more details, see [7].

The main emphasis in the present work is to investigate the existence criteria for the solutions to a coupled system of nonlinear sequential fractional differential equations equipped with multipoint Riemann–Stieltjes integral-type boundary conditions. Here, one can see that the coupled boundary conditions relate the value of the unknown function X (*t*) (Y(*t*)) at *t* = 1 with the distributions of the unknown function Y(*t*) (X (*t*)) on the segments [0, *ρ*] and [*ν*, 1] in the sense of Riemann–Stieltjes integrals, together with the sum of its discrete values at *σi*, *i* = 1, 2, ··· , *n* − 2. The present study is novel in the given configuration and enriches the literature on boundary value problems of sequential fractional differential equations.

Concerning our strategy when studying the problem (1)–(2), we use the fixed-point approach, which is based on the idea of converting the given problem into a fixed-point problem, followed by the application of appropriate fixed-point theorems to show the existence of the fixed points for the operator involved in the problem at hand. We make use of the Leray–Schauder alternative to show the existence of a solution to the given problem, while the uniqueness result for the given problem is derived with the aid of the contraction mapping principle due to Banach.

The rest of this paper is organized as follows. In Section 2, we present some basic definitions of fractional calculus and prove an auxiliary lemma concerning the linear variant of the problem (1)–(2), helping to convert it into a fixed-point problem. Section 3 establishes the existence and uniqueness results for the given problem, whereas Section 4 contains an example illustrating the main results. The paper ends with a discussion in Section 5, where some special cases and possible future works are indicated.

#### **2. Preliminary Material**

First, we outline some basic concepts of fractional calculus [24].

**Definition 1.** *The Riemann–Liouville fractional integral of order <sup>ϑ</sup>* <sup>∈</sup> <sup>R</sup> (*<sup>ϑ</sup>* <sup>&</sup>gt; <sup>0</sup>) *for a locally integrable, real-valued function U on* <sup>−</sup><sup>∞</sup> <sup>≤</sup> *<sup>a</sup>* <sup>&</sup>lt; *<sup>z</sup>* <sup>&</sup>lt; *<sup>b</sup>* <sup>≤</sup> <sup>+</sup>∞*, denoted by I<sup>ϑ</sup> <sup>a</sup> U*(*z*), *is defined by*

$$I\_a^{\theta} \mathcal{U}(z) = \frac{1}{\Gamma(\theta)} \int\_a^z (z - s)^{\theta - 1} \mathcal{U}(s) ds.$$

*Here,* Γ(·) *is the familiar Gamma function.*

**Definition 2.** *The Caputo derivative of fractional order ϑ for an (r* − 1*)-times absolutely continuous function U* : [*a*, <sup>∞</sup>) −→ <sup>R</sup> *is defined as*

$$\, ^cD^\theta \mathcal{U}(z) = \frac{1}{\Gamma(r-\theta)} \int\_a^z (z-s)^{r-\theta-1} \mathcal{U}^{(r)}(s) ds, \; r-1 < \theta < r, \; r = \left[\theta\right] + 1, \; \theta < r$$

*where* [*ϑ*] *denotes the integer part of the real number ϑ.*

**Lemma 1.** *The general solution of the fractional differential equation cD<sup>ϑ</sup>*<sup>X</sup> (*z*) = 0, *<sup>r</sup>* <sup>−</sup><sup>1</sup> <sup>&</sup>lt; *<sup>ϑ</sup>* <sup>&</sup>lt; *r, z* ∈ [*a*, *b*], *is*

$$\mathcal{X}(z) = \varrho\_0 + \varrho\_1(z - a) + \varrho\_2(z - a)^2 + \dots + \varrho\_{r-1}(z - a)^{r-1},$$

*where <sup>i</sup>* <sup>∈</sup> <sup>R</sup>, *<sup>i</sup>* <sup>=</sup> 0, 1, ··· ,*<sup>r</sup>* <sup>−</sup> 1. *Furthermore,*

$$I^{\theta \cdot c} D^{\theta} \mathcal{X}(z) = \mathcal{X}(z) + \sum\_{i=0}^{r-1} \varrho\_i (z - a)^i.$$

**Lemma 2.** *Let <sup>ψ</sup>*, *<sup>φ</sup>* <sup>∈</sup> (*C*[0, 1], <sup>R</sup>) *and* <sup>Δ</sup> <sup>=</sup> 0. *Then the unique solution of the linear system of fractional differential*

$$\begin{cases} \begin{aligned} (^cD^{q+1} + ^cD^q) \mathcal{X}(t) = \psi(t), & 2 < q \le 3, \ t \in [0, 1], \\ (^cD^{p+1} + ^cD^p) \mathcal{Y}(t) = \phi(t), & 2 < p \le 3, \ t \in [0, 1]. \end{aligned} \end{cases} \tag{3}$$

*supplemented with the boundary conditions (2), can be expressed in the following formulas:*

$$\mathcal{X}(t) \quad = \int\_0^t e^{-(t-s)} I\_{0^+}^q \psi(s) ds + \sum\_{i=1}^4 Q\_i(t)\mathcal{E}\_{i\prime} \text{ i } = 1, 2, 3, 4,\tag{4}$$

$$\mathcal{Y}(t) \quad = \int\_0^t e^{-(t-s)} I\_{0^+}^p \phi(s) ds + \sum\_{j=1}^4 \mathcal{P}\_j(t) \mathcal{E}\_j, \ j = 1, 2, 3, 4,\tag{5}$$

*where*

$$\begin{array}{rcl} \mathcal{E}\_{1} &=& \int\_{0}^{1} e^{-(1-s)} I\_{0^{+}}^{q} \psi(s) ds - I\_{0^{+}}^{q} \psi(1), \\ \mathcal{E}\_{2} &=& \int\_{0}^{1} e^{-(1-s)} I\_{0^{+}}^{p} \phi(s) ds - I\_{0^{+}}^{p} \phi(1), \\ \mathcal{E}\_{3} &=& k \int\_{0}^{\rho} \left( \int\_{0}^{s} e^{-(s-z)} I\_{0^{+}}^{p} \phi(z) dz \right) dA(s) + \sum\_{i=1}^{n-2} a\_{i} \int\_{0}^{\sigma\_{i}} e^{-(\sigma\_{i}-s)} I\_{0^{+}}^{p} \phi(s) ds \\ &+ k\_{1} \int\_{\nu}^{1} \left( \int\_{0}^{s} e^{-(s-z)} I\_{0^{+}}^{p} \phi(z) dz \right) dA(s) - \int\_{0}^{1} e^{-(1-s)} I\_{0^{+}}^{q} \psi(s) ds \\ \mathcal{E}\_{4} &=& h \int\_{0}^{\rho} \left( \int\_{0}^{s} e^{-(s-z)} I\_{0^{+}}^{q} \psi(z) dz \right) dA(s) + \sum\_{i=1}^{n-2} \beta\_{i} \int\_{0}^{\sigma\_{i}} e^{-(\sigma\_{i}-s)} I\_{0^{+}}^{q} \psi(s) ds \end{array}$$

$$+h\_1 \int\_{\nu}^{1} \left( \int\_0^s e^{-(s-z)} I\_{0^+}^q \psi(z) dz \right) dA(s) - \int\_0^1 e^{-(1-s)} I\_{0^+}^p \phi(s) ds,\tag{6}$$

$$\begin{array}{rcl}\mathcal{Q}\_{i}(t) &=& (e^{-t} + t - 1)\lambda\_{i} + (-2e^{-t} + t^{2} - 2t + 2)\nu\_{i}, i = 1,2,3,4, \\\mathcal{P}\_{\bar{j}}(t) &=& (e^{-t} + t - 1)\rho\_{\bar{j}} + (-2e^{-t} + t^{2} - 2t + 2)\omega\_{\bar{j}}, \; j = 1,2,3,4,\end{array} \tag{7}$$

$$\begin{array}{rcl} \upsilon\_1 &=& \frac{e + (1 - e)\lambda\_1}{2}, \ \upsilon\_2 = \frac{(1 - e)\lambda\_2}{2}, \ \upsilon\_3 = \frac{(1 - e)\lambda\_3}{2}, \ \upsilon\_4 = \frac{(1 - e)\lambda\_4}{2}, &\\ & & (1 - e)\alpha\_1 & \ & (1 - e)\alpha\_2 \end{array} \tag{8}$$

$$
\omega\_1 = \frac{(1-e)\rho\_1}{2}, \omega\_2 = \frac{e + (1-e)\rho\_2}{2}, \omega\_3 = \frac{(1-e)\rho\_3}{2}, \omega\_4 = \frac{(1-e)\rho\_4}{2}, \tag{9}
$$

$$
\text{and}
\quad
\omega\_1 \wedge \omega\_2 \wedge \omega\_3 = \frac{(1-e)\rho\_3}{2}, \omega\_4 \wedge \omega\_5 = \frac{(1-e)\rho\_4}{2}, \tag{10}
$$

$$
\lambda\_1 \lambda\_1 = \frac{(2 - e)\gamma\_1 - A\_4 \gamma\_2 e}{2\Delta}, \\
\lambda\_2 = \frac{A\_2 \gamma\_1 e - (2 - e)\gamma\_2}{2\Delta}, \\
\lambda\_3 = \frac{\gamma\_1}{\Delta}, \\
\lambda\_4 = \frac{-\gamma\_2}{\Delta}, \quad \text{(10)}
$$

$$\rho\_1 \quad = \frac{A\_4 \gamma\_1 e - (2 - e)\gamma\_3}{2\Delta}, \rho\_2 = \frac{(2 - e)\gamma\_1 - A\_2 \gamma\_3 e}{2\Delta}, \rho\_3 = \frac{-\gamma\_3}{\Delta}, \rho\_4 = \frac{\gamma\_1}{\Delta}, \tag{11}$$

$$
\Delta = \gamma\_1^2 - \gamma\_2 \gamma\_3, \ \gamma\_1 = \frac{3-e}{2}, \ \gamma\_2 = -A\_1 - A\_2 \frac{(1-e)}{2}, \ \gamma\_3 = -A\_3 - A\_4 \frac{(1-e)}{2}, \tag{12}
$$

$$\begin{split} A\_{1} &= \quad k \int\_{0}^{\rho} (e^{-s} + s - 1) dA(s) + \sum\_{i=1}^{n-2} a\_{i} (e^{-\sigma\_{i}} + \sigma\_{i} - 1) + k\_{1} \int\_{\nu}^{1} (e^{-s} + s - 1) dA(s), \\ A\_{2} &= \quad k \int\_{0}^{\rho} (-2e^{-s} + s^{2} - 2s + 2) dA(s) + \sum\_{i=1}^{n-2} a\_{i} (-2e^{-\sigma\_{i}} + \sigma\_{i}^{2} - 2\sigma\_{i} + 2) \\ &\quad + k\_{1} \int\_{\nu}^{1} (-2e^{-s} + s^{2} - 2s + 2) dA(s), \\ A\_{3} &= \quad h \int\_{0}^{\rho} (e^{-s} + s - 1) dA(s) + \sum\_{i=1}^{n-2} \beta\_{i} (e^{-\sigma\_{i}} + \sigma\_{i} - 1) + h\_{1} \int\_{\nu}^{1} (e^{-s} + s - 1) dA(s), \\ A\_{4} &= \quad h \int\_{0}^{\rho} (-2e^{-s} + s^{2} - 2s + 2) dA(s) + \sum\_{i=1}^{n-2} \beta\_{i} (-2e^{-\sigma\_{i}} + \sigma\_{i}^{2} - 2\sigma\_{i} + 2) \\ &\quad + h\_{1} \int\_{\nu}^{1} (-2e^{-s} + s^{2} - 2s + 2) dA(s). \end{split} \tag{13}$$

**Proof.** Rewriting the first equation in (3) as *cDq*(*<sup>D</sup>* <sup>+</sup> <sup>1</sup>)<sup>X</sup> (*t*) = *<sup>ψ</sup>*(*t*) and then applying the integral operator *I q* <sup>0</sup><sup>+</sup> to it, we obtain

$$\begin{aligned} \mathcal{X}(t) &= (-e^{-t} + 1)c\_1 + (e^{-t} + t - 1)c\_2 + (-2e^{-t} + t^2 - 2t + 2)c\_3 + e^{-t}c\_4 \\ &+ \int\_0^t e^{-(t-s)} I\_{0^+}^q \psi(s) ds, \end{aligned} \tag{14}$$

where *ci* <sup>∈</sup> <sup>R</sup>, *<sup>i</sup>* <sup>=</sup> 1, 2, 3, 4 are unknown arbitrary constants. In a similar manner, applying the integral operator *I p* <sup>0</sup><sup>+</sup> to the second equation in (3), we get

$$\begin{aligned} \mathcal{Y}(t) &= (-e^{-t} + 1)b\_1 + (e^{-t} + t - 1)b\_2 + (-2e^{-t} + t^2 - 2t + 2)b\_3 + e^{-t}b\_4 \\ &+ \int\_0^t e^{-(t-s)} I\_{0^+}^p \phi(s) ds, \end{aligned} \tag{15}$$

where *bi* <sup>∈</sup> <sup>R</sup>, *<sup>i</sup>* <sup>=</sup> 1, 2, 3, 4 are unknown arbitrary constants. From (14) and (15), we have

$$\begin{aligned} \mathcal{X}'(t) &= -e^{-t}c\_1 + (-e^{-t} + 1)c\_2 + (2e^{-t} + 2t - 2)c\_3 - e^{-t}c\_4 \\ &- \int\_0^t e^{-(t-s)} I\_{0^+}^q \psi(s) ds + I\_{0^+}^q \psi(t), \end{aligned} \tag{16}$$

$$\mathcal{O}'(t) = -e^{-t}b\_1 + (-e^{-t} + 1)b\_2 + (2e^{-t} + 2t - 2)b\_3 - e^{-t}b\_4$$

$$-\int\_0^t e^{-(t-s)}I\_{0^+}^p \phi(s)ds + I\_{0^+}^p \phi(t). \tag{17}$$

Using the conditions X (0) = 0, Y(0) = 0, X (0) = 0, Y (0) = 0 in Equations (14)–(17), we obtain *c*<sup>1</sup> = *c*<sup>4</sup> = 0 and *b*<sup>1</sup> = *b*<sup>4</sup> = 0. Then (14)–(17) become

$$\mathcal{X}(t) \;=\; \left(\varepsilon^{-t} + t - 1\right)c\_2 + \left(-2\varepsilon^{-t} + t^2 - 2t + 2\right)c\_3 + \int\_0^t e^{-(t-s)} I\_{0^+}^{\emptyset} \psi(s) ds,\tag{18}$$

$$\mathcal{X}'(t) = \begin{array}{c} (-e^{-t} + 1)c\_2 + (2e^{-t} + 2t - 2)c\_3 - \int\_0^t e^{-(t-s)} I\_{0^+}^q \psi(s)ds + I\_{0^+}^q \psi(t), \end{array} \tag{19}$$

$$\mathcal{Y}(t) \quad = \begin{array}{c} (e^{-t} + t - 1)b\_2 + (-2e^{-t} + t^2 - 2t + 2)b\_3 + \int\_0^t e^{-(t-s)}I\_{0^+}^p \phi(s)ds, \end{array} \tag{20}$$

$$\mathcal{Y}'(t) = (-e^{-t} + 1)b\_2 + (2e^{-t} + 2t - 2)b\_3 - \int\_0^t e^{-(t-s)} I\_{0^+}^p \phi(s)ds + I\_{0^+}^p \phi(t). \tag{21}$$

Using (18)–(21) in the rest of the boundary conditions given by (2), together with notation (13), yields

$$(-e^{-1} + 1)c\_2 + 2e^{-1}c\_3 \quad = \quad \mathcal{E}\_{1\prime} \tag{22}$$

$$(-e^{-1} + 1)b\_2 + 2e^{-1}b\_3 = \,\_2\mathcal{E}\_2\tag{23}$$

$$-e^{-1}c\_2 + (-2e^{-1} + 1)c\_3 - A\_1b\_2 - A\_2b\_3 = \\\left. \mathcal{E}\_{3\prime} \right| \tag{24}$$

$$e^{-1}b\_2 + (-2e^{-1} + 1)b\_3 - A\_3c\_2 - A\_4c\_3 \ = \ \mathcal{E}\_{4\prime} \tag{25}$$

where *Ai*, *i* = 1, 2, 3, 4 are given by (13) and E*i*, *i* = 1, 2, 3, 4 are defined by (6). Inserting the values of *c*<sup>3</sup> and *b*<sup>3</sup> from (22) and (23) into (24) and (25), we obtain

$$
\gamma\_1 c\_2 + \gamma\_2 b\_2 = \frac{(2-e)}{2} \mathcal{E}\_1 + \frac{A\_2 e}{2} \mathcal{E}\_2 + \mathcal{E}\_3 \tag{26}
$$

$$
\gamma\_3 c\_2 + \gamma\_1 b\_2 = \frac{A\_4 e}{2} \mathcal{E}\_1 + \frac{(2 - e)}{2} \mathcal{E}\_2 + \mathcal{E}\_{4\prime} \tag{27}
$$

where *γi*, *i* = 1, 2, 3 are given by (12). Solving (26) and (27) for *c*<sup>2</sup> and *b*2, we obtain

$$c\_2 = \sum\_{i=1}^4 \lambda\_i \mathcal{E}\_{i\prime} \quad b\_2 = \sum\_{j=1}^4 \rho\_j \mathcal{E}\_{j\prime}$$

where *λ<sup>i</sup>* (*i* = 1, 2, 3, 4) and *ρ<sup>j</sup>* (*j* = 1, 2, 3, 4) are given in (10) and (11), respectively. Substituting the values of *c*<sup>2</sup> and *b*<sup>2</sup> into (22) and (23) respectively, we find that

$$c\_3 = \sum\_{i=1}^4 \nu\_i \mathcal{E}\_{i\prime} \quad b\_3 = \sum\_{j=1}^4 \omega\_j \mathcal{E}\_{j\prime}$$

where *νi*, *i* = 1, 2, 3, 4, and *ωj*, *j* = 1, 2, 3, 4 are given by (8) and (9) respectively. Inserting the values of *c*2, *c*3, *b*<sup>2</sup> and *b*<sup>3</sup> in (18) and (20), together with the notation (7), we obtain the solution (4) and (5). One can obtain the converse of this lemma by direct computation. This completes the proof.

For computational convenience, we introduce the following lemma:

**Lemma 3.** *For <sup>ψ</sup>*, *<sup>φ</sup>* <sup>∈</sup> *<sup>C</sup>*([0, 1], <sup>R</sup>)*, we have*

$$\text{(i)}\quad \left| \int\_0^t e^{-(t-s)} I\_{0^+}^q \psi(s) ds \right| \le \frac{1}{\Gamma(q+1)} (1 - e^{-1}) ||\psi||\_{\mathcal{H}}$$

 *<sup>t</sup>* 0 *e* <sup>−</sup>(*t*−*s*)*I p* <sup>0</sup><sup>+</sup> *φ*(*s*)*ds* ≤ 1 Γ(*p* + 1) (1 − *e* <sup>−</sup>1) *φ* . *(ii)* <sup>1</sup> 0 *e* <sup>−</sup>(1−*s*)*I q* <sup>0</sup><sup>+</sup> *ψ*(*s*)*ds* ≤ 1 Γ(*q* + 1) (1 − *e* <sup>−</sup>1) *ψ* , <sup>1</sup> 0 *e* <sup>−</sup>(1−*s*)*I p* <sup>0</sup><sup>+</sup> *φ*(*s*)*ds* ≤ 1 Γ(*p* + 1) (1 − *e* <sup>−</sup>1) *φ* . *(iii) n*−2 ∑ *i*=1 *αi <sup>σ</sup><sup>i</sup>* 0 *e* <sup>−</sup>(*σi*−*s*)*I p* <sup>0</sup><sup>+</sup> *φ*(*s*)*ds* ≤ 1 Γ(*p* + 1) *n*−2 ∑ *i*=1 <sup>|</sup>*αi*|*σ<sup>p</sup> <sup>i</sup>* (1 − *e* <sup>−</sup>*σi*) *φ* , *n*−2 ∑ *i*=1 *βi <sup>σ</sup><sup>i</sup>* 0 *e* <sup>−</sup>(*σi*−*s*)*I q* <sup>0</sup><sup>+</sup> *ψ*(*s*)*ds* ≤ 1 Γ(*q* + 1) *n*−2 ∑ *i*=1 <sup>|</sup>*βi*|*σ<sup>q</sup> <sup>i</sup>* (1 − *e* <sup>−</sup>*σi*) *ψ* . *(iv) <sup>ρ</sup>* 0 *<sup>s</sup>* 0 *e* <sup>−</sup>(*s*−*z*)*I p* <sup>0</sup><sup>+</sup> *<sup>φ</sup>*(*z*)*dz dA*(*s*) ≤ *<sup>ρ</sup>* 0 *sp* Γ(*p* + 1) (1 − *e* −*s* )*dA*(*s*) *φ* , *<sup>ρ</sup>* 0 *<sup>s</sup>* 0 *e* <sup>−</sup>(*s*−*z*)*I q* <sup>0</sup><sup>+</sup> *<sup>ψ</sup>*(*z*)*dz dA*(*s*) ≤ *<sup>ρ</sup>* 0 *sq* Γ(*q* + 1) (1 − *e* −*s* )*dA*(*s*) *ψ* . *(v)* <sup>1</sup> *ν <sup>s</sup>* 0 *e* <sup>−</sup>(*s*−*z*)*I p* <sup>0</sup><sup>+</sup> *<sup>φ</sup>*(*z*)*dz dA*(*s*) ≤ <sup>1</sup> *ν sp* Γ(*p* + 1) (1 − *e* −*s* )*dA*(*s*) *φ* , <sup>1</sup> *ν <sup>s</sup>* 0 *e* <sup>−</sup>(*s*−*z*)*I q* <sup>0</sup><sup>+</sup> *<sup>ψ</sup>*(*z*)*dz dA*(*s*) ≤ <sup>1</sup> *ν sq* Γ(*q* + 1) (1 − *e* −*s* )*dA*(*s*) *ψ* .

**Proof.** To prove (i), we have

$$\begin{aligned} \left| \int\_0^t e^{-(t-s)} I\_{0+}^q \psi(s) ds \right| &= \left| \int\_0^t e^{-(t-s)} \left( \int\_0^s \frac{(s-z)^{q-1}}{\Gamma(q)} \psi(z) dz \right) ds \right| \\ &\leq \quad \frac{t^q}{\Gamma(q+1)} (1 - e^{-t}) \|\psi\| \\ &\leq \quad \frac{1}{\Gamma(q+1)} (1 - e^{-1}) \|\psi\|. \end{aligned}$$

The other cases are similar. Therefore, we omit the details.

#### **3. Main Results**

Let (X, · ) be a Banach space equipped with the norm X = sup{|X (*t*)|, *<sup>t</sup>* <sup>∈</sup> [0, 1]}, where <sup>X</sup> <sup>=</sup> {X (*t*)|X (*t*) <sup>∈</sup> (*C*[0, 1], <sup>R</sup>)}. Then (<sup>X</sup> <sup>×</sup> <sup>X</sup>, (·, ·) ) is also a Banach space endowed with norm (X , Y) = X + Y , X , Y ∈ X.

By Lemma 2, we introduce an operator *T* : X × X → X × X defined by

$$T(\mathcal{X}, \mathcal{Y})(t) = \begin{pmatrix} T\_1(\mathcal{X}, \mathcal{Y})(t) \\ T\_2(\mathcal{X}, \mathcal{Y})(t) \end{pmatrix} \tag{28}$$

where

$$\begin{aligned} T\_1(\mathcal{X}, \mathcal{Y})(t) &= \int\_0^t e^{-(t-s)} I\_{0^+}^q f(s, \mathcal{X}(s), \mathcal{Y}(s)) ds \\ &+ \mathcal{Q}\_1(t) \left[ \int\_0^1 e^{-(1-s)} I\_{0^+}^q f(s, \mathcal{X}(s), \mathcal{Y}(s)) ds - I\_{0^+}^q f(s, \mathcal{X}(s), \mathcal{Y}(s))(1) \right] \\ &+ \mathcal{Q}\_2(t) \left[ \int\_0^1 e^{-(1-s)} I\_{0^+}^p \mathfrak{g}(s, \mathcal{X}(s), \mathcal{Y}(s)) ds - I\_{0^+}^p \mathfrak{g}(s, \mathcal{X}(s), \mathcal{Y}(s))(1) \right] \end{aligned}$$

$$\begin{aligned} &+\mathcal{Q}\_{3}(t)\Big[k\int\_{0}^{\rho}\Big(\int\_{0}^{s}e^{-(s-z)}I\_{0}^{p}\cdot\mathfrak{g}(z,\mathcal{X}(z),\mathcal{Y}(z))dz\Big)dA(s) \\ &+\sum\_{i=1}^{n-2}a\_{i}\int\_{0}^{c\_{i}}e^{-(\rho\_{i}-s)}I\_{0}^{p}\cdot\mathfrak{g}(s,\mathcal{X}(s),\mathcal{Y}(s))ds \\ &+k\_{1}\int\_{\nu}^{1}\Big(\int\_{0}^{s}e^{-(s-z)}I\_{0}^{p}\cdot\mathfrak{g}(z,\mathcal{X}(z),\mathcal{Y}(z))dz\Big)dA(s) \\ &-\int\_{0}^{1}e^{-(1-s)}I\_{0}^{q}\cdot\mathfrak{f}(s,\mathcal{X}(s),\mathcal{Y}(s))ds\Big] \\ &+\mathcal{Q}\_{4}(t)\Big[h\int\_{0}^{\rho}\Big(\int\_{0}^{s}e^{-(s-z)}I\_{0}^{q}\cdot\mathfrak{f}(z,\mathcal{X}(z),\mathcal{Y}(z))dz\Big)dA(s) \\ &+\sum\_{i=1}^{n-2}\beta\_{i}\int\_{0}^{c\_{i}}e^{-(c\_{i}-z)}I\_{0+}^{q}\cdot\mathfrak{f}(s,\mathcal{X}(s),\mathcal{Y}(s))ds \\ &+h\_{1}\int\_{0}^{1}\Big(\int\_{0}^{s}e^{-(s-z)}I\_{0+}^{q}\cdot\mathfrak{f}(z,\mathcal{X}(z),\mathcal{Y}(z))dz\Big)dA(s) \\ &-\int\_{0}^{1}e^{-(1-s)}I\_{0+}^{p}\cdot\mathfrak{g}(s,\mathcal{X}(s),\mathcal{Y}(s))ds\Big), \end{aligned} \tag{29}$$

*<sup>T</sup>*2(<sup>X</sup> , <sup>Y</sup>)(*t*) = *<sup>t</sup>* 0 *e* <sup>−</sup>(*t*−*s*)*I p* <sup>0</sup><sup>+</sup> g(*s*, X (*s*), Y(*s*))*ds* +P1(*t*) <sup>1</sup> 0 *e* <sup>−</sup>(1−*s*)*I q* <sup>0</sup><sup>+</sup> f(*s*, X (*s*), Y(*s*))*ds* − *I q* <sup>0</sup><sup>+</sup> f(*s*, X (*s*), Y(*s*))(1) + P2(*t*) <sup>1</sup> 0 *e* <sup>−</sup>(1−*s*)*I p* <sup>0</sup><sup>+</sup> g(*s*, X (*s*), Y(*s*))*ds* − *I p* <sup>0</sup><sup>+</sup> g(*s*, X (*s*), Y(*s*))(1) + P3(*t*) *k <sup>ρ</sup>* 0 *<sup>s</sup>* 0 *e* <sup>−</sup>(*s*−*z*)*I p* <sup>0</sup><sup>+</sup> <sup>g</sup>(*z*, <sup>X</sup> (*z*), <sup>Y</sup>(*z*))*dz dA*(*s*) + *n*−2 ∑ *i*=1 *αi <sup>σ</sup><sup>i</sup>* 0 *e* <sup>−</sup>(*σi*−*s*)*I p* <sup>0</sup><sup>+</sup> *g*(*s*, X (*s*), Y(*s*))*ds* +*k*<sup>1</sup> <sup>1</sup> *ν <sup>s</sup>* 0 *e* <sup>−</sup>(*s*−*z*)*I p* <sup>0</sup><sup>+</sup> <sup>g</sup>(*z*, <sup>X</sup> (*z*), <sup>Y</sup>(*z*))*dz dA*(*s*) − <sup>1</sup> 0 *e* <sup>−</sup>(1−*s*)*I q* <sup>0</sup><sup>+</sup> <sup>f</sup>(*s*, <sup>X</sup> (*s*), <sup>Y</sup>(*s*))*ds* + P4(*t*) *h <sup>ρ</sup>* 0 *<sup>s</sup>* 0 *e* <sup>−</sup>(*s*−*z*)*I q* <sup>0</sup><sup>+</sup> <sup>f</sup>(*z*, <sup>X</sup> (*z*), <sup>Y</sup>(*z*))*dz dA*(*s*) + *n*−2 ∑ *i*=1 *βi <sup>σ</sup><sup>i</sup>* 0 *e* <sup>−</sup>(*σi*−*s*)*I q* <sup>0</sup><sup>+</sup> f(*s*, X (*s*), Y(*s*))*ds* +*h*<sup>1</sup> <sup>1</sup> *ν <sup>s</sup>* 0 *e* <sup>−</sup>(*s*−*z*)*I q* <sup>0</sup><sup>+</sup> <sup>f</sup>(*z*, <sup>X</sup> (*z*), <sup>Y</sup>(*z*))*dz dA*(*s*) − <sup>1</sup> 0 *e* <sup>−</sup>(1−*s*)*I p* <sup>0</sup><sup>+</sup> <sup>g</sup>(*s*, <sup>X</sup> (*s*), <sup>Y</sup>(*s*))*ds* , (30)

where Q*i*(*t*), *i* = 1, 2, 3, 4 and P*j*(*t*), *j* = 1, 2, 3, 4 are given in (7).

In the forthcoming analysis, we assume that <sup>f</sup>, <sup>g</sup> : [0, 1] <sup>×</sup> <sup>R</sup> <sup>×</sup> <sup>R</sup> <sup>→</sup> <sup>R</sup> are continuous functions satisfying the following conditions:

**(**F1**)** There are real constants *ηi*, *ζ<sup>i</sup>* ≥ 0, *i* = 1, 2, *η*0, *ζ*<sup>0</sup> > 0 such that

$$|\mathfrak{f}(t, \mathcal{X}, \mathcal{Y})| \le \eta\_0 + \eta\_1 |\mathcal{X}| + \eta\_2 |\mathcal{Y}|\prime$$

$$|\mathfrak{g}(t, \mathcal{X}, \mathcal{Y})| \le \zeta\_0 + \zeta\_1 |\mathcal{X}| + \zeta\_2 |\mathcal{Y}|\prime$$
 $\forall t \in [0, 1], \mathcal{X}, \mathcal{Y} \in \mathbb{R}.$ 

**(**F2**)** There are positive real constants *L*<sup>1</sup> and *L*2, such that

$$\left|\mathfrak{f}(t,\mathcal{X}\_{1},\mathcal{Y}\_{1}) - \mathfrak{f}(t,\mathcal{X}\_{2},\mathcal{Y}\_{2})\right| \leq L\_{1}(|\mathcal{X}\_{1} - \mathcal{X}\_{2}| + |\mathcal{Y}\_{1} - \mathcal{Y}\_{2}|),$$

$$\left|\mathfrak{g}(t,\mathcal{X}\_{1},\mathcal{Y}\_{1}) - \mathfrak{g}(t,\mathcal{X}\_{2},\mathcal{Y}\_{2})\right| \leq L\_{2}(|\mathcal{X}\_{1} - \mathcal{X}\_{2}| + |\mathcal{Y}\_{1} - \mathcal{Y}\_{2}|),$$
 $\forall t \in [0,1], \mathcal{X}\_{1}, \mathcal{X}\_{2}, \mathcal{Y}\_{1}, \mathcal{Y}\_{2} \in \mathbb{R};$ 

In the sequel, we use the notation:

$$
\Theta = \Lambda\_1 L\_1 + \Lambda\_2 L\_2 \quad \overline{\Theta} = \overline{\Lambda}\_1 L\_1 + \overline{\Lambda}\_2 L\_2 \tag{31}
$$

$$\mathcal{M} = \Lambda\_1 \mathcal{N}\_1 + \Lambda\_2 \mathcal{N}\_2 \cdot \overline{\mathcal{M}} = \overline{\Lambda}\_1 \mathcal{N}\_1 + \overline{\Lambda}\_2 \mathcal{N}\_2 \tag{32}$$

<sup>Λ</sup><sup>1</sup> <sup>=</sup> <sup>1</sup> Γ(*q* + 1) (1 − *e* <sup>−</sup>1)+(<sup>2</sup> <sup>−</sup> *<sup>e</sup>* <sup>−</sup>1)<sup>Q</sup><sup>1</sup> + (<sup>1</sup> <sup>−</sup> *<sup>e</sup>* <sup>−</sup>1)<sup>Q</sup><sup>3</sup> <sup>+</sup> <sup>Q</sup><sup>4</sup> |*h*| *<sup>ρ</sup>* 0 *s <sup>q</sup>*(<sup>1</sup> <sup>−</sup> *<sup>e</sup>* −*s* )*dA*(*s*) + *n*−2 ∑ *i*=1 <sup>|</sup>*βi*|*σ<sup>q</sup> <sup>i</sup>* (1 − *e* <sup>−</sup>*σi*) + <sup>|</sup>*h*1<sup>|</sup> <sup>1</sup> *ν s <sup>q</sup>*(<sup>1</sup> <sup>−</sup> *<sup>e</sup>* −*s* )*dA*(*s*) , <sup>Λ</sup><sup>2</sup> <sup>=</sup> <sup>1</sup> Γ(*p* + 1) (2 − *e* <sup>−</sup>1)<sup>Q</sup><sup>2</sup> <sup>+</sup> <sup>Q</sup><sup>3</sup> |*k*| *<sup>ρ</sup>* 0 *<sup>s</sup>p*(<sup>1</sup> <sup>−</sup> *<sup>e</sup>* −*s* )*dA*(*s*) (33) + *n*−2 ∑ *i*=1 <sup>|</sup>*αi*|*σ<sup>p</sup> <sup>i</sup>* (1 − *e* <sup>−</sup>*σi*) + <sup>|</sup>*k*1<sup>|</sup> <sup>1</sup> *ν <sup>s</sup>p*(<sup>1</sup> <sup>−</sup> *<sup>e</sup>* −*s* )*dA*(*s*) + (1 − *e* <sup>−</sup>1)<sup>Q</sup><sup>4</sup> , <sup>Λ</sup><sup>1</sup> <sup>=</sup> <sup>1</sup> Γ(*q* + 1) (2 − *e* <sup>−</sup>1)<sup>P</sup><sup>1</sup> + (<sup>1</sup> <sup>−</sup> *<sup>e</sup>* <sup>−</sup>1)<sup>P</sup><sup>3</sup> <sup>+</sup> <sup>P</sup><sup>4</sup> |*h*| *<sup>ρ</sup>* 0 *s <sup>q</sup>*(<sup>1</sup> <sup>−</sup> *<sup>e</sup>* −*s* )*dA*(*s*) + *n*−2 ∑ *i*=1 <sup>|</sup>*βi*|*σ<sup>q</sup> <sup>i</sup>* (1 − *e* <sup>−</sup>*σi*) + <sup>|</sup>*h*1<sup>|</sup> <sup>1</sup> *ν s <sup>q</sup>*(<sup>1</sup> <sup>−</sup> *<sup>e</sup>* −*s* )*dA*(*s*) ,

$$\begin{split} \overline{\Lambda}\_{2} &= \quad \frac{1}{\Gamma(p+1)} \Big\{ (1-e^{-1}) + (2-e^{-1})\widetilde{\mathcal{P}}\_{2} + \widetilde{\mathcal{P}}\_{3} \Big[ |k| \int\_{0}^{p} \mathbf{s}^{p} (1-e^{-s}) dA(s) \\ &+ \sum\_{i=1}^{n-2} |a\_{i}| \boldsymbol{\sigma}\_{i}^{p} (1-e^{-\sigma\_{i}}) + |k\_{1}| \int\_{V}^{1} \mathbf{s}^{p} (1-e^{-s}) dA(s) \Big] + (1-e^{-1})\widetilde{\mathcal{P}}\_{4} \Big\}, \end{split} \tag{34}$$

$$\mathcal{N}\_1 = \sup\_{t \in [0,1]} |\mathfrak{f}(t,0,0)| < \infty,\\ \mathcal{N}\_2 = \sup\_{t \in [0,1]} |\mathfrak{g}(t,0,0\_\prime)| < \infty,\tag{35}$$

$$\text{where } \tilde{Q}\_i = \sup\_{t \in [0,1]} |\mathcal{Q}\_i(t)|, \ i = 1, 2, 3, 4 \text{ and } \tilde{\mathcal{P}}\_j = \sup\_{t \in [0,1]} |\mathcal{P}\_j(t)|, \ j = 1, 2, 3, 4.$$

$$\begin{array}{rcl}\Omega\_0 &=& (\Lambda\_1 + \overline{\Lambda}\_1)\eta\_0 + (\Lambda\_2 + \overline{\Lambda}\_2)\zeta\_0\\\Omega\_1 &=& (\Lambda\_1 + \overline{\Lambda}\_1)\eta\_1 + (\Lambda\_2 + \overline{\Lambda}\_2)\zeta\_1\\\Omega\_2 &=& (\Lambda\_1 + \overline{\Lambda}\_1)\eta\_2 + (\Lambda\_2 + \overline{\Lambda}\_2)\zeta\_2\end{array} \tag{36}$$

and

$$
\Omega = \max\{\Omega\_1, \Omega\_2\}.\tag{37}
$$

The following result shows the existence of a solution for the coupled system (1)–(2) and is based on the Leray–Schauder alternative [6].

**Theorem 1.** *Assume that the condition (*F1*) holds and* Ω < 1, *where* Ω *is given by (37). Then, the problem (1) and (2) has at least one solution on* [0, 1].

**Proof.** In the first step, it will be shown that the operator *T* : X × X → X × X is completely continuous. Note that the operator *T* is continuous in view of the continuity of the functions f and g. Let V ⊂ X × X be bounded. Then, we can find positive constants *M*<sup>1</sup> and *M*<sup>2</sup> such that |f(*t*, X (*t*), Y(*t*))| ≤ *M*<sup>1</sup> and |g(*t*, X (*t*), Y(*t*))| ≤ *M*2, ∀(X , Y) ∈ V. Therefore, for any (X , X ) ∈ V, we have

<sup>|</sup>*T*1(<sup>X</sup> , <sup>Y</sup>)(*t*)| ≤ *<sup>t</sup>* 0 *e* <sup>−</sup>(*t*−*s*)*I q* <sup>0</sup><sup>+</sup> |f(*s*, X (*s*), Y(*s*))|*ds* +|Q1(*t*)| <sup>1</sup> 0 *e* <sup>−</sup>(1−*s*)*I q* <sup>0</sup><sup>+</sup> |f(*s*, X (*s*), Y(*s*))|*ds* + *I q* <sup>0</sup><sup>+</sup> |f(*s*, X (*s*), Y(*s*))|(1) +|Q2(*t*)| <sup>1</sup> 0 *e* <sup>−</sup>(1−*s*)*I p* <sup>0</sup><sup>+</sup> |g(*s*, X (*s*), Y(*s*))|*ds* + *I p* <sup>0</sup><sup>+</sup> |g(*s*, X (*s*), Y(*s*))|(1) +|Q3(*t*)| |*k*| *<sup>ρ</sup>* 0 *<sup>s</sup>* 0 *e* <sup>−</sup>(*s*−*z*)*I p* <sup>0</sup><sup>+</sup> <sup>|</sup>g(*z*, <sup>X</sup> (*z*), <sup>Y</sup>(*z*))|*dz dA*(*s*) + *n*−2 ∑ *i*=1 |*αi*| *<sup>σ</sup><sup>i</sup>* 0 *e* <sup>−</sup>(*σi*−*s*)*I p* <sup>0</sup><sup>+</sup> |g(*s*, X (*s*), Y(*s*))|*ds* +|*k*1| <sup>1</sup> *ν <sup>s</sup>* 0 *e* <sup>−</sup>(*s*−*z*)*I p* <sup>0</sup><sup>+</sup> <sup>|</sup>g(*z*, <sup>X</sup> (*z*), <sup>Y</sup>(*z*))|*dz dA*(*s*) + <sup>1</sup> 0 *e* <sup>−</sup>(1−*s*)*I q* <sup>0</sup><sup>+</sup> <sup>|</sup>f(*s*, <sup>X</sup> (*s*), <sup>Y</sup>(*s*))|*ds* +|Q4(*t*)| |*h*| *<sup>ρ</sup>* 0 *<sup>s</sup>* 0 *e* <sup>−</sup>(*s*−*z*)*I q* <sup>0</sup><sup>+</sup> <sup>|</sup>f(*z*, <sup>X</sup> (*z*), <sup>Y</sup>(*z*))|*dz dA*(*s*) + *n*−2 ∑ *i*=1 |*βi*| *<sup>σ</sup><sup>i</sup>* 0 *e* <sup>−</sup>(*σi*−*s*)*I q* <sup>0</sup><sup>+</sup> |f(*s*, X (*s*), Y(*s*))|*ds* +|*h*1| <sup>1</sup> *ν <sup>s</sup>* 0 *e* <sup>−</sup>(*s*−*z*)*I q* <sup>0</sup><sup>+</sup> <sup>|</sup>f(*z*, <sup>X</sup> (*z*), <sup>Y</sup>(*z*))|*dz dA*(*s*) + <sup>1</sup> 0 *e* <sup>−</sup>(1−*s*)*I p* <sup>0</sup><sup>+</sup> <sup>|</sup>g(*s*, <sup>X</sup> (*s*), <sup>Y</sup>(*s*))|*ds* ≤ *M*<sup>1</sup> 1 Γ(*q* + 1) (1 − *e* <sup>−</sup>1) + <sup>Q</sup><sup>1</sup> *M*<sup>1</sup> 1 Γ(*q* + 1) (1 − *e* <sup>−</sup>1) + *M*<sup>1</sup> 1 Γ(*q* + 1) <sup>+</sup><sup>Q</sup><sup>2</sup> *M*<sup>2</sup> 1 Γ(*p* + 1) (1 − *e* <sup>−</sup>1) + *M*<sup>2</sup> 1 Γ(*p* + 1) <sup>+</sup><sup>Q</sup><sup>3</sup> |*k*|*M*<sup>2</sup> *<sup>ρ</sup>* 0 *sp* Γ(*p* + 1) (1 − *e* −*s* )*dA*(*s*) +*M*<sup>2</sup> *n*−2 ∑ *i*=1 <sup>|</sup>*αi*<sup>|</sup> <sup>1</sup> Γ(*p* + 1) *σp <sup>i</sup>* (1 − *e* <sup>−</sup>*σi*) +*M*2|*k*1| <sup>1</sup> *ν sp* Γ(*p* + 1) (1 − *e* −*s* )*dA*(*s*) + *M*<sup>1</sup> 1 Γ(*q* + 1) (1 − *e* <sup>−</sup>1) <sup>+</sup><sup>Q</sup><sup>4</sup> *M*1|*h*| *<sup>ρ</sup>* 0 *sq* Γ(*q* + 1) (1 − *e* −*s* )*dA*(*s*) + *M*<sup>1</sup> *n*−2 ∑ *i*=1 <sup>|</sup>*βi*|*σ<sup>q</sup> <sup>i</sup>* (1 − *e* <sup>−</sup>*σi*) +*M*1|*h*1| <sup>1</sup> *ν sp* Γ(*p* + 1) (1 − *e* −*s* )*dA*(*s*) + *M*<sup>2</sup> 1 Γ(*p* + 1) (1 − *e* <sup>−</sup>1) <sup>≤</sup> *<sup>M</sup>*<sup>1</sup> Γ(*q* + 1) (1 − *e* <sup>−</sup>1)+(<sup>2</sup> <sup>−</sup> *<sup>e</sup>* <sup>−</sup>1)<sup>Q</sup><sup>1</sup> + (<sup>1</sup> <sup>−</sup> *<sup>e</sup>* <sup>−</sup>1)<sup>Q</sup><sup>3</sup> <sup>+</sup><sup>Q</sup><sup>4</sup> |*h*| *<sup>ρ</sup>* 0 *s <sup>q</sup>*(<sup>1</sup> <sup>−</sup> *<sup>e</sup>* −*s* )*dA*(*s*) + *n*−2 ∑ *i*=1 <sup>|</sup>*βi*|*σ<sup>q</sup> <sup>i</sup>* (1 − *e* <sup>−</sup>*σi*)

$$\begin{aligned} &+|k\_1|\int\_{\nu}^{1}s^q(1-e^{-s})dA(s)\Big|\Big) + \frac{M\_2}{\Gamma(p+1)}\Big{(}(2-e^{-1})\tilde{\mathcal{Q}}\_2 \\ &+\tilde{\mathcal{Q}}\_3\Big{[}|k|\int\_0^p s^p(1-e^{-s})dA(s) + \sum\_{i=1}^{n-2}|a\_i|\sigma\_i^p(1-e^{-\sigma\_i}) \\ &+|k\_1|\int\_{\nu}^{1}s^p(1-e^{-s})dA(s)\Big{]} + (1-e^{-1})\tilde{\mathcal{Q}}\_4\Big{)} \\ &=\quad\Lambda\_1\mathcal{M}\_1 + \Lambda\_2\mathcal{M}\_2.\end{aligned}$$

Thus,

$$\|\|T\_1(\mathcal{X}, \mathcal{Y})\|\|\quad \leq \quad \Lambda\_1 M\_1 + \Lambda\_2 M\_2. \tag{38}$$

Similarly, we have

$$\|\|T\_2(\mathcal{X}, \mathcal{Y})\|\|\leq \quad \overline{\Lambda}\_1 M\_1 + \overline{\Lambda}\_2 M\_2. \tag{39}$$

Hence, (38) and (39) imply that the operator *T* uniformly bounded. Now, we establish that the operator *T* is equicontinuous. For *t*1, *t*<sup>2</sup> ∈ [0, 1] with *t*<sup>1</sup> < *t*2, we obtain

 *T*1(X , Y)(*t*2) − *T*1(X , Y)(*t*1) ≤ *<sup>t</sup>*<sup>1</sup> 0 [*e* <sup>−</sup>(*t*2−*s*) <sup>−</sup> *<sup>e</sup>* −(*t*1−*s*) ]*I q* <sup>0</sup><sup>+</sup> f(*s*, X (*s*), Y(*s*))*ds* + *<sup>t</sup>*<sup>2</sup> *t*1 *e* <sup>−</sup>(*t*2−*s*)*I q* <sup>0</sup><sup>+</sup> f(*s*, X (*s*), Y(*s*))*ds* + Q1(*t*2) − Q1(*t*1) ! <sup>1</sup> 0 *e* <sup>−</sup>(1−*s*)*I q* <sup>0</sup><sup>+</sup> f(*s*, X (*s*), Y(*s*))*ds* + *I q* <sup>0</sup><sup>+</sup> f(*s*, X (*s*), Y(*s*))(1) " + Q2(*t*2) − Q2(*t*1) ! <sup>1</sup> 0 *e* <sup>−</sup>(1−*s*)*I p* <sup>0</sup><sup>+</sup> g(*s*, X (*s*), Y(*s*))*ds* + *I p* <sup>0</sup><sup>+</sup> g(*s*, X (*s*), Y(*s*))(1) " + Q3(*t*2) − Q3(*t*1) ! |*k*| *<sup>ρ</sup>* 0 *<sup>s</sup>* 0 *e* <sup>−</sup>(*s*−*z*)*I p* <sup>0</sup><sup>+</sup> <sup>g</sup>(*z*, <sup>X</sup> (*z*), <sup>Y</sup>(*z*))*dz dA*(*s*) + *n*−2 ∑ *i*=1 |*αi*| *<sup>σ</sup><sup>i</sup>* 0 *e* <sup>−</sup>(*σi*−*s*)*I p* <sup>0</sup><sup>+</sup> g(*s*, X (*s*), Y(*s*))*ds* +|*k*1| <sup>1</sup> *ν <sup>s</sup>* 0 *e* <sup>−</sup>(*s*−*z*)*I p* <sup>0</sup><sup>+</sup> <sup>g</sup>(*z*, <sup>X</sup> (*z*), <sup>Y</sup>(*z*))*dz dA*(*s*) + <sup>1</sup> 0 *e* <sup>−</sup>(1−*s*)*I q* <sup>0</sup><sup>+</sup> f(*s*, X (*s*), Y(*s*))*ds* " + Q4(*t*2) − Q4(*t*1) ! |*h*| *<sup>ρ</sup>* 0 *<sup>s</sup>* 0 *e* <sup>−</sup>(*s*−*z*)*I q* <sup>0</sup><sup>+</sup> <sup>f</sup>(*z*, <sup>X</sup> (*z*), <sup>Y</sup>(*z*))*dz dA*(*s*) + *n*−2 ∑ *i*=1 |*βi*| *<sup>σ</sup><sup>i</sup>* 0 *e* <sup>−</sup>(*σi*−*s*)*I q* <sup>0</sup><sup>+</sup> f(*s*, X (*s*), Y(*s*))*ds* +|*h*1| <sup>1</sup> *ν <sup>s</sup>* 0 *e* <sup>−</sup>(*s*−*z*)*I q* <sup>0</sup><sup>+</sup> <sup>f</sup>(*z*, <sup>X</sup> (*z*), <sup>Y</sup>(*z*))*dz dA*(*s*) + <sup>1</sup> 0 *e* <sup>−</sup>(1−*s*)*I p* <sup>0</sup><sup>+</sup> g(*s*, X (*s*), Y(*s*))*ds* " <sup>≤</sup> *<sup>M</sup>*<sup>1</sup> Γ(*q* + 1) ! *t q* 1 *e* <sup>−</sup>(*t*2−*t*1) <sup>−</sup> <sup>1</sup> <sup>−</sup> *<sup>e</sup>* <sup>−</sup>*t*<sup>2</sup> + *e* −*t*<sup>1</sup> + *t q* 2 1 − *e* −(*t*2−*t*1) "

$$\begin{split} & \quad + \frac{M\_1}{\Gamma(q+1)} \left\{ (2-e^{-1}) \left| \mathcal{Q}\_1(t\_2) - \mathcal{Q}\_1(t\_1) \right| + (1-e^{-1}) \left| \mathcal{Q}\_3(t\_2) - \mathcal{Q}\_3(t\_1) \right| \right. \\ & \left. + \left| \mathcal{Q}\_4(t\_2) - \mathcal{Q}\_4(t\_1) \right| \left| \left| h \right| \right| \int\_0^\rho s^q (1-e^{-s}) dA(s) \right| \\ & \left. + \left| \sum\_{i=1}^{n-2} \beta\_i \sigma\_i^q (1-e^{-\sigma\_i}) \right| + |h\_1| \left| \int\_\nu s^q (1-e^{-s}) dA(s) \right| \right) \right. \\ & \left. + \frac{M\_2}{\Gamma(p+1)} \left\{ (2-e^{-1}) \left| \mathcal{Q}\_2(t\_2) - \mathcal{Q}\_2(t\_1) \right| \right. \\ & \left. + \left| \mathcal{Q}\_3(t\_2) - \mathcal{Q}\_3(t\_1) \right| \left| \left| k \right| \right| \int\_0^\rho s^p (1-e^{-s}) dA(s) \right| \\ & \left. + \left| \sum\_{i=1}^{n-2} a\_i \sigma\_i^p (1-e^{-\sigma\_i}) \right| + |k\_1| \left| \int\_\nu s^p (1-e^{-s}) dA(s) \right| \right. \\ & \left. + \left| \sum\_{i=1}^{n-2} \sigma\_i \sigma\_i^p (1-e^{-\sigma\_i}) \right| + |k\_1| \left| \int\_\nu s^p (1-e^{-s}) dA(s) \right| \right. \end{split}$$

and

 *T*2(X , Y)(*t*2) − *T*2(X , Y)(*t*1) <sup>≤</sup> *<sup>M</sup>*<sup>2</sup> Γ(*p* + 1) ! *t p* 1 *e* <sup>−</sup>(*t*2−*t*1) <sup>−</sup> <sup>1</sup> <sup>−</sup> *<sup>e</sup>* <sup>−</sup>*t*<sup>2</sup> + *e* −*t*<sup>1</sup> + *t p* 2 1 − *e* −(*t*2−*t*1) " + *M*<sup>1</sup> Γ(*q* + 1) (2 − *e* <sup>−</sup>1) P1(*t*2) − P1(*t*1) + (<sup>1</sup> <sup>−</sup> *<sup>e</sup>* <sup>−</sup>1) P3(*t*2) − P3(*t*1) + P4(*t*2) − P4(*t*1) |*h*| *<sup>ρ</sup>* 0 *s <sup>q</sup>*(<sup>1</sup> <sup>−</sup> *<sup>e</sup>* −*s* )*dA*(*s*) + *n*−2 ∑ *i*=1 *<sup>β</sup>iσ<sup>q</sup> <sup>i</sup>* (1 − *e* <sup>−</sup>*σi*) <sup>+</sup> <sup>|</sup>*h*1<sup>|</sup> <sup>1</sup> *ν s <sup>q</sup>*(<sup>1</sup> <sup>−</sup> *<sup>e</sup>* −*s* )*dA*(*s*) # + *M*<sup>2</sup> Γ(*p* + 1) (2 − *e* <sup>−</sup>1) P2(*t*2) − P2(*t*1) + P3(*t*2) − P3(*t*1) |*k*| *<sup>ρ</sup>* 0 *<sup>s</sup>p*(<sup>1</sup> <sup>−</sup> *<sup>e</sup>* −*s* )*dA*(*s*) + *n*−2 ∑ *i*=1 *<sup>α</sup>iσ<sup>p</sup> <sup>i</sup>* (1 − *e* <sup>−</sup>*σi*) <sup>+</sup> <sup>|</sup>*k*1<sup>|</sup> <sup>1</sup> *ν <sup>s</sup>p*(<sup>1</sup> <sup>−</sup> *<sup>e</sup>* −*s* )*dA*(*s*) + (1 − *e* <sup>−</sup>1) P4(*t*2) − P4(*t*1) .

Clearly, |*T*1(X , Y)(*t*2) − *T*1(X , Y)(*t*1)| → 0 and |*T*2(X , Y)(*t*2) − *T*2(X , Y)(*t*1)| → 0 as *t*<sup>2</sup> → *t*<sup>1</sup> independent of (X , Y) ∈ V. In consequence, the operator *T*(X , Y) is equicontinuous. Hence, it follows, according to Arzelá-Ascoli theorem, that *T*(X , Y) is completely continuous. In the second step, we consider a set

$$\mathcal{U} = \{ (\mathcal{X}, \mathcal{Y}) \in \mathfrak{X} \times \mathfrak{X} | (\mathcal{X}, \mathcal{Y}) = \sigma T(\mathcal{X}, \mathcal{Y}), 0 < \sigma < 1 \}$$

and show that it is bounded. Let (X , Y) ∈ U, then (X , Y) = *σT*(X , Y) and for any *t* ∈ [0, 1], we have

$$\mathcal{X}(t) = \sigma T\_1(\mathcal{X}, \mathcal{Y})(t), \; \mathcal{Y}(t) = \sigma T\_2(\mathcal{X}, \mathcal{Y})(t).$$

In consequence, we have

$$|\mathcal{X}(t)| \le \Lambda\_1(\eta\_0 + \eta\_1|\mathcal{X}| + \eta\_2|\mathcal{Y}|) + \Lambda\_2(\mathcal{Z}\_0 + \mathcal{Z}\_1|\mathcal{X}| + \mathcal{Z}\_2|\mathcal{Y}|),$$

which leads to

$$\|\mathcal{X}\| \le \Lambda\_1(\eta\_0 + \eta\_1 \|\mathcal{X}\| + \eta\_2 \|\mathcal{Y}\|) + \Lambda\_2(\zeta\_0 + \zeta\_1 \|\mathcal{X}\| + \zeta\_2 \|\mathcal{Y}\|). \tag{40}$$

Likewise, one can obtain that

$$\|\|\mathcal{Y}\|\| \le \overline{\Lambda}\_1(\eta\_0 + \eta\_1 \|\|\mathcal{X}\|\| + \eta\_2 \|\|\mathcal{Y}\|\|) + \overline{\Lambda}\_2(\zeta\_0 + \zeta\_1 \|\|\mathcal{X}\|\| + \zeta\_2 \|\|\mathcal{Y}\|\|). \tag{41}$$

From (40) and (41), together with notations (36) and (37), we obtain

$$\begin{split} \|\|\mathcal{X}\|\| + \|\mathcal{Y}\|\| &\leq \left[ (\Lambda\_1 + \overline{\Lambda}\_1)\eta\_0 + (\Lambda\_2 + \overline{\Lambda}\_2)\zeta\_0 \right] + \left[ (\Lambda\_1 + \overline{\Lambda}\_1)\eta\_1 + (\Lambda\_2 + \overline{\Lambda}\_2)\zeta\_1 \right] \|\|\mathcal{X}\| \\ &+ \left[ (\Lambda\_1 + \overline{\Lambda}\_1)\eta\_2 + (\Lambda\_2 + \overline{\Lambda}\_2)\zeta\_2 \right] \|\|\mathcal{Y}\|\| \end{split}$$

which implies that

$$\|\|(\mathcal{X},\mathcal{Y})\|\| \le \Omega\_0 + \max\{\Omega\_1 + \Omega\_2\} \|(\mathcal{X},\mathcal{Y})\|\| \le \Omega\_0 + \Omega \|(\mathcal{X},\mathcal{Y})\|\|.$$

Thus

$$\|(\mathcal{X}, \mathcal{Y})\| \le \frac{\Omega\_0}{1 - \Omega} \prime$$

which shows that U is bounded. In view of the foregoing steps, we deduce that the hypothesis of the Leray–Schauder alternative [6] is satisfied; hence, its conclusion implies that the operator *T* has at least one fixed point. Thus, there is at least one solution to the problem (1) and (2) on [0, 1].

Our next result deals with the uniqueness of solutions for the problem (1) and (2) and relies on Banach's fixed point theorem.

**Theorem 2.** *Let the condition* (F2) *hold, and that*

$$
\Theta + \overline{\Theta} < 1,\tag{42}
$$

*where* Θ *and* Θ *are given in (31). Then, there is a unique solution to the problem (1) and (2) on* [0, 1]*.*

**Proof.** Let us first establish that *T*U*<sup>ε</sup>* ⊂ U*ε*, where the operator *T* is given by (28) and

$$\mathcal{U}\_{\varepsilon} = \{ (\mathcal{X}, \mathcal{Y}) \in \mathfrak{X} \times \mathfrak{X} : \| (\mathcal{X}, \mathcal{Y}) \| \le \varepsilon \} \,,$$

with *ε* > M + M 1 − (Θ + Θ) , Θ, Θ and M,M are respectively given by (31) and (32). By the assumption (F2) and (35), for (X , Y) ∈ U*ε*, *t* ∈ [0, 1], we have

$$\begin{aligned} |\left|\mathfrak{f}(t,\mathcal{X}(t),\mathcal{Y}(t))\right| &\leq& |\mathfrak{f}(t,\mathcal{X}(t),\mathcal{Y}(t)) - \mathfrak{f}(t,0,0)| + |\mathfrak{f}(t,0,0)| \\ &\leq& L\_1(|\mathcal{X}(t)| + |\mathcal{Y}(t)|) + \mathcal{N}\_1 \leq L\_1(|\mathcal{X}| + |\mathcal{Y}|) + \mathcal{N}\_1 \leq L\_1\varepsilon + \mathcal{N}\_1. \end{aligned}$$

Similarly, one can show that |g(*t*, X (*t*), Y(*t*))| ≤ *L*2*ε* + N2. Taking into account (31) and (32), we obtain

$$|T\_1(\mathcal{X}, \mathcal{Y})(t)| \le \quad (\Lambda\_1 L\_1 + \Lambda\_2 L\_2)\varepsilon + (\Lambda\_1 \mathcal{N}\_1 + \Lambda\_2 \mathcal{N}\_2) = \Theta \varepsilon + \mathcal{M}\_1$$

which yields

$$\|\|T\_1(\mathcal{X}, \mathcal{Y})\|\|\leq \quad \Theta \varepsilon + \mathcal{M}.\tag{43}$$

In a similar manner, we obtain

$$\left\|\left|T\_2(\mathcal{X}, \mathcal{Y})\right\|\right\| \leq \left\|\overline{\Theta}\varepsilon + \overline{\mathcal{M}}.\tag{44}$$

It then follows from (43) and (44) that

$$\|T(\mathcal{X}, \mathcal{Y})\| \le (\Theta \varepsilon + \mathcal{M}) + (\overline{\Theta} \varepsilon + \overline{\mathcal{M}}) = (\Theta + \overline{\Theta})\varepsilon + (\mathcal{M} + \overline{\mathcal{M}}) \le \varepsilon.$$

Consequently, *T*U*<sup>ε</sup>* ⊂ U*ε*. Next, we show that the operator *T* is a contraction. Using conditions (F2) and (31), we get

 *T*1(X1, Y1) − *T*1(X2, Y2) = sup *t*∈[0,1] |*T*1(X1, Y1)(*t*) − *T*1(X2, Y2)(*t*)| ≤ sup *t*∈[0,1] *<sup>t</sup>* 0 *e* −(*t*−*s*) *I q* <sup>0</sup><sup>+</sup> f(*s*, X1(*s*), Y1(*s*)) − *I q* <sup>0</sup><sup>+</sup> f(*s*, X2(*s*), Y2(*s*)) *ds* +|Q1(*t*)| <sup>1</sup> 0 *e* −(1−*s*) *I q* <sup>0</sup><sup>+</sup> f(*s*, X1(*s*), Y1(*s*)) − *I q* <sup>0</sup><sup>+</sup> f(*s*, X2(*s*), Y2(*s*)) *ds* + *I q* <sup>0</sup><sup>+</sup> f(*s*, X1(*s*), Y1(*s*)) − *I q* <sup>0</sup><sup>+</sup> f(*s*, X2(*s*), Y2(*s*)) (1) + |Q2(*t*)| <sup>1</sup> 0 *e* −(1−*s*) *I p* <sup>0</sup><sup>+</sup> g(*s*, X1(*s*), Y1(*s*)) − *I p* <sup>0</sup><sup>+</sup> g(*s*, X2(*s*), Y2(*s*)) *ds* + *I p* <sup>0</sup><sup>+</sup> g(*s*, X1(*s*), Y1(*s*)) − *I p* <sup>0</sup><sup>+</sup> g(*s*, X2(*s*), Y2(*s*)) (1) +|Q3(*t*)| |*k*| *<sup>ρ</sup>* 0 *<sup>s</sup>* 0 *e* −(*s*−*z*) *I p* <sup>0</sup><sup>+</sup> g(*z*, X1(*z*), Y1(*z*)) − *I p* <sup>0</sup><sup>+</sup> g(*z*, X2(*z*), Y2(*z*)) *dz dA*(*s*) + *n*−2 ∑ *i*=1 |*αi*| *<sup>σ</sup><sup>i</sup>* 0 *e* −(*σi*−*s*) *I p* <sup>0</sup><sup>+</sup> g(*s*, X1(*s*), Y1(*s*)) − *I p* <sup>0</sup><sup>+</sup> g(*s*, X2(*s*), Y2(*s*)) *ds* +|*k*1| <sup>1</sup> *ν <sup>s</sup>* 0 *e* −(*s*−*z*) *I p* <sup>0</sup><sup>+</sup> g(*z*, X1(*z*), Y1(*z*)) − *I p* <sup>0</sup><sup>+</sup> g(*z*, X2(*z*), Y2(*z*)) *dz dA*(*s*) + <sup>1</sup> 0 *e* −(1−*s*) *I q* <sup>0</sup><sup>+</sup> f(*s*, X1(*s*), Y1(*s*)) − *I q* <sup>0</sup><sup>+</sup> f(*s*, X2(*s*), Y2(*s*)) *ds* +|Q4(*t*)| |*h*| *<sup>ρ</sup>* 0 *<sup>s</sup>* 0 *e* −(*s*−*z*) *I q* <sup>0</sup><sup>+</sup> f(*z*, X1(*z*), Y1(*z*)) − *I q* <sup>0</sup><sup>+</sup> f(*z*, X2(*z*), Y2(*z*)) *dz dA*(*s*) + *n*−2 ∑ *i*=1 |*βi*| *<sup>σ</sup><sup>i</sup>* 0 *e* −(*σi*−*s*) *I q* <sup>0</sup><sup>+</sup> f(*s*, X1(*s*), Y1(*s*)) − *I q* <sup>0</sup><sup>+</sup> f(*s*, X2(*s*), Y2(*s*)) *ds* +|*h*1| <sup>1</sup> *ν <sup>s</sup>* 0 *e* −(*s*−*z*) *I q* <sup>0</sup><sup>+</sup> f(*z*, X1(*z*), Y1(*z*)) − *I q* <sup>0</sup><sup>+</sup> f(*z*, X2(*z*), Y2(*z*)) *dz dA*(*s*) + <sup>1</sup> 0 *e* −(1−*s*) *I p* <sup>0</sup><sup>+</sup> g(*s*, X1(*s*), Y1(*s*)) − *I p* <sup>0</sup><sup>+</sup> g(*s*, X2(*s*), Y2(*s*)) *ds* ≤ Λ1*L*<sup>1</sup> X1 − X2 + Y1 − Y2 + Λ2*L*<sup>2</sup> X1 − X2 + Y1 − Y2 = Λ1*L*<sup>1</sup> + Λ2*L*<sup>2</sup> X1 − X2 + Y1 − Y2 = Θ X1 − X2 + Y1 − Y2 .

Similarly, we can find that

$$\begin{aligned} \|T\_2(\mathcal{X}\_1, \mathcal{Y}\_1) - T\_2(\mathcal{X}\_2, \mathcal{Y}\_2)\| &= \sup\_{t \in [0, 1]} |T\_2(\mathcal{X}\_1, \mathcal{Y}\_1)(t) - T\_2(\mathcal{X}\_2, \mathcal{Y}\_2)(t)| \\ &\le \quad \left(\overline{\Lambda}\_1 L\_1 + \overline{\Lambda}\_2 L\_2\right) \left(\|\mathcal{X}\_1 - \mathcal{X}\_2\| + \|\mathcal{Y}\_1 - \mathcal{Y}\_2\|\right) \\ &= \quad \overline{\Theta} \left(\|\mathcal{X}\_1 - \mathcal{X}\_2\| + \|\mathcal{Y}\_1 - \mathcal{Y}\_2\|\right). \end{aligned}$$

Hence we obtain

$$\left\| \left| T(\mathcal{X}\_1, \mathcal{Y}\_1) - T(\mathcal{X}\_2, \mathcal{Y}\_2) \right| \right\| \le (\Theta + \overline{\Theta}) (\left\| \mathcal{X}\_1 - \mathcal{X}\_2 \right\| + \left\| \mathcal{Y}\_1 - \mathcal{Y}\_2 \right\|).$$

which, in view of the condition (42), shows that *T* is a contraction. Thus, the conclusion of Banach's fixed-point theorem applies and, hence, the problem (1) and (2) has a unique solution on [0, 1]. The proof is finished.

#### **4. An Example**

**Example 1.** *Consider a coupled system of fractional differential equations*

$$\begin{cases} (^cD^{26/7} + ^cD^{19/7})\mathcal{X}(t) = \frac{135\mathcal{X}(t)}{225 + t} + \frac{3\sin\mathcal{Y}(t)}{13 + t^2} + \frac{3}{13\sqrt{9 + t^2}},\\ (^cD^{17/5} + ^cD^{12/5})\mathcal{Y}(t) = \frac{\sqrt{16 - t^2}}{\pi(40 + t)}\sin(2\pi\mathcal{X}(t)) + \frac{24|\tan^{-1}\mathcal{Y}(t)|}{\pi(t^2 + 120)} + \frac{\ln 5}{2}, \ t \in [0, 1], \end{cases} \tag{45}$$

*equipped with the coupled boundary conditions*

$$\begin{cases} \begin{aligned} \mathcal{X}(0) = 0, \mathcal{X}'(0) = 0, \mathcal{X}'(1) = 0, \mathcal{X}(1) = k \int\_0^\rho \mathcal{Y}(s) dA(s) + \sum\_{l=1}^3 a\_l \mathcal{Y}(\sigma\_l) + k\_1 \int\_\nu^1 \mathcal{Y}(s) dA(s), \\ \mathcal{Y}(0) = 0, \mathcal{Y}'(0) = 0, \mathcal{Y}'(1) = 0, \mathcal{Y}(1) = h \int\_0^\rho \mathcal{X}(s) dA(s) + \sum\_{l=1}^3 \beta\_l \mathcal{X}(\sigma\_l) + h\_1 \int\_\nu^1 \mathcal{X}(s) dA(s). \end{aligned} \end{cases} \tag{46}$$

*Here <sup>q</sup>* <sup>=</sup> 19/7, *<sup>p</sup>* <sup>=</sup> 12/5, *<sup>k</sup>* <sup>=</sup> 3/16, *<sup>k</sup>*<sup>1</sup> <sup>=</sup> 2/175, *<sup>h</sup>* <sup>=</sup> 5/88, *<sup>h</sup>*<sup>1</sup> <sup>=</sup> 3/104, *<sup>A</sup>*(*s*) = <sup>1</sup> <sup>+</sup> *<sup>s</sup>r*+<sup>1</sup> *<sup>r</sup>*+<sup>1</sup> , *<sup>r</sup>* <sup>∈</sup> <sup>N</sup>*, <sup>ρ</sup>* <sup>=</sup> 2/7, *<sup>ν</sup>* <sup>=</sup> 6/7, *<sup>σ</sup>*<sup>1</sup> <sup>=</sup> 3/7, *<sup>σ</sup>*<sup>2</sup> <sup>=</sup> 4/7, *<sup>σ</sup>*<sup>3</sup> <sup>=</sup> 5/7, *<sup>α</sup>*<sup>1</sup> <sup>=</sup> 1/10, *<sup>α</sup>*<sup>2</sup> <sup>=</sup> 1/414, *α*<sup>3</sup> = 3/313, *β*<sup>1</sup> = 1/3, *β*<sup>2</sup> = 1/41, *β*<sup>3</sup> = 7/121. *Clearly*

$$\begin{aligned} |\mathfrak{f}(t, \mathcal{X}(t), \mathcal{Y}(t))| &\leq \frac{1}{13} + \frac{3}{5} \|\mathcal{X}\| + \frac{3}{13} \|\mathcal{Y}\|, \\ |\mathfrak{g}(t, \mathcal{X}(t), \mathcal{Y}(t))| &\leq \frac{\ln 5}{2} + \frac{1}{5} \|\mathcal{X}\| + \frac{1}{10} \|\mathcal{Y}\|. \end{aligned}$$

*and hence η*<sup>0</sup> = 1/13, *η*<sup>1</sup> = 3/5, *η*<sup>2</sup> = 3/13, *ζ*<sup>0</sup> = (ln 5)/2, *ζ*<sup>1</sup> = 1/5, *ζ*<sup>2</sup> = 1/10. *Using (36) and (37) with the given data and r* = 2, *we find that* Ω<sup>1</sup> 0.331501, Ω<sup>2</sup> 0.138843 *and* Ω = max{Ω1, Ω2} 0.331501 < 1. *Therefore, by Theorem 1, the problem (45) and (46) has at least one solution on* [0, 1].

*To explain Theorem 2, we consider the following system of sequential fractional differential equations supplemented with the boundary conditions (46):*

$$\begin{cases} \begin{aligned} \left(^{c}D^{26/7} + ^{c}D^{19/7}\right)\mathcal{X}(t) &= \frac{3e^{-t}}{\sqrt{(t^{4}+25)}} \frac{|\mathcal{X}(t)|}{(1+|\mathcal{X}(t)|)} + \frac{18}{(t^{2}+30)} \sin(\mathcal{Y}(t)) + \frac{9}{2\sqrt{5+t}},\\ \left(^{c}D^{17/5} + ^{c}D^{12/5}\right)\mathcal{Y}(t) &= \frac{1}{(t+10)} \tan^{-1}\mathcal{X}(t) + \frac{e^{-t}}{10} \frac{|\mathcal{Y}(t)|^{3}}{(1+|\mathcal{Y}(t)|^{3})} + \frac{\cos(t+1)}{(9+t)},\end{aligned} \end{cases} \end{cases}$$

*t* ∈ [0, 1]. *It is easy to check whether* |f(*t*, X1, Y1) − f(*t*, X2, Y2)| ≤ *L*1( X1 − X2 + Y1 − Y2 ) *with L*<sup>1</sup> = 3/5 *and* |g(*t*, X1, Y1) −g(*t*, X2, Y2)| ≤ *L*2( X1 − X2 + Y1 − Y2 ) *with L*<sup>2</sup> = 1/10*. Additionally,* Θ + Θ 0.282351 < 1. *Therefore, the hypothesis of Theorem 2 is satisfied. Hence, by the conclusion of Theorem 2, there is a unique solution to the system (47) equipped with the boundary conditions (46) on* [0, 1]*.*

#### **5. Discussion**

We have presented the criteria ensuring the existence and uniqueness of solutions for a coupled system of higher-order sequential Caputo fractional differential equations complemented with Riemann–Stieltjes integro-multipoint boundary conditions on the interval [0, 1]. A characteristic of the method employed in the present study is its generality, as it can be applied to a variety of boundary value problems. As a special case, our results become associated with multipoint boundary conditions:

$$\begin{cases} \begin{aligned} \mathcal{X}(0) = 0, \, \mathcal{X}'(0) = 0, \, \mathcal{X}'(1) = 0, \, \mathcal{X}(1) = \sum\_{i=1}^{n-2} a\_i \mathcal{Y}(\sigma\_i), \\ \mathcal{Y}(0) = 0, \, \mathcal{Y}'(0) = 0, \, \mathcal{Y}'(1) = 0, \, \mathcal{Y}(1) = \sum\_{i=1}^{n-2} \beta\_i \mathcal{X}(\sigma\_i), \end{aligned} \end{cases} \tag{48}$$

if we take *k* = *k*<sup>1</sup> = *h* = *h*<sup>1</sup> = 0 in (2). In this case, the corresponding operators take the form:

$$\begin{split} \widehat{T}\_{1}(\mathcal{X},\mathcal{Y})(t) &= \int\_{0}^{t} e^{-(t-s)} I\_{0+}^{q} \mathfrak{f}(s, \mathcal{X}(s), \mathcal{Y}(s)) ds \\ &+ \mathcal{Q}\_{1}(t) \left[ \int\_{0}^{1} e^{-(1-s)} I\_{0+}^{q} \mathfrak{f}(s, \mathcal{X}(s), \mathcal{Y}(s)) ds - I\_{0+}^{q} \mathfrak{f}(s, \mathcal{X}(s), \mathcal{Y}(s))(1) \right] \\ &+ \mathcal{Q}\_{2}(t) \left[ \int\_{0}^{1} e^{-(1-s)} I\_{0+}^{p} \mathfrak{g}(s, \mathcal{X}(s), \mathcal{Y}(s)) ds - I\_{0+}^{p} \mathfrak{g}(s, \mathcal{X}(s), \mathcal{Y}(s))(1) \right] \\ &+ \mathcal{Q}\_{3}(t) \left[ \sum\_{i=1}^{n-2} \alpha\_{i} \int\_{0}^{\sigma\_{i}} e^{-(\sigma\_{i}-s)} I\_{0+}^{p} \mathfrak{g}(s, \mathcal{X}(s), \mathcal{Y}(s)) ds \\ &- \int\_{0}^{1} e^{-(1-s)} I\_{0+}^{q} \mathfrak{f}(s, \mathcal{X}(s), \mathcal{Y}(s)) ds \right] \\ &+ \mathcal{Q}\_{4}(t) \left[ \sum\_{i=1}^{n-2} \beta\_{i} \int\_{0}^{\sigma\_{i}} e^{-(\sigma\_{i}-s)} I\_{0+}^{q} \mathfrak{f}(s, \mathcal{X}(s), \mathcal{Y}(s)) ds \right] \\ &- \int\_{0}^{1} e^{-(1-s)} I\_{0+}^{p} \mathfrak{g}(s, \mathcal{X}(s), \mathcal{Y}(s)) ds \Big], \end{split}$$

$$\begin{split} \widehat{T}\_{2}(\mathcal{X},\mathcal{Y})(t) &= \int\_{0}^{t} e^{-(t-s)} I\_{0^{+}}^{p} \mathfrak{g}(s,\mathcal{X}(s),\mathcal{Y}(s)) ds \\ &\quad + \mathcal{P}\_{1}(t) \left[ \int\_{0}^{1} e^{-(1-s)} I\_{0^{+}}^{q} \mathfrak{f}(s,\mathcal{X}(s),\mathcal{Y}(s)) ds - I\_{0^{+}}^{q} \mathfrak{f}(s,\mathcal{X}(s),\mathcal{Y}(s))(1) \right] \\ &\quad + \mathcal{P}\_{2}(t) \left[ \int\_{0}^{1} e^{-(1-s)} I\_{0^{+}}^{p} \mathfrak{g}(s,\mathcal{X}(s),\mathcal{Y}(s)) ds - I\_{0^{+}}^{p} \mathfrak{g}(s,\mathcal{X}(s),\mathcal{Y}(s))(1) \right] \\ &\quad + \mathcal{P}\_{3}(t) \left[ \sum\_{i=1}^{n-2} a\_{i} \int\_{0}^{\sigma\_{i}} e^{-(\sigma\_{i}-s)} I\_{0^{+}}^{p} \mathfrak{g}(s,\mathcal{X}(s),\mathcal{Y}(s)) ds \\ &\quad - \int\_{0}^{1} e^{-(1-s)} I\_{0^{+}}^{q} \mathfrak{f}(s,\mathcal{X}(s),\mathcal{Y}(s)) ds \right] \\ &\quad + \mathcal{P}\_{4}(t) \left[ \sum\_{i=1}^{n-2} \beta\_{i} \int\_{0}^{\sigma\_{i}} e^{-(\sigma\_{i}-s)} I\_{0^{+}}^{q} \mathfrak{f}(s,\mathcal{X}(s),\mathcal{Y}(s)) ds \\ &\quad - \int\_{0}^{1} e^{-(1-s)} I\_{0^{+}}^{p} \mathfrak{g}(s,\mathcal{X}(s),\mathcal{Y}(s)) ds \right]. \end{split}$$

In future, we plan to develop the existence theory for the multivalued analogue of the problem (1) and (2). Moreover, the boundary value problem considered in this paper can be studied for other kinds of derivatives, such as Hadamard, Caputo–Hadamard, Hilfer, Hilfer–Hadamard, etc.

**Author Contributions:** Conceptualization, Y.A., B.A. and S.K.N.; methodology, Y.A., B.A., S.K.N. and A.S.M.A.; validation, Y.A., B.A., S.K.N. and A.S.M.A.; formal analysis, Y.A., B.A., S.K.N. and A.S.M.A.; writing—original draft preparation, Y.A., B.A., S.K.N. and A.S.M.A.; funding acquisition, A.S.M.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** Taif University Researchers Supporting Project number (TURSP-2020/303), Taif University, Taif, Saudi Arabia.

**Acknowledgments:** Taif University Researchers Supporting Project number (TURSP-2020/303), Taif University, Taif, Saudi Arabia. The authors thank the reviewers for their constructive remarks on their work.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Mawhin's Continuation Technique for a Nonlinear BVP of Variable Order at Resonance via Piecewise Constant Functions**

**Shahram Rezapour 1,2,† , Mohammed Said Souid 3,†, Sina Etemad 1,† , Zoubida Bouazza 4,†, Sotiris K. Ntouyas 5,6,† , Suphawat Asawasamrit 7,† and Jessada Tariboon 7,\*,†**


**Abstract:** In this paper, we establish the existence of solutions to a nonlinear boundary value problem (BVP) of variable order at resonance. The main theorem in this study is proved with the help of generalized intervals and piecewise constant functions, in which we convert the mentioned Caputo BVP of fractional variable order to an equivalent standard Caputo BVP at resonance of constant order. In fact, to use the Mawhin's continuation technique, we have to transform the variable order BVP into a constant order BVP. We prove the existence of solutions based on the existing notions in the coincidence degree theory and Mawhin's continuation theorem (MCTH). Finally, an example is provided according to the given variable order BVP to show the correctness of results.

**Keywords:** piecewise constant function; Mawhin's continuation technique; variable order; resonance; existence

#### **1. Introduction**

The initial idea of fractional calculus is taken from the powers of real or complex numbers in the order of differentiation and integration operators. In recent decades, fractional operators of variable order are appeared extensively in a vast domain of sciences including chaotic dynamical systems, fractal theory, rheology, signal processing, mathematical modeling, control theory, and biomedical applications. This range of applications is due to the fact that fractional derivatives provide a strong tool in the mathematics to describe the memory and hereditary properties of processes and various materials; see, for example [1–3].

Before the variable order systems, discussion of boundary value problems with fractional constant orders has attracted the attention of most researchers, and valuable findings have been established. Various researches have been conducted to study the behaviors of different fractional BVPs by means of some known methods such as fixed point theorems, numerical methods, monotone iterative methods, variational methods, and etc. [4–12].

Nevertheless, in addition to numerous published papers on fractional constant order problems, few studies on the existence theory have been done in relation to variable order problems [13–19]. Hence, investigation of this interesting and general topic makes all our findings worthy.

**Citation:** Rezapour, S.; Souid, M.S.; Etemad, S.; Bouazza, Z.; Ntouyas, S.K.; Asawasamrit, S.; Tariboon, J. Mawhin's Continuation Technique for a Nonlinear BVP of Variable Order at Resonance via Piecewise Constant Functions. *Fractal Fract.* **2021**, *5*, 216. https://doi.org/10.3390/fractalfract 5040216

Academic Editor: Rodica Luca

Received: 17 October 2021 Accepted: 10 November 2021 Published: 12 November 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

In 1970, Gaines and Mawhin [20] introduced the theory of coincidence degree for analysis of differential and functional equations. Mawhin has made important contributions since then, and the mentioned theory is also famous to the Mawhin's coincidence theory. Coincidence theory is considered as a powerful technique, especially with regard to questions about the existence of solutions for nonlinear differential equations. Mawhin's theory permits the use of a method based on the topological degree notion for some problems which can be written as an abstract operator equation of the form Θ*x* = *Wx*, where Θ is a linear non-invertible operator and *W* is a nonlinear operator acting on a Banach space.

In 1972, Mawhin extended a technique to solve this operator equation in his famous paper [21]. He assumed that Θ is a Fredholm operator of index zero. Then, he developed a new theory of topological degree known as the degree of coincidence for (Θ, *W*), that is also known as Mawhin's coincidence degree theory in honor of him.

A given boundary value problem is said to be at resonance if the corresponding linear homogeneous BVP has a non-trivial solution. Many authors studied ordinary BVPs at resonance using Mawhin's coincidence degree theory; we can cite some works done by Feng and Webb [22], Guezane-Lakoud and Frioui [23], Mawhin and Ward [24], Infante [25], and references therein.

Based on the aforementioned technique in relation to Mawhin's method, in this paper, we shall investigate a nonlinear boundary value problem of variable order at resonance which takes the form as follows

$$\begin{cases} \ ^cD\_{0^+}^{\iota(t)}\phi(t) = \emptyset(t, \phi(t)), \ t \in A\_\iota \\\\ \phi(0) = \phi(T), \end{cases} \tag{1}$$

where *A* = [0, *T*], *T* ∈ (0, ∞), the function *u*(*t*) : *A* → (0, 1] is the order of the existing derivative in the above boundary problem, *cD<sup>u</sup>*(*t*) <sup>0</sup><sup>+</sup> is the variable order Caputo derivative, and also *<sup>g</sup>* <sup>∈</sup> *<sup>C</sup>*(*<sup>A</sup>* <sup>×</sup> <sup>R</sup>, <sup>R</sup>).

The important aim of this research is to investigate some qualitative properties of solutions of the given Caputo boundary value problem of variable order (1). The main novelty of this paper is that we use the Mawhin's continuation technique for the first time for proving the existence of solutions of a Caputo boundary value problem at resonance equipped with variable order. Most papers apply this technique on the constant order systems, while we here try to derive the necessary conditions on a variable order system. In comparison to variable order partial systems, a linear analogue of this problem can be observed in the framework of partial differential equation [26] and this shows another version of such problems and specify our main contribution in this work. It is notable for young researchers that they can implement and investigate this methods and techniques on hidden-memory variable-order fractional problems introduced in [27,28] in the future.

The structure of the paper is organized as follows: Initially, some auxiliary definitions and remarks are collected for recalling the required notions in Section 2. Further, in Section 3, based on coincidence degree theory, a partition of the given interval *A* is applied, and by defining the relevant piecewise constant functions, the existence results are derived for an equivalent constant-order BVP at resonance and accordingly, for the given Caputo BVP of variable order (1). This proof is completed in some steps. In Section 4, we give an example to illustrate the theoretical existence theorems. The paper is completed with conclusions in Section 5.

#### **2. Auxiliary Concepts**

At first, some needed concepts about our study are collected from different sources. Here, the Banach space *<sup>C</sup>*(*A*, <sup>R</sup>) consisting of continuous functions like *<sup>φ</sup>* : *<sup>A</sup>* <sup>→</sup> <sup>R</sup> is equipped with the sup-norm *φ* = sup{|*φ*(*t*)| : *t* ∈ *A*}.

**Definition 1** ([29,30])**.** *The Riemann-Liouville fractional integral (RLFI) of variable order u*(*t*) *for the function φ is defined by*

$$I\_{0^{+}}^{
u(t)}\phi(t) = \frac{1}{\Gamma(
u(t))} \int\_{0}^{t} (t-s)^{
u(t)-1} \phi(s)ds, \ t \in A,\tag{2}$$

*where* Γ(*z*) = % <sup>∞</sup> <sup>0</sup> *<sup>x</sup>z*−1*e*−*xdx, and the left Caputo fractional derivative (CFD) of variable order u*(*t*) *for φ*(*t*) *is defined by*

$${}^{c}D\_{0^{+}}^{
u(t)}\phi(t) = \frac{1}{1 - \Gamma(\mu(t))} \int\_{0}^{t} (t - s)^{-\mu(t)} \phi'(s) ds, \ t \in A. \tag{3}$$

**Remark 1.** *Notice that in* (2)*, we have specified the variable order as the function u* : *A* → (0, 1]*, while for defining RLFI, we can consider it as a function with extended values like u* : *A* → (0, ∞)*.*

**Remark 2** ([30])**.** *When we define the variable order u as a constant-valued function in both* (2) *and* (3)*, then the variable order RLFI and CFD operators are the same as the usual RLFI and CFD operators, respectively.*

**Remark 3** ([29])**.** *As we know, the semigroup property is satisfied for the standard RLFI operators equipped with constant order, while it is not valid for extended case of variable orders β*1(*t*) *and <sup>β</sup>*2(*t*)*. In other words, Iβ*1(*t*) <sup>0</sup><sup>+</sup> ( *I β*2(*t*) <sup>0</sup><sup>+</sup> )*φ*(*t*) = *I β*1(*t*)+*β*2(*t*) <sup>0</sup><sup>+</sup> *φ*(*t*).

To see this problem, we give the following example.

**Example 1.** *Let A* = [0, 3] *and φ*(*t*) ≡ 1, ∀ *t* ∈ *A. The variable orders of RLFI operator can be taken as: <sup>β</sup>*1(*t*) = *<sup>t</sup>* <sup>2</sup> *and <sup>β</sup>*2(*t*) = - 1, *t* ∈ [0, 1] 2, *<sup>t</sup>* <sup>∈</sup> [1, 3] *.*

*Then for all t* ∈ *A, and according to Definition* (2)*, we compute*

$$\begin{array}{rcl} I\_{0^{+}}^{\beta\_{1}(t)}\left(I\_{0^{+}}^{\beta\_{2}(t)}\Phi(t)\right) &=& \int\_{0}^{t} \frac{(t-s)^{\beta\_{1}(t)-1}}{\Gamma(\beta\_{1}(t))} \int\_{0}^{s} \frac{(s-\tau)^{\beta\_{2}(s)-1}}{\Gamma(\beta\_{2}(s))} \Phi(\tau)d\tau ds \\\\ &=& \int\_{0}^{t} \frac{(t-s)^{\beta\_{1}(t)-1}}{\Gamma(u(t))} [\int\_{0}^{1} \frac{(s-\tau)^{0}}{\Gamma(1)} d\tau + \int\_{1}^{s} \frac{(s-\tau)}{\Gamma(2)} d\tau] ds \\\\ &=& \int\_{0}^{t} \frac{(t-s)^{\beta\_{1}(t)-1}}{\Gamma(\beta\_{1}(t))} [\frac{s^{2}}{2} - s + \frac{3}{2}] ds, \end{array}$$

*and*

$$I\_{0^{+}}^{\\\\\beta\_1(t) + \beta\_2(t)}\phi(t) \;=\int\_0^t \frac{(t-s)^{\beta\_1(t) + \beta\_2(t) - 1}}{\Gamma(\beta\_1(t) + \beta\_2(t))}\phi(s)ds.$$

*For t* = 2*, it becomes*

$$\begin{array}{rcl} \left.I\_{0^{+}}^{\beta\_{1}(t)}\left(I\_{0^{+}}^{\beta\_{2}(t)}\phi(t)\right)\right|\_{t=2} &=& \int\_{0}^{2} \frac{(2-s)^{0}}{\Gamma(1)} [\frac{s^{2}}{2} - s + \frac{3}{2}] ds \\\\ &=& \int\_{0}^{2} (\frac{s^{2}}{2} - s + \frac{3}{2}) ds \\\\ &=& \frac{7}{3^{\prime}} \end{array}$$

*and*

$$\begin{array}{rcl} I\_{0^{+}}^{\beta\_{1}(t) + \beta\_{2}(t)} \phi(t)|\_{t=2} & = & \int\_{0}^{2} \frac{(2-s)^{\beta\_{1}(t) + \beta\_{2}(t) - 1}}{\Gamma(\beta\_{1}(t) + \beta\_{2}(t))} \phi(s) ds \\\\ & = & \int\_{0}^{1} \frac{(2-s)^{1}}{\Gamma(2)} ds + \int\_{1}^{2} \frac{(2-s)^{2}}{\Gamma(3)} ds \\\\ & = & \frac{3}{2} + \frac{1}{6} \\\\ & = & \frac{5}{3} .\end{array}$$

*Hence, it is simply seen that the mentioned property is not correct for the generalized RLFI operators with respect to variable orders.*

The following expansion is key for our argument.

**Lemma 1** ([31])**.** *Let a*1, *α*<sup>1</sup> > 0 *and n* = 1 + [*α*1]*. Then*

$$I\_{a\_1^{+}}^{
\alpha\_1}(^{c}D\_{a\_1^{+}}^{
\alpha\_1}
\phi(t)) = \phi(t) - \sum\_{k=0}^{n-1} \frac{\phi^{(k)}(a\_1)}{k!} t^k.$$

**Lemma 2** ([32])**.** *Let α*1, *α*<sup>2</sup> > 0*, φ*, *<sup>c</sup> Dα*<sup>1</sup> *a*+ 1 *<sup>φ</sup>* <sup>∈</sup> *<sup>L</sup>*1(*a*1, *<sup>a</sup>*2)*. Then, the differential equation*

$$^cD\_{a\_1^+}^{\alpha\_1}\phi(t) = 0,$$

*has unique solution*

$$\phi(t) = r\_0 + r\_1(t - a\_1) + r\_2(t - a\_1)^2 + \dots + r\_{n-1}(t - a\_1)^{n-1},$$

*and we have*

$$T\_{a\_1^+}^{a\_1}(^cD\_{a\_1^+}^{a\_1})\phi(t) = \phi(t) + r\_0 + r\_1(t - a\_1) + r\_2(t - a\_1)^2 + \dots + r\_{n-1}(t - a\_1)^{n-1}$$

*such that n* <sup>−</sup> <sup>1</sup> <sup>&</sup>lt; *<sup>α</sup>*<sup>1</sup> <sup>≤</sup> *n, rj* <sup>∈</sup> <sup>R</sup>, *<sup>j</sup>* <sup>=</sup> 1, 2, ... , *n.*

*Furthermore, we have*

$$^cD\_{a\_1^+}^{\alpha\_1}(I\_{a\_1^+}^{\alpha\_1})\phi(t) = \phi(t)\_{,t}$$

*and*

$$I\_{a\_1^+}^{\alpha\_1}(I\_{a\_1^+}^{\alpha\_2})\phi(t) = I\_{a\_1^+}^{\alpha\_2}(I\_{a\_1^+}^{\alpha\_1})\phi(t) = I\_{a\_1^+}^{\alpha\_1 + \alpha\_2}\phi(t).$$

We recall some properties of variable order RLFI operator formulated by (2) which will be used in the sequel.

**Lemma 3** ([33])**.** *If u* : *A* → (0, 1] *has the continuity property, then for*

$$h \in \mathbb{C}\_{\delta}(A, \mathbb{R}) = \{ h(t) \in \mathbb{C}(A, \mathbb{R}), \ t^{\delta}h(t) \in \mathbb{C}(A, \mathbb{R}) \}, \ 0 \le \delta \le 1\_{\Delta}$$

*the integral Iu*(*t*) <sup>0</sup><sup>+</sup> *h*(*t*) *admits a finite value for all t* ∈ *A.*

**Lemma 4** ([33])**.** *Assume that u* : *A* → (0, 1] *has the continuity property. Then*

$$I\_{0^+}^{
u(t)}h(t) \in \mathcal{C}(A, \mathbb{R}) \text{ for } h \in \mathcal{C}(A, \mathbb{R}).$$

**Definition 2** ([34,35])**.** *An interval <sup>J</sup>* <sup>⊆</sup> <sup>R</sup> *is termed as a generalized interval if <sup>I</sup> is either an interval, or* {*a*1}*, or* ∅*. A finite set* F *is defined to be a partition of J if every x* ∈ *J belongs to exactly one and one generalized interval* <sup>I</sup> *in* <sup>F</sup>*. Finally, <sup>w</sup>* : *<sup>J</sup>* <sup>→</sup> <sup>R</sup> *is piecewise constant w.r.t* <sup>F</sup> *as a partition of J, if for each* <sup>I</sup> ∈ F*, w is constant on* <sup>I</sup>*.*

The next definitions and basic lemmas from coincidence degree theory are fundamental in the proof of theorems which we will establish them later.

**Definition 3** ([20,36])**.** *Consider two normed spaces* S<sup>1</sup> *and* S2*. A Fredholm operator of index zero is a linear operator like* <sup>Θ</sup> : *Dom*(Θ) <sup>⊂</sup> <sup>S</sup><sup>1</sup> <sup>→</sup> <sup>S</sup><sup>2</sup> *satisfying:*


In view of Definition 3, it is followed the existence of continuous projections Ψ : <sup>S</sup><sup>1</sup> <sup>→</sup> <sup>S</sup><sup>1</sup> and <sup>Φ</sup> : <sup>S</sup><sup>2</sup> <sup>→</sup> <sup>S</sup><sup>2</sup> such that IMG(Ψ) = KER(Θ), KER(Φ) = IMG(Θ), <sup>S</sup><sup>1</sup> <sup>=</sup> KER(Θ) <sup>⊕</sup> KER(Ψ), and <sup>S</sup><sup>2</sup> <sup>=</sup> IMG(Θ) <sup>⊕</sup> IMG(Φ).

It is known that the restriction of <sup>Θ</sup> to *Dom*(Θ) <sup>∩</sup> KER(Ψ), which we shall represent by ΘΨ, will be an isomorphism onto its image [20,36].

**Definition 4** ([20,36])**.** *Let* <sup>Θ</sup> *be a Fredholm operator of index zero and* <sup>Ω</sup> <sup>⊆</sup> <sup>S</sup><sup>1</sup> *be bounded with Dom*(Θ) <sup>∩</sup> <sup>Ω</sup> <sup>=</sup> <sup>∅</sup>*. We say W* : <sup>Ω</sup> <sup>→</sup> <sup>S</sup><sup>2</sup> *has the* <sup>Θ</sup>*-compactness property in* <sup>Ω</sup> *whenever: (H1)* <sup>Φ</sup>*<sup>W</sup>* : <sup>Ω</sup> <sup>→</sup> <sup>S</sup><sup>2</sup> *is continuous, and* <sup>Φ</sup>*W*(Ω) <sup>⊆</sup> <sup>S</sup><sup>2</sup> *is bounded, (H2)* (ΘΨ)−1(*<sup>I</sup>* <sup>−</sup> <sup>Φ</sup>)*<sup>W</sup>* : <sup>Ω</sup> <sup>→</sup> <sup>S</sup><sup>1</sup> *is completely continuous.*

The next theorem entitled *Mawhin's Continuation Theorem* is our main criterion in the present study which proves the existence of solution.

**Theorem 1** ([37])**.** *Assume that* <sup>S</sup><sup>1</sup> *and* <sup>S</sup><sup>2</sup> *are two Banach spaces and* <sup>Ω</sup> <sup>⊂</sup> <sup>S</sup><sup>1</sup> *is an open, bounded and symmetric set with* 0 ∈ Ω*. Also, assume that:*

*(A1) the Fredholm operator* <sup>Θ</sup> : *Dom*(Θ) <sup>⊂</sup> <sup>S</sup><sup>1</sup> <sup>→</sup> <sup>S</sup><sup>2</sup> *of index zero is such that*

*Dom*(Θ) ∩ Ω = ∅,

*(A2) the operator W* : <sup>S</sup><sup>1</sup> <sup>→</sup> <sup>S</sup><sup>2</sup> *is* <sup>Θ</sup>*-compact on* <sup>Ω</sup>*,*

*(A3)* ∀ *x* ∈ *Dom*(Θ) ∩ *∂*Ω *and* ∀ *λ* ∈ (0, 1]*,*

Θ*x* − *Wx* = −*λ*(Θ*x* + *W*(−*x*)),

*where ∂*Ω *denotes the boundary of* Ω *w.r.t.* S1*. Then the operator equation* Θ*x* = *Wx has at least one solution on Dom*(Θ) ∩ Ω*.*

#### **3. Existence of Solutions**

To begin the desired analysis, we consider the following assumptions:

**(AS1)** Consider a sequence of finite many points {*Tk*}*<sup>n</sup> <sup>k</sup>*=<sup>0</sup> so that 1 = *T*<sup>0</sup> < *Tk* < *Tn* = *T*, *<sup>k</sup>* <sup>∈</sup> <sup>N</sup>*n*−<sup>1</sup> <sup>1</sup> . For *<sup>k</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>1</sup> , denote the subintervals *Ak* as *Ak* := (*Tk*−1, *Tk*]. Then <sup>P</sup> <sup>=</sup> &*<sup>n</sup> <sup>k</sup>*=<sup>1</sup> *Ak* is a partition of *A*.

**(AS2)** Let *<sup>g</sup>* <sup>∈</sup> *<sup>C</sup>*(*Aj* <sup>×</sup> <sup>R</sup>, <sup>R</sup>) and there exists *<sup>δ</sup>* <sup>∈</sup> (0, 1) such that *<sup>t</sup> <sup>δ</sup><sup>g</sup>* <sup>∈</sup> *<sup>C</sup>*(*Aj* <sup>×</sup> <sup>R</sup>, <sup>R</sup>) and there exists *<sup>K</sup>* <sup>&</sup>gt; 0 with *<sup>K</sup>* <sup>&</sup>lt; min 1, Γ(*uj* + 1) (*Tj* − *Tj*−1) *uj* # so that *t <sup>δ</sup>*|*g*(*t*, *<sup>φ</sup>*1) <sup>−</sup> *<sup>g</sup>*(*t*, *<sup>φ</sup>*2)| ≤

$$K|\phi\_1 - \phi\_2| \text{, for any } \phi\_1, \phi\_2 \in \mathbb{R} \text{ and } \hat{t} \in A\_j.$$

For each *<sup>j</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>1</sup> , the notation *Ej* <sup>=</sup> *<sup>C</sup>*(*Aj*, <sup>R</sup>) denotes the Banach space of continuous functions *<sup>φ</sup>* : *Aj* <sup>→</sup> <sup>R</sup> with the sup-norm *φ Ej* <sup>=</sup> sup*t*∈*Aj* |*φ*(*t*)|.

On the other side, consider the piecewise constant mapping *u*(*t*) : *A* → (0, 1] w.r.t. P, i.e.,

$$u(t) = \sum\_{j=1}^{n} u\_j I\_j(t)\_j$$

where 0 <sup>&</sup>lt; *uj* <sup>≤</sup> 1 are real numbers, and *Ij* denotes the indicator of *Aj*, *<sup>j</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>1</sup> ; that is, *Ij*(*t*) = 1 if *t* ∈ *Aj* and *Ij*(*t*) = 0 otherwise. In this case, the left CFD of variable order *u*(*t*) for *<sup>φ</sup>*(*t*) <sup>∈</sup> *<sup>C</sup>*(*A*, <sup>R</sup>), defined as (3), can be formulated as a sum of the left CFD operators of constant orders *uk* <sup>∈</sup> <sup>R</sup> which takes the form

$${}^{c}D\_{0^{+}}^{
u(t)}\phi(t) = \sum\_{k=1}^{j-1} \int\_{T\_{k-1}}^{T\_{k}} \frac{(t-s)^{-\mu\_{k}}}{\Gamma(1-\mu\_{k})} \phi'(s)ds + \int\_{T\_{j-1}}^{t} \frac{(t-s)^{-\mu\_{j}}}{\Gamma(1-\mu\_{j})} \phi'(s)ds.\tag{4}$$

Thus, the given Caputo BVP of variable order (1) can be reformulated for each *t* ∈ *Aj*, *<sup>j</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>1</sup> in the following structure

$$\sum\_{k=1}^{j-1} \int\_{T\_{k-1}}^{T\_k} \frac{(t-s)^{-\mu\_k}}{\Gamma(1-\mu\_k)} \phi'(s)ds + \int\_{T\_{j-1}}^t \frac{(t-s)^{-\mu\_j}}{\Gamma(1-\mu\_j)} \phi'(s)ds = g(t, \phi(t)).\tag{5}$$

Let the function *<sup>φ</sup>*˜ <sup>∈</sup> *Ej* be so that *<sup>φ</sup>*˜(*t*) <sup>≡</sup> 0 on *<sup>t</sup>* <sup>∈</sup> [0, *Tj*−1] and it satisfies the above integral Equation (5). In such a situation, (5) is converted to the standard constant order fractional differential equation (FDE) as

$$\prescript{c}{}{D}\_{T^+\_{j-1}}^{\mu\_j} \tilde{\phi}(t) = \gpharpoonright \tilde{\phi}(t) \urcorner \overleftarrow{\phi}(t) \urcorner , \ t \in A\_j .$$

In accordance with above equation, for each *<sup>j</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>1</sup> , we have the auxiliary FBVP equipped with Caputo constant order CFD operator

$$\begin{cases} \ ^cD\_{T\_{j-1}^+}^ {\mu\_j} \phi(t) = \g(t\_\prime \phi(t)), \ t \in A\_{j\prime} \\ \ \ \_\phi(T\_{j-1}) = \phi(T\_j). \end{cases} \tag{6}$$

A resonance problem is a boundary problem in which the corresponding homogeneous BVP has a non–trivial solution. Hence, we consider the homogeneous version of the given equivalent constant order FBVP (6) by

$$\begin{cases} \ ^cD\_{T\_{j-1}^+}^ {\iota\_j} \phi(t) = 0, \ t \in A\_{j\prime} \\ \ \ \phi(T\_{j-1}) = \phi(T\_j). \end{cases} \tag{7}$$

By Lemma 2, the homogeneous constant order FBVP (7) has nontrivial solution *φ*(*t*) = *c* which converts the equivalent constant order FBVP (6) to a resonance FBVP.

As well as, on the given subintervals, let <sup>S</sup><sup>1</sup> <sup>=</sup> {*<sup>φ</sup>* <sup>∈</sup> *Ej* :*φ*(*t*) = *<sup>I</sup> uj T*+ *j*−1 *v*(*t*) : *v* ∈ *Ej*,*t* ∈ *Aj*}

with the norm

$$\|\|\phi\|\|\_{\mathbb{S}\_1} = \|\|\phi\|\|\_{E\_j}.$$

The linear operator <sup>Θ</sup> : *Dom*(Θ) <sup>⊆</sup> <sup>S</sup><sup>1</sup> <sup>→</sup> *Ej* along with the operator *<sup>W</sup>* : <sup>S</sup><sup>1</sup> <sup>→</sup> *Ej* are defined as

$$\Theta[\phi(t)] := \, ^cD\_{T\_{j-1}^+}^ {\iota\_j} \phi(t) \, , \tag{8}$$

and

$$\mathcal{W}[\phi(t)] := \mathcal{g}(t, \phi(t)), \quad t \in A\_{j\prime} \tag{9}$$

where

$$Dom(\Theta) = \{ \phi \in \mathbb{S}\_1 : {}^c D\_{T\_{j-1}^+}^ {\mu\_j} \phi \in E\_j \quad \text{and} \quad \phi(T\_{j-1}) = \phi(T\_j) \}.$$

Then the equivalent constant order resonance FBVP (6) can be reformulated by the equation Θ*φ* = *Wφ*.

The first theorem on the existence of solutions for the equivalent constant order resonance FBVP (6) is established in this position.

**Theorem 2.** *If the condition* (AS2) *holds, then the equivalent constant order resonance FBVP* (6) *has at least one solution.*

**Proof.** The proof will be followed in a sequence of claims.

**Claim 1.** We show that

$$\mathbb{K}\mathbb{E}\mathbb{R}(\Theta) = \{c: c \in \mathbb{R}\},$$

and

$$\mathbb{IMG}(\Theta) = \left\{ \phi \in E\_{\bar{\jmath}} : \int\_{T\_{\bar{\jmath}-1}}^{T\_{\bar{\jmath}}} (T\_{\bar{\jmath}} - s)^{\iota\_{\bar{\jmath}} - 1} \phi(s) ds = 0 \right\}.$$

Let Θ (defined by (8)) be such that for *t* ∈ *Aj* and by Lemma 2, the equation Θ[*φ*(*t*)] = *cDuj T*+ *j*−1 *<sup>φ</sup>*(*t*) = 0 has the solution *<sup>φ</sup>*(*t*) = *<sup>c</sup>*, *<sup>c</sup>* <sup>∈</sup> <sup>R</sup>. Then

$$\mathbb{K}\mathbb{E}\mathbb{R}(\Theta) = \{\phi(t) = c: c \in \mathbb{R}\}.$$

On the other hand, for *<sup>v</sup>* <sup>∈</sup> IMG(Θ), there exits *<sup>φ</sup>* <sup>∈</sup> *Dom*(Θ) such that *<sup>v</sup>* <sup>=</sup> <sup>Θ</sup>*<sup>φ</sup>* <sup>∈</sup> *Ej*. By Lemma 1, for any *t* ∈ *Aj*, we have

$$
\phi(t) = \phi(T\_{j-1}) + \frac{1}{\Gamma(\mu\_j)} \int\_{T\_{j-1}}^t (t-s)^{\mu\_j - 1} v(s)ds.
$$

Since *φ* ∈ *Dom*(Θ), *v* satisfies

$$\frac{1}{\Gamma(\mu\_j)} \int\_{T\_{j-1}}^{T\_j} (T\_j - s)^{\mu\_j - 1} v(s) ds = 0.$$

Also, assume that *v* ∈ *Ej* satisfies

$$\int\_{T\_{j-1}}^{T\_j} (T\_j - s)^{\mu\_j - 1} v(s) ds = 0.$$

Let *φ*(*t*) = *I uj T*+ *j*−1 *<sup>v</sup>*(*t*). Then *<sup>v</sup>*(*t*) = *cDuj T*+ *j*−1 *<sup>φ</sup>*(*t*) and so *<sup>φ</sup>* <sup>∈</sup> *Dom*(Θ). Hence, *<sup>v</sup>* <sup>∈</sup> IMG(Θ),

so

$$\mathbb{IIIG}(\Theta) = \left\{ \phi \in \mathbb{S}\_2 : \int\_{T\_{j-1}}^{T\_j} (T\_j - s)^{\iota\_j - 1} \phi(s) ds = 0 \right\}.$$

**Claim 2.** Θ is a Fredholm operator of index zero.

The linear continuous projector operators <sup>Ψ</sup> : <sup>S</sup><sup>1</sup> <sup>→</sup> <sup>S</sup><sup>1</sup> and <sup>Φ</sup> : *Ej* <sup>→</sup> *Ej* can be considered by the following forms

$$\,\_1\Psi\phi = \phi(T\_{j-1})\_\prime \quad \Phi v(t) = \frac{u\_{\dot{j}}}{(T\_{\dot{j}} - T\_{j-1})^{u\_{\dot{j}}}} \int\_{T\_{\dot{j}-1}}^{T\_{\dot{j}}} (T\_{\dot{j}} - s)^{u\_{\dot{j}} - 1} v(s) ds.$$

Clearly, IMG(Ψ) = KER(Θ) and <sup>Ψ</sup><sup>2</sup> <sup>=</sup> <sup>Ψ</sup>. It follows that for any *<sup>φ</sup>* <sup>∈</sup> <sup>S</sup>1,

$$\phi = (\phi - \Psi \phi) + \Psi \phi\_\star$$

i.e., <sup>S</sup><sup>1</sup> <sup>=</sup> KER(Ψ) + KER(Θ). A simple computation shows that KER(Ψ) <sup>∩</sup> KER(Θ) = 0. Therefore, <sup>S</sup><sup>1</sup> <sup>=</sup> KER(Ψ) <sup>⊕</sup> KER(Θ). A similar argument shows that for every *<sup>v</sup>* <sup>∈</sup> *Ej*, <sup>Φ</sup>2*<sup>v</sup>* <sup>=</sup> <sup>Φ</sup>*<sup>v</sup>* and *<sup>v</sup>* = (*<sup>v</sup>* <sup>−</sup> <sup>Φ</sup>(*v*)) + <sup>Φ</sup>(*v*), where (*<sup>v</sup>* <sup>−</sup> <sup>Φ</sup>(*v*)) <sup>∈</sup> KER(Φ) = IMG(Θ). From IMG(Θ) = KER(Φ) and Φ<sup>2</sup> = Φ, we have

$$\mathbb{IIIG}(\Phi) \cap \mathbb{IIIG}(\Theta) = 0.$$

Then, *Ej* <sup>=</sup> IMG(Θ) <sup>⊕</sup> IMG(Φ). In this case,

$$\dim(\mathbb{K}\mathbb{E}\mathbb{R}(\Theta) = \dim \mathbb{I}\mathbb{M}\mathbb{G}(\Phi) = \text{codim}\mathbb{I}\mathbb{M}\mathbb{G}(\Theta).$$

The obtained result shows that Θ is a Fredholm operator of index zero.

**Claim 3.** Θ−<sup>1</sup> <sup>Ψ</sup> = (Θ|*Dom*(Θ)∩KER(Ψ))−<sup>1</sup> (the inverse of <sup>Θ</sup>|*Dom*(Θ)∩KER(Ψ)). Clearly, Θ−<sup>1</sup> <sup>Ψ</sup> : IMG(Θ) <sup>→</sup> <sup>S</sup><sup>1</sup> <sup>∩</sup> KER(Ψ) satisfies

$$
\Theta^{-1}\_{\Psi}(v)(t) = I^{\mu\_j}\_{T^+\_{j-1}}v(t).
$$

Let *<sup>v</sup>* <sup>∈</sup> IMG(Θ). Then

$$\Theta \Theta\_{\Psi}^{-1}(v) = \,^c D\_{T\_{j-1}^+}^{\mu\_j}(I\_{T\_{j-1}^+}^{\mu\_j}v) = v. \tag{10}$$

Furthermore, for *<sup>φ</sup>* <sup>∈</sup> *Dom*(Θ) <sup>∩</sup> KER(Ψ), we get

$$(\Theta\_\Psi^{-1}(\Theta(\phi(t))) = I\_{T\_{j-1}^+}^{\mu\_j}(^cD\_{T\_{j-1}^+}^{\mu\_j}\phi(t)) = \phi(t) - \phi(T\_{j-1}).$$

Since *<sup>φ</sup>* <sup>∈</sup> *Dom*(Θ) <sup>∩</sup> KER(Ψ), we know that *<sup>φ</sup>*(*Tj*−1) = 0. Thus

$$
\Theta^{-1}\_{\Psi}(\Theta(\phi(t))) = \phi(t). \tag{11}
$$

Combining (10) and (11) shows that ΘΨ−<sup>1</sup> = (Θ|*Dom*(Θ)∩KER(Ψ))−1.

**Claim 4.** On every bounded and open set <sup>Ω</sup> <sup>⊂</sup> <sup>S</sup>1, *<sup>W</sup>* is <sup>Θ</sup>-compact.

Define <sup>Ω</sup> <sup>=</sup> {*<sup>φ</sup>* <sup>∈</sup> <sup>S</sup><sup>1</sup> : *φ* <sup>S</sup><sup>1</sup> < *M*} as a bounded and open set, where *M* > 0.

The proof of this claim will be done in three steps.

**Step 1.** Φ*W* is continuous.

This property for Φ*W* is derived due to the imposed conditions on the nonlinear function *g* and the Lebesgue dominated convergence criterion, immediately.

**Step 2.** Φ*W*(Ω) is bounded.

Now, for each *φ* ∈ Ω and for all *t* ∈ *Aj*, we have

$$\begin{split} |\Phi \mathcal{W}(\boldsymbol{\phi})(t)| &\leq \frac{u\_{j}}{(T\_{j}-T\_{j-1})^{u\_{j}}} \int\_{T\_{j-1}}^{T\_{j}} (T\_{j}-s)^{u\_{j}-1} |\boldsymbol{g}(s,\boldsymbol{\phi}(s))| ds \\ &\leq \frac{u\_{j}}{(T\_{j}-T\_{j-1})^{u\_{j}}} \int\_{T\_{j-1}}^{T\_{j}} (T\_{j}-s)^{u\_{j}-1} |\boldsymbol{g}(s,\boldsymbol{\phi}(s)) - \boldsymbol{g}(s,0)| ds \\ &\quad + \frac{u\_{j}}{(T\_{j}-T\_{j-1})^{u\_{j}}} \int\_{T\_{j-1}}^{T\_{j}} (T\_{j}-s)^{u\_{j}-1} |\boldsymbol{g}(s,0)| ds \\ &\leq \boldsymbol{g}^{\*} + \frac{u\_{j}}{(T\_{j}-T\_{j-1})^{u\_{j}}} \int\_{T\_{j-1}}^{T\_{j}} (T\_{j}-s)^{u\_{j}-1} \boldsymbol{s}^{-\delta}(K|\boldsymbol{\phi}(s)|) ds \\ &\leq \boldsymbol{g}^{\*} + MKT\_{j-1}^{-\delta} \end{split}$$

by assuming *g*∗ = sup *t*∈*Aj* |*g*(*t*, 0)|. Thus,

$$\|\!\!\!\!\Phi\mathcal{W}(\phi)\|\!\!\!\!\_{E\_{\vec{j}}} \le \mathcal{g}^\* + MKT\_{\vec{j}-1}^{-\delta} := \mathcal{R} > 0.$$

This shows that Φ*W*(Ω) ⊆ *Ej* is bounded.

**Step 3.** Θ−<sup>1</sup> <sup>Ψ</sup> (*<sup>I</sup>* <sup>−</sup> <sup>Φ</sup>)*<sup>W</sup>* : <sup>Ω</sup> <sup>→</sup> <sup>S</sup><sup>1</sup> is completely continuous.

By considering the existing hypotheses in relation to Ascoli-Arzelà theorem, it is necessary that we prove two properties of the boundedness and equi-continuity for Θ−<sup>1</sup> <sup>Ψ</sup> (*<sup>I</sup>* <sup>−</sup> <sup>Φ</sup>)*W*(Ω) <sup>⊂</sup> <sup>S</sup>1. At first, for each *<sup>φ</sup>* <sup>∈</sup> <sup>Ω</sup> and for all *<sup>t</sup>* <sup>∈</sup> *Aj*, we have

$$\begin{split} \Theta\_{\mathbf{T}}^{-1}(I-\Phi)\mathcal{W}\boldsymbol{\phi}(t) &= \Theta\_{\mathbf{T}}^{-1}(\mathcal{W}\boldsymbol{\phi}(t) - \Phi\mathcal{W}\boldsymbol{\phi}(t)) \\ &= I\_{T\_{j-1}^{+}}^{\boldsymbol{u}\_{j}^{\boldsymbol{u}\_{j}}} \Big[ \mathcal{g}(t, \boldsymbol{\phi}(t)) - \frac{\boldsymbol{u}\_{j}}{(T\_{j} - T\_{j-1})\_{j}^{\boldsymbol{u}}} \int\_{T\_{j-1}}^{T\_{j}} (T\_{j} - s)^{\boldsymbol{u}\_{j} - 1} \boldsymbol{g}(s, \boldsymbol{\phi}(s)) \Big] ds \\ &= \frac{1}{\Gamma(\boldsymbol{u}\_{j})} \int\_{T\_{j-1}}^{t} (t - s)^{\boldsymbol{u}\_{j} - 1} \boldsymbol{g}(s, \boldsymbol{\phi}(s)) ds \\ &- \frac{t^{\boldsymbol{u}\_{j}}}{(T\_{j} - T\_{j-1})^{\boldsymbol{u}\_{j}} \Gamma(\boldsymbol{u}\_{j})} \int\_{T\_{j-1}}^{T\_{j}} (T\_{j} - s)^{\boldsymbol{u}\_{j} - 1} \boldsymbol{g}(s, \boldsymbol{\phi}(s)) ds. \end{split}$$

Further, for each *φ* ∈ Ω and for all *t* ∈ *Aj*, we get

$$\begin{aligned} \left| \Theta\_{\mathbf{F}}^{-1}(I-\Phi)\mathcal{W}\phi(t) \right| &\leq \frac{2}{\Gamma(u\_{\vec{j}})} \int\_{T\_{\vec{j}-1}}^{T\_{\vec{j}}} (T\_{\vec{j}}-s)^{u\_{\vec{j}}-1} |\mathcal{g}(s,\boldsymbol{\phi}(s)) - \mathcal{g}(t,0)| ds \\ &\quad + \frac{2}{\Gamma(u\_{\vec{j}})} \int\_{T\_{\vec{j}-1}}^{T\_{\vec{j}}} (T\_{\vec{j}}-s)^{u\_{\vec{j}}-1} |\mathcal{g}(t,0)| ds \\ &\leq [\mathcal{g}^{\*} + MKT\_{\vec{j}-1}^{-\delta}] \frac{2(T\_{\vec{j}}-T\_{\vec{j}-1})^{u\_{\vec{j}}}}{\Gamma(u\_{\vec{j}}+1)} := B\_{1}. \end{aligned}$$

so

$$||\Theta^{-1}\_{\Psi}(I-\Phi)\mathcal{W}\phi||\_{E\_{\bar{j}}} \le B\_{1\prime}$$

which gives the uniform boundedness of Θ−<sup>1</sup> <sup>Ψ</sup> (*<sup>I</sup>* <sup>−</sup> <sup>Φ</sup>)*W*(Ω) in <sup>S</sup>1.

To prove the equi-continuity of Θ−<sup>1</sup> <sup>Ψ</sup> (*<sup>I</sup>* − <sup>Φ</sup>)*W*(Ω), notice that for *Tj*−<sup>1</sup> ≤ *<sup>t</sup>*<sup>1</sup> ≤ *<sup>t</sup>*<sup>2</sup> ≤ *Tj* and *φ* ∈ Ω, we get

$$\begin{split} \left| \Theta\_{\mathbf{F}}^{-1}(I-\Phi)\mathcal{W}\phi(t\_{2}) - \Theta\_{\mathbf{F}}^{-1}(I-\Phi)\mathcal{W}\phi(t\_{1}) \right| &\leq \frac{\mathsf{g}^{\*} + T\_{j-1}^{-\delta}MK}{\Gamma(u\_{j})} \Big[ \int\_{t\_{1}}^{t\_{2}} (t\_{2}-s)^{u\_{j}-1}ds \\ &+ \int\_{T\_{j-1}}^{t\_{1}} \left| (t\_{2}-s)^{u\_{j}-1} - (t\_{1}-s)^{u\_{j}-1}|ds \right| + \left[ \frac{T\_{j-1}^{-\delta}MK + g^{\*}}{\Gamma(u\_{j}+1)} \right] (t\_{2}^{u\_{j}}-t\_{1}^{u\_{j}}). \end{split}$$

The right-hand side of above inequality tends to zero as *<sup>t</sup>*<sup>1</sup> <sup>→</sup> *<sup>t</sup>*2. Thus, <sup>Θ</sup>−<sup>1</sup> <sup>Ψ</sup> (*I* − Φ)*W*(Ω) is equicontinuous in S1. On the basis of the Ascoli-Arzelà theorem, *L*−<sup>1</sup> <sup>Ψ</sup> (*I* − Φ)*W*(Ω) is relatively compact. In accordance with the steps 1 to 3, we can follow that *W* is Θ-compact in Ω.

**Claim 5.** There exists > 0 (not depending on *λ*) so that if

$$
\Theta(\phi) - \mathcal{W}(\phi) = -\lambda[\Theta(\phi) + \mathcal{W}(-\phi)], \quad \lambda \in (0, 1], \tag{12}
$$

then *φ* <sup>S</sup><sup>1</sup> <sup>≤</sup> . By the condition (AS2) and for each *<sup>φ</sup>* <sup>∈</sup> <sup>S</sup><sup>1</sup> satisfying (12), we get

$$
\Theta(\phi) - \mathcal{W}(\phi) = -\lambda \Theta(\phi) - \lambda \mathcal{W}(-\phi).
$$

So

$$\Theta(\phi) = \frac{1}{1+\lambda} \mathcal{W}(\phi) - \frac{\lambda}{1+\lambda} \mathcal{W}(-\phi). \tag{13}$$

By (13), and for all *t* ∈ *Aj*, we get

$$\phi(t) = \frac{1}{1+\lambda} \Theta\_{\Psi}^{-1} \mathcal{W} \phi(t) - \frac{\lambda}{1+\lambda} \Theta\_{\Psi}^{-1} \mathcal{W} (-\phi(t)), \dots$$

and so we estimate

$$\begin{split} |\phi(t)| &\leq \frac{1}{(1+\lambda)\Gamma(u\_{j})} \int\_{T\_{j-1}}^{t} (t-s)^{u\_{j}-1} |g(s,\phi(s)) - g(s,0)| ds \\ &\quad + \frac{\lambda}{(1+\lambda)\Gamma(u\_{j})} \int\_{T\_{j-1}}^{t} (t-s)^{u\_{j}-1} |g(s,-\phi(s)) - g(s,0)| ds \\ &\quad + \frac{\mathcal{g}^{\*}\left(T\_{j}-T\_{j-1}\right)^{u\_{j}}}{(1+\lambda)\Gamma(u\_{j}+1)} + \frac{\lambda \mathcal{g}^{\*}\left(T\_{j}-T\_{j-1}\right)^{u\_{j}}}{(1+\lambda)\Gamma(u\_{j}+1)} \\ &\leq \left(\frac{1}{1+\lambda} + \frac{\lambda}{1+\lambda}\right) \frac{T\_{j-1}^{-\delta} \left(T\_{j}-T\_{j-1}\right)^{u\_{j}}}{\Gamma(u\_{j}+1)} (K\|\phi\|\_{E\_{j}}) \\ &\quad + \left(\frac{1}{1+\lambda} + \frac{\lambda}{1+\lambda}\right) \frac{\mathcal{g}^{\*}\left(T\_{j}-T\_{j-1}\right)^{u\_{j}}}{\Gamma(u\_{j}+1)} \\ &= \frac{KT\_{j-1}^{-\delta} (T\_{j}-T\_{j-1})^{u\_{j}}}{\Gamma(u\_{j}+1)} \|\phi\|\_{E\_{j}} + \frac{\mathcal{g}^{\*}\left(T\_{j}-T\_{j-1}\right)^{u\_{j}}}{\Gamma(u\_{j}+1)}. \end{split}$$

Hence,

$$\|\|\phi\|\|\_{E\_j} \le \left(\mathcal{g}^\* + KT\_{j-1}^{-\delta} \|\|\phi\|\|\_{E\_j}\right) \frac{(T\_j - T\_{j-1})^{u\_j}}{\Gamma(u\_j + 1)},\tag{14}$$

and so

$$||\phi||\_{\mathbb{S}\_1} \le \frac{\mathcal{S}^\*}{\frac{\Gamma(\mu\_j+1)}{\left(T\_j - T\_{j-1}\right)^{\mu\_j}} - KT\_{j-1}^{-\delta}} := \epsilon.$$

**Claim 6.** There exists a bounded and open set <sup>Ω</sup> <sup>⊂</sup> <sup>S</sup><sup>1</sup> such that

$$
\Theta(\phi) - \mathcal{W}(\phi) \neq -\lambda [\Theta(\phi) + \mathcal{W}(-\phi)]\_{\prime\prime}
$$

for all *φ* ∈ *∂*Ω and all *λ* ∈ (0, 1].

By the condition (AS2) and Claim 5, there exits > 0 (independent of *λ*) such that if *φ* solves

$$\Theta(\phi) - \mathcal{W}(\phi) = -\lambda[\Theta(\phi) + \mathcal{W}(-\phi)], \ \lambda \in (0, 1]\_\lambda$$

then *φ* <sup>S</sup><sup>1</sup> ≤ . Consequently, if

$$\Omega = \{ \phi \in \mathbb{S}\_1 : ||\phi||\_{\mathbb{S}\_1} < B \}, \tag{15}$$

then from the condition (AS2), it is immediately obtained that the set Ω introduced by (15), is symmetric, 0 <sup>∈</sup> <sup>Ω</sup>, and <sup>S</sup><sup>1</sup> <sup>∩</sup> <sup>Ω</sup> <sup>=</sup> <sup>Ω</sup> <sup>=</sup> <sup>∅</sup>.

Furthermore, it is obtained that

$$
\Theta(\phi) - \mathcal{W}(\phi) \neq -\lambda [\Theta(\phi) - \mathcal{W}(-\phi)]\_{\prime\prime}
$$

for all *<sup>φ</sup>* <sup>∈</sup> *<sup>∂</sup>*<sup>Ω</sup> <sup>=</sup> {*<sup>φ</sup>* <sup>∈</sup> <sup>S</sup><sup>1</sup> : *φ* <sup>S</sup><sup>1</sup> = *B*} and for all *λ* ∈ (0, 1], where *B* > . This together with Theorem 1 imply that the equivalent constant order resonance FBVP (6) has at least one solution, and this completes the proof.

Now, we complete our deduction on the existence property for solutions of the given Caputo FBVP of variable order (1).

**Theorem 3.** *Let the conditions* (AS1) *and* (AS2) *be satisfied for all <sup>j</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>1</sup> *. Then, the given Caputo FBVP of variable order* (1) *has at least a solution in C*(*A*, R)*.*

**Proof.** We know that for all *<sup>j</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>1</sup> , and according to Theorem 2, the equivalent constant order resonance FBVP (6) has at least one solution *<sup>φ</sup><sup>j</sup>* <sup>∈</sup> *Ej*. For each *<sup>j</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>1</sup> , and on the existing subintervals, define

$$\phi\_{\vec{\jmath}} = \begin{cases} \ 0, & t \in [0, T\_{\vec{\jmath}-1}], \\ \ \widetilde{\phi}\_{\vec{\jmath}\prime} & t \in A\_{\vec{\jmath}}. \end{cases}$$

In such a case, *<sup>φ</sup><sup>j</sup>* <sup>∈</sup> *<sup>C</sup>*([0, *Tj*], <sup>R</sup>) satisfies the integral equation (5) for *<sup>t</sup>* <sup>∈</sup> *Aj*, which means that *<sup>φ</sup>j*(0) = 0, *<sup>φ</sup>j*(*Tj*) = *<sup>φ</sup><sup>j</sup>*(*Tj*) = 0 and satisfies (5) for *<sup>t</sup>* <sup>∈</sup> *Aj*, *<sup>j</sup>* <sup>∈</sup> <sup>N</sup>*<sup>n</sup>* <sup>1</sup> . Therefore, the piecewise function

$$\phi(t) = \begin{cases} \phi\_1(t), & t \in A\_{1\prime} \\\\ \phi\_2(t), & t \in A\_{2\prime} \\\\ \dots & \dots & \dots & \dots \\\\ \phi\_n(t), & t \in A\_n = [0, T] \end{cases}$$

is a solution to the given Caputo FBVP of variable order (1) in *C*(*A*, R).

#### **4. Example**

**Example 2.** *Consider the following FBVP (based on the given Caputo FBVP of variable order* (1)*) as follows*

$$\begin{cases} \, \, ^c D\_{0^+}^{\mu(t)} \phi(t) = \frac{\sin \phi(t) - \phi(t) \cos t}{5 \sqrt{1+t}}, & t \in A := [0, 2], \\\ \, \, \phi(0) = \phi(2). \end{cases} \tag{16}$$

*Let*

$$g(t,\phi) = \frac{\sin\phi - \phi\cos t}{5\sqrt{1+t}},\ (t,\phi) \in [0,2] \times [0,+\infty) \omega$$

*and*

$$u(t) = \begin{cases} \frac{7}{5}, & t \in A\_1 := [0, 1], \\ \frac{3}{2}, & t \in A\_2 := [1, 2]. \end{cases} \tag{17}$$

*In this case,*

$$\begin{split} |t|^{\frac{1}{2}} |g(t, \phi\_1) - g\_1(t, \phi\_2)| &= \left| \frac{t^{\frac{1}{2}} (\sin \phi\_1 - \phi\_1 \cos t)}{5 \sqrt{1 + t}} - \frac{t^{\frac{1}{2}} (\sin \phi\_2 - \phi\_2 \cos t)}{5 \sqrt{1 + t}} \right| \\ &\leq \frac{1}{5} \sqrt{\frac{t}{1 + t}} \left( |\sin \phi\_1 - \sin \phi\_2| + |\cos t| |\phi\_1 - \phi\_2| \right) \\ &\leq \frac{2}{5} |\phi\_1 - \phi\_2|. \end{split}$$

*By* (17) *and* (6)*, on every subintervals A*<sup>1</sup> *and A*2*, two auxiliary constant order resonance FBVPs are considered as*

$$\begin{cases} \ ^c D\_{0^+}^5 \phi(t) = \frac{\sin \phi(t) - \phi(t) \cos t}{5 \sqrt{1+t}}, & t \in A\_{1\prime} \\ \ \phi(0) = \phi(1), & \end{cases} \tag{18}$$

*and*

$$\begin{cases} \ ^c D\_{1^+}^{\frac{3}{2}} \phi(t) = \frac{\sin \phi(t) - \phi(t) \cos t}{5 \sqrt{1+t}}, & t \in A\_{2\prime} \\ \ \phi(1) = \phi(2). \end{cases} \tag{19}$$

*Evidently, the condition* (AS2) *is satisfied for j* <sup>=</sup> <sup>1</sup> *with <sup>δ</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> *and K* <sup>=</sup> <sup>2</sup> 5 *, and*

$$0 < K = \frac{2}{5} < \min\left\{ 1, \frac{\Gamma(u\_j + 1)}{(T\_j - T\_{j-1})^{u\_j}} \right\} = \min\left\{ 1, \Gamma(\frac{12}{5}) \right\} = 1.$$

*According to Theorem 2, the constant order resonance FBVP* (18) *has a solution like <sup>φ</sup>*<sup>1</sup> <sup>∈</sup> *<sup>E</sup>*1*. Next, the condition* (AS2) *is also valid for j* <sup>=</sup> <sup>2</sup> *with <sup>δ</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> *and K* <sup>=</sup> <sup>2</sup> 5 *, and*

$$0 < K = \frac{2}{5} < \min\left\{1, \frac{\Gamma(u\_j + 1)}{(T\_j - T\_{j-1})^{u\_j}}\right\} = \min\left\{1, \Gamma(\frac{5}{2})\right\} = 1.$$

*According to Theorem 2, the constant order resonance FBVP* (19) *has a solution like <sup>φ</sup>*<sup>2</sup> <sup>∈</sup> *<sup>E</sup>*2*. Then, by Theorem 3, the given Caputo FBVP of variable order* (16) *has a solution as*

$$\phi(t) = \begin{cases} \quad \widetilde{\phi}\_1(t), & t \in A\_{1\prime} \\\\ \quad \phi\_2(t), & t \in A\_{2\prime} \end{cases}$$

*where*

$$\phi\_2(t) = \begin{cases} \ 0, & t \in A\_{1\prime} \\ \ \ \tilde{\phi}\_2(t), & t \in A\_{2\prime} \end{cases}$$

*and this shows the correctness of our results.*

#### **5. Conclusions**

In this paper, a theoretical study was done for the given Caputo BVP of variable order (1) at resonance. To conduct this research, we defined some generalized subintervals as a partition of the main interval, and then on each subinterval, the piecewise constant functions were defined. With the help of these notions, we converted the given variableorder system to a constant-order system at resonance. In this case, we implemented the conditions of the Mawhin's continuation theorem for proving the existence criterion for solutions of the corresponding BVP. Finally, an example was simulated numerically to show the correctness of our results. This technique on a variable-order BVP is new and determines the novelty of this work compared with other limited published papers in the form of variable orders. In relation to next studies, we aim to work on hidden-memory variable order systems and analyze the qualitative behaviors of their solutions such as existence, stability, and numerical solutions.

**Author Contributions:** Conceptualization, M.S.S. and Z.B.; Formal analysis, S.R., M.S.S., S.E., Z.B., S.K.N., S.A. and J.T.; Funding acquisition, J.T.; Methodology, S.R., M.S.S., S.E., Z.B., S.K.N. and S.A.; Software, S.E. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by King Mongkut's University of Technology North Bangkok. Contract No. KMUTNB-62-KNOW-29.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

**Acknowledgments:** The first and third authors would like to thank Azarbaijan Shahid Madani University.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:

BVP Boundary Value Problem

#### **References**


## *Article* **Solvability of Some Nonlocal Fractional Boundary Value Problems at Resonance in** R*<sup>n</sup>*

**Yizhe Feng † and Zhanbing Bai \*,†**

College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China; yzfeng2021@163.com

**\*** Correspondence: zhanbingbai@163.com

† These authors contributed equally to this work.

**Abstract:** In this paper, the solvability of a system of nonlinear Caputo fractional differential equations at resonance is considered. The interesting point is that the state variable *<sup>x</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* and the effect of the coefficient matrices matrices *B* and *C* of boundary value conditions on the solvability of the problem are systematically discussed. By using Mawhin coincidence degree theory, some sufficient conditions for the solvability of the problem are obtained.

**Keywords:** coincidence degree theory; four-point boundary value problem system; at resonance

#### **1. Introduction**

In partial differential equations theory, multipoint boundary conditions are those which the solutions of multiple-parameter differential equations should satisfy. In recent decades, more and more mathematicians turned their attention to nonlinear boundary value problems (BVPs) in resonance cases and non-resonance cases. For some non-resonance cases, we recommend readers to [1–4], and for resonance cases to [5–12] and the references therein. In [8], Feng first obtained the existence of one solution of semilinear three-point BVPs at resonance by making use of the coincidence degree theory of Mawhin. Then, as an extension of [8], Ma [9] first developed the upper and lower solution method to obtain some multiplicity results. Motivated by [9], Bai [6] researched a four-point boundary value problem, and proved the existence and multiplicity results by making use of the method of upper and lower solutions established by the coincidence degree theorem. Subsequently, various boundary value conditions were studied.

V.A. Il'in and E.I. Moiseev in [1] studied Sturm–Liouville operator of the first kind of nonlocal boundary value problem, which originated from the famous work of A. V. Bitsadze and A. A. Samarskogo [3]: In the Euclidean *n*-dimensional space with orthogonal Cartesian coordinates *x*1, *x*2, ..., *xn*, the elliptic linear differential equation on the (*n* − 1) -dimensional piecewise smooth Lyapunov surface is transformed into a nonlocal problem of an ordinary differential equation when solving a partial differential equation by the separation of variables method. When the state variable is *n*-dimensional, consideration of the general fractional model will naturally involve the model of the problem considered in this paper.

To our best knowledge, before P.D. Phung [13], almost all articles on resonance BVPs were focused on a single second-order equation with the dimension of Ker *L* ∈ [0, 2]. For a second-order equation boundary value problem system with *<sup>x</sup>* <sup>∈</sup> <sup>R</sup>*n*, the dimension of Ker *L* will be between 0 and 2*n*; it will not be as easy as dim Ker *L* = 1 to establish projections *Q* for matrices *B* and *C* with different properties. For the case of *n* = 2, Zhang in [12] considered a three-point BVP at resonance for nonlinear fractional differential equations:

> ⎧ ⎪⎨ ⎪⎩ *D<sup>α</sup>* <sup>0</sup>+*u*(*t*) = *<sup>f</sup>*(*t*, *<sup>v</sup>*(*t*), *<sup>D</sup>β*−<sup>1</sup> <sup>0</sup><sup>+</sup> *v*(*t*)), 0 < *t* < 1, *Dβ* <sup>0</sup>+*v*(*t*) = *<sup>g</sup>*(*t*, *<sup>u</sup>*(*t*), *<sup>D</sup>α*−<sup>1</sup> <sup>0</sup><sup>+</sup> *u*(*t*)), 0 < *t* < 1, *u*(0) = *v*(0) = 0, *u*(1) = *σ*1*u*(*η*1), *v*(1) = *σ*2*v*(*η*2),

**Citation:** Feng, Y.; Bai, Z. Solvability of Some Nonlocal Fractional Boundary Value Problems at Resonance in R*<sup>n</sup>*. *Fractal Fract.* **2022**, *6*, 25. https://doi.org/10.3390/ fractalfract6010025

Academic Editor: Rodica Luca

Received: 12 December 2021 Accepted: 31 December 2021 Published: 1 January 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

and obtained two existence results using the coincidence degree theory. In [13]. P.D. Phung first researched the following resonant three-point BVPs in R*n*:

$$\begin{cases} \mathfrak{x}''(t) = f(t, \mathfrak{x}, \mathfrak{x}'), & t \in (0, 1), \\ \mathfrak{x}'(0) = \theta, & \mathfrak{x}(1) = A\mathfrak{x}(\eta), \end{cases}$$

where *θ* is an *n*-order zero vector, the matrix *A* satisfies one of the following conditions:

$$\begin{cases} A^2 = I \left( \text{ stands for } n - \text{order identity matrix} \right), \\ A^2 = A. \end{cases}$$

In [14], P.D. Phung removed the restriction on matrix *A* and studied the solvability of the same problem as in [13]. Then, P.D. Phung [15] used similar methods to study the following three-point boundary conditions in the fractional differential equations at resonance:

$$\begin{aligned} D^{\alpha} \mathfrak{x}(t) &= f(t, \mathfrak{x}(t), D^{\alpha - 1} \mathfrak{x}(t)) \\ \mathfrak{x}(0) &= \theta \text{ } D^{\alpha - 1} \mathfrak{x}(1) = A D^{\alpha - 1} \mathfrak{x}(\eta). \end{aligned}$$

Recently, the solvability of integer or fractional differential equations with a wide range of boundary value conditions at resonance in R*<sup>n</sup>* has been researched. We direct readers to [13–21] for details.

For nearly a decade, the resonant boundary value problem with *n* equations has been studied by an increasing number of mathematicians. However, we found that the following two problems have not been addressed. First, Zhang in [12] studied the resonance boundary value problem of two equations, but used the same boundary value conditions for different state variables *u* and *v*, so the study was similar to that of a single equation and could not be easily extended to the case of *n* dimensions. Therefore, in this study we consider the characterization of different constraints on different state variables, in other words, we introduce matrices to control the constraints on state variables so that the expression of the equation can be richer. However, other works [13–16,20,21] under the condition of zero boundary value (similar to *u*(0) = 0) studied *n* equations of the problem. Gupta in [10] proposed that many multi-point boundary value problems can be transformed into four-point boundary value problems under certain conditions, so studying four-point BVPs is more meaningful. The four-point boundary value condition does not contain zero boundary value, which makes the structure of irreversible operators and the construction of projection *P* and *Q* more complicated than that of three-point BVPs. Therefore, it is more meaningful to introduce a matrix to study four-point boundary value problems in mathematics.

Motivated by the above ideas, we consider the following fractional-order equations with a new boundary value condition in R*n*:

$$\prescript{c}{}{D}\_{0+}^{a}u(t) = f(t, u(t), \prescript{c}{}{D}\_{0+}^{a-1}u), \quad t \in (0, 1), \tag{1}$$

$$u(0) = Bu(\xi), \quad u(1) = \mathbb{C}u(\eta), \tag{2}$$

where 0 < *η*, *ξ* < 1, 1 < *α* - 2; *B*, *C* are two *n*-order nonzero square matrices, *cD<sup>α</sup>* 0+ represents the Caputo differentiation, and *<sup>f</sup>* : [0, 1] <sup>×</sup> <sup>R</sup>2*<sup>n</sup>* <sup>→</sup> <sup>R</sup>*<sup>n</sup>* satisfies Carathéodory conditions. In this situation, Ker *L* may become a polynomial set with vector coefficients and the construction of projectors will be somewhat complex. We say *<sup>f</sup>* : [0, 1] <sup>×</sup> <sup>R</sup>2*<sup>n</sup>* <sup>→</sup> <sup>R</sup>*<sup>n</sup>* satisfies Carathéodory conditions, that is,


The problem in (1) and (2) is in resonance, meaning that the following linear homogeneous boundary value problem has nontrivial solutions:

$${}^{\mathbb{C}}D\_{0+}^{\alpha}u(t) = \theta, \quad 0 < t < 1,\tag{3}$$

$$u(0) = Bu(\xi), \quad u(1) = Cu(\eta). \tag{4}$$

By (3), there is *<sup>u</sup>*(*t*) = *<sup>c</sup>*<sup>1</sup> <sup>+</sup> *<sup>c</sup>*2*t*, *<sup>c</sup>*1, *<sup>c</sup>*<sup>2</sup> <sup>∈</sup> <sup>R</sup>*n*. Combining with (4), we can get the following equations:

$$\begin{cases} (I - \eta \mathcal{C})c\_1 + (I - \mathcal{C})c\_2 = \theta\_\prime \\ -\xi B c\_1 + (I - B)c\_2 = \theta. \end{cases}$$

Clearly, the resonance condition is

$$
\Delta = \begin{vmatrix} I - \eta \mathbf{C} & I - \mathbf{C} \\ -\xi B & I - B \end{vmatrix} = \mathbf{0}.
$$

From the calculation formula of block matrix determinant, we can know that Δ = 0 if and only if

$$|(I - \eta \mathbb{C})(I - B) + \xi B(I - \mathbb{C})| = 0. \tag{5}$$

Condition (5) can be divided into three cases:

*Case* (1) *B* = *I*, *C* = *I*, |(*I* − *ηC*)(*I* − *B*) + *ξB*(*I* − *C*)| = 0; *Case* (2) *B* = *I*, |*I* − *C*| = 0; *Case* (3) *B* = *I*, *C* = *I*, |*I* − *B*| = 0.

The paper is organized as follows. In Section 2, we state several notations and definitions. In Sections 3 and 4, two main theorems (see Theorem 2 and 3) are established for the solvability of problem (1) and (2) under resonance cases (1) and (2), respectively. It is worth mentioning that, inspired by [14], in Section 4, we remove the restriction on the matrix *C*, and give the existence theorem of the solution of the problem only under the most basic resonance conditions (refer to *case* (2)).

#### **2. Preliminaries**

First, we recall some related definitions and lemmas of fractional calculus; we refer the readers to [22] for more properties.

**Definition 1.** *The α–order* (*α* > 0) *Riemann–Liouville fractional integral of function u is defined as*

$$d\_{0+}^{a}u(t) = \frac{1}{\Gamma(a)} \int\_{0}^{t} \frac{u(s)}{(t-s)^{1-a}} ds,\tag{6}$$

*and the right side of the equation is defined at* (0, ∞)*.*

**Definition 2.** *The α–order* (*α* > 0) *Caputo fractional derivative of function u* : *R*<sup>+</sup> → *R is defined as*

$$\, \_C^C D\_{0+}^{\mu} u(t) = I\_{0+}^{\eta - a} D^n u(t) = \frac{1}{\Gamma(n-a)} \int\_0^t \frac{u^{(n)}(s)}{(t-s)^{1+a-n}} ds \tag{7}$$

*as long as the right side of the equation is defined at* (0, ∞)*.*

**Lemma 1** ([22])**.** *If u* <sup>∈</sup> *<sup>C</sup>n*−1(0, 1) <sup>∩</sup> *<sup>L</sup>*[0, 1]*, then the fractional differential equation*

$$\, ^\mathbb{C} D\_{0+}^\alpha u(t) = 0\tag{8}$$

*has a unique solution*

$$
\mu(t) = \sum\_{i=0}^{n-1} \frac{\mu^{(i)}(0)}{i!} t^{k}. \tag{9}
$$

The following lemma is also very important for subsequent research.

**Lemma 2** ([22])**.** *Let α* > 0 *and n* − 1 < *α n.*

*(1) Let α* > *θ* > 0 *and u be a continuous function, then*

$$\prescript{C}{}{D}\_{0+}^{\theta}l\_{0+}^{\alpha}u(t) = l\_{0+}^{\alpha-\theta}u.\tag{10}$$

*(2) Let u be an absolute continuous function of n* − 1 *times differentiable, then*

$$d\_{0+}^{a}{}^{C}D\_{0+}^{a}u(t) = u(t) - \sum\_{i=0}^{n-1} \frac{D^{i}u(0)}{i!}t^{i}.\tag{11}$$

Let *X*, *Y* be two Banach spaces, we call *L* : dom *L* ⊂ *X* → *Y* a Fredholm mapping of index zero if

(E1) Im *L* is closed in *Y* and has codimension of finite dimension;

(E2) Tthe dimension of Ker *L* is equal to the codimension of Im *L*.

If *L* satisfies (E1) and (E2), then there will be two projectors *Q* : *Y* → *Y*, *P* : *X* → *X* satisfies Ker *Q* = Im *L*, Im *P* = Ker *L*. Therefore, we can get the straight-sum decomposition: *Y* = Im *L* ⊕ Im *Q*, X = Ker *L* ⊕ Ker *P*. Here, by *KP* we denote the inverse of *L*|Ker *<sup>P</sup>*∩dom *<sup>L</sup>* : Ker *P* ∩ dom *L* → Im *L* and by *KP*,*<sup>Q</sup>* := *KP*(*Id* − *Q*) the generalized inverse of *L*.

We call *N L*-compact on Ω (Ω is an open bounded subset of *X* with dom *L* ∩ Ω = ∅, when it satisfies


**Theorem 1** ([23])**.** *Let L be a Fredholm operator of index zero and N(*Ω*) be L-compact. Suppose the following conditions are satisfied:*


*Then, the equation Lu* = *Nu has at least one solution in* dom *L* ∩ Ω*.*

By *u* = *max*{ *u* <sup>∞</sup>, *cD<sup>α</sup>*−<sup>1</sup> <sup>0</sup><sup>+</sup> *u* <sup>∞</sup>} we denote the norm of space *<sup>X</sup>* <sup>=</sup> *<sup>C</sup>*1([0, 1]; <sup>R</sup>*n*), where · <sup>∞</sup> is the maximum norm. Additionally, by *y* <sup>1</sup> we denote the Lebesgue norm of *Y* = *L*1([0, 1]; R*n*). Set

$$X\_1 := \{ \mu : [0,1] \to \mathbb{R}^n \mid \mu \in \mathbb{C}^2([0,1]; \mathbb{R}^n) \}.$$

Then, define map *L* : dom *L* → *Y* by setting

$$\text{dom}\ L = \{\iota \in X\_1 : \iota(0) = Bu(\xi), \; \iota(1) = \mathbb{C}u(\eta)\}\_{\mathcal{A}}$$

for *u* ∈ dom *L*,

$$Lu := {}^C D\_{0+}^\alpha u. \tag{12}$$

#### **3. Existence Results for Case (1)**

Now, we show the solvability of BVP (1), (2) when *B* = *I*, *C* = *I*, |(*I* − *ηC*)(*I* − *B*) + *ξB*(*I* − *C*)| = 0. Furthermore, suppose the matrices *B*, *C* satisfy the following conditions:

(H1) *I* − *B* is reversible;


where Θ is an *n*-order zero matrix. From (12) we can know

$$\text{Ker } L = \{c\_2t + \mathcal{C}\_0c\_2, \ c\_2 \in \mathbb{R}^n\}\_{\mathsf{V}}$$

where *<sup>C</sup>*<sup>0</sup> <sup>=</sup> *<sup>ξ</sup>*(*<sup>I</sup>* <sup>−</sup> *<sup>B</sup>*)−1*B*, and from (H3) we have (*<sup>I</sup>* <sup>−</sup> *<sup>C</sup>*)*C*<sup>0</sup> = (*η<sup>C</sup>* <sup>−</sup> *<sup>I</sup>*). Let

$$G(s) = \begin{cases} (\sharp - s)^{a-1} (I - \mathbb{C}) (I - \mathbb{B})^{-1} B - (\eta - s)^{a-1} \mathbb{C} + (1 - s)^{a-1} I, & 0 < s < \sharp; \\ -(\eta - s)^{a-1} \mathbb{C} + (1 - s)^{a-1} I, & \mathbb{S} < s < \eta; \\ (1 - s)^{a-1} I, & \eta \le s \le 1, \end{cases}$$

then

$$\operatorname{Im} L = \left\{ y \in Y \, \Big| \, \frac{1}{\Gamma(\alpha)} \int\_0^1 G(s) y(s) ds = \theta \right\}.$$

Define a mapping *Q* : *Y* → *Y* as

$$Qy = \gamma \int\_0^1 G(s)y(s)ds,\tag{13}$$

where

$$\gamma = \alpha \{ (\eta \mathcal{C} - I) \mathfrak{f}^{\kappa - 1} - \eta^{\kappa} \mathcal{C} + I \}^{-1}.$$

**Lemma 3.** *The operator L is a Fredholm operator with an index of zero.*

**Proof.** For *y* ∈ *Y*, ∀*t* ∈ [0, 1]

$$\begin{aligned} Q^2y(t) &= \gamma \int\_0^1 G(s)Qy(s)ds \\ &= \frac{\gamma}{\alpha} \{ (\eta \mathcal{C} - I)\xi^{\alpha - 1} - \eta^\alpha \mathcal{C} + I \} Qy(t) \\ &= Qy(t), \end{aligned}$$

so linear operator *Q* is a continuous projector. For *y* ∈ Im *L*, one has *Qy*(*t*) = *θ*; this shows that *y* ∈ Ker *Q*. In fact, Im *L* = Ker *Q*.

Let *y* ∈ *Y* and it is easy to verify *y* − *Qy* ∈ Im *L*. Thus, *Y* = Im *L* + Im *Q*. For every *<sup>y</sup>* <sup>∈</sup> Im *<sup>Q</sup>* have the form *<sup>y</sup>* <sup>=</sup> *<sup>c</sup>*, *<sup>c</sup>* <sup>∈</sup> <sup>R</sup>*n*. At this time, if *<sup>y</sup>* <sup>∈</sup> Im *<sup>L</sup>*, then *<sup>y</sup>* <sup>=</sup> *<sup>θ</sup>*. Hence, *<sup>Y</sup>* <sup>=</sup> Im *L* ⊕ Im *Q*. Combine with codim Im *L* = dim Im *Q* = dim Ker *L*, so *L* satisfies (E1) and (E2), and the index of the Fredholm operator *L* is zero.

Define another projector *P* : *X* → *X* by

$$Pu = u'(0)t + \mathcal{C}\_0 u'(0). \tag{14}$$

For *v* ∈ Ker *L*, one has

and

$$v(t) = c\_2 t + \mathcal{C}\_0 c\_2, \ c\_2 \in \mathbb{R}^n.$$

$$Pv(t) = c\_2t + \mathcal{C}\_0c\_2 = v(t).$$

This shows that *v* ∈ Im *P*. Conversely, for every *v* ∈ Im *P*, there is *x* ∈ *X* such that *v*(*t*) = *Px*(*t*). Thus,

$$v(t) = P\mathbf{x}(t) = \mathbf{x}'(0)t + c\_0\mathbf{x}'(0) \in \text{Ker}L.\tag{15}$$

Hence, Ker *L* = Im *P*. Clearly, *X* = Ker *P* ⊕ Ker *L*. In fact, Ker *P* ∩ Ker *L* = {*θ*}. Define a mapping *KP* : Im *L* → Ker *P* ∩ dom *L* as

$$K\_P y(t) = (I - B)^{-1} B I\_{0+}^a y(\xi) + I\_{0+}^a y(t), \ 0 \le t \le 1. \tag{16}$$

**Lemma 4.** *KP is the inverse o f the mapping L*|Ker *<sup>P</sup>*∩dom *<sup>L</sup> and*

$$\|\|K\_P y\|\| \lesssim D \|\|y\|\|\_{1} \tag{17}$$

*where D* = 1 + *ξ* (*<sup>I</sup>* <sup>−</sup> *<sup>B</sup>*)−1*<sup>B</sup>* <sup>∗</sup>*,* · <sup>∗</sup> *stand for the max-norm of matrices.*

**Proof.** Let *y* ∈ Im *L*. It is clear that *KPy*(0) = *BKPy*(*ξ*) and *KPy*(1) = *CKPy*(*η*), such that *KPy* ∈ dom *L*. Furthermore

$$PK\_{P}y(t) = \left(K\_{P}y\right)'(t)|\_{t=0}t + c\_{0}(K\_{P}y)'(t)|\_{t=0} = \theta.\tag{18}$$

This shows that *KPy* ∈ Ker *P*. So, the definition of *KP* is reasonable. For *u* ∈ Ker *P* ∩ dom *L*, from (11), one has

$$\begin{aligned} K\_P L u &= \left(I - B\right)^{-1} B I\_{0+}^{\underline{a}} \,^c D\_{0+}^{\underline{a}} u(\mathfrak{f}) + I\_{0+}^{\underline{a}} \,^c D\_{0+}^{\underline{a}} u(t) \\ &= \left(I - B\right)^{-1} B \left[u(\mathfrak{f}) - u(0) - u'(0)\mathfrak{f}\right] - I(u(0) - u(t) + u'(0)t) \\ &= u. \end{aligned}$$

Conversely, for *<sup>y</sup>* <sup>∈</sup> Im *<sup>L</sup>*, one has *LKPy* <sup>=</sup> *<sup>y</sup>*. Thus, *KP* = (*L*|dom *<sup>L</sup>*∩Ker *<sup>P</sup>*)−1. Again, since

$$\begin{aligned} \| |^\mathcal{C} D\_{0+}^{a-1} (K\_P y)(t) | \|\_{\infty} &= \| (I - B)^{-1} B I\_{0+}^1 y(\mathfrak{f}) | \|\_{\infty} + \| |I\_{0+}^1 y(t)| |\_{\infty} \\ &\leqslant (1 + \mathfrak{f} \| (I - B)^{-1} B \| \_{\ast}) \| y(t) \|\_{1 \prime} \end{aligned}$$

combining with (16), one has

$$||K\_P y||\_\infty \le \frac{D}{\Gamma(\alpha)}||y||\_1.$$

Thus, we have *KPy* - *D y* 1.

Define an operator *N* : *X* → *Y* by

$$Nu(t) = f(t, u(t), \,^cD\_{0+}^{
u-1}u(t)), \, 0 \lessapprox t \lessapprox 1. \tag{19}$$

**Lemma 5.** *N is L-compact.*

**Proof.** We divide the proof into two parts. The first part is bounded continuous. The second part is completely continuous. Indeed, for *f*(*t*, *u*(*t*), *cD<sup>α</sup>*−<sup>1</sup> <sup>0</sup><sup>+</sup> *u*(*t*)), there exists a function *gW*(*t*) : *R* → *Y* s.t. for every *u* ∈ *W* ⊂ *X* and a.e. 0 *t* -1

$$\|\|f(t, u\_\prime^c D\_{0+}^{\alpha-1} u)\|\|\_{\infty} \leqslant g\_W. \tag{20}$$

Combining with (13), one has

$$\|\|Qy\|\|\_{1} \leqslant \|\|G(s)\|\|\_{\*} \|\|\gamma\|\|\_{\*} \|\|y\|\|\_{1} \tag{21}$$

where

$$\begin{aligned} \|\gamma\|\|\_\ast &= \alpha \|\| \left\{ (\eta \mathbf{C} - I) \mathfrak{f}^{\ast -1} - \eta^\ast \mathbf{C} + I \right\}^{-1} \|\|\_\ast, \\ \|\| G(\mathbf{s}) \|\|\_\ast &= (1 + \|\mathbf{C} \|\|\_\ast + \|(I - \mathbf{C})(I - B)^{-1} B\|\|\_\ast). \end{aligned}$$

Thus, *QN*(*W*) is bounded. Obviously, *QN*(*W*) is continuous.

For all *u* ∈ *W* ⊂ *X*, one has

$$\begin{aligned} \mathcal{K}\_{P,\mathcal{Q}}Nu &= \mathcal{K}\_P(I-\mathcal{Q})Nu \\ &= (I-B)^{-1}BI\_{0+}^\alpha Nu(\mathcal{\zeta}) + I\_{0+}^\alpha Nu(t) - (I-B)^{-1}BI\_{0+}^\alpha QNu(\mathcal{\zeta}) - I\_{0+}^\alpha QNu(t) \\ &= (I-B)^{-1}BI\_{0+}^\alpha Nu(\mathcal{\zeta}) + I\_{0+}^\alpha Nu(t) \end{aligned}$$

$$-\frac{\gamma}{\Gamma(\alpha)} \left\{ \tilde{\xi}^{\alpha} (Id - B)^{-1} B \int\_{0}^{1} G(s) N u(s) ds + t^{\alpha} \int\_{0}^{1} G(s) N u(s) ds \right\}.\tag{22}$$

$$\prescript{c}{}{D}\_{0+}^{\mu-1}K\_{P,Q}Nu = \prescript{c}{}{D}\_{0+}^{\mu-1}K\_{P}(Id-Q)Nu = I\_{0+}^{1}Nu(t) - \gamma\Gamma(\mu+1)t\int\_{0}^{1}G(s)Nu(s)ds. \tag{23}$$

Combining (20), (22), and (23), we have

$$\begin{aligned} \left| \mathcal{K}\_{P,Q} \mathcal{N}u(t) \right| &\leqslant (1 + ||(I - B)^{-1}B||\_\*) ||\mathcal{N}u||\_1 \\ &+ \frac{||\gamma||\_\*}{\Gamma(\alpha)} (1 + ||(I - B)^{-1}B||\_\*) ||G(s)||\_\* ||\mathcal{N}u||\_1 \\\\ \left| ^c D\_{0+}^{\alpha - 1} \mathcal{K}\_{P,Q} \mathcal{N}u(t) \right| &\leqslant (1 + \Gamma(\alpha + 1) ||\gamma||\_\* ||G(s)||\_\*) ||\mathcal{N}u||\_1. \end{aligned}$$

That is, *KP*,*QNu*(*W*) is uniformly bounded in *X*. Now we only need to prove *KP*,*QNu*(*W*) is equicontinuous in *X* to end the proof of Lemma 5. For 0 *t*<sup>1</sup> < *t*<sup>2</sup> -1, one has

$$\begin{split} & \left| \langle K\_{\mathcal{Q},\mathcal{Q}} Nu(t\_2) - K\_{\mathcal{P},\mathcal{Q}} Nu(t\_1) \rangle \right| \\ & \leq \frac{1}{\Gamma(\alpha)} \left| \int\_{t\_1}^{t\_2} (t\_2 - s)^{\alpha - 1} Nu(s) ds + \int\_0^{t\_1} ((t\_2 - s)^{\alpha - 1} - (t\_1 - s)^{\alpha - 1}) Nu(s) ds \right| \\ & \quad + \frac{\gamma}{\alpha \Gamma(\alpha)} \| \mathcal{G}(s) \| \_\* \| Nu \| \_1 \| t\_2^{\alpha} - t\_1^{\alpha} \| \\ & \leq \frac{1}{\Gamma(\alpha)} \left( \int\_0^{t\_1} (t\_2 - t\_1)^{\alpha - 1} g\_{\mathcal{W}}(s) ds + \int\_{t\_1}^{t\_2} g\_{\mathcal{W}}(s) ds \right) \\ & + \frac{\gamma}{\alpha \Gamma(\alpha)} \| \mathcal{G}(s) \| \_\* \| \mathcal{G}\_{\mathcal{W}}(t) \| \_1 \| t\_2^{\alpha} - t\_1^{\alpha} \| \_ . \end{split}$$

and

$$\left| \left| \,^c D\_{0+}^{\mathfrak{a}-1} \mathcal{K}\_{P,\mathcal{Q}} \mathcal{N} \mathfrak{u}(t\_2) - \,^c D\_{0+}^{\mathfrak{a}-1} \mathcal{K}\_{P,\mathcal{Q}} \mathcal{N} \mathfrak{u}(t\_1) \right| \right| \lesssim \int\_{t\_1}^{t\_2} \mathcal{g}\_W(\mathbf{s}) d\mathbf{s} + \gamma \left\| \mathcal{G}(\mathbf{s}) \right\|\_\ast \left\| \mathcal{g}\_W(t) \right\| \left\| \mathbf{1}\_{\mathfrak{l}} \left| t\_2 - t\_1 \right|.$$

Thus, *KP*,*QNu*(*W*) is equicontinuous in X. In summary, *N* is *L*-compact.

We will use the following assumptions:

(M1) For all *<sup>t</sup>* <sup>∈</sup> [0, 1], *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> <sup>R</sup>*n*, there exist three functions *<sup>a</sup>*1, *<sup>b</sup>*1, *<sup>c</sup>* <sup>∈</sup> *<sup>Y</sup>*, s.t.

$$(1 + \|c\_0\|\_\* + D)(\|a\_1\|\_1 + \|b\_1\|\_1) < 1,\tag{24}$$

and

$$|f(t, \mathbf{x}, y)| \le a\_1(t)|\mathbf{x}| + b\_1(t)|y| + c(t),\tag{25}$$

where *D* is the constant given in (17).

(M2) For *u* ∈ dom *L*, if there exist *σ*<sup>1</sup> ∈ *R*+, s.t.

$$|\,^cD\_{0+}^{\alpha-1}u(\nu)| > \sigma\_1 \,\,\forall \nu \in [0,1]\_\prime$$

then

$$\begin{aligned} \mathbb{C} \int\_0^\eta (\eta - s)^{a-1} f(\nu, u(\nu), ^c D\_{0+}^{a-1} u(\nu)) d\nu \\ - I \int\_0^1 (1 - s)^{a-1} f(\nu, u(\nu) d\nu, ^c D\_{0+}^{a-1} u(\nu)) d\nu \in \text{Im} \ (I - \mathbb{C}). \end{aligned}$$

$$\text{(M3) Let } q(t) := (tI + \mathbb{C}\_0)\_\prime \text{ } \mathbb{C}\_0 = \mathfrak{F} (I - B)^{-1} B \text{,} and$$

$$q(t)\tau = (q\_1, \dots, q\_n)^\top \ , \ q\_i \in \mathbb{R}.$$

If there exist *σ*<sup>2</sup> ∈ *R*+, s.t. ∀*t* ∈ [0, 1],

<sup>|</sup>*qi*<sup>|</sup> <sup>&</sup>gt; *<sup>σ</sup>*2, <sup>∀</sup>*<sup>τ</sup>* <sup>∈</sup> <sup>R</sup>*n*, *<sup>i</sup>* <sup>=</sup> 1, ..., *<sup>n</sup>*,

then either

$$
\langle q(t)\tau, \mathbb{Q}N(q(t)\tau)\rangle \leqslant 0 \; or \; \langle q(t)\tau, \mathbb{Q}N(q(t)\tau)\rangle \geqslant 0,\tag{26}
$$

·, · stands for the scalar product in <sup>R</sup>*n*.

**Theorem 2.** *If assumptions (M1)–(M3) are satisfied, then Problem* (1)*,* (2) *has at least one solution in X.*

**Proof.** Set Ω<sup>1</sup> = {*x* ∈ dom *L* \ Ker *L* : *Lx* = *λNx*, 0 < *λ* < 1}. For *u* ∈ Ω1, one has *Nu* ∈ Im *L* = Ker *Q*. Thus,

$$\begin{aligned} \text{C} \int\_0^\eta (\eta - s)^{a-1} f(s, u(s), {}^c D\_{0+}^{a-1} u(s)) ds - I \int\_0^1 (1 - s)^{a-1} f(s, u(s), {}^c D\_{0+}^{a-1} u(s)) ds \\ = (I - \text{C})(I - B)^{-1} B \int\_0^\xi (\tilde{\xi} - s)^{a-1} f(s, u(s), {}^c D\_{0+}^{a-1} u(s)) ds \in \text{Im} \,(I - \text{C}). \end{aligned}$$

From (M2), there exist *t*<sup>0</sup> ∈ [0, 1], s.t. | *cD<sup>α</sup>*−<sup>1</sup> <sup>0</sup><sup>+</sup> *u*(*t*0)| *σ*1, thus

$$|\,^cD\_{0+}^{a-1}u(0)| = \left| \,^cD\_{0+}^{a-1}u(t\_0) - \int\_0^{t\_0} \,^cD\_{0+}^a u(s)ds \right| \leqslant \sigma\_1 + ||\,^cD\_{0+}^a u(t)||\_1.$$

Furthermore

$$\|\|Pu(t)\|\| = \|\mu'(0)t + \mathbb{C}\_0\mu'(0)\| \leqslant (\|\|Nu\|\|\_1 + \sigma\_1)(1 + \|\mathbb{C}\_0\|\_\*).\tag{27}$$

Note that *Id* is the identity operator. Combining with (27), one has

$$\begin{aligned} \|\boldsymbol{u}(t)\| &= \|P\boldsymbol{u} + (Id - P)\boldsymbol{u}\| \\ &\leqslant \|P\boldsymbol{u}\| + \|K\_{P}\boldsymbol{L}(Id - P)\boldsymbol{u}\| \\ &\leqslant (\|\boldsymbol{N}\boldsymbol{u}\|\_{1} + \sigma\_{1})(1 + \|\mathbb{C}\_{0}\|\_{\*}) + D\|\boldsymbol{N}\boldsymbol{u}\|\_{1} \\ &= (1 + \|\mathbb{C}\_{0}\|\_{\*} + D)\|\boldsymbol{N}\boldsymbol{u}\|\_{1} + (1 + \|\mathbb{C}\_{0}\|\_{\*})\sigma\_{1} \end{aligned} \tag{28}$$

where *D* was given in (16). Combining (19), (28), and (M1), we get

$$\begin{split} \|\|Nu\|\|\_{1} &\leqslant \int\_{0}^{1} |f(s, u(s), {}^{c}D\_{0+}^{a-1}u(s))| ds \\ &\leqslant \|\|a\_{1}\|\|\_{1} \|\|u\|\|\_{\infty} + \|\|b\_{1}\|\|\_{1} \|{{}^{c}D\_{0+}^{a-1}u}\|\|\_{\infty} + \|\|c\|\|\_{1} \\ &\leqslant \left(\|\|a\_{1}\|\|\_{1} + \|\|b\_{1}\|\|\_{1}\right) \|u\| + \|c\|\|\_{1} \\ &\leqslant \left(\|\|a\_{1}\|\|\_{1} + \|\|b\_{1}\|\|\_{1}\right) \left[\left(1 + \|\mathbb{C}\_{0}\|\right|\_{\*} + D\right) \|Nu\|\|\_{1} + \left(1 + \|\mathbb{C}\_{0}\|\|\_{\*}\right) \sigma\_{1}\right) + \|c\|\|\_{1}. \end{split}$$

Therefore, it can be obtained that

$$||Nu||\_1 \leqslant \frac{(||a\_1||\_1 + ||b\_1||\_1)(1 + ||\mathbb{C}\_0||\_\*)\sigma\_1| + ||\mathbb{c}||\_1}{1 - (1 + ||\mathbb{C}\_0||\_\* + D)(||a\_1||\_1 + ||b\_1||\_1)}.\tag{29}$$

From (29) and (28), one has

$$\sup\_{\mu \in \Omega\_1} \|\mu\| = \sup\_{\mu \in \Omega\_1} \max \{ \||\mu||\_{\infty}, \||^c D\_{0+}^{\alpha-1} \mu||\_{\infty} \} < +\infty.$$

Hence Ω<sup>1</sup> is bounded in *X*.

Set Ω<sup>2</sup> = {*u* ∈ Ker *L* | *Nu* ∈ Im *L*}. Assuming *u* ∈ Ω2, one has *u* = *c*2*t* + *C*0*c*2, *<sup>c</sup>*<sup>2</sup> <sup>∈</sup> <sup>R</sup>*n*. Thus

$$\begin{aligned} \text{C} \int\_0^\eta (\eta - s)^{a-1} f(s, c\_2s + \mathbb{C}\_0 c\_2, c\_2) ds - I \int\_0^1 (1 - s)^{a-1} f(s, c\_2s + \mathbb{C}\_0 c\_2, c\_2) ds \\ = (I - \mathbb{C})(I - B)^{-1} B \int\_0^\xi (\xi - s)^{a-1} f(s, c\_2s + \mathbb{C}\_0 c\_2, c\_2) ds \in \text{Im} \,(I - \mathbb{C}). \end{aligned}$$

Then, from assumption (M2), one has

$$\begin{aligned} ||u|| &= \max\{||u||\_{\infty}, ||^{c}D\_{0+}^{\alpha-1}u||\_{\infty}\} \\ &= \max\{||c\_{2}t + \mathbb{C}\_{0}c\_{2}||\_{\infty}, ||c\_{2}||\_{\infty}\} \\ &< \max\{(1 + ||\mathbb{C}\_{0}||\_{\*})\sigma\_{1}, \sigma\_{1}\} \\ &\leq (1 + ||\mathbb{C}\_{0}||\_{\*})\sigma\_{1} < +\infty. \end{aligned}$$

Therefore, Ω<sup>2</sup> is a bounded subset.

Set Ω± <sup>3</sup> = {*u* ∈ Ker *L* : ±*λ*1*u* + (1 − *λ*1)*QNu* = *θ*, 0 *λ*<sup>1</sup> - 1}. We divide the proof into the following two steps:

**Step 1** : For *<sup>u</sup>* <sup>=</sup> *<sup>c</sup>*2*<sup>t</sup>* <sup>+</sup> *<sup>C</sup>*0*c*<sup>2</sup> <sup>∈</sup> <sup>Ω</sup><sup>+</sup> <sup>3</sup> , one has

$$
\lambda\_1(\mathbf{c\_2}t + \mathbf{C\_0c\_2}) + (1 - \lambda\_1)\mathbf{Q}N(\mathbf{c\_2}t + \mathbf{C\_0c\_2}) = \theta.
$$

**Case 1** : If *λ*<sup>1</sup> = 0, then *QN*(*c*2*t* + *C*0*c*2) = *θ*, such that *N*(*c*2*t* + *C*0*c*2) ∈ Ker *Q* = Im *L*. Thus we have *N*(*c*2*t* + *C*0*c*2) ∈ Ω2, so *u* - (1 + *C*0 <sup>∗</sup>)*σ*1.

**Case 2** : If *λ*<sup>1</sup> ∈ (0, 1], suppose *u* > *nσ*2. Then, from (M3) obtain that

$$0 > -\lambda\_1|\mu|^2 = (1 - \lambda\_1)\langle \mu, QNu \rangle \gg 0.1$$

So, we have a contradiction. Thus *u σ*2.

**Step 2** : For *u* ∈ Ω<sup>−</sup> <sup>3</sup> , using same arguments as in Step 1 above, we can deduce that *u σ*2. Thus we can show that Ω<sup>−</sup> <sup>3</sup> , <sup>Ω</sup><sup>+</sup> <sup>3</sup> ⊂ *X* are two bounded subsets.

Now, let <sup>Ω</sup> <sup>⊂</sup> *<sup>Y</sup>* and &<sup>3</sup> *<sup>i</sup>*=<sup>1</sup> Ω*<sup>i</sup>* ⊂ Ω. According to the above arguments, we know that both conditions (*i*) and (*ii*) of Theorem 1 are satisfied. In order to prove (*iii*), we use isomorphic mapping J to construct the homotopy operator by

$$H(\mathfrak{x}(t), \lambda) = \pm \lambda \mathfrak{x}(t) + (1 - \lambda) \mathcal{J} \mathcal{Q} \mathcal{N} \mathfrak{x}(t).$$

Hence

$$\begin{aligned} \deg(\mathcal{J}\mathbb{Q}N|\_{\text{Ker }L^{\prime}}\Omega \cap \text{Ker }L,\theta) &= \deg(H(\cdot,0),\Omega \cap \text{Ker }L,\theta) \\ &= \deg(\pm Id,\Omega \cap \text{Ker }L,\theta) \neq 0. \end{aligned}$$

Therefore, (*iii*) of Theorem 1 is satisfied. Theorem 2 is proved.

#### **4. Existence Results for Case (2)**

Now, we show the solvability of BVP (1), (2) when *B* = *I*, |*I* − *C*| = 0. In this case, the boundary value condition degenerates to

$$\mathfrak{x}(0) = \mathfrak{x}(\mathfrak{f}), \quad \mathfrak{x}(1) = \mathbb{C}\mathfrak{x}(\eta). \tag{30}$$

Unlike Section 3, this section removes the restriction on matrix *C* and uses the generalized inverse to conduct research under the most basic resonance conditions, inspired by [14]. Now we study the BVP (1) and (30) using Theorem 1. We use the same notations as in Section 3. *L*, *N*, J . In this case,

$$\text{dom}\,L = \{\mathfrak{x} \in X\_1 : \mathfrak{x} \text{ satisfies (30)}\}.$$

Let <sup>T</sup> <sup>=</sup> *<sup>I</sup>* <sup>−</sup> *<sup>C</sup>* and <sup>T</sup> <sup>+</sup> be the *Moore*–*Penrose pseudoinverse matrix* of <sup>T</sup> . From [24] we can get the following conclusions, which are necessary for our subsequent research:


From (12), we have

$$\text{Ker } L = \{c\_1^\* \in \mathbb{R}^n : \mathcal{T}c\_1^\* = \theta\}.$$

Define a linear operator *H*∗ by

$$H^\*y(t) = \frac{\eta \mathcal{C} - I}{\mathfrak{F}} I\_{0+}^\alpha y(\mathfrak{f}) - \mathcal{C}I\_{0+}^\alpha y(\eta) + I\_{0+}^\alpha y(1).$$

Then

$$\operatorname{Im} L = \{ y \in \mathcal{Y} \mid H^\* y(t) \in \operatorname{Im} \mathcal{T} \}.$$

Define an operator *Q*<sup>∗</sup> : *Y* → *Y* as

$$\mathbb{Q}^\* y = \gamma^\* H^\* y(t),\tag{31}$$

where

$$\gamma^\* = \frac{\mathfrak{Z}\alpha\Gamma(\alpha)}{\eta\mathfrak{Z}^\alpha - \mathfrak{Z}^\alpha + \mathfrak{z} - \mathfrak{z}\eta^\alpha} (I - \mathcal{T}\mathcal{T}^+).$$

Then for *y* ∈ *Y*, we can get

$$\begin{split} \mathbb{Q}^{\*2}y &= \gamma^\* H^\* \mathbb{Q}^\* y \\ &= \frac{\mathfrak{Z}a \Gamma(a)}{\eta \mathfrak{Z}^a - \mathfrak{Z}^a + \mathfrak{Z} - \mathfrak{Z}\eta^a} (I - \mathcal{T}\mathcal{T}^+) \frac{(\eta \mathbb{C} - I)\mathfrak{Z}^a + \mathfrak{Z}I - \mathfrak{Z}\eta^a \mathbb{C}}{a \mathfrak{Z}\Gamma(a)} \mathbb{Q}^\* y \\ &= \mathbb{Q}^\* y. \end{split}$$

In fact

$$\begin{split} & (I - \mathcal{T}\mathcal{T}^{+}) (\eta \mathcal{C} - I) \mathfrak{z}^{\mathfrak{a}} + \mathfrak{z}^{\mathfrak{z}} I - \mathfrak{z} \mathfrak{y}^{\mathfrak{a}} \mathcal{C} \\ &= (I - \mathcal{T}\mathcal{T}^{+}) \{ \eta (\mathcal{C} - I) \mathfrak{z}^{\mathfrak{a}} + (\eta - 1) \mathfrak{z}^{\mathfrak{a}} I + \eta^{\mathfrak{a}} (I - \mathcal{C}) + (1 - \eta^{\mathfrak{a}}) I \} \\ &= (\eta - 1) \mathfrak{z}^{\mathfrak{a}} I + \mathfrak{z} (1 - \eta^{\mathfrak{a}}) (I - \mathcal{T}\mathcal{T}^{+}) . \end{split}$$

By similar arguments to Lemma 2.5 in [14], we have that the index of the Fredholm operator *L* is zero.

Define an operator *P*<sup>∗</sup> : *X* → *X* as

$$P^\*\mathbf{x}(t) = (I - \mathcal{T}^+\mathcal{T})\mathbf{x}(0). \tag{32}$$

If *v* ∈ Ker *L*, one has *v* = *c*<sup>∗</sup> <sup>1</sup>, *c*<sup>∗</sup> <sup>1</sup> <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* <sup>∩</sup> Ker(<sup>T</sup> ) = Im (*<sup>I</sup>* − T <sup>+</sup><sup>T</sup> ), thus there exists *<sup>d</sup>*<sup>∗</sup> <sup>1</sup> <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* suct that

$$c\_1^\* = (I - \mathcal{T}^+ \mathcal{T})d\_1^\*.$$

So, *v* ∈ Im *P*∗. Conversely, if *v* ∈ Im *P*∗, from *(I*2) we can know that *v* ∈ Ker T . Again, since Ker *P*<sup>∗</sup> ∩ Ker *L* = {*θ*}, then *X* = Ker *P*<sup>∗</sup> ⊕ Ker *L*.

Define a mapping *K*∗ *<sup>P</sup>* : Im *L* → Ker *P*<sup>∗</sup> ∩ dom *L* as

$$K\_P^\* y(s) = \mathcal{T}^+ H^\* y + I\_{0+}^a y(s) - \frac{s}{\tilde{\xi}} I\_{0+}^a y(\tilde{\xi}).\tag{33}$$

Through checking calculation, we can get *K*∗ *<sup>P</sup>y* ∈ dom *L* and *K*<sup>∗</sup> *<sup>P</sup>y* ∈ Ker *P*∗. Thus the definition of *K*∗ *<sup>P</sup>* is reasonable.

Letting *u* ∈ Ker *P*<sup>∗</sup> ∩ dom *L*, one has

$$\begin{split} K\_{\mathbb{P}}^{\*}\boldsymbol{L}\boldsymbol{u}(t) &= \boldsymbol{\mathcal{T}}^{+} \boldsymbol{H}^{\*c} \boldsymbol{D}\_{0+}^{\boldsymbol{a}} \boldsymbol{u} + \boldsymbol{I}\_{0+}^{\boldsymbol{a}} \boldsymbol{D}\_{0+}^{\boldsymbol{a}} \boldsymbol{u}(t) - \frac{\boldsymbol{t}}{\boldsymbol{\xi}} \boldsymbol{I}\_{0+}^{\boldsymbol{a}} \boldsymbol{C} \boldsymbol{D}\_{0+}^{\boldsymbol{a}} \boldsymbol{u}(\boldsymbol{\xi}) \\ &= -\boldsymbol{\mathcal{T}}^{+}(\boldsymbol{\eta}\boldsymbol{\mathsf{C}} - \boldsymbol{I}) \boldsymbol{u}^{'}(0) \boldsymbol{\xi} - \boldsymbol{\mathcal{T}}^{+} \boldsymbol{\mathsf{C}}(\boldsymbol{u}(0) + \boldsymbol{u}^{'}(0)\boldsymbol{\eta}) + \boldsymbol{\mathcal{T}}^{+}(\boldsymbol{u}(0) + \boldsymbol{u}^{'}(0)) + \boldsymbol{u}(t) - \boldsymbol{u}(0) \\ &= -\boldsymbol{\mathcal{T}}^{+} \boldsymbol{\mathsf{C}} \boldsymbol{u}^{'}(0) \boldsymbol{\eta} + \boldsymbol{\mathcal{T}}^{+} \boldsymbol{u}^{'}(0) + \boldsymbol{\mathcal{T}}^{+}((\boldsymbol{\eta}\boldsymbol{\mathsf{C}} - \boldsymbol{I})) \boldsymbol{u}^{'}(0) - (\boldsymbol{I} - \boldsymbol{\mathcal{T}}^{+} \boldsymbol{\mathcal{T}}) \boldsymbol{u}(0) + \boldsymbol{u}(t) \\ &= \boldsymbol{u}(t). \end{split}$$

Similarly, for *y* ∈ Im *L*, we have *LK*<sup>∗</sup> *<sup>P</sup>y* = *y*. Then we can deduce that *K*<sup>∗</sup> *<sup>P</sup>* = (*L*|dom *<sup>L</sup>*∩Ker *<sup>P</sup>*)−1. Denote

$$D^\* = \mathfrak{Z} + \|\mathcal{T}^+\|\_\* ( (\eta + 1) \|\mathbb{C}\|\_\* + \mathfrak{Z} ). \tag{34}$$

By the similar proof process as in Lemma 4 and Lemma 5, we know that *K*∗ *Py* - *D*∗ *y* 1, and *K*∗ *<sup>P</sup>*(*I* − *Q*)*N* is completely continuous.

Now we give the following assumptions:

(M1∗) For all *<sup>s</sup>* <sup>∈</sup> [0, 1], *<sup>u</sup>*, *<sup>v</sup>* <sup>∈</sup> <sup>R</sup>*n*, we have

$$|f(s,\mu,v)| \ll a|\mu| + b|v| + c,\tag{35}$$

where *a*, *b*, *c* ∈ *Y* are three positive functions satisfying ( *<sup>I</sup>* −TT <sup>+</sup> <sup>∗</sup> + *D*∗)( *a* <sup>1</sup> + *b* <sup>1</sup>) < 1, and *D*<sup>∗</sup> is the constant given in (34). (M2∗) For all *u* ∈ dom *L*, if

$$H^\* f(s, u(t), \, ^c D\_{0+}^{a-1} u(t)) \in \text{Im}(\mathcal{T}).\tag{36}$$

Then, there exist *σ*∗ <sup>1</sup> ∈ *R*<sup>+</sup> and *s*<sup>0</sup> ∈ [0, 1], s.t. |*u*(*s*0)| *σ*∗ 1 . (M3∗) There exist *σ*∗ <sup>2</sup> <sup>∈</sup> *<sup>R</sup>*+, s.t. for every *<sup>ν</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* with *<sup>ν</sup>* <sup>=</sup> *<sup>C</sup><sup>ν</sup>* and <sup>|</sup>*ν*<sup>|</sup> <sup>&</sup>gt; *<sup>σ</sup>*<sup>∗</sup> <sup>2</sup> , either

$$<\langle \upsilon, Q^\* N(\upsilon) \rangle \leqslant 0 \; or \; \langle \upsilon, Q^\* N(\upsilon) \rangle \geqslant 0,\tag{37}$$

where ·, · stands for scalar product in <sup>R</sup>*n*.

**Theorem 3.** *If assumptions (M*1∗*)–(M*3∗*) are satisfied, BVP* (1) *and* (30) *has at least one solution in X .*

**Proof.** We use the same definitions of Ω1, Ω2, and Ω<sup>3</sup> as in Theorem 2. For *x* ∈ Ω1, we have that *Nx* ∈ Im *L* = Ker *Q*∗. Similarly, we can show

$$H^\* f(s, \mu(t), {}^c D\_{0+}^{\alpha-1} \mu(t)) \in \text{Im}(\mathcal{T}).$$

In fact,

$$\begin{aligned} \mathcal{H}^\* f(\mathbf{s}, \boldsymbol{u}, \mathbf{c}^c D\_{0+}^{\underline{a}-1} \boldsymbol{u}) &= \boldsymbol{H}^{\*c} D\_{0+}^{\underline{a}} \boldsymbol{u} \\ &= (\eta \mathsf{C} - \mathsf{I}) \boldsymbol{u}'(0) + \mathsf{C} \boldsymbol{u}(\eta) + \mathsf{u}(0) - \mathsf{C} \boldsymbol{u}(0) + (\mathsf{I} - \mathsf{C}\eta) \boldsymbol{u}'(0) - \mathsf{u}(1) \\ &= \mathcal{T} \boldsymbol{u}(0) \in \mathrm{Im}(\mathcal{T}). \end{aligned}$$

Using assumption (M2∗), we can deduce that

$$|u(0)| = \left| u(t\_0) - \int\_0^{t\_0} c^\gamma D\_{0+}^{\alpha-1} u(s) ds \right| \leqslant \sigma\_1^\* + ||^c D\_{0+}^{\alpha-1} u||\_{\infty \prime}$$

and

$$|\,^cD\_{0+}^{\alpha-1}u(t)| \leqslant \int\_0^t |\,^cD\_{0+}^{\alpha}u(s)|ds \leqslant ||Lu||\_1.$$

Then with the similar proof process in Theorem 2 we can know that

$$\|\|u(t)\|\| \leqslant \left(\|I - \mathcal{T}^+ \mathcal{T}\|\|\_\ast + D^\*\right) \|\|Nu\|\|\_1 + \sigma\_1^\* \|\|I - \mathcal{T}^+ \mathcal{T}\|\|\_\ast \tag{38}$$

and

$$||Nu||\_1 \leqslant \frac{\sigma\_1^\* \left( ||a||\_1 + ||b||\_1 \right) ||I - \mathcal{T}^+ \mathcal{T}||\_\* + ||\mathcal{c}||\_1}{- (||I - \mathcal{T}^+ \mathcal{T}||\_\* + D^\*) (||a||\_1 + ||b||\_1) + 1}. \tag{39}$$

Combining (38) and (39) we can deduce that

$$\sup\_{\mu \in \Omega\_1} ||u|| = \sup\_{\mu \in \Omega\_1} \max \{ ||u||\_{\infty}, ||^c D\_{0+}^{\alpha - 1} u||\_{\infty} \} < +\infty.$$

Hence Ω<sup>1</sup> is a bounded subset of *X*.

For *u* ∈ Ω2, one has *u* = *c*<sup>∗</sup> <sup>1</sup>, *c*<sup>∗</sup> <sup>1</sup> <sup>∈</sup> <sup>R</sup>*n*. Combining with *Nu* <sup>∈</sup> Im *<sup>L</sup>*, we can get

$$H^\*Nu \in \text{Im}(\mathcal{T}).$$

From assumptions (M2∗), we get

$$||u|| = \max\{||u||\_{\infty}, ||{\cal cD}\_{0+}^{\alpha-1}u||\_{\infty}\} = ||\mathcal{c}||\_{\infty} = |u(t\_0)| \leqslant \sigma\_1^\* < +\infty.$$

Such that Ω<sup>2</sup> is bounded in *X*.

In order to prove both Ω− <sup>3</sup> and <sup>Ω</sup><sup>+</sup> <sup>3</sup> are bounded, we also divide the proof process into two steps:

**Step 1** : Assuming *u* ∈ Ω<sup>−</sup> , one has *u* = *c*<sup>∗</sup> , where *c*<sup>∗</sup> <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* <sup>∩</sup> Ker(<sup>T</sup> ). Thus we have

$$-\lambda c\_1^\* + (1 - \lambda) Q N(c\_1^\*) = \theta.$$

**Case 1** : If *λ* = 0, then *QN*(*c*∗ <sup>1</sup> ) = *θ*, such that *N*(*c*<sup>∗</sup> <sup>1</sup> ) ∈ Ker *Q* = Im *L*. Thus we have *Nx* ∈ Ω2, so *x σ*∗ 1 .

**Case 2** : If *λ* ∈ (0, 1], suppose *u* > *σ*2. From (B3) we get

$$0 < \lambda |c\_1^\*|^2 = (1 - \lambda) \langle c\_1^\*, QNc\_1^\* \rangle \leqslant 0.$$

Therefore, we have *u σ*∗ 2 .

**Step 2** : For *<sup>u</sup>* <sup>∈</sup> <sup>Ω</sup><sup>+</sup> <sup>3</sup> , through a similar proof process as in Step 1, we can deduce that *u σ*∗ 2 .

Thus, Ω− <sup>3</sup> and <sup>Ω</sup><sup>+</sup> <sup>3</sup> are two bounded subsets in *X*.

Let the definitions of bounded open subset Ω and homotopy *H*(*u*, *λ*) be the same as in Theorem 2. Then we can deduce that (*iii*) of Theorem 1 is also satisfied. By Theorem 1, Equations (1) and (30) must have a solution in dom *L* ∩ Ω.

#### **5. Examples**

In this section, we present two examples to illustrate our main results in Sections 3 and 4.

**Example 1.** Consider the following boundary value problem:

$$\begin{cases} ^\mathbb{C}D\_{0+}^{a}\mathbf{x}(t) = f\_1(t, \mathbf{x}(t), \mathbf{y}(t), \, ^\mathbb{C}D\_{0+}^{a-1}\mathbf{x}(t), \, ^\mathbb{C}D\_{0+}^{a-1}\mathbf{y}(t)), \quad t \in (0, 1), \\ ^\mathbb{C}D\_{0+}^{a}\mathbf{y}(t) = f\_2(t, \mathbf{x}(t), \mathbf{y}(t), \, ^\mathbb{C}D\_{0+}^{a-1}\mathbf{x}(t), \, ^\mathbb{C}D\_{0+}^{a-1}\mathbf{y}(t)), \quad t \in (0, 1), \\ \mathbf{x}(0) = \mathbf{5}\mathbf{x}(\frac{1}{4}), \, \mathbf{y}(0) = 0, \\ \mathbf{x}(1) = \frac{1}{2}\mathbf{x}(\frac{3}{4}), \, \mathbf{y}(1) = \frac{4}{3}\mathbf{y}(\frac{3}{4}), \end{cases} \tag{40}$$

*where α* = <sup>3</sup> <sup>2</sup> , *fi* : [0, 1] <sup>×</sup> <sup>R</sup><sup>4</sup> <sup>→</sup> <sup>R</sup>*, i* <sup>=</sup> 1, 2, *are defined as*

$$f\_1(t, \mathbf{x}\_1, \mathbf{x}\_2, y\_1, y\_2) = -\frac{\mathbf{x}\_1 + y\_1}{40},\tag{41}$$

$$f\_2(t, x\_1, x\_2, y\_1, y\_2) = \frac{|x\_2| + |y\_2| + 1}{60},\tag{42}$$

*for all t* ∈ [0, 1]*.*

Clearly, *ξ* = <sup>1</sup> <sup>4</sup> , *<sup>η</sup>* <sup>=</sup> <sup>3</sup> 4 ,

$$B = \begin{bmatrix} 5 & 0 \\ 0 & 0 \end{bmatrix}, \; \mathbb{C} = \begin{bmatrix} \frac{1}{2} & 0 \\ 0 & \frac{4}{3} \end{bmatrix}, \; (I - \mathbb{C})(I - B)^{-1}B = \begin{bmatrix} -\frac{5}{8} & 0 \\ 0 & 0 \end{bmatrix}, \; \mathbb{C}\_0 = \begin{bmatrix} -\frac{5}{16} & 0 \\ 0 & 0 \end{bmatrix},$$

and *<sup>I</sup>* <sup>−</sup> *<sup>η</sup><sup>C</sup>* <sup>+</sup> *<sup>ξ</sup>*(*<sup>I</sup>* <sup>−</sup> *<sup>C</sup>*)(*<sup>I</sup>* <sup>−</sup> *<sup>B</sup>*)−1*<sup>B</sup>* <sup>=</sup> <sup>Θ</sup>. Denote *<sup>u</sup>*<sup>1</sup> = (*x*1, *<sup>x</sup>*2), *<sup>u</sup>*<sup>2</sup> = (*y*1, *<sup>y</sup>*2) <sup>∈</sup> <sup>R</sup>2, define function *<sup>f</sup>* : [0, 1] <sup>×</sup> <sup>R</sup><sup>2</sup> <sup>×</sup> <sup>R</sup><sup>2</sup> <sup>→</sup> <sup>R</sup><sup>2</sup>

$$f(t, \mu\_1, \mu\_2) = (f\_1(t, \mu\_1, \mu\_2), f\_2(t, \mu\_1, \mu\_2)) \mid \,\,\forall t \in [0, 1].$$

By (41), (42), and (43), *f* satisfies Carathéodory conditions.

Now we show that the other conditions of Theorem 3 hold. Choose positive integrable functions

$$a(t) = b(t) = c(t) = \frac{1}{40}.$$

Then we have

$$|f(t,u,v)| \lesssim a(t)|u| + b(t)|v| + c(t)\_{\prime}$$

By some simple computation, we get

$$(1 + ||\mathbb{C}\_0||\_\* + D)(||a||\_1 + ||b||\_1) = \frac{25}{384} < 1.5$$

Hence, (M1) is satisfied.

In order to check (M2), one has

$$f\_2(t, u(t)) \prime D\_{0+}^{\alpha - 1} u(t) \rangle > \frac{1}{60} \prime$$

for all *<sup>u</sup>* <sup>∈</sup> *<sup>C</sup>*1([0, 1]; <sup>R</sup>2) and all *<sup>t</sup>* <sup>∈</sup> [0, 1]. Letting *<sup>f</sup>*2(*t*, *<sup>u</sup>*(*t*), *<sup>C</sup> Dα*−<sup>1</sup> <sup>0</sup><sup>+</sup> *u*(*t*)) = *f*<sup>2</sup> be a positive constant, we have

$$\begin{aligned} &C\int\_0^\eta (\eta - s)^{a-1} f(t, u(t), ^\mathbb{C}D\_{0+}^{a-1} u(t)) dt \\ &- I \int\_0^1 (1 - s)^{a-1} f(t, u(t), ^\mathbb{C}D\_{0+}^{a-1} u(t)) dt = \begin{bmatrix} \frac{1}{2} f\_1^\* + f\_2^\* \\\ \frac{260}{2911} f\_2^\* \end{bmatrix} \end{aligned}$$

where *f* ∗ <sup>1</sup> = *<sup>I</sup><sup>α</sup>* <sup>0</sup><sup>+</sup> *f*1(*η*), *f* <sup>∗</sup> <sup>2</sup> = *<sup>I</sup><sup>α</sup>* <sup>0</sup><sup>+</sup> *<sup>f</sup>*1(1). If *<sup>f</sup>*<sup>2</sup> = <sup>1</sup> <sup>60</sup> , there is

$$\begin{aligned} \mathbb{C} \int\_0^\eta (\eta - s)^{a-1} f(t, u(t) \prescript{\mathbb{C}}{}{D\_{0+}^{a-1}} u(t)) dt \\ - \prescript{}{I}{\int\_0^1 (1 - s)^{a-1} f(t, u(t) \prescript{\mathbb{C}}{}{D\_{0+}^{a-1}} u(t)) dt} &= \begin{bmatrix} \frac{1}{2} f\_1^\* + f\_2^\* \\ \frac{13}{8733} \end{bmatrix} .\end{aligned}$$

This shows that when *f*2(*t*, *u*(*t*), *<sup>C</sup> Dα*−<sup>1</sup> <sup>0</sup><sup>+</sup> *<sup>u</sup>*(*t*)) <sup>&</sup>gt; *<sup>f</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>60</sup> , one has

$$\begin{aligned} \mathbb{C} \int\_0^\eta (\eta - s)^{a-1} f(t, u(t) \prime \, ^\mathbb{C} D\_{0+}^{a-1} u(t)) dt \\ - I \int\_0^1 (1 - s)^{a-1} f(t, u(t) \prime \, ^\mathbb{C} D\_{0+}^{a-1} u(t)) dt \notin \text{Im}((I - \mathbb{C})(I - B)^{-1} B) \end{aligned}$$

because Im((*<sup>I</sup>* <sup>−</sup> *<sup>C</sup>*)(*<sup>I</sup>* <sup>−</sup> *<sup>B</sup>*)−1*B*) = {(*p*, 0) : *<sup>p</sup>* <sup>∈</sup> <sup>R</sup>}.

Finally, we check (M3). Let *q*(*t*)=(*tI* + *C*0). Denote *τ* = (*τ*1, *τ*2). So

$$q(t)\tau = ((t + \frac{5}{16})\tau\_1, t\tau\_2)^\top$$

$$^C D\_{0+}^{\alpha-1} q(t) = (\frac{2\tau\_1\sqrt{t}}{\sqrt{\pi}}, \frac{2\tau\_2\sqrt{t}}{\sqrt{\pi}})^\top.$$

Then there is

$$Nq(t)\tau = \left(-\frac{(t+\frac{5}{16})\tau\_1}{40} - \frac{2\tau\_1\sqrt{t}}{40\sqrt{\pi}}, \frac{t|\tau\_2| + \frac{2\sqrt{t}}{\sqrt{\pi}}|\tau\_2| + 1}{60}\right)^\top.$$

So

$$QN(q(t)\tau) = \alpha \begin{bmatrix} -\frac{31}{5234}\tau\_1\\ \frac{14}{34946}|\tau\_2| + \frac{25}{181937} \end{bmatrix} / $$

and

$$
\langle q(t)\tau, \mathbb{Q}N(q(t)\tau)\rangle = \alpha(-\frac{31}{5234}\tau\_1^2 + \frac{15}{34946}|\tau\_2|\tau\_2 + \frac{25}{181937}\tau\_2) \leqslant 0.1
$$

In fact, if *τ*<sup>2</sup> - 0, this is obviously true. If *<sup>τ</sup>*<sup>2</sup> <sup>&</sup>gt; 0, letting <sup>|</sup>*τ*2| ≥ 1, one has *<sup>τ</sup>*<sup>2</sup> <sup>2</sup> > *τ*2. Again, since

$$\frac{15}{34946} > \frac{25}{181937}.$$

So, the formula above has no real root, which means <sup>−</sup> <sup>31</sup> <sup>5234</sup> *<sup>τ</sup>*<sup>2</sup> <sup>1</sup> + <sup>15</sup> <sup>34946</sup> <sup>|</sup>*τ*2|*τ*<sup>2</sup> <sup>+</sup> <sup>25</sup> <sup>181937</sup> *τ*<sup>2</sup> < 0. Thus, by Theorem 2, BVP (40) has at least one solution.

**Example 2.** Consider the following boundary value problem:

$$\begin{cases} ^\mathbb{C}D\_{0+}^{a}\mathbf{x}(t) = f\_{1}(t, \mathbf{x}(t), \mathbf{y}(t), \, ^\mathbb{C}D\_{0+}^{a-1}\mathbf{x}(t), \, ^\mathbb{C}D\_{0+}^{a-1}\mathbf{y}(t)), \quad t \in (0, 1), \\ ^\mathbb{C}D\_{0+}^{a}\mathbf{y}(t) = f\_{2}(t, \mathbf{x}(t), \mathbf{y}(t), \, ^\mathbb{C}D\_{0+}^{a-1}\mathbf{x}(t), \, ^\mathbb{C}D\_{0+}^{a-1}\mathbf{y}(t)), \quad t \in (0, 1), \\ \mathbf{x}(0) = \mathbf{x}(\frac{1}{4}), \, \mathbf{y}(0) = \mathbf{y}(\frac{1}{4}), \\ \mathbf{x}(1) = \mathbf{y}(\frac{3}{4}), \, \mathbf{y}(1) = \mathbf{y}(\frac{3}{4}). \end{cases} \tag{43}$$

We use the same *<sup>α</sup>*, *<sup>f</sup>* , *<sup>ξ</sup>*, *<sup>η</sup>*, *<sup>a</sup>*(*t*), *<sup>b</sup>*(*t*), and *<sup>c</sup>*(*t*) as in Example 1 and *fi* : [0, 1] <sup>×</sup> <sup>R</sup><sup>4</sup> <sup>→</sup> R, *i* = 1, 2 are defined as

$$f\_1(t, x\_1, x\_2, y\_1, y\_2) = \frac{x\_2 + y\_2}{40}.$$

$$f\_2(t, x\_1, x\_2, y\_1, y\_2) = \begin{cases} \frac{\sqrt{y\_1^2 + y\_2^2}}{40}, & \text{if } |u\_2| > 1;\\ f\_1(t, x\_1, x\_2, y\_1, y\_2), & \text{otherwise.} \end{cases}$$

$$C = \begin{bmatrix} 0 & 1\\ 0 & 1 \end{bmatrix}, \mathcal{T} = \begin{bmatrix} 1 & -1\\ 0 & 0 \end{bmatrix}, \mathcal{T}^+ = \begin{bmatrix} \frac{1}{2} & 0\\ -\frac{1}{2} & 0 \end{bmatrix}.$$

We can easily check that assumption (M1∗) is satisfied. When |*u*2| > 1, from the definition of *f* , one has *y*<sup>2</sup> <sup>1</sup> + *<sup>y</sup>*<sup>2</sup> <sup>2</sup> > 1 and *<sup>H</sup>*<sup>∗</sup> *<sup>f</sup>*<sup>2</sup> > *<sup>H</sup>*<sup>∗</sup> <sup>1</sup> <sup>40</sup> <sup>=</sup> <sup>51</sup> <sup>13571</sup> > 0. According to a similar proof process as Example 1, one has

$$H^\*f(t, u(t), \prescript{C}{}{D}\_{0+}^{a-1}u(t)) = \begin{bmatrix} H^\*f\_1 \\ H^\*f\_2 \end{bmatrix} \notin \text{Im}(\mathcal{T}),$$

because Im(<sup>T</sup> ) = {(*p*, 0) : *<sup>p</sup>* <sup>∈</sup> <sup>R</sup>}. Finally, we check (M3∗). Letting *<sup>τ</sup>* = (*τ*0, *<sup>τ</sup>*0) <sup>∈</sup> Ker(T ), one has

$$N\tau = \begin{pmatrix} f\_1(t,\tau,\theta), f\_2(t,\tau,\theta) \end{pmatrix}^\top = \begin{cases} \left(\frac{\tau\_0}{40}, 0\right)^\top, & \text{if } |u\_2| > 1; \\\left(\frac{\tau\_0}{40}, \frac{\tau\_0}{40}\right)^\top, & \text{otherwise.} \end{cases}$$

So

$$=\gamma^\* \begin{cases} \begin{bmatrix} \frac{\pi\_0}{120} \\ 0 \end{bmatrix}, & \text{if } |u\_2| > 1; \\\\ \begin{bmatrix} (7\pi\_0)/480 - (3^{1/2}\pi\_0)/160 \\ (7\pi\_0)/480 - (3^{1/2}\pi\_0)/160 \end{bmatrix}, & \text{otherwise} \end{cases}$$

and

$$\langle \tau, QN\tau \rangle = \begin{cases} \frac{1}{\text{T} \text{20}} \tau\_0^2 c\_\prime & \text{if} |u\_2| > 1; \\ \frac{7 - 3 \ast 3^{1/2}}{480} \tau\_0^2 c\_\prime & \text{otherwise}. \end{cases}$$

where *c* = *ξα*Γ(*α*) *ηξα*−*ξα*+*ξ*−*ξη<sup>α</sup>* <sup>=</sup> <sup>5302</sup>*π*1/2 <sup>1594</sup> > 0. Thus, *τ*, *QNτ* > 0, by Theorem 3, (43) has at least one solution.

#### **6. Conclusions**

This paper mainly studied a class of second-order nonlocal boundary value problem systems at resonance which state variable *<sup>x</sup>* <sup>∈</sup> <sup>R</sup>*n*, and gave two new theorems on the existence of solutions in different kernel spaces by using the Mawhin coincidence degree theorem.

In the future, we could consider studying resonance boundary value problems under less-restricted conditions or under more complicated boundary value conditions.

**Author Contributions:** Conceptualization, Y.F. and Z.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by NSFC grant number 11571207 and Shandong Provincial Natural Science Foundation number ZR2021MA064 and the Taishan Scholar project.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **The Mixed Boundary Value Problems and Chebyshev Collocation Method for Caputo-Type Fractional Ordinary Differential Equations**

**Jun-Sheng Duan <sup>1</sup> , Li-Xia Jing <sup>1</sup> and Ming Li 2,3,\***


**\*** Correspondence: mli@ee.ecnu.edu.cn or mli15@zju.edu.cn

**Abstract:** The boundary value problem (BVP) for the varying coefficient linear Caputo-type fractional differential equation subject to the mixed boundary conditions on the interval 0 ≤ *x* ≤ 1 was considered. First, the BVP was converted into an equivalent differential–integral equation merging the boundary conditions. Then, the shifted Chebyshev polynomials and the collocation method were used to solve the differential–integral equation. Varying coefficients were also decomposed into the truncated shifted Chebyshev series such that calculations of integrals were only for polynomials and can be carried out exactly. Finally, numerical examples were examined and effectiveness of the proposed method was verified.

**Keywords:** fractional calculus; fractional differential equation; boundary value problem; Chebyshev polynomial; collocation method

#### **1. Introduction**

In recent decades, the theory of fractional calculus has been attracting much attention partly due to its ability for describing memory and hereditary properties of various materials and processes [1–7]. Fractional calculus has been applied to different fields such as viscoelastic constitutive equations and related mechanical models [6–11], anomalous diffusion phenomena [4,6,12,13], hydrology [14], control and optimization theory [3,15], etc. It is worthwhile to mention that fractional calculus can be used to describe not only viscoelasticity, but also viscoinertia by different values of order [16,17]. The applications of fractional calculus lead to fractional differential equations (FDEs) in theory [2–5,18].

Let us recall some related definitions of fractional calculus used in this article. Let *f*(*x*) be piecewise continuous on (0, +∞) and integrable on any finite subinterval of (0, +∞). Then, for *x* > 0, the Riemann–Liouville fractional integral of *f*(*x*) is defined as

$$I\_x^{\beta}f(\mathbf{x}) = \int\_0^{\chi} \frac{(\mathbf{x} - \tau)^{\beta - 1}}{\Gamma(\beta)} f(\tau) d\tau,\tag{1}$$

for *β* > 0, and *I*<sup>0</sup> *<sup>x</sup> f*(*x*) = *f*(*x*) for *β* = 0, where Γ(·) is the gamma function. The fractional integral satisfies the following equalities:

$$I\_\mathbf{x}^\beta I\_\mathbf{x}^\upsilon f(\mathbf{x}) = I\_\mathbf{x}^{\beta+\upsilon} f(\mathbf{x}), \ \beta \ge 0, \ \upsilon \ge 0,\tag{2}$$

$$I\_x^{\nu} x^{\mu} = \frac{\Gamma(\mu+1)}{\Gamma(\mu+\nu+1)} x^{\mu+\nu}, \; \nu \ge 0, \; \mu > -1. \tag{3}$$

**Citation:** Duan, J.-S.; Jing, L.-X.; Li, M. The Mixed Boundary Value Problems and Chebyshev Collocation Method for Caputo-Type Fractional Ordinary Differential Equations. *Fractal Fract.* **2022**, *6*, 148. https:// doi.org/10.3390/fractalfract6030148

Academic Editor: Rodica Luca

Received: 11 February 2022 Accepted: 7 March 2022 Published: 9 March 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Let *<sup>α</sup>* be a positive real number, *<sup>m</sup>* <sup>−</sup> <sup>1</sup> <sup>&</sup>lt; *<sup>α</sup>* <sup>≤</sup> *<sup>m</sup>*, *<sup>m</sup>* <sup>∈</sup> <sup>N</sup>+, and *<sup>f</sup>* (*m*)(*x*) be piecewise continuous on (0, +∞) and integrable on any finite subinterval of (0, +∞). Then, the Caputo fractional derivative of *f*(*x*) of order *α* is defined as

$$D\_{\mathbf{x}}^{\mathfrak{a}}f(\mathbf{x}) = I\_{\mathbf{x}}^{m-\mathfrak{a}}f^{(m)}(\mathbf{x}), \; m-1 < \mathfrak{a} \le m. \tag{4}$$

For the power function *<sup>x</sup>μ*, *<sup>μ</sup>* <sup>&</sup>gt; 0, if 0 <sup>≤</sup> *<sup>m</sup>* <sup>−</sup> <sup>1</sup> <sup>&</sup>lt; *<sup>α</sup>* <sup>≤</sup> *<sup>m</sup>* <sup>&</sup>lt; *<sup>μ</sup>* <sup>+</sup> 1, then we have

$$D\_x^{\alpha} x^{\mu} = \frac{\Gamma(\mu + 1)}{\Gamma(\mu - \alpha + 1)} x^{\mu - \alpha}, \ge 0. \tag{5}$$

The *α*-order integral of the *α*-order Caputo fractional derivative requires the knowledge of the initial values of the function and its integer-ordered derivatives,

$$I\_x^a D\_x^a f(\mathbf{x}) = f(\mathbf{x}) - \sum\_{k=0}^{m-1} f^{(k)}(0) \frac{\mathbf{x}^k}{k!}, \ m - 1 < a \le m. \tag{6}$$

This property enables the Caputo fractional derivative to be conveniently applied and analyzed.

In the earlier monograph [1], the Grünwald definition and the Riemann–Liouville definition of fractional calculus were introduced, where numerical differentiation and integration were considered and semi-integration was introduced by a designed electrical circuit model and semi-differentiation was applied to diffusion problems. The Weyl fractional calculus was introduced in [2] beside the Grünwald definition and the Riemann–Liouville definition. In [3], FDEs and fractional-order system and controllers were considered, where the Caputo fractional derivative was introduced. The existence, uniqueness and analytical methods of solutions for FDEs were investigated in [4]. In [18], the Caputo-type fractional derivative and FDEs were emphasized. In [6], fractional viscoelastic models and fractional wave models in viscoelastic media were introduced. In [5], numerical methods and fractional variational principle were reviewed.

Damping, deformation, vibration and dissipation arising from viscoelastic material can be modeled by FDEs [3,4,6,7]. The method of variable separation for fractional partial differential equation describing anomalous diffusion [4,6,12,14] can lead to a boundary value problem (BVP) for a fractional ordinary differential equation (ODE) [19]. The theorem of existence and uniqueness of solutions for fractional ODEs was presented in [3,4,18,20]. Some analytical and numerical methods were proposed to solve FDEs, e.g., see [3–5,21–25]. BVPs for fractional ODEs were considered in [19,26–29] by using the Adomian decomposition method, wavelet method, the method of upper and lower solutions, orthogonal polynomial method, etc. However, a fractional BVP with varying coefficients and mixed boundary conditions has hardly been considered.

In this work, we consider the BVP for the varying coefficient linear Caputo fractional ODE

$$D\_x^{\lambda}u(\mathbf{x}) + c\_1(\mathbf{x})u'(\mathbf{x}) + c\_0(\mathbf{x})u(\mathbf{x}) = g(\mathbf{x}),\ 0 < \mathbf{x} < 1,\ 1 < \lambda \le 2,\tag{7}$$

Subject to the mixed boundary conditions

$$p\_0\mu(0) - q\_0\mu'(0) = b\_{0\prime} \tag{8}$$

$$p\_1 u(1) + q\_1 u'(1) = b\_1 \tag{9}$$

where the coefficients *c*1(*x*), *c*0(*x*), *g*(*x*) are specified continuous functions, the boundary parameters satisfy *p*0, *q*0, *p*1, *q*<sup>1</sup> ≥ 0 and *p*<sup>0</sup> *p*<sup>1</sup> + *p*0*q*<sup>1</sup> + *q*<sup>0</sup> *p*<sup>1</sup> = 0. In the next Section 2, some preliminaries about the shifted Chebyshev polynomials are presented. In Section 3, we first convert the BVP, (7)–(9), into an equivalent fractional differential–integral equation merging the boundary conditions, then introduce the collocation method using the shifted Chebyshev polynomials of the first kind to solve the fractional differential–integral equation. Next, three numerical examples are solved by using the proposed method. Section 4 summarizes our conclusions.

#### **2. The Shifted Chebyshev Polynomials of the First Kind**

The Chebyshev polynomials of the first kind are defined by the formulae [30]

$$T\_n(\mathbf{x}) = \cos(n \arccos \mathbf{x}), \ -1 \le \mathbf{x} \le 1, \ n = 0, 1, \ldots \tag{10}$$

They take on the explicit expressions as

$$T\_0(\mathbf{x}) = 1, \ T\_n(\mathbf{x}) = \frac{n}{2} \sum\_{k=0}^{\left[n/2\right]} (-1)^k \frac{(n-k-1)!}{k!(n-2k)!} (2\mathbf{x})^{n-2k}, \ n \ge 1. \tag{11}$$

It is well-known that the Chebyshev polynomials of the first kind are orthogonal on the interval [−1, 1] with the weight function *ρ*(*x*) = <sup>√</sup> 1 <sup>1</sup>−*x*<sup>2</sup> , and *Tn*(*x*) has exactly *<sup>n</sup>* zeros within the interval (−1, 1): *<sup>ξ</sup><sup>i</sup>* <sup>=</sup> cos <sup>2</sup>*i*+<sup>1</sup> <sup>2</sup>*<sup>n</sup> π* , *i* = 0, 1, ... , *n* − 1. The Chebyshev polynomials of the first kind satisfy the recurrence relation

$$T\_0(\mathbf{x}) = 1, \ T\_1(\mathbf{x}) = \mathbf{x}, \ T\_n(\mathbf{x}) = 2\mathbf{x}T\_{n-1}(\mathbf{x}) - T\_{n-2}(\mathbf{x}), \ n = 2, 3, \dots \tag{12}$$

It is well-known that if *f*(*x*) is *L*<sup>2</sup> integrable on [−1, 1] with the weight function *ρ*(*x*), then its Chebyshev series expansion is *L*<sup>2</sup> convergent with respect to its weight function *ρ*(*x*). If *f*(*x*) has better smoothness, then stronger convergence can be attained for its Chebyshev series. If the function *f*(*x*) has *n* + 1 continuous derivatives on [−1, 1], then <sup>|</sup> *<sup>f</sup>*(*x*) <sup>−</sup> *Sm <sup>f</sup>*(*x*)<sup>|</sup> <sup>=</sup> *<sup>O</sup>*(*m*−*n*) for all *<sup>x</sup>* <sup>∈</sup> [−1, 1], where *Sm <sup>f</sup>*(*x*) is the (*<sup>m</sup>* <sup>+</sup> <sup>1</sup>)-term truncation of the Chebyshev series expansion of *f*(*x*). For more details for convergence, see [30].

In order to deal with the BVP on the interval [0, 1], we consider the shifted Chebyshev polynomials

$$T\_n^\*(\mathbf{x}) = T\_n(2\mathbf{x} - \mathbf{1}), \; \mathbf{x} \in [0, 1], \; n = 0, 1, \dots \tag{13}$$

They are orthogonal on the interval [0, 1] with the weight function *ρ*∗(*x*) = √ 1 *<sup>x</sup>*−*x*<sup>2</sup> , and the zeros of *T*∗ *<sup>n</sup>* (*x*) are *xi* = <sup>1</sup> <sup>2</sup> <sup>+</sup> <sup>1</sup> <sup>2</sup> cos <sup>2</sup>*i*+<sup>1</sup> <sup>2</sup>*<sup>n</sup> π* , *i* = 0, 1, ... , *n* − 1. As a complement to Equation (13), the shifted Chebyshev polynomials satisfy the relationship *T*∗ *<sup>n</sup>* (*x*) = *T*2*n*( <sup>√</sup>*x*). So, the explicit expressions of the shifted Chebyshev polynomials are conveniently obtained:

$$T\_0^\*(\mathbf{x}) = 1, \ T\_n^\*(\mathbf{x}) = n \sum\_{k=0}^n (-1)^k \frac{(2n - k - 1)!}{k!(2n - 2k)!} (4\mathbf{x})^{n-k}, \ n \ge 1. \tag{14}$$

Finally, we mention the shifted Chebyshev polynomials of the second kind, which will also be used in the next section for the representation of solutions, *U*∗ *<sup>n</sup>*(*x*) = *Un*(2*x* − 1), 0 ≤ *x* ≤ 1, *n* = 0, 1, . . . , where *Un*(*x*) is the Chebyshev polynomials of the second kind.

#### **3. The Equivalent Fractional Differential-Integral Equation and Chebyshev Collocation Method**

First, we derive an equivalent differential–integral equation to the BVP (7)–(9). Applying the integral operator *I<sup>λ</sup> <sup>x</sup>* (·) to both sides of Equation (7) and using Equation (6) yields

$$u(\mathbf{x}) - u(0) - u'(0)\mathbf{x} + I\_\mathbf{x}^\lambda(\varepsilon\_1(\mathbf{x})u'(\mathbf{x}) + \varepsilon\_0(\mathbf{x})u(\mathbf{x})) = I\_\mathbf{x}^\lambda g(\mathbf{x}).\tag{15}$$

Our aim is to solve for *u*(0) and *u* (0) from the boundary conditions (8) and (9), and then obtain an equation about the solution *u*(*x*) without any undetermined constants. Substituting *x* = 1 in Equation (15) yields

$$u(1) = u(0) + u'(0) - I\_{\mathbf{x},1}^{\lambda} (c\_1(\mathbf{x}) u'(\mathbf{x}) + c\_0(\mathbf{x}) u(\mathbf{x})) + I\_{\mathbf{x},1}^{\lambda} g(\mathbf{x}),\tag{16}$$

where the value of the fractional integral is defined for a general *β*th order integral of a function *v*(*x*) at *x* = *ξ* as

$$I\_{\chi\_{\mathfrak{F}}^{\mathfrak{F}}}^{\mathfrak{F}}v(\mathfrak{x}) = \int\_{0}^{\mathfrak{z}} \frac{(\mathfrak{f} - \mathfrak{r})^{\mathfrak{f} - 1}}{\Gamma(\mathfrak{z})} v(\mathfrak{r}) d\mathfrak{r}.\tag{17}$$

Calculating the first order derivative on the both sides of Equation (15) leads to

$$u'(\mathbf{x}) - u'(0) + I\_\mathbf{x}^{\lambda - 1}(c\_1(\mathbf{x})u'(\mathbf{x}) + c\_0(\mathbf{x})u(\mathbf{x})) = I\_\mathbf{x}^{\lambda - 1}\mathbf{g}(\mathbf{x}).\tag{18}$$

Substituting *x* = 1 yields

$$u'(1) = u'(0) - I\_{\mathbf{x},1}^{\lambda - 1} (c\_1(\mathbf{x})u'(\mathbf{x}) + c\_0(\mathbf{x})u(\mathbf{x})) + I\_{\mathbf{x},1}^{\lambda - 1} \mathbf{g}(\mathbf{x}).\tag{19}$$

Substituting Equations (16) and (19) into Equation (9) yields

$$p\_1 u(0) + (p\_1 + q\_1)u'(0) = b\_1^\* \tag{20}$$

where

$$b\_1^\* = b\_1 + p\_1 I\_{\mathbf{x},1}^{\lambda} (c\_1(\mathbf{x}) u'(\mathbf{x}) + c\_0(\mathbf{x}) u(\mathbf{x})) - p\_1 I\_{\mathbf{x},1}^{\lambda} \mathbf{g}(\mathbf{x}) + q\_1 I\_{\mathbf{x},1}^{\lambda - 1} (c\_1(\mathbf{x}) u'(\mathbf{x}) + c\_0(\mathbf{x}) u(\mathbf{x})) - q\_1 I\_{\mathbf{x},1}^{\lambda - 1} \mathbf{g}(\mathbf{x}).\tag{21}$$

Equations (8) and (20) constitute a system of algebraic equations about *u*(0) and *u* (0). The coefficient determinant is

$$P = p\_0 p\_1 + p\_0 q\_1 + q\_0 p\_1 \,\tag{22}$$

which is positive by our assumptions. Thus, we can solve the system of algebraic Equations (8) and (20) and obtain

$$
\mu(0) \quad = \quad \frac{(p\_1 + q\_1)b\_0}{P} + \frac{q\_0 b\_1^\*}{P} \, , \tag{23}
$$

$$
\mu'(0) = \quad -\frac{p\_1 b\_0}{P} + \frac{p\_0 b\_1^\*}{P}.\tag{24}
$$

Substituting Equations (23) and (24) into Equation (15), we obtain

$$u(\mathbf{x}) - \frac{(p\_1 + q\_1)b\_0}{P} + \frac{p\_1 b\_0}{P}\mathbf{x} - \frac{p\_0 \mathbf{x} + q\_0}{P} b\_1^\* + I\_\mathbf{x}^\lambda (c\_1(\mathbf{x})u'(\mathbf{x}) + c\_0(\mathbf{x})u(\mathbf{x})) = I\_\mathbf{x}^\lambda g(\mathbf{x}). \tag{25}$$

Replacing *b*∗ <sup>1</sup> by using Equation (21) and reorganizing the equation yield

$$\begin{aligned} u(\mathbf{x}) - \frac{p\_1(p\_0 \mathbf{x} + q\_0)}{P} I\_{\mathbf{x},1}^{\lambda}(c\_1(\mathbf{x})u'(\mathbf{x}) + c\_0(\mathbf{x})u(\mathbf{x})) \\ - \frac{q\_1(p\_0 \mathbf{x} + q\_0)}{P} I\_{\mathbf{x},1}^{\lambda - 1}(c\_1(\mathbf{x})u'(\mathbf{x}) + c\_0(\mathbf{x})u(\mathbf{x})) + I\_x^{\lambda}(c\_1(\mathbf{x})u'(\mathbf{x}) + c\_0(\mathbf{x})u(\mathbf{x})) = h(\mathbf{x}), \end{aligned} \tag{26}$$

where

$$h(\mathbf{x}) = \frac{(p\_1 + q\_1)b\_0 - p\_1 b\_0 \mathbf{x}}{P} + \frac{p\_0 \mathbf{x} + q\_0}{P} \left( b\_1 - p\_1 I\_{\mathbf{x},1}^{\lambda} \mathbf{g}(\mathbf{x}) - q\_1 I\_{\mathbf{x},1}^{\lambda - 1} \mathbf{g}(\mathbf{x}) \right) + I\_{\mathbf{x}}^{\lambda} \mathbf{g}(\mathbf{x}), \quad \text{(27)}$$

Only involves the known boundary parameters and the known input function *g*(*x*). Equation (26) is the equivalent differential–integral equation to the BVP (7)–(9). In the sequel, we seek for the solution to the differential-integral Equation (26).

We approximate the solution by an (*m* +1)-term truncation of the shifted Chebyshev series,

$$\varphi\_m(\mathbf{x}) = \sum\_{n=0}^{m} a\_n T\_n^\*(\mathbf{x}) \,\tag{28}$$

where *an*, *n* = 0, 1, ... , *m*, are undetermined coefficients. Inserting *ϕm*(*x*) into Equation (26), we obtain the linear equation about *an*, *n* = 0, 1, . . . , *m*,

$$\begin{split} \sum\_{n=0}^{m} a\_{n} \left( T\_{n}^{\*} (\mathbf{x}) - \frac{p\_{1} (p\_{0} \mathbf{x} + q\_{0})}{P} I\_{\mathbf{x},1}^{\lambda} \left( c\_{1}(\mathbf{x}) T\_{n}^{\*\prime} (\mathbf{x}) + c\_{0}(\mathbf{x}) T\_{n}^{\*} (\mathbf{x}) \right) \\ - \frac{q\_{1} (p\_{0} \mathbf{x} + q\_{0})}{P} I\_{\mathbf{x},1}^{\lambda - 1} \left( c\_{1}(\mathbf{x}) T\_{n}^{\*\prime} (\mathbf{x}) + c\_{0}(\mathbf{x}) T\_{n}^{\*} (\mathbf{x}) \right) + I\_{\mathbf{x}}^{\lambda} \left( c\_{1}(\mathbf{x}) T\_{n}^{\*\prime} (\mathbf{x}) + c\_{0}(\mathbf{x}) T\_{n}^{\*} (\mathbf{x}) \right) \right) = h(\mathbf{x}). \end{split} \tag{29}$$

We note that in Equation (29), *I<sup>λ</sup> x*,1 *c*1(*x*)*T*<sup>∗</sup> *n* (*x*) + *c*0(*x*)*T*<sup>∗</sup> *<sup>n</sup>* (*x*)  and *I <sup>λ</sup>*−<sup>1</sup> *<sup>x</sup>*,1 *c*1(*x*)*T*<sup>∗</sup> *n* (*x*) + *c*0(*x*)*T*<sup>∗</sup> *<sup>n</sup>* (*x*) are constants, represent the values of fractional integrals.

The collocation method may be applied to determine the coefficients *an*. The collocation points are taken as the zeroes of the *m* +1 degree shifted Chebyshev polynomial *T*∗ *<sup>m</sup>*+1(*x*),

$$x\_i = \frac{1}{2} + \frac{1}{2}\cos\left(\frac{2i+1}{2m+2}\pi\right), \ i = 0, 1, \ldots, m. \tag{30}$$

Thus, the collocation equation system is

$$\sum\_{n=0}^{m} a\_n \left( T\_n^\*(\mathbf{x}\_i) - \frac{p\_1(p\_0 \mathbf{x}\_i + q\_0)}{P} I\_{\mathbf{x},1}^\lambda \left( c\_1(\mathbf{x}) T\_n^{\*\prime}(\mathbf{x}) + c\_0(\mathbf{x}) T\_n^\*(\mathbf{x}) \right) \right)$$

$$\begin{split} -\frac{q\_1(p\_0 \mathbf{x}\_i + q\_0)}{P} I\_{\mathbf{x},1}^{\lambda - 1} \left( c\_1(\mathbf{x}) T\_n^{\*\prime}(\mathbf{x}) + c\_0(\mathbf{x}) T\_n^\*(\mathbf{x}) \right) \\ + I\_{\mathbf{x}, \mathbf{x}\_i}^\lambda \left( c\_1(\mathbf{x}) T\_n^{\*\prime}(\mathbf{x}) + c\_0(\mathbf{x}) T\_n^\*(\mathbf{x}) \right) \end{split} \tag{31}$$

where

$$h(\mathbf{x}\_i) = \frac{(p\_1 + q\_1)b\_0 - p\_1 b\_0 \mathbf{x}\_i}{P} + \frac{p\_0 \mathbf{x}\_i + q\_0}{P} \left( b\_1 - p\_1 I\_{x,1}^{\lambda} \mathbf{g}(\mathbf{x}) - q\_1 I\_{x,1}^{\lambda - 1} \mathbf{g}(\mathbf{x}) \right) + I\_{x, \mathbf{x}\_i}^{\lambda} \mathbf{g}(\mathbf{x}), \ i = 0, 1, \ldots, m. \tag{32}$$

The matrix form of the collocation equation system (31) is

$$\mathcal{W}\vec{a} = \vec{h},\tag{33}$$

where

$$\vec{a} = (a\_0, a\_1, \dots, a\_m)^T,\\ \vec{h} = (h(\mathbf{x}\_0), h(\mathbf{x}\_1), \dots, h(\mathbf{x}\_m))^T,\tag{34}$$

and the entries of the matrix *W* are

$$\begin{array}{ll} w\_{lj} &=& T\_{\uparrow}^{\*}(\mathbf{x}\_{i}) - \frac{p\_{1}(p\_{0}\mathbf{x}\_{i} + q\_{0})}{P} I\_{\mathbf{x},1}^{\lambda} \left( c\_{1}(\mathbf{x})T\_{\uparrow}^{\*\prime}(\mathbf{x}) + c\_{0}(\mathbf{x})T\_{\uparrow}^{\*\ast}(\mathbf{x}) \right) \\ &- \frac{q\_{1}(p\_{0}\mathbf{x}\_{i} + q\_{0})}{P} I\_{\mathbf{x},1}^{\lambda-1} \left( c\_{1}(\mathbf{x})T\_{\uparrow}^{\*\prime}(\mathbf{x}) + c\_{0}(\mathbf{x})T\_{\uparrow}^{\*\ast}(\mathbf{x}) \right) + I\_{\mathbf{x},2}^{\lambda} \left( c\_{1}(\mathbf{x})T\_{\downarrow}^{\*\prime}(\mathbf{x}) + c\_{0}(\mathbf{x})T\_{\uparrow}^{\*\ast}(\mathbf{x}) \right), \\ &i, j = 0, 1, \ldots, m. \end{array} \tag{35}$$

The solution of the linear algebraic equation system (31) or (33) gives the coefficients *an* in Equation (28).

For the Dirichlet boundary conditions *u*(0) = *b*0, *u*(1) = *b*1, the boundary parameters are simplified as *p*<sup>0</sup> = *p*<sup>1</sup> = 1 and *q*<sup>0</sup> = *q*<sup>1</sup> = 0, and thus Equation (31) degenerates to

$$\sum\_{\mathbf{n}=0}^{\mathbf{m}} a\_{\mathbf{n}} \left( T\_{\mathbf{n}}^{\ast}(\mathbf{x}\_{\mathbf{i}}) - \mathbf{x}\_{\mathbf{i}} I\_{\mathbf{x},1}^{\lambda} \left( \mathbf{c}\_{1}(\mathbf{x}) T\_{\mathbf{n}}^{\ast \prime}(\mathbf{x}) + \mathbf{c}\_{0}(\mathbf{x}) T\_{\mathbf{n}}^{\ast}(\mathbf{x}) \right) + I\_{\mathbf{x},\mathbf{i}\_{\mathbf{i}}}^{\lambda} \left( \mathbf{c}\_{1}(\mathbf{x}) T\_{\mathbf{n}}^{\ast \prime}(\mathbf{x}) + \mathbf{c}\_{0}(\mathbf{x}) T\_{\mathbf{n}}^{\ast}(\mathbf{x}) \right) \right) = h(\mathbf{x}\_{\mathbf{i}}),\tag{36}$$

where *<sup>h</sup>*(*xi*) = *<sup>b</sup>*<sup>0</sup> + (*b*<sup>1</sup> <sup>−</sup> *<sup>b</sup>*0)*xi* <sup>−</sup> *xiI<sup>λ</sup> <sup>x</sup>*,1*g*(*x*) + *<sup>I</sup><sup>λ</sup> x*,*xi g*(*x*), *i* = 0, 1, . . . , *m*.

We remark that by the relationship of the first-kind and second-kind Chebyshev polynomials *Tn* (*x*) = *nUn*−1(*x*), we have the relationship of the shifted Chebyshev polynomials of the two kinds

$$T\_{n}^{\*\prime}(\mathbf{x}) = \mathfrak{D}n\mathcal{U}\_{n-1}^{\*}(\mathbf{x}).\tag{37}$$

So, the derivative *T*∗ *n* (*x*) in Equations (31), (35) and (36) may be replaced by the second-kind Chebyshev polynomials.

The operators *I<sup>λ</sup> <sup>x</sup>*,1(·), *I <sup>λ</sup>*−<sup>1</sup> *<sup>x</sup>*,1 (·) and *<sup>I</sup><sup>λ</sup> x*,*xi* (·) in Equations (31), (32) and (35) represent the values of fractional integrals of the known functions. Since the appearance of the varying coefficients *g*(*x*) and *ck*(*x*), manual computations for these integrals are laborious in general. Here we approximate the varying coefficients again using their truncated shifted Chebyshev series as

$$\text{g}(\mathbf{x}) = \sum\_{n=0}^{M} \text{g}\_{n} T\_{n}^{\*}(\mathbf{x}), \ c\_{k}(\mathbf{x}) = \sum\_{n=0}^{M} c\_{k,n} T\_{n}^{\*}(\mathbf{x}), \ k = 0, 1, \ 0 \le \mathbf{x} \le 1,\tag{38}$$

where

$$\mathbf{g}\_n = \frac{2}{\pi} \int\_0^1 \frac{1}{\sqrt{\mathbf{x} - \mathbf{x}^2}} \mathbf{g}(\mathbf{x}) T\_n^\*(\mathbf{x}) d\mathbf{x}, \ n = 0, 1, \dots, M,\tag{39}$$

$$c\_{k,n} = \frac{2}{\pi} \int\_0^1 \frac{1}{\sqrt{\mathbf{x} - \mathbf{x}^2}} c\_k(\mathbf{x}) T\_n^\*(\mathbf{x}) d\mathbf{x}, \ k = 0, 1, \ n = 0, 1, \dots, M. \tag{40}$$

and the superscript of ∑ denotes that the first term in the sum is halved. We note that there is no need of connections between the values of *m* and *M* in Equations (28) and (38). Utilizing the Gauss–Chebyshev quadrature formula we derive the numerical formulae for *gn* and *ck*,*<sup>n</sup>* as

$$g\_{\boldsymbol{\pi}} = \frac{2}{M+1} \sum\_{i=0}^{M} g(\boldsymbol{\pi}\_i) T\_n^\*(\boldsymbol{\pi}\_i), \; n = 0, 1, \dots, M,\tag{41}$$

$$c\_{k,n} = \frac{2}{M+1} \sum\_{i=0}^{M} c\_k(\mathbf{x}\_i) T\_n^\*(\mathbf{x}\_i), \; k = 0, 1, \; n = 0, 1, \dots, M,\tag{42}$$

where *xi* are the zeroes of the *M* + 1 degree shifted Chebyshev polynomial *T*<sup>∗</sup> *<sup>M</sup>*+1(*x*),

$$x\_i = \frac{1}{2} + \frac{1}{2}\cos\left(\frac{2i+1}{2M+2}\pi\right), \ i = 0, 1, \ldots, M. \tag{43}$$

Thus, making use of the decompositions in (38), the calculation of the integrals *I<sup>λ</sup> <sup>x</sup>*,1(·), *I <sup>λ</sup>*−<sup>1</sup> *<sup>x</sup>*,1 (·) and *<sup>I</sup><sup>λ</sup> x*,*xi* (·) in Equations (31), (32) and (35) only involves integrals of polynomials, so can be carried out exactly.

In the following three examples, we take *M* = 5 in Equation (38) to truncate the decompositions of the coefficients *g*(*x*) and *ck*(*x*) and to calculate the involved integrals *Iλ <sup>x</sup>*,1(·), *I <sup>λ</sup>*−<sup>1</sup> *<sup>x</sup>*,1 (·) and *<sup>I</sup><sup>λ</sup> x*,*xi* (·). Collocation equation systems are solved by using Mathematica command "LinearSolve". Figures of approximate analytical solutions and errors are generated by using Mathematica.

**Example 1.** *Consider the BVP for the linear FDE*

$$D\_x^{1.5}u(\mathbf{x}) - \frac{\mathbf{x}\sin(\mathbf{x})}{\mathfrak{Z}}u'(\mathbf{x}) + \sin(\mathbf{x})u(\mathbf{x}) = \mathbf{g}(\mathbf{x}),\ 0 < \mathbf{x} < 1,\tag{44}$$

$$
u'(0) = -1, \; \mathfrak{u}(1) = 1,\tag{45}$$

*where g*(*x*) = <sup>3</sup> √*<sup>π</sup>* <sup>4</sup> <sup>+</sup> <sup>8</sup>*x*1.5 <sup>√</sup>*<sup>π</sup>* <sup>−</sup> <sup>2</sup> <sup>3</sup> *<sup>x</sup>* sin(*x*) + <sup>1</sup> <sup>2</sup> *<sup>x</sup>*1.5 sin(*x*).

The BVP has the exact solution *<sup>u</sup>*∗(*x*) = <sup>−</sup>*<sup>x</sup>* <sup>+</sup> *<sup>x</sup>*1.5 <sup>+</sup> *<sup>x</sup>*3. The boundary parameters are *p*<sup>0</sup> = 0, *q*<sup>0</sup> = −1, *b*<sup>0</sup> = −1, *p*<sup>1</sup> = 1, *q*<sup>1</sup> = 0, *b*<sup>1</sup> = 1. The collocation equation system in (31) is

$$\sum\_{n=0}^{m} a\_n \left( T\_n^\*(\mathbf{x}\_i) - I\_{\mathbf{x},1}^{1.5} \left( \frac{-\mathbf{x}\sin(\mathbf{x})}{3} T\_n^{\*\prime}(\mathbf{x}) + \sin(\mathbf{x}) T\_n^\*(\mathbf{x}) \right) \right)$$

$$+ I\_{\mathbf{x}, \mathbf{x}\_i}^{1.5} \left( \frac{-\mathbf{x}\sin(\mathbf{x})}{3} T\_n^{\*\prime}(\mathbf{x}) + \sin(\mathbf{x}) T\_n^\*(\mathbf{x}) \right) \Big) = h(\mathbf{x}\_i), \tag{46}$$

where *<sup>h</sup>*(*xi*) = <sup>2</sup> <sup>−</sup> *xi* <sup>−</sup> *<sup>I</sup>*1.5 *<sup>x</sup>*,1 *<sup>g</sup>*(*x*) + *<sup>I</sup>*1.5 *x*,*xi g*(*x*), *i* = 0, 1, . . . , *m*. Take *m* = 2, 3, 4 and 5, respectively, the solution approximations *ϕm*(*x*) are calculated as

$$\begin{aligned} \varphi\_2(\mathbf{x}) &= -0.0175774 - 1.10039\mathbf{x} + 2.05832\mathbf{x}^2/\mathbf{z} \\ \varphi\_3(\mathbf{x}) &= -0.00586106 - 0.689478\mathbf{x} + 0.925577\mathbf{x}^2 + 0.768798\mathbf{x}^3/\mathbf{z} \\ \varphi\_4(\mathbf{x}) &= -0.00288415 - 0.76024\mathbf{x} + 1.24242\mathbf{x}^2 + 0.293418\mathbf{x}^3 + 0.22758\mathbf{x}^4/\mathbf{z} \\ \varphi\_5(\mathbf{x}) &= -0.00167028 - 0.8038\mathbf{x} + 1.54387\mathbf{x}^2 - 0.47843\mathbf{x}^3 + 1.05648\mathbf{x}^4 - 0.316564\mathbf{x}^5. \end{aligned}$$

The error function and maximum error of the approximate solution *ϕm*(*x*) are defined as

$$ER\_{\mathfrak{m}}(\mathbf{x}) = |\varphi\_{\mathfrak{m}}(\mathbf{x}) - u^\*(\mathbf{x})| \text{ and } \; ME\_{\mathfrak{m}} = \max\_{0 \le \mathbf{x} \le 1} ER\_{\mathfrak{m}}(\mathbf{x}). \tag{47}$$

In Figure 1, the error functions *ERm*(*x*) for *m* = 2, 3, 4, 5 are depicted, where at the *m* + 1 collocation points of *ϕm*(*x*), errors are zero. The maximum errors of the four approximate solutions are 0.028696, 0.005861, 0.002884, and 0.001670, respectively.

**Figure 1.** The error functions *ERm*(*x*) for *m* = 2 (solid line), *m* = 3 (dot line), *m* = 4 (dash line) and *m* = 5 (dot-dash line).

**Example 2.** *Consider the BVP for the linear FDE*

$$D\_{\mathbf{x}}^{\lambda}u(\mathbf{x}) - u(\mathbf{x}) = -4\mathbf{x}e^{\mathbf{x}},\ 0 < \mathbf{x} < 1,\ 1 < \lambda \le 2,\tag{48}$$

$$
\mu(0) - \mu'(0) = -1,\\
\mu(1) + \mu'(1) = -e.\tag{49}
$$

If *<sup>λ</sup>* <sup>=</sup> 2, the BVP has the exact solution *<sup>u</sup>*∗(*x*) = *<sup>x</sup>*(<sup>1</sup> <sup>−</sup> *<sup>x</sup>*)*ex*.

For this example, the coefficients and parameters are *c*1(*x*) = 0, *c*0(*x*) = −1, *g*(*x*) = <sup>−</sup>4*xex*, *<sup>p</sup>*<sup>0</sup> <sup>=</sup> *<sup>q</sup>*<sup>0</sup> <sup>=</sup> *<sup>p</sup>*<sup>1</sup> <sup>=</sup> *<sup>q</sup>*<sup>1</sup> <sup>=</sup> 1, *<sup>b</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>1 and *<sup>b</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>*e*. The collocation equation system in Equation (31) becomes

$$\begin{aligned} &\sum\_{n=0}^{m} a\_n \left( T\_n^\*(\mathbf{x}\_i) - \frac{\boldsymbol{\chi}\_i + 1}{3} I\_{\mathbf{x},1}^\lambda (-T\_n^\*(\boldsymbol{\chi})) \right) \\ &- \frac{\boldsymbol{\chi}\_i + 1}{3} I\_{\mathbf{x},1}^{\lambda - 1} (-T\_n^\*(\boldsymbol{\chi})) + I\_{\mathbf{x}, \mathbf{x}\_i}^\lambda (-T\_n^\*(\boldsymbol{\chi})) \right) = h(\mathbf{x}\_i)\_i \end{aligned}$$

where *<sup>h</sup>*(*xi*) = *xi*−2−*exi*−*<sup>e</sup>* <sup>3</sup> <sup>+</sup> *xi*+<sup>1</sup> 3 <sup>−</sup>*I<sup>λ</sup> <sup>x</sup>*,1*g*(*x*) − *I <sup>λ</sup>*−<sup>1</sup> *<sup>x</sup>*,1 *<sup>g</sup>*(*x*) + *I<sup>λ</sup> x*,*xi g*(*x*), *i* = 0, 1, . . . , *m*.

For the case of *λ* = 2, the error functions *ERm*(*x*) = |*ϕm*(*x*) − *u*∗(*x*)| are depicted in Figure 2 for *m* = 2–5. The maximum errors of the approximate solutions are 0.069103, 0.007877, 0.000620, and 0.000038, respectively. For the case of *λ* = 1.5, the solution approximations *ϕm*(*x*), *m* = 2–5, are calculated as

*<sup>ϕ</sup>*2(*x*) = 0.578503 <sup>+</sup> 3.00467*<sup>x</sup>* <sup>−</sup> 2.45143*x*2, *<sup>ϕ</sup>*3(*x*) = 0.659109 <sup>+</sup> 1.56065*<sup>x</sup>* <sup>+</sup> 1.43569*x*<sup>2</sup> <sup>−</sup> 2.61289*x*3, *<sup>ϕ</sup>*4(*x*) = 0.651626 <sup>+</sup> 1.81823*<sup>x</sup>* <sup>+</sup> 0.108085*x*<sup>2</sup> <sup>−</sup> 0.448696*x*<sup>3</sup> <sup>−</sup> 1.09621*x*4, *<sup>ϕ</sup>*5(*x*) = 0.653112 <sup>+</sup> 1.75369*<sup>x</sup>* <sup>+</sup> 0.599869*x*<sup>2</sup> <sup>−</sup> 1.78975*x*<sup>3</sup> <sup>+</sup> 0.411041*x*<sup>4</sup> <sup>−</sup> 0.596018*x*5.

The condition numbers of the coefficient matrices *W* in the derivations of the four solution approximations are 2.85, 3.29, 3.67 and 4.02, respectively. These values show that the coefficient matrices *W* are well conditioned. We note that the condition number is based on the *l*2-matrix norm. The four solution approximations are plotted in Figure 3, where a fast convergence is shown.

**Figure 2.** For *λ* = 2, the error functions *ERm*(*x*) for *m* = 2 (solid line), *m* = 3 (dot line), *m* = 4 (dash line) and *m* = 5 (dot-dash line).

**Figure 3.** For *λ* = 1.5, the solution approximations *ϕm*(*x*) for *m* = 2 (solid line), *m* = 3 (dot line), *m* = 4 (dash line) and *m* = 5 (dot-dash line).

**Example 3.** *Consider the BVP for the linear FDE*

$$D\_{\mathbf{x}}^{\lambda}u(\mathbf{x}) - \frac{\mathbf{x}}{1+\mathbf{x}}u'(\mathbf{x}) - \frac{1}{1+\mathbf{x}}u(\mathbf{x}) = 0, \ 0 < \mathbf{x} < 1, \ 1 < \lambda \le 2,\tag{50}$$

$$
u(0) - 2\boldsymbol{u}'(0) = -1,\\
\boldsymbol{u}(1) + 2\boldsymbol{u}'(1) = 3\boldsymbol{e}.\tag{51}$$

If *λ* = 2, the BVP has the exact solution *u*∗(*x*) = *ex*.

The coefficients and parameters are *<sup>c</sup>*1(*x*) = <sup>−</sup> *<sup>x</sup>* <sup>1</sup>+*<sup>x</sup>* , *<sup>c</sup>*0(*x*) = <sup>−</sup> <sup>1</sup> <sup>1</sup>+*<sup>x</sup>* , *g*(*x*) = 0, *p*<sup>0</sup> = *p*<sup>1</sup> = 1, *q*<sup>0</sup> = *q*<sup>1</sup> = 2, *b*<sup>0</sup> = −1, *b*<sup>1</sup> = 3*e*. The collocation equation system in Equation (31) becomes

$$\begin{split} &\sum\_{n=0}^{m} a\_{n} \left( T\_{n}^{\*} (\mathbf{x}\_{i}) - \frac{\mathbf{x}\_{i} + 2}{5} I\_{\mathbf{x},1}^{\lambda} \left( \frac{-\mathbf{x}}{1+\mathbf{x}} T\_{n}^{\*\prime} (\mathbf{x}) + \frac{-1}{1+\mathbf{x}} T\_{n}^{\*} (\mathbf{x}) \right) \\ &- \frac{2(\mathbf{x}\_{i} + 2)}{5} I\_{\mathbf{x},1}^{\lambda-1} \left( \frac{-\mathbf{x}}{1+\mathbf{x}} T\_{n}^{\*\prime} (\mathbf{x}) + \frac{-1}{1+\mathbf{x}} T\_{n}^{\*} (\mathbf{x}) \right) + I\_{\mathbf{x},\mathbf{x}\_{i}}^{\lambda} \left( \frac{-\mathbf{x}}{1+\mathbf{x}} T\_{n}^{\*\prime} (\mathbf{x}) + \frac{-1}{1+\mathbf{x}} T\_{n}^{\*} (\mathbf{x}) \right) \end{split}$$

where *h*(*xi*) = <sup>1</sup> <sup>5</sup> (6*e* − 3 + 3*exi* + *xi*), *i* = 0, 1, . . . , *m*.

For the case of *λ* = 2, the error functions *ERm*(*x*) = |*ϕm*(*x*) − *u*∗(*x*)| for *m* = 2, 3, 4, 5, are depicted in Figure 4. The maximum errors of the approximate solutions are 0.011605, 0.000742, 0.000037, and 0.000002, respectively. For the case of *λ* = 1.5, the solution approximations *ϕm*(*x*) for *m* = 2, 3, 4, 5 are calculated as

$$\begin{array}{rcl}\varphi\_2(\mathbf{x}) &=& 0.627196 + 0.801952\mathbf{x} + 0.961626\mathbf{x}^2/2 \end{array}$$


The condition numbers of the coefficient matrices *W* in the derivations of the four solution approximations are 4.87, 7.83, 11.90 and 17.05, respectively. So the coefficient matrices *W* are well conditioned. The four solution approximations are plotted in Figure 5.

In the three examples, fast convergent rates are shown only using the minor term number with *M* = 5 in Equation (38) for the integral computation of the known functions, and the minor term number with *m* = 2, 3, 4 and 5 in Equation (28) for the truncated Chebyshev series of the unknown function.

**Figure 4.** For *λ* = 2, the error functions *ERm*(*x*) for *m* = 2 (solid line), *m* = 3 (dot line), *m* = 4 (dash line) and *m* = 5 (dot-dash line).

**Figure 5.** For *λ* = 1.5, the solution approximations *ϕm*(*x*) for *m* = 2 (solid line), *m* = 3 (dot line), *m* = 4 (dash line) and *m* = 5 (dot-dash line).

#### **4. Conclusions**

We considered the BVP for the varying coefficient linear Caputo-type fractional ODE subject to the mixed boundary conditions on the interval 0 ≤ *x* ≤ 1. The BVP was conveniently converted into an equivalent differential–integral equation merging the boundary conditions. Then, the solution was decomposed into a truncated shifted Chebyshev series. The collocation method was used to determine the solution. In order to deal with the involved integrations, the varying coefficients were again decomposed into the truncated shifted Chebyshev series. Thus, the calculations of the integrals are only for polynomials and can be carried out exactly. Three numerical examples were solved by using the proposed method, where fast convergent rates are shown only using the minor term number with *M* = 5 in Equation (38) for the integral computation of the known functions, and the minor term number with *m* = 2, 3, 4 and 5 in Equation (28) for the truncated Chebyshev series of the unknown function.

In the presented method, there is no need to divide the interval commonly used in numerical methods. The collocation points or the zeros of the Chebyshev polynomials have exact explicit expressions. Approximate analytical solutions in the polynomial forms are obtained, which are different from a discrete numerical solution. The obtained approximate analytical solutions in the polynomial forms can be directly checked by substitution. The convergence and effectiveness of solutions can be examined by remainder errors. Convergence order of the approximate solutions could be further consideration in this field.

**Author Contributions:** Conceptualization, J.-S.D. and M.L.; data curation, L.-X.J. and M.L.; formal analysis, J.-S.D., L.-X.J. and M.L.; funding acquisition, J.-S.D. and M.L.; investigation, J.-S.D. and L.-X.J.; methodology, J.-S.D. and M.L.; software, J.-S.D. and L.-X.J.; supervision, J.-S.D.; validation, M.L.; visualization, L.-X.J.; writing–original draft, J.-S.D. and L.-X.J.; writing–review and editing, J.-S.D. and M.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Natural Science Foundation of China (Nos. 11772203; 61672238).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** Jun-Sheng Duan acknowledges the supports in part by the National Natural Science Foundation of China under the project grant number 11772203. Ming Li acknowledges the supports in part by the National Natural Science Foundation of China under the project grant number 61672238. The authors show their appreciation for the valuable comments from the reviewers on the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Positive Solutions of a Singular Fractional Boundary Value Problem with** *r***-Laplacian Operators**

**Alexandru Tudorache <sup>1</sup> and Rodica Luca 2,\***


**Abstract:** We investigate the existence and multiplicity of positive solutions for a system of Riemann– Liouville fractional differential equations with *r*-Laplacian operators and nonnegative singular nonlinearities depending on fractional integrals, supplemented with nonlocal uncoupled boundary conditions which contain Riemann–Stieltjes integrals and various fractional derivatives. In the proof of our main results we apply the Guo–Krasnosel'skii fixed point theorem of cone expansion and compression of norm type.

**Keywords:** Riemann–Liouville fractional differential equations; nonlocal boundary conditions; singular functions; positive solutions; multiplicity

**MSC:** 34A08; 34B10; 34B16; 34B18

#### **1. Introduction**

We consider the system of fractional differential equations with *r*1-Laplacian and *r*2-Laplacian operators

$$\begin{cases} D\_{0+}^{\gamma\_1} \left( \varphi\_{\mathbb{P}\_1} \big( D\_{0+}^{\delta\_1} u(\tau) \big) \right) = f \big( \tau, u(\tau), v(\tau), I\_{0+}^{\sigma\_1} u(\tau), I\_{0+}^{\sigma\_2} v(\tau) \big), \ \tau \in (0, 1), \\\ D\_{0+}^{\gamma\_2} \left( \varphi\_{\mathbb{P}\_2} \big( D\_{0+}^{\delta\_2} v(\tau) \big) \right) = g \big( \tau, u(\tau), v(\tau), I\_{0+}^{\xi\_1} u(\tau), I\_{0+}^{\xi\_2} v(\tau) \big), \ \tau \in (0, 1), \end{cases} \tag{1}$$

subject to the uncoupled nonlocal boundary conditions

$$\begin{cases} \begin{aligned} \boldsymbol{u}^{(i)}(0) = 0, \; \boldsymbol{i} = 0, \; \boldsymbol{\dots}, \; p - 2, \; \boldsymbol{D}\_{0+}^{\delta\_1} \boldsymbol{u}(0) = 0, \\ \boldsymbol{\varrho}\_{\boldsymbol{r}\_1} (\boldsymbol{D}\_{0+}^{\delta\_1} \boldsymbol{u}(1)) = \int\_0^1 \boldsymbol{\varrho}\_{\boldsymbol{r}\_1} (\boldsymbol{D}\_{0+}^{\delta\_1} \boldsymbol{u}(\eta)) \, d\mathcal{H}\_0(\eta), \; \boldsymbol{D}\_{0+}^{\delta\_0} \boldsymbol{u}(1) = \sum\_{k=1}^n \int\_0^1 \boldsymbol{D}\_{0+}^{\delta\_k} \boldsymbol{u}(\eta) \, d\mathcal{H}\_k(\eta), \\ \boldsymbol{v}^{(i)}(0) = 0, \; \boldsymbol{i} = 0, \; \boldsymbol{i} = 0, \; \boldsymbol{r} - 2, \; \boldsymbol{D}\_{0+}^{\delta\_2} \boldsymbol{v}(0) = 0, \\ \boldsymbol{\varrho}\_{\boldsymbol{r}\_2} (\boldsymbol{D}\_{0+}^{\delta\_2} \boldsymbol{v}(1)) = \int\_0^1 \boldsymbol{\varrho}\_{\boldsymbol{r}\_2} (\boldsymbol{D}\_{0+}^{\delta\_2} \boldsymbol{v}(\eta)) \, d\mathcal{K}\_0(\eta), \; \boldsymbol{D}\_{0+}^{\delta\_0} \boldsymbol{v}(1) = \sum\_{k=1}^m \int\_0^1 \boldsymbol{D}\_{0+}^{\delta\_k} \boldsymbol{v}(\eta) \, d\mathcal{K}\_k(\eta), \end{aligned} \end{cases} \tag{2}$$

where *<sup>γ</sup>*1, *<sup>γ</sup>*<sup>2</sup> <sup>∈</sup> (1, 2], *<sup>δ</sup>*<sup>1</sup> <sup>∈</sup> (*<sup>p</sup>* <sup>−</sup> 1, *<sup>p</sup>*], *<sup>p</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>p</sup>* <sup>≥</sup> 3, *<sup>δ</sup>*<sup>2</sup> <sup>∈</sup> (*<sup>q</sup>* <sup>−</sup> 1, *<sup>q</sup>*], *<sup>q</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>q</sup>* <sup>≥</sup> 3, *<sup>n</sup>*, *<sup>m</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>σ</sup>*1, *<sup>σ</sup>*2, *<sup>ς</sup>*1, *<sup>ς</sup>*<sup>2</sup> <sup>&</sup>gt; 0, *<sup>α</sup><sup>k</sup>* <sup>∈</sup> <sup>R</sup>, *<sup>k</sup>* <sup>=</sup> 0, ... , *<sup>n</sup>*, 0 <sup>≤</sup> *<sup>α</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>α</sup>*<sup>2</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *<sup>α</sup><sup>n</sup>* <sup>≤</sup> *<sup>α</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>δ</sup>*<sup>1</sup> <sup>−</sup> 1, *<sup>α</sup>*<sup>0</sup> <sup>≥</sup> 1, *<sup>β</sup><sup>k</sup>* <sup>∈</sup> <sup>R</sup>, *<sup>k</sup>* <sup>=</sup> 0, ... , *<sup>m</sup>*, 0 <sup>≤</sup> *<sup>β</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>β</sup>*<sup>2</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *<sup>β</sup><sup>m</sup>* <sup>≤</sup> *<sup>β</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>δ</sup>*<sup>2</sup> <sup>−</sup> 1, *<sup>β</sup>*<sup>0</sup> <sup>≥</sup> 1, *<sup>ϕ</sup>ri* (*η*) = |*η*| *ri*−2*η*, *<sup>ϕ</sup>*−<sup>1</sup> *ri* <sup>=</sup> *ϕ<sup>i</sup>* , *<sup>i</sup>* = *ri ri*−<sup>1</sup> , *<sup>i</sup>* <sup>=</sup> 1, 2, *ri* <sup>&</sup>gt; 1, *<sup>i</sup>* <sup>=</sup> 1, 2, *<sup>f</sup>* , *<sup>g</sup>* : (0, 1) <sup>×</sup> <sup>R</sup><sup>4</sup> <sup>+</sup> <sup>→</sup> <sup>R</sup><sup>+</sup> are continuous functions, singular at *τ* = 0 and/or *τ* = 1, (R<sup>+</sup> = [0, ∞)), *I<sup>κ</sup>* <sup>0</sup><sup>+</sup> is the Riemann–Liouville fractional integral of order *κ* (for *κ* = *σ*1, *σ*2, *ς*1, *ς*2), *D<sup>κ</sup>* <sup>0</sup><sup>+</sup> is the Riemann-Liouville fractional derivative of order *κ* (for *κ* = *γ*1, *δ*1, *γ*2, *δ*2, *α*0, ... , *αn*, *β*0, ... , *βm*), and the integrals from the boundary conditions (2) are Riemann–Stieltjes integrals with <sup>H</sup>*<sup>i</sup>* : [0, 1] <sup>→</sup> <sup>R</sup>, *<sup>i</sup>* <sup>=</sup> 0, ... , *<sup>n</sup>* and <sup>K</sup>*<sup>i</sup>* : [0, 1] <sup>→</sup> <sup>R</sup>, *<sup>i</sup>* <sup>=</sup> 0, . . . , *<sup>m</sup>* functions of bounded variation.

**Citation:** Tudorache, A.; Luca, R. Positive Solutions of a Singular Fractional Boundary Value Problem with *r*-Laplacian Operators. *Fractal Fract.* **2022**, *6*, 18. https://doi.org/ 10.3390/fractalfract6010018

Academic Editor: Maria Rosaria Lancia

Received: 18 November 2021 Accepted: 29 December 2021 Published: 30 December 2021

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We give in this paper various conditions for the functions *f* and *g* such that problems (1) and (2) have at least one or two positive solutions. From a positive solution of (1) and (2) we understand a pair of functions (*u*, *<sup>v</sup>*) <sup>∈</sup> (*C*([0, 1], <sup>R</sup>+))<sup>2</sup> satisfying the system (1) and the boundary conditions (2), with *u*(*τ*) > 0 for all *τ* ∈ (0, 1] or *v*(*τ*) > 0 for all *τ* ∈ (0, 1]. In the proof of our main results we use the Guo–Krasnosel'skii fixed point theorem of cone expansion and compression of norm type. We now present some recent results which are connected with our problem. In [1], the authors studied the existence of multiple positive solutions for the system of nonlinear fractional differential equations with a *p*-Laplacian operator

$$\begin{cases} D\_{0+}^{\beta\_1}(\mathcal{q}\_{p\_1}(D\_{0+}^{\kappa\_1}\mathfrak{x}(\tau))) = f(\tau, \mathfrak{x}(\tau), y(\tau)), & \tau \in (0,1), \\ D\_{0+}^{\beta\_2}(\mathcal{q}\_{p\_2}(D\_{0+}^{\kappa\_2}y(\tau))) = g(\tau, \mathfrak{x}(\tau), y(\tau)), & \tau \in (0,1). \end{cases}$$

supplemented with the uncoupled boundary conditions

$$\begin{cases} \varkappa(0) = 0, \ D\_{0+}^{\gamma\_1} \varkappa(1) = \sum\_{i=1}^{m-2} \zeta\_{1i} D\_{0+}^{\gamma\_1} \varkappa(\eta\_{1i}), \\\ D\_{0+}^{a\_1} \varkappa(0) = 0, \ \wp\_{p\_1}(D\_{0+}^{a\_1} \varkappa(1)) = \sum\_{i=1}^{m-2} \zeta\_{1i} \wp\_{p\_1}(D\_{0+}^{a\_1} \varkappa(\eta\_{1i})), \\\ y(0) = 0, \ D\_{0+}^{\gamma\_2} y(1) = \sum\_{i=1}^{m} \zeta\_{2i} D\_{0+}^{\gamma\_2} y(\eta\_{2i}), \\\ D\_{0+}^{a\_2} y(0) = 0, \ \wp\_{p\_2}(D\_{0+}^{a\_2} y(1)) = \sum\_{i=1}^{m-2} \zeta\_{2i} \wp\_{p\_2}(D\_{0+}^{a\_2} y(\eta\_{2i})). \end{cases}$$

where *αi*, *β<sup>i</sup>* ∈ (1, 2], *γ<sup>i</sup>* ∈ (0, 1], *α<sup>i</sup>* + *β<sup>i</sup>* ∈ (3, 4], *α<sup>i</sup>* > *γ<sup>i</sup>* + 1, *i* = 1, 2, *ξ*1*i*, *η*1*i*, *ζ*1*i*, *ξ*2*i*, *η*2*i*, *ζ*2*<sup>i</sup>* ∈ (0, 1) for *i* = 1, ... , *m* − 2, and *f* and *g* are nonnegative and nonsingular functions. In the proof of the existence results they use the Leray–Schauder alternative theorem, the Leggett–Williams fixed point theorem and the Avery–Henderson fixed point theorem. In [2], the authors investigated the existence and multiplicity of positive solutions for the system of fractional differential equations with 1-Laplacian and 2-Laplacian operators

$$\begin{cases} D\_{0+}^{\gamma\_1} (\varrho\_{\ell\_1} (D\_{0+}^{\delta\_1} \mathfrak{x}(\tau))) + f(\tau, \mathfrak{x}(\tau), y(\tau)) = 0, & \tau \in (0, 1), \\\ D\_{0+}^{\gamma\_2} (\varrho\_{\ell\_2} (D\_{0+}^{\delta\_2} y(\tau))) + g(\tau, \mathfrak{x}(\tau), y(\tau)) = 0, & \tau \in (0, 1), \end{cases} \tag{3}$$

subject to the uncoupled nonlocal boundary conditions

$$\begin{cases} \mathbf{x}^{(j)}(0) = 0, \; j = 0, \ldots, p - 2; \; D\_{0+}^{\delta\_1} \mathbf{x}(0) = 0, \; D\_{0+}^{\kappa\_0} \mathbf{x}(1) = \sum\_{i=1}^{n} \int\_0^1 D\_{0+}^{\kappa\_i} \mathbf{x}(\tau) \, d\mathcal{H}\_i(\tau), \\\ y^{(j)}(0) = 0, \; j = 0, \ldots, q - 2; \; D\_{0+}^{\delta\_2} y(0) = 0, \; D\_{0+}^{\delta\_0} y(1) = \sum\_{i=1}^{m} \int\_0^1 D\_{0+}^{\delta\_i} y(\tau) \, d\mathcal{K}\_i(\tau), \end{cases}$$

where *<sup>γ</sup>*1, *<sup>γ</sup>*<sup>2</sup> <sup>∈</sup> (0, 1], *<sup>δ</sup>*<sup>1</sup> <sup>∈</sup> (*<sup>p</sup>* <sup>−</sup> 1, *<sup>p</sup>*], *<sup>δ</sup>*<sup>2</sup> <sup>∈</sup> (*<sup>q</sup>* <sup>−</sup> 1, *<sup>q</sup>*], *<sup>p</sup>*, *<sup>q</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>p</sup>*, *<sup>q</sup>* <sup>≥</sup> 3, *<sup>n</sup>*, *<sup>m</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>α</sup><sup>i</sup>* <sup>∈</sup> <sup>R</sup> for all *<sup>i</sup>* <sup>=</sup> 0, 1, ... , *<sup>n</sup>*, 0 <sup>≤</sup> *<sup>α</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>α</sup>*<sup>2</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *<sup>α</sup><sup>n</sup>* <sup>≤</sup> *<sup>α</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>δ</sup>*<sup>1</sup> <sup>−</sup> 1, *<sup>α</sup>*<sup>0</sup> <sup>≥</sup> 1, *<sup>β</sup><sup>i</sup>* <sup>∈</sup> <sup>R</sup> for all *i* = 0, 1, ... , *m*, 0 ≤ *β*<sup>1</sup> < *β*<sup>2</sup> < ··· < *β<sup>m</sup>* ≤ *β*<sup>0</sup> < *δ*<sup>2</sup> − 1, *β*<sup>0</sup> ≥ 1, 1, <sup>2</sup> > 1, the functions *f* and *g* are nonnegative and continuous, and they may be singular at *τ* = 0 and/or *τ* = 1, and H*i*, *i* = 1, ... , *n* and K*j*, *j* = 1, ... , *m* are functions of bounded variation. In the proof of the main existence results they applied the Guo–Krasnosel'skii fixed point theorem. In [3], the authors studied the existence and nonexistence of positive solutions for the system (3) with two positive parameters *λ* and *μ*, supplemented with the coupled nonlocal boundary conditions

$$\begin{cases} \mathbf{x}^{(j)}(0) = 0, \; j = 0, \dots, p - 2; \; \mathcal{D}\_{0+}^{\delta\_1} \mathbf{x}(0) = \mathbf{0}, \; \mathcal{D}\_{0+}^{a\_0} \mathbf{x}(1) = \sum\_{i=1}^{n} \int\_0^1 D\_{0+}^{a\_i} y(\tau) \, d\mathcal{H}\_i(\tau), \\\ y^{(j)}(0) = 0, \; j = 0, \dots, q - 2; \; \mathcal{D}\_{0+}^{\delta\_2} y(0) = \mathbf{0}, \; \mathcal{D}\_{0+}^{\delta\_0} y(1) = \sum\_{i=1}^{m} \int\_0^1 D\_{0+}^{\delta\_i} \mathbf{x}(\tau) \, d\mathcal{K}\_i(\tau), \end{cases} (4)$$

where *<sup>n</sup>*, *<sup>m</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>α</sup><sup>i</sup>* <sup>∈</sup> <sup>R</sup> for all *<sup>i</sup>* <sup>=</sup> 0, ... , *<sup>n</sup>*, 0 <sup>≤</sup> *<sup>α</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>α</sup>*<sup>2</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *<sup>α</sup><sup>n</sup>* <sup>≤</sup> *<sup>β</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>δ</sup>*<sup>2</sup> <sup>−</sup> 1, *<sup>β</sup>*<sup>0</sup> <sup>≥</sup> 1, *<sup>β</sup><sup>i</sup>* <sup>∈</sup> <sup>R</sup> for all *<sup>i</sup>* <sup>=</sup> 0, ... , *<sup>m</sup>*, 0 <sup>≤</sup> *<sup>β</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>β</sup>*<sup>2</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *<sup>β</sup><sup>m</sup>* <sup>≤</sup> *<sup>α</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>δ</sup>*<sup>1</sup> <sup>−</sup> 1, *<sup>α</sup>*<sup>0</sup> <sup>≥</sup> 1, the functions *<sup>f</sup>* , *<sup>g</sup>* <sup>∈</sup> *<sup>C</sup>*([0, 1] <sup>×</sup> <sup>R</sup><sup>+</sup> <sup>×</sup> <sup>R</sup>+, <sup>R</sup>+), and the functions <sup>H</sup>*i*, *<sup>i</sup>* <sup>=</sup> 1, ... , *<sup>n</sup>* and <sup>K</sup>*j*, *<sup>j</sup>* <sup>=</sup> 1, ... , *<sup>m</sup>* are bounded variation functions. They presented sufficient conditions on the functions *f* and *g*, and intervals for the parameters *λ* and *μ* such that the problem (3) with these parameters and (4) has positive solutions. In [4], by using the Guo–Krasnosel'skii fixed point theorem, the authors investigated the existence and multiplicity of positive solutions for the nonlinear singular fractional differential equation

$$D\_{0+}^{a}w(\tau) + f(\tau, w(\tau), D\_{0+}^{a\_1}w(\tau), \dots, D\_{0+}^{a\_{n-2}}w(\tau)) = 0, \ \tau \in (0, 1),$$

with the nonlocal boundary conditions

$$\begin{cases} w(0) = D\_{0+}^{\gamma\_1} w(0) = \dots = D\_{0+}^{\gamma\_{n-2}} w(0) = 0, \\\ D\_{0+}^{\beta\_1} w(1) = \int\_0^{\eta} h(\tau) D\_{0+}^{\beta\_2} w(\tau) \, dA(\tau) + \int\_0^1 a(\tau) D\_{0+}^{\beta\_3} w(\tau) \, dA(\tau) \end{cases}$$

where *α* ∈ (*n* − 1, *n*], *n* ≥ 3, *αk*, *γ<sup>k</sup>* ∈ (*k* − 1, *k*], *k* = 1, ... , *n* − 2, *α* − *γ<sup>j</sup>* ∈ (*n* − *j* − 1, *n* − *j*], *j* = 1, ... , *n* − 2, *α* − *αn*−<sup>2</sup> − 1 ∈ (1, 2], *γn*−<sup>2</sup> ≥ *αn*−2, *β*<sup>1</sup> ≥ *β*2, *β*<sup>1</sup> ≥ *β*3, *α* ≥ *β<sup>i</sup>* + 1, *<sup>β</sup><sup>i</sup>* <sup>≥</sup> *<sup>α</sup>n*−<sup>2</sup> <sup>+</sup> 1, *<sup>i</sup>* <sup>=</sup> 1, 2, 3, *<sup>β</sup>*<sup>1</sup> <sup>≤</sup> *<sup>n</sup>* <sup>−</sup> 1, the function *<sup>f</sup>* : (0, 1) <sup>×</sup> <sup>R</sup>*n*−<sup>1</sup> <sup>+</sup> <sup>→</sup> <sup>R</sup><sup>+</sup> is continuous, *<sup>a</sup>*, *<sup>h</sup>* <sup>∈</sup> *<sup>C</sup>*((0, 1), <sup>R</sup>+), and *<sup>A</sup>* is a function of bounded variation. In [5], the authors studied the existence of a unique positive solution for a system of three Caputo fractional equations with (*p*, *q*,*r*)-Laplacian operators subject to two-point boundary conditions, by using an *n*-fixed point theorem of ternary operators in partially ordered complete metric spaces. By relying on the properties of the Kuratowski noncompactness measure and the Sadovskii fixed point theorem; in [6], the authors obtained new existence results for the solutions of a Riemann–Liouville fractional differential equation with a *p*-Laplacian operator in a Banach space, supplemented with multi-point boundary conditions with fractional derivatives. In [7], the authors investigated the existence of solutions for a mixed fractional differential equation with *p*(*t*)-Laplacian operator and two-point boundary conditions at resonance, by applying the continuation theorem of coincidence degree theory. By using the Leggett–Williams fixed-point theorem, the authors studied in [8] the multiplicity of positive solutions for a Riemann–Liouville fractional differential equation with a *p*-Laplacian operator, subject to four-point boundary conditions. In [9], the authors established suitable criteria for the existence of positive solutions for a Riemann–Liouville fractional equation with a *p*-Laplacian operator and infinite-point boundary value conditions, by using the Krasnosel'skii fixed point theorem and Avery–Peterson fixed point theorem. By applying the Guo–Krasnosel'skii fixed point theorem the authors investigated in [10] the existence, multiplicity and the nonexistence of positive solutions for a mixed fractional differential equation with a generalized *p*-Laplacian operator and a positive parameter, supplemented with two-point boundary conditions. We also mention some recent monographs devoted to the investigation of boundary value problems for fractional differential equations and systems with many examples and applications, namely [11–15].

So in comparison with the above papers, the new characteristics of our problem (1) and (2) consist in a combination between the fractional orders *γ*1, *γ*<sup>2</sup> ∈ (1, 2] with the arbitrary fractional orders *δ*1, *δ*2, the existence of the fractional integral terms in equations of (1), and the general uncoupled nonlocal boundary conditions with Riemann–Stieltjes integrals and fractional derivatives. In addition, one of its special feature is the singularity of the nonlinearities from the system (1), that is *f* , *g* become unbounded in the vicinity of 0 and/or 1 in the first variable (see Assumption (*I*2) in Section 3).

The structure of this paper is as follows. In Section 2, some preliminary results including the properties of the Green functions associated to our problem (1) and (2) are presented. In Section 3 we discuss the existence and multiplicity of positive solutions for (1) and (2). Then two examples to illustrate our obtained theorems are given in Section 4, and Section 5 contains the conclusions for this paper.

#### **2. Preliminary Results**

We consider the fractional differential equation

$$D\_{0+}^{\gamma\_1} \left( \varphi\_{r\_1} \left( D\_{0+}^{\delta\_1} \mu(\tau) \right) \right) = \mathfrak{x}(\tau), \ \tau \in (0,1), \tag{5}$$

where *<sup>x</sup>* <sup>∈</sup> *<sup>C</sup>*(0, 1) <sup>∩</sup> *<sup>L</sup>*1(0, 1), with the boundary conditions

$$\begin{cases} \boldsymbol{u}^{(i)}(0) = 0, \; i = 0, \dots, p - 2, \; D\_{0+}^{\delta\_1} \boldsymbol{u}(0) = 0, \\\ \boldsymbol{\varrho}\_{r\_1}(D\_{0+}^{\delta\_1} \boldsymbol{u}(1)) = \int\_0^1 \boldsymbol{\varrho}\_{r\_1}(D\_{0+}^{\delta\_1} \boldsymbol{u}(\eta)) \, d\mathcal{H}\_0(\eta), \; D\_{0+}^{\mu\_0} \boldsymbol{u}(1) = \sum\_{k=1}^n \int\_0^1 D\_{0+}^{\mu\_k} \boldsymbol{u}(\eta) \, d\mathcal{H}\_k(\eta). \end{cases} \tag{6}$$

We denote by

$$\mathfrak{a}\_1 = 1 - \int\_0^1 \eta^{\gamma\_1 - 1} \, d\mathcal{H}\_0(\eta), \; \mathfrak{a}\_2 = \frac{\Gamma(\delta\_1)}{\Gamma(\delta\_1 - \mathfrak{a}\_0)} - \sum\_{i=1}^n \frac{\Gamma(\delta\_1)}{\Gamma(\delta\_1 - \mathfrak{a}\_i)} \int\_0^1 \eta^{\delta\_1 - \mathfrak{a}\_i - 1} \, d\mathcal{H}\_i(\eta). \tag{7}$$

**Lemma 1.** *If* a<sup>1</sup> = 0 *and* a<sup>2</sup> = 0*, then the unique solution u* ∈ *C*[0, 1] *of problem (5) and (6) is given by*

$$u(\boldsymbol{\pi}) = \int\_0^1 \mathcal{G}\_2(\boldsymbol{\pi}, \boldsymbol{\eta}) \, \boldsymbol{\varrho}\_{\ell\_1} \left( \int\_0^1 \mathcal{G}\_1(\boldsymbol{\eta}, \boldsymbol{\theta}) \boldsymbol{x}(\boldsymbol{\theta}) \, d\boldsymbol{\theta} \right) d\boldsymbol{\eta}, \; \boldsymbol{\pi} \in [0, 1], \tag{8}$$

*where*

$$\mathcal{G}\_{1}(\tau,\eta) = \mathfrak{g}\_{1}(\tau,\eta) + \frac{\tau^{\gamma\_{1}-1}}{\mathfrak{a}\_{1}} \int\_{0}^{1} \mathfrak{g}\_{1}(\theta,\eta) \, d\mathcal{H}\_{0}(\theta), \ (\tau,\eta) \in [0,1] \times [0,1], \tag{9}$$

*with*

$$\mathfrak{g}\_{\mathbb{I}}(\tau,\eta) = \frac{1}{\Gamma(\gamma\_1)} \begin{cases} \tau^{\gamma\_1 - 1} (1 - \eta)^{\gamma\_1 - 1} - (\tau - \eta)^{\gamma\_1 - 1}, & 0 \le \eta \le \tau \le 1, \\\ \tau^{\gamma\_1 - 1} (1 - \eta)^{\gamma\_1 - 1}, & 0 \le \tau \le \eta \le 1, \end{cases} \tag{10}$$

*and*

$$\mathcal{G}\_2(\tau,\eta) = \mathfrak{g}\_2(\tau,\eta) + \frac{\tau^{\delta\_1 - 1}}{\mathfrak{a}\_2} \sum\_{i=1}^n \left( \int\_0^1 \mathfrak{g}\_{2i}(\theta,\eta) \, d\mathcal{H}\_i(\theta) \right), \ (\tau,\eta) \in [0,1] \times [0,1], \tag{11}$$

*with*

$$\begin{array}{lcl} \mathfrak{g}\_{2}(\tau,\eta) = \frac{1}{\Gamma(\delta\_{1})} \begin{cases} \tau^{\delta\_{1}-1}(1-\eta)^{\delta\_{1}-a\_{0}-1} - (\tau-\eta)^{\delta\_{1}-1}, & 0 \le \eta \le \tau \le 1, \\\tau^{\delta\_{1}-1}(1-\eta)^{\delta\_{1}-a\_{0}-1}, & 0 \le \tau \le \eta \le 1, \\\eta\_{2i}(\tau,\eta) = \frac{1}{\Gamma(\delta\_{1}-a\_{i})} \begin{cases} \tau^{\delta\_{1}-a\_{i}-1}(1-\eta)^{\delta\_{1}-a\_{0}-1} - (\tau-\eta)^{\delta\_{1}-a\_{i}-1}, & 0 \le \eta \le \tau \le 1, \\\tau^{\delta\_{1}-a\_{i}-1}(1-\eta)^{\delta\_{1}-a\_{0}-1}, & 0 \le \tau \le \eta \le 1, \\\end{cases} & \end{array} \end{array} \tag{12}$$

**Proof.** We denote by *<sup>ϕ</sup>r*<sup>1</sup> (*Dδ*<sup>1</sup> <sup>0</sup>+*u*(*τ*)) = *φ*1(*τ*), *τ* ∈ (0, 1). Hence problems (5) and (6) are equivalent to the following two boundary value problems

$$(I) \qquad \qquad \begin{cases} \ D\_{0+}^{\gamma\_1} \phi\_1(\tau) = \mathfrak{x}(\tau), & \tau \in (0,1), \\ \ \phi\_1(0) = 0, \ \phi\_1(1) = \int\_0^1 \phi\_1(\eta) \, d\mathcal{H}\_0(\eta) \, d\eta \end{cases}$$

and

$$\begin{aligned} (II) \qquad \left\{ \begin{aligned} &D\_{0+}^{\delta\_1} u(\tau) = \varphi\_{\ell\_1}(\phi\_1(\tau)), & \tau \in (0,1), \\ &u^{(j)}(0) = 0, & j = 0, \dots, p-2, \ D\_{0+}^{a\_0} u(1) = \sum\_{k=1}^n \int\_0^1 D\_{0+}^{a\_k} u(\eta) \, d\mathcal{H}\_k(\eta). \end{aligned} \right. \end{aligned}$$

By using Lemma 4.1.5 from [14], the unique solution *φ*<sup>1</sup> ∈ *C*[0, 1] of problem (*I*) is

$$\phi\_1(\tau) = -\int\_0^1 \mathcal{G}\_1(\tau, \theta) \mathbf{x}(\theta) \, d\theta, \ \tau \in [0, 1], \tag{13}$$

where G<sup>1</sup> is given by (9). By using Lemma 2.4.2 from [12], the unique solution *u* ∈ *C*[0, 1] of problem (*I I*) is

$$\mu(\tau) = -\int\_0^1 \mathcal{G}\_2(\tau, \eta) \, \wp\_{\mathbb{Q}\_1}(\phi\_1(\eta)) \, d\eta, \quad \tau \in [0, 1], \tag{14}$$

where G<sup>2</sup> is given by (11). Combining the relations (13) and (14) we obtain the solution *u* of problem (5) and (6) which is given by relation (8).

We consider now the fractional differential equation

$$D\_{0+}^{\gamma\_2} \left( \varphi\_{r\_2} \left( D\_{0+}^{\delta\_2} v(\tau) \right) \right) = y(\tau), \ \tau \in (0,1), \tag{15}$$

where *<sup>y</sup>* <sup>∈</sup> *<sup>C</sup>*(0, 1) <sup>∩</sup> *<sup>L</sup>*1(0, 1), with the boundary conditions

$$\begin{cases} \boldsymbol{v}^{(i)}(0) = 0, \; i = 0, \dots, q-2, \; D\_{0+}^{\delta\_2} \boldsymbol{v}(0) = 0, \\\ \boldsymbol{\varrho}\_{\boldsymbol{\tau}\_2}(D\_{0+}^{\delta\_2} \boldsymbol{v}(1)) = \int\_0^1 \boldsymbol{\varrho}\_{\boldsymbol{\tau}\_2}(D\_{0+}^{\delta\_2} \boldsymbol{v}(\eta)) \, d\mathcal{K}\_0(\eta), \; D\_{0+}^{\delta\_0} \boldsymbol{v}(1) = \sum\_{k=1}^m \int\_0^1 D\_{0+}^{\delta\_k} \boldsymbol{v}(\eta) \, d\mathcal{K}\_k(\eta). \end{cases} \tag{16}$$

We denote by

$$\mathfrak{b}\_1 = 1 - \int\_0^1 \eta^{\gamma\_2 - 1} \, d\mathcal{K}\_0(\eta), \ \mathfrak{b}\_2 = \frac{\Gamma(\delta\_2)}{\Gamma(\delta\_2 - \beta\_0)} - \sum\_{i=1}^m \frac{\Gamma(\delta\_2)}{\Gamma(\delta\_2 - \beta\_i)} \int\_0^1 \eta^{\delta\_2 - \beta\_i - 1} \, d\mathcal{K}\_i(\eta). \tag{17}$$

Similar to Lemma 1 we obtain the next result.

**Lemma 2.** *If* b<sup>1</sup> = 0 *and* b<sup>2</sup> = 0*, then the unique solution v* ∈ *C*[0, 1] *of problem (15) and (16) is given by*

$$v(\tau) = \int\_0^1 \mathcal{G}\_4(\tau, \eta) \, \wp\_{\ell 2} \left( \int\_0^1 \mathcal{G}\_3(\eta, \theta) y(\theta) \, d\theta \right) d\eta, \quad \tau \in [0, 1], \tag{18}$$

*where*

$$\mathcal{G}\_3(\tau,\eta) = \mathfrak{g}\_3(\tau,\eta) + \frac{\tau^{\gamma\_2 - 1}}{\mathfrak{b}\_1} \int\_0^1 \mathfrak{g}\_3(\theta,\eta) \, d\mathcal{K}\_0(\theta), \ (\tau,\eta) \in [0,1] \times [0,1], \tag{19}$$

*with*

$$\mathfrak{g}\_{\mathfrak{D}}(\tau,\eta) = \frac{1}{\Gamma(\gamma\_2)} \begin{cases} \tau^{\gamma\_2 - 1} (1 - \eta)^{\gamma\_2 - 1} - (\tau - \eta)^{\gamma\_2 - 1}, & 0 \le \eta \le \tau \le 1, \\\ t^{\gamma\_2 - 1} (1 - \eta)^{\gamma\_2 - 1}, & 0 \le \tau \le \eta \le 1, \end{cases} \tag{20}$$

*and*

$$\mathcal{G}\_{4}(\boldsymbol{\tau},\boldsymbol{\eta}) = \mathfrak{g}\_{4}(\boldsymbol{\tau},\boldsymbol{\eta}) + \frac{\mathsf{r}^{\delta\_{2}-1}}{\mathsf{b}\_{2}} \sum\_{i=1}^{m} \left( \int\_{0}^{1} \mathfrak{g}\_{4i}(\boldsymbol{\theta},\boldsymbol{\eta}) \, d\boldsymbol{\mathcal{K}}\_{i}(\boldsymbol{\theta}) \right)\_{i} (\boldsymbol{\tau},\boldsymbol{\eta}) \in [0,1] \times [0,1],\tag{21}$$

*with*

$$\begin{array}{lcl} \mathfrak{g}\_{4}(\tau,\eta) = \frac{1}{\Gamma(\delta\_{2})} \begin{cases} \tau^{\delta\_{2}-1}(1-\eta)^{\delta\_{2}-\beta\_{0}-1}-(\tau-\eta)^{\delta\_{2}-1}, & 0 \le \eta \le \tau \le 1, \\\tau^{\delta\_{2}-1}(1-\eta)^{\delta\_{2}-\beta\_{0}-1}, & 0 \le \tau \le \eta \le 1, \\\mathfrak{g}\_{4i}(\tau,\eta) = \frac{1}{\Gamma(\delta\_{2}-\beta\_{i})} \begin{cases} \tau^{\delta\_{2}-\beta\_{i}-1}(1-\eta)^{\delta\_{2}-\beta\_{0}-1}-(\tau-\eta)^{\delta\_{2}-\beta\_{i}-1}, & 0 \le \eta \le \tau \le 1, \\\tau^{\delta\_{2}-\beta\_{i}-1}(1-\eta)^{\delta\_{2}-\beta\_{0}-1}, & 0 \le \tau \le \eta \le 1, \\\end{cases} & \end{array} \end{array} \tag{22}$$

**Lemma 3.** *We assume that* a1, a2, b1, b<sup>2</sup> > 0*,* H*i*, *i* = 0, ... , *n, and* K*j*, *j* = 0, ... , *m are nondecreasing functions. Then the functions* G*i*, *i* = 1, ... , 4 *given by (9), (11), (19) and (21) have the properties*

*(a)* G*<sup>i</sup>* : [0, 1] × [0, 1] → [0, ∞)*, i* = 1, . . . , 4 *are continuous functions; (b)* G1(*τ*, *η*) ≤ J1(*η*), ∀ (*τ*, *η*) ∈ [0, 1] × [0, 1]*, where*

$$\mathcal{J}\_1(\eta) = \mathfrak{h}\_1(\eta) + \frac{1}{\mathfrak{a}\_1} \int\_0^1 \mathfrak{g}\_1(\theta, \eta) \, d\mathcal{H}\_0(\vartheta), \,\,\forall \,\eta \in [0, 1].$$

*with* h1(*η*) = <sup>1</sup> <sup>Γ</sup>(*γ*1)(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)*γ*1<sup>−</sup>1, *<sup>η</sup>* <sup>∈</sup> [0, 1]*; (c)* G2(*τ*, *η*) ≤ J2(*η*), ∀ (*τ*, *η*) ∈ [0, 1] × [0, 1]*, where*

$$\mathcal{J}\_2(\eta) = \mathfrak{h}\_2(\eta) + \frac{1}{\mathfrak{a}\_2} \sum\_{i=1}^n \int\_0^1 \mathfrak{g}\_{2i}(\mathfrak{d}, \eta) \, d\mathcal{H}\_i(\mathfrak{d}), \,\,\forall \,\eta \in [0, 1].$$

$$\begin{array}{c} \text{with } \mathfrak{h}\_{2}(\eta) = \frac{1}{\Gamma(\delta\_{1})} (1 - \eta)^{\delta\_{1} - \mathfrak{a}\_{0} - 1} (1 - (1 - \eta)^{\mathfrak{a}\_{0}}), \ \eta \in [0, 1];\\\ (d) \; \mathcal{G}\_{2}(\mathsf{r}, \eta) \ge \mathsf{r}^{\delta\_{1} - 1} \mathcal{J}\_{2}(\eta), \ \forall \ (\mathsf{r}, \eta) \in [0, 1] \times [0, 1];\\\ (e) \; \mathcal{G}\_{3}(\mathsf{r}, \eta) \le \mathcal{J}\_{3}(\eta), \ \forall \ (\mathsf{r}, \eta) \in [0, 1] \times [0, 1], where \end{array}$$

$$\mathcal{J}\_3(\eta) = \mathfrak{h}\_3(\eta) + \frac{1}{\mathfrak{b}\_1} \int\_0^1 \mathfrak{g}\_3(\theta, \eta) \, d\mathcal{K}\_0(\theta), \,\,\forall \,\eta \in [0, 1].$$

*with* h3(*η*) = <sup>1</sup> <sup>Γ</sup>(*γ*2)(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)*γ*2<sup>−</sup>1, *<sup>η</sup>* <sup>∈</sup> [0, 1]*; (f)* G4(*τ*, *η*) ≤ J4(*η*), ∀ (*τ*, *η*) ∈ [0, 1] × [0, 1]*, where*

$$\mathcal{J}\_4(\eta) = \mathfrak{h}\_4(\eta) + \frac{1}{\mathfrak{b}\_2} \sum\_{i=1}^m \int\_0^1 \mathfrak{g}\_{4i}(\theta, \eta) \, d\mathcal{K}\_i(\theta), \,\,\forall \,\eta \in [0, 1].$$

$$\begin{aligned} \text{with } \mathfrak{h}\_4(\eta) &= \frac{1}{\Gamma(\delta\_2)} (1 - \eta)^{\delta\_2 - \beta\_0 - 1} (1 - (1 - \eta)^{\beta\_0}), \ \eta \in [0, 1];\\ \text{(g) } \mathcal{G}\_4(\tau, \eta) &\ge \tau^{\delta\_2 - 1} \mathcal{J}\_4(\eta), \ \forall \ (\tau, \eta) \in [0, 1] \times [0, 1]. \end{aligned}$$

**Proof.** (a) Based on the continuity of functions g1, g2, g2*i*, *i* = 1, ... , *n*, g3, g4, g4*i*, *i* = 1, ... , *m* (given by (10), (12), (20) and (22)), we obtain that the functions G*i*, *i* = 1, ... , 4 are continuous.

(b) By the definition of g<sup>1</sup> we find

$$\begin{split} \mathcal{G}\_{1}(\boldsymbol{\tau},\boldsymbol{\eta}) &\leq \frac{1}{\Gamma(\gamma\_{1})}(1-\boldsymbol{\eta})^{\gamma\_{1}-1} + \frac{1}{\mathfrak{a}\_{1}} \int\_{0}^{1} \mathfrak{g}\_{1}(\boldsymbol{\theta},\boldsymbol{\eta}) \, d\mathcal{H}\_{0}(\boldsymbol{\theta}) \\ &= \mathfrak{h}\_{1}(\boldsymbol{\eta}) + \frac{1}{\mathfrak{a}\_{1}} \int\_{0}^{1} \mathfrak{g}\_{1}(\boldsymbol{\theta},\boldsymbol{\eta}) \, d\mathcal{H}\_{0}(\boldsymbol{\theta}) = \mathcal{J}\_{1}(\boldsymbol{\eta}), \ \forall \ \boldsymbol{\tau}, \boldsymbol{\eta} \in [0,1]. \end{split}$$

(c–d) Using our assumptions and the properties of function g<sup>2</sup> from Lemma 2.1.3 from [12], namely <sup>g</sup>2(*τ*, *<sup>η</sup>*) <sup>≤</sup> <sup>1</sup> <sup>Γ</sup>(*δ*1)(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)*δ*1−*α*0−1(<sup>1</sup> <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)*α*<sup>0</sup> ) = <sup>h</sup>2(*η*) and <sup>g</sup>2(*τ*, *<sup>η</sup>*) <sup>≥</sup> *<sup>τ</sup>δ*1−1h2(*η*) for all *<sup>τ</sup>*, *<sup>η</sup>* <sup>∈</sup> [0, 1], we deduce

$$\begin{split} \mathcal{G}\_{2}(\boldsymbol{\tau},\boldsymbol{\eta}) &\leq \mathfrak{h}\_{2}(\boldsymbol{\eta}) + \frac{1}{\mathfrak{a}\_{2}} \sum\_{i=1}^{n} \int\_{0}^{1} \mathfrak{g}\_{2i}(\boldsymbol{\theta},\boldsymbol{\eta}) \, d\mathcal{H}\_{i}(\boldsymbol{\theta}) = \mathcal{J}\_{2}(\boldsymbol{\eta}), \\ \mathcal{G}\_{2}(\boldsymbol{\tau},\boldsymbol{\eta}) &\geq \tau^{\delta\_{1}-1} \Big( \mathfrak{h}\_{2}(\boldsymbol{\eta}) + \frac{1}{\mathfrak{a}\_{2}} \sum\_{i=1}^{n} \mathfrak{g}\_{2i}(\boldsymbol{\theta},\boldsymbol{\eta}) \, d\mathcal{H}\_{i}(\boldsymbol{\theta}) \Big) = \tau^{\delta\_{1}-1} \mathcal{J}\_{2}(\boldsymbol{\eta}), \ \forall \ \boldsymbol{\tau}, \boldsymbol{\eta} \in [0,1]. \end{split}$$

(e) By the definition of g<sup>3</sup> we obtain

$$\begin{split} \mathcal{G}\_{3}(\boldsymbol{\tau},\boldsymbol{\eta}) &\leq \frac{1}{\Gamma(\gamma\_{2})}(1-\boldsymbol{\eta})^{\gamma\_{2}-1} + \frac{1}{\mathfrak{b}\_{1}} \int\_{0}^{1} \mathfrak{g}\_{3}(\boldsymbol{\theta},\boldsymbol{\eta}) \, d\boldsymbol{\mathcal{K}}\_{0}(\boldsymbol{\vartheta}) \\ &= \mathfrak{h}\_{3}(\boldsymbol{\eta}) + \frac{1}{\mathfrak{b}\_{1}} \int\_{0}^{1} \mathfrak{g}\_{3}(\boldsymbol{\theta},\boldsymbol{\eta}) \, d\boldsymbol{\mathcal{K}}\_{0}(\boldsymbol{\vartheta}) = \mathcal{J}\_{3}(\boldsymbol{\eta}), \ \forall \ \boldsymbol{\tau}, \boldsymbol{\eta} \in [0,1]. \end{split}$$

(f–g) Using the assumptions of this lemma and the properties of function g<sup>4</sup> from Lemma 2.1.3 from [12], namely <sup>g</sup>4(*τ*, *<sup>η</sup>*) <sup>≤</sup> <sup>1</sup> <sup>Γ</sup>(*δ*2)(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)*δ*2−*β*0−1(<sup>1</sup> <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)*β*<sup>0</sup> ) = <sup>h</sup>4(*η*) and <sup>g</sup>4(*τ*, *<sup>η</sup>*) <sup>≥</sup> *<sup>τ</sup>δ*2−1h4(*η*) for all *<sup>τ</sup>*, *<sup>η</sup>* <sup>∈</sup> [0, 1], we find

$$\begin{aligned} \mathcal{G}\_{4}(\tau,\eta) &\leq \mathfrak{h}\_{4}(\eta) + \frac{1}{\mathfrak{b}\_{2}} \sum\_{i=1}^{m} \int\_{0}^{1} \mathfrak{g}\_{4i}(\theta,\eta) \, d\mathcal{K}\_{i}(\theta) = \mathcal{J}\_{4}(\eta),\\ \mathcal{G}\_{4}(\tau,\eta) &\geq \tau^{\delta\_{2}-1} \Big( \mathfrak{h}\_{4}(\eta) + \frac{1}{\mathfrak{b}\_{2}} \sum\_{i=1}^{m} \mathfrak{g}\_{4i}(\vartheta,\eta) \, d\mathcal{K}\_{i}(\theta) \Big) = \tau^{\delta\_{2}-1} \mathcal{J}\_{4}(\eta), \ \forall \ \tau, \eta \in [0,1]. \end{aligned}$$

**Lemma 4.** *We assume that* a1, a2, b1, b<sup>2</sup> > 0*,* H*i*, *i* = 0, ... , *n, and* K*j*, *j* = 0, ... , *m are nondecreasing functions, <sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> *<sup>C</sup>*(0, 1) <sup>∩</sup> *<sup>L</sup>*1(0, 1) *with <sup>x</sup>*(*τ*) <sup>≥</sup> 0, *<sup>y</sup>*(*τ*) <sup>≥</sup> <sup>0</sup> *for all <sup>τ</sup>* <sup>∈</sup> (0, 1)*. Then the solutions u and v of problems (5), (6) and (15), (16), respectively, satisfy the inequalities u*(*τ*) ≥ 0*, <sup>v</sup>*(*τ*) <sup>≥</sup> <sup>0</sup> *for all <sup>τ</sup>* <sup>∈</sup> [0, 1] *and u*(*τ*) <sup>≥</sup> *<sup>τ</sup>δ*1−1*u*(*s*) *and v*(*τ*) <sup>≥</sup> *<sup>τ</sup>δ*2−1*v*(*s*) *for all <sup>τ</sup>*,*<sup>s</sup>* <sup>∈</sup> [0, 1]*.*

**Proof.** Based on the assumptions of this lemma, we obtain that the solutions *u* and *v* of problems (5), (6) and (15), (16), respectively, are nonnegative, that is *u*(*τ*) ≥ 0, *v*(*τ*) ≥ 0 for all *τ* ∈ [0, 1]. In addition, by using Lemma 3, we deduce

$$\begin{split} u(\boldsymbol{\tau}) &\geq \tau^{\delta\_1 - 1} \int\_0^1 \mathcal{G}\_2(\boldsymbol{\eta}) \, \boldsymbol{\varrho}\_{\boldsymbol{\varrho}\_1} \Big( \int\_0^1 \mathcal{G}\_1(\boldsymbol{\eta}, \boldsymbol{\theta}) \boldsymbol{x}(\boldsymbol{\vartheta}) \, d\boldsymbol{\theta} \Big) d\boldsymbol{\eta} \\ &\geq \tau^{\delta\_1 - 1} \int\_0^1 \mathcal{G}\_2(\boldsymbol{s}, \boldsymbol{\eta}) \, \boldsymbol{\varrho}\_{\boldsymbol{\varrho}\_1} \Big( \int\_0^1 \mathcal{G}\_1(\boldsymbol{\eta}, \boldsymbol{\theta}) \boldsymbol{x}(\boldsymbol{\vartheta}) \, d\boldsymbol{\vartheta} \Big) d\boldsymbol{\eta} \\ &= \tau^{\delta\_1 - 1} \boldsymbol{u}(\boldsymbol{s}), \\ v(\boldsymbol{\tau}) &\geq \tau^{\delta\_2 - 1} \int\_0^1 \mathcal{J}\_4(\boldsymbol{\eta}) \, \boldsymbol{\varrho}\_{\boldsymbol{\varrho}\_2} \Big( \int\_0^1 \mathcal{G}\_3(\boldsymbol{\eta}, \boldsymbol{\theta}) \boldsymbol{y}(\boldsymbol{\vartheta}) \, d\boldsymbol{\vartheta} \Big) d\boldsymbol{\eta} \\ &\geq \tau^{\delta\_2 - 1} \int\_0^1 \mathcal{G}\_4(\boldsymbol{s}, \boldsymbol{\eta}) \, \boldsymbol{\varrho}\_{\boldsymbol{\varrho}\_2} \Big( \int\_0^1 \mathcal{G}\_3(\boldsymbol{\eta}, \boldsymbol{\theta}) \boldsymbol{y}(\boldsymbol{\vartheta}) \, d\boldsymbol{\vartheta} \Big) d\boldsymbol{\eta} \\ &= \tau^{\delta\_2 - 1} \boldsymbol{v}(\boldsymbol{s}), \end{split}$$

for all *τ*, *s* ∈ [0, 1].

We present finally in this section the Guo–Krasnosel'skii fixed point theorem, which we will use in the proofs of our main results.

**Theorem 1.** *([16]). Let* X *be a real Banach space with the norm* · *, and let* C ⊂ *X be a cone in* X *. Assume* Ω<sup>1</sup> *and* Ω<sup>2</sup> *are bounded open subsets of* X *with* 0 ∈ Ω1*,* Ω<sup>1</sup> ⊂ Ω<sup>2</sup> *and let* A : C ∩ (Ω<sup>2</sup> \ Ω1) → C *be a completely continuous operator such that, either (i)* A*u* ≤ *u* , ∀ *u* ∈C∩ *∂*Ω1*, and* A*u* ≥ *u* , ∀ *u* ∈C∩ *∂*Ω2*; or (ii)* A*u* ≥ *u* , ∀ *u* ∈C∩ *∂*Ω1, *and* A*u* ≤ *u* , ∀ *u* ∈C∩ *∂*Ω2*. Then* A *has at least one fixed point in* C ∩ (Ω<sup>2</sup> \ Ω1)*.*

#### **3. Existence of Positive Solutions**

According to Lemmas 1 and 2, the pair of functions (*u*, *v*) is a solution of problem (1) and (2) if and only if (*u*, *v*) is a solution of the system

$$\begin{cases} \begin{aligned} u(\tau) &= \int\_0^1 \mathcal{G}\_2(\tau,\zeta) \, \boldsymbol{\varrho}\_{\boldsymbol{\varrho}\_1} \Big( \int\_0^1 \mathcal{G}\_1(\zeta,\theta) f \Big( \theta, u(\vartheta), v(\vartheta), I\_{0+}^{\sigma\_1} u(\vartheta), I\_{0+}^{\sigma\_2} v(\vartheta) \Big) d\theta \Big) d\zeta, \\\ v(\tau) &= \int\_0^1 \mathcal{G}\_4(\tau,\zeta) \, \boldsymbol{\varrho}\_{\boldsymbol{\varrho}\_2} \Big( \int\_0^1 \mathcal{G}\_3(\zeta,\theta) g \Big( \theta, u(\vartheta), v(\vartheta), I\_{0+}^{\zeta\_1} u(\vartheta), I\_{0+}^{\zeta\_2} v(\vartheta) \Big) d\theta \Big) d\zeta, \end{aligned} \end{cases}$$

for all *τ* ∈ [0, 1]. We introduce the Banach space X = *C*[0, 1] with supreme norm *u* = sup*τ*∈[0,1] <sup>|</sup>*u*(*τ*)|, and the Banach space <sup>Y</sup> <sup>=</sup> X ×X with the norm (*u*, *v*) <sup>Y</sup> = *u* + *v* . We define the cone

$$\mathcal{P} = \{(u, v) \in \mathcal{Y} \: \: \: u(\tau) \ge 0, \ v(\tau) \ge 0, \ \forall \ \tau \in [0, 1] \}.$$

We also define the operators A1, A<sup>2</sup> : Y→X and A : Y→Y by

$$\begin{split} \mathcal{A}\_{1}(\boldsymbol{u},\boldsymbol{v})(\boldsymbol{\tau}) &= \int\_{0}^{1} \mathcal{G}\_{2}(\boldsymbol{\tau},\boldsymbol{\zeta}) \boldsymbol{\varrho}\_{\boldsymbol{\varrho}\_{1}} \Big( \int\_{0}^{1} \mathcal{G}\_{1}(\boldsymbol{\zeta},\boldsymbol{\theta}) f\Big(\boldsymbol{\theta},\boldsymbol{u}(\boldsymbol{\theta}),\boldsymbol{v}(\boldsymbol{\theta}),I\_{0+}^{\sigma\_{1}}\boldsymbol{u}(\boldsymbol{\theta}),I\_{0+}^{\sigma\_{2}}\boldsymbol{v}(\boldsymbol{\theta})\Big) d\boldsymbol{\theta} \Big) d\boldsymbol{\zeta} \\ \mathcal{A}\_{2}(\boldsymbol{u},\boldsymbol{v})(\boldsymbol{\tau}) &= \int\_{0}^{1} \mathcal{G}\_{4}(\boldsymbol{\tau},\boldsymbol{\zeta}) \boldsymbol{\varrho}\_{\boldsymbol{\varrho}\_{2}} \Big( \int\_{0}^{1} \mathcal{G}\_{3}(\boldsymbol{\zeta},\boldsymbol{\theta}) \boldsymbol{g}\Big(\boldsymbol{\theta},\boldsymbol{u}(\boldsymbol{\theta}),\boldsymbol{v}(\boldsymbol{\theta}),I\_{0+}^{\xi\_{1}}\boldsymbol{u}(\boldsymbol{\theta}),I\_{0+}^{\xi\_{2}}\boldsymbol{v}(\boldsymbol{\theta})\Big) d\boldsymbol{\theta} \Big) d\boldsymbol{\zeta} \end{split}$$

for *τ* ∈ [0, 1] and (*u*, *v*) ∈ Y, and A(*u*, *v*)=(A1(*u*, *v*), A2(*u*, *v*)), (*u*, *v*) ∈ Y. We see that (*u*, *v*) is a solution of problem (1) and (2) if and only if (*u*, *v*) is a fixed point of operator A. We introduce now the basic assumptions that we will use in this section.


$$f(\mathbf{r}, z\_1, z\_2, z\_3, z\_4) \le \psi\_1(\mathbf{r}) \chi\_1(\mathbf{r}, z\_1, z\_2, z\_3, z\_4), \\ g(\mathbf{r}, z\_1, z\_2, z\_3, z\_4) \le \psi\_2(\mathbf{r}) \chi\_2(\mathbf{r}, z\_1, z\_2, z\_3, z\_4), $$

for any *<sup>τ</sup>* <sup>∈</sup> (0, 1), *zi* <sup>∈</sup> <sup>R</sup>+, *<sup>i</sup>* <sup>=</sup> 1, . . . , 4.

**Lemma 5.** *We assume that assumptions* (*I*1) *and* (*I*2) *are satisfied. Then operator* A : P→P *is completely continuous.*

**Proof.** We denote by *<sup>M</sup>*<sup>1</sup> = % <sup>1</sup> <sup>0</sup> <sup>J</sup>1(*η*)*ψ*1(*η*) *<sup>d</sup>η*, *<sup>M</sup>*<sup>2</sup> <sup>=</sup> % <sup>1</sup> <sup>0</sup> J3(*η*)*ψ*2(*η*) *dη*. By using (*I*2) and Lemma 3, we deduce that *M*<sup>1</sup> > 0 and *M*<sup>2</sup> > 0. In addition we find

*M*<sup>1</sup> = <sup>1</sup> 0 J1(*η*)*ψ*1(*η*) *dη* = <sup>1</sup> 0 <sup>h</sup>1(*η*) + <sup>1</sup> a1 <sup>1</sup> 0 g1(*ζ*, *η*) *d*H0(*ζ*) *ψ*1(*η*) *dη* ≤ 1 Γ(*γ*1) <sup>1</sup> 0 (<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)*γ*1−1*ψ*1(*η*) *<sup>d</sup><sup>η</sup>* <sup>+</sup> 1 a1 <sup>1</sup> 0 <sup>1</sup> 0 1 Γ(*γ*1) *<sup>ζ</sup>γ*1−1(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)*γ*1−<sup>1</sup> *<sup>d</sup>*H0(*ζ*) *ψ*1(*η*)*dη* = \* 1 + 1 a1 <sup>1</sup> 0 *<sup>ζ</sup>γ*1−<sup>1</sup> *<sup>d</sup>*H0(*ζ*) + 1 Γ(*γ*1) <sup>1</sup> 0 (<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)*γ*1−1*ψ*1(*η*) *<sup>d</sup><sup>η</sup>* <sup>&</sup>lt; <sup>∞</sup>, *M*<sup>2</sup> = <sup>1</sup> 0 J3(*η*)*ψ*2(*η*) *dη* = <sup>1</sup> 0 <sup>h</sup>3(*η*) + <sup>1</sup> b1 <sup>1</sup> 0 g3(*ζ*, *η*) *d*K0(*ζ*) *ψ*2(*η*) *dη* ≤ 1 Γ(*γ*2) <sup>1</sup> 0 (<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)*γ*2−1*ψ*2(*η*) *<sup>d</sup><sup>η</sup>* <sup>+</sup> 1 b1 <sup>1</sup> 0 <sup>1</sup> 0 1 Γ(*γ*2) *<sup>ζ</sup>γ*2−1(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)*γ*2−<sup>1</sup> *<sup>d</sup>*K0(*ζ*) *ψ*2(*η*)*dη* = \* 1 + 1 b1 <sup>1</sup> 0 *<sup>ζ</sup>γ*2−<sup>1</sup> *<sup>d</sup>*K0(*ζ*) + 1 Γ(*γ*2) <sup>1</sup> 0 (<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)*γ*2−1*ψ*2(*η*) *<sup>d</sup><sup>η</sup>* <sup>&</sup>lt; <sup>∞</sup>.

Also, by Lemma 3 we conclude that A maps P into P.

We will prove that A maps bounded sets into relatively compact sets. Let E⊂P be an arbitrary bounded set. Then there exists Ξ<sup>1</sup> > 0 such that (*u*, *v*) <sup>Y</sup> ≤ Ξ<sup>1</sup> for all (*u*, *v*) ∈ E. By the continuity of *χ*<sup>1</sup> and *χ*2, we deduce that there exists Ξ<sup>2</sup> > 0 such that Ξ<sup>2</sup> = max{sup*τ*∈[0,1],*zi*∈[0,*ω*], *<sup>i</sup>*=1,...,4 *<sup>χ</sup>*1(*τ*, *<sup>z</sup>*1, *<sup>z</sup>*2, *<sup>z</sup>*3, *<sup>z</sup>*4), sup*τ*∈[0,1],*zi*∈[0,*ω*], *<sup>i</sup>*=1,...,4 *<sup>χ</sup>*2(*τ*, *<sup>z</sup>*1, *<sup>z</sup>*2, *<sup>z</sup>*3, *<sup>z</sup>*4)}, where *<sup>ω</sup>* <sup>=</sup> <sup>Ξ</sup><sup>1</sup> max 1, <sup>1</sup> <sup>Γ</sup>(*σ*1+1), <sup>1</sup> <sup>Γ</sup>(*σ*2+1), <sup>1</sup> <sup>Γ</sup>(*ς*1+1), <sup>1</sup> Γ(*ς*2+1) . Based on the inequality |*I ξ* <sup>0</sup>+*w*(*η*)| ≤ *w* <sup>Γ</sup>(*ξ*+1), for *ξ* > 0 and *w* ∈ *C*[0, 1], and by Lemma 3, we find for any (*u*, *v*) ∈ E and *η* ∈ [0, 1]

$$\begin{split} \mathcal{A}\_{1}(u,v)(\eta) &\leq \int\_{0}^{1} \mathcal{I}\_{2}(\xi) \, \varrho\_{\theta\_{1}} \Big( \int\_{0}^{1} \mathcal{I}\_{1}(\tau) \, \psi\_{1}(\tau) \chi\_{1}(\tau), u(\tau), v(\tau), I\_{0+}^{\mathcal{I}\_{1}} u(\tau), I\_{0+}^{\mathcal{I}\_{2}} v(\tau) \big) d\mathcal{I}\_{1} \\ &\leq \Xi\_{2}^{\varrho\_{1}-1} \, \varrho\_{\theta\_{1}} \Big( \int\_{0}^{1} \mathcal{I}\_{1}(\tau) \, \psi\_{1}(\tau) \, d\tau \Big) \int\_{0}^{1} \mathcal{I}\_{2}(\zeta) \, d\zeta = M\_{1}^{\varrho\_{1}-1} \Xi\_{2}^{\varrho\_{1}-1} M\_{3} \\ &\mathcal{A}\_{2}(u,v)(\eta) \le \int\_{0}^{1} \mathcal{I}\_{4}(\zeta) \, \varrho\_{\theta\_{2}} \Big( \int\_{0}^{1} \mathcal{I}\_{3}(\tau) \, \psi\_{2}(\tau) \chi\_{2}(\tau), v(\tau), I\_{0+}^{\mathcal{I}\_{1}} u(\tau), I\_{0+}^{\mathcal{I}\_{2}} v(\tau) \big) d\zeta \\ &\leq \Xi\_{2}^{\varrho\_{2}-1} \, \varrho\_{\theta\_{2}} \Big( \int\_{0}^{1} \mathcal{I}\_{3}(\tau) \, \psi\_{2}(\tau) \, d\tau \Big) \int\_{0}^{1} \mathcal{I}\_{4}(\zeta) \, d\zeta = M\_{2}^{\varrho\_{2}-1} \Xi\_{2}^{\varrho\_{2}-1} M\_{4} \end{split}$$

where *<sup>M</sup>*<sup>3</sup> = % <sup>1</sup> <sup>0</sup> <sup>J</sup>2(*ζ*) *<sup>d</sup><sup>ζ</sup>* and *<sup>M</sup>*<sup>4</sup> <sup>=</sup> % <sup>1</sup> <sup>0</sup> J4(*ζ*) *dζ*.

Then A1(*u*, *v*) <sup>≤</sup> *<sup>M</sup>*1−<sup>1</sup> <sup>1</sup> <sup>Ξ</sup>1−<sup>1</sup> <sup>2</sup> *M*3, A2(*u*, *v*) <sup>≤</sup> *<sup>M</sup>*2−<sup>1</sup> <sup>2</sup> <sup>Ξ</sup>2−<sup>1</sup> <sup>2</sup> *M*<sup>4</sup> for all (*u*, *v*) ∈ E, and A(*u*, *v*) <sup>Y</sup> <sup>≤</sup> *<sup>M</sup>*1−<sup>1</sup> <sup>1</sup> <sup>Ξ</sup>1−<sup>1</sup> <sup>2</sup> *<sup>M</sup>*<sup>3</sup> <sup>+</sup> *<sup>M</sup>*2−<sup>1</sup> <sup>2</sup> <sup>Ξ</sup>2−<sup>1</sup> <sup>2</sup> *M*<sup>4</sup> for all (*u*, *v*) ∈ E, that is A1(E), A2(E) and A(E) are bounded.

We will show that A(E) is equicontinuous. By using Lemma 1, for (*u*, *v*) ∈ E and *η* ∈ [0, 1] we obtain

<sup>A</sup>1(*u*, *<sup>v</sup>*)(*η*) = <sup>1</sup> 0 , <sup>g</sup>2(*η*, *<sup>ζ</sup>*) + *<sup>η</sup>δ*1−<sup>1</sup> a2 *n* ∑ *i*=1 <sup>1</sup> 0 g2*i*(*τ*, *ζ*) *d*H*i*(*τ*) *ϕ*<sup>1</sup> <sup>1</sup> 0 G1(*ζ*, *ϑ*) ×*f*(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I σ*1 <sup>0</sup>+*u*(*ϑ*), *I σ*2 <sup>0</sup>+*v*(*ϑ*)) *dϑ dζ* = *<sup>η</sup>* 0 1 Γ(*δ*1) [*ηδ*1−1(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)*δ*1−*α*0−<sup>1</sup> <sup>−</sup> (*<sup>η</sup>* <sup>−</sup> *<sup>ζ</sup>*)*δ*1−1] ×*ϕ*<sup>1</sup> <sup>1</sup> 0 G1(*ζ*, *ϑ*)*f*(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I σ*1 <sup>0</sup>+*u*(*ϑ*), *I σ*2 <sup>0</sup>+*v*(*ϑ*)) *dϑ dζ* + <sup>1</sup> *η* 1 Γ(*δ*1) *<sup>η</sup>δ*1−1(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)*δ*1−*α*0−1*ϕ*<sup>1</sup> <sup>1</sup> 0 G1(*ζ*, *ϑ*)*f*(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I σ*1 <sup>0</sup>+*u*(*ϑ*), *I σ*2 <sup>0</sup>+*v*(*ϑ*)) *dϑ dζ* +*ηδ*1−<sup>1</sup> a2 <sup>1</sup> 0 *n* ∑ *i*=1 <sup>1</sup> 0 g2*i*(*τ*, *ζ*) *d*H*i*(*τ*) *ϕ*<sup>1</sup> <sup>1</sup> 0 G1(*ζ*, *ϑ*)*f*(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I σ*1 <sup>0</sup>+*u*(*ϑ*), *I σ*2 <sup>0</sup>+*v*(*ϑ*)) *dϑ dζ*.

Then for any *η* ∈ (0, 1) we deduce

(A1(*u*, *v*)) (*η*) = *<sup>η</sup>* 0 1 Γ(*δ*1) [(*δ*<sup>1</sup> <sup>−</sup> <sup>1</sup>)*ηδ*1−2(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)*δ*1−*α*0−<sup>1</sup> <sup>−</sup> (*δ*<sup>1</sup> <sup>−</sup> <sup>1</sup>)(*<sup>η</sup>* <sup>−</sup> *<sup>ζ</sup>*)*δ*1−2] ×*ϕ*<sup>1</sup> <sup>1</sup> 0 G1(*ζ*, *ϑ*)*f*(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I σ*1 <sup>0</sup>+*u*(*ϑ*), *I σ*2 <sup>0</sup>+*v*(*ϑ*)) *dϑ dζ* + <sup>1</sup> *η* 1 Γ(*δ*1) (*δ*<sup>1</sup> <sup>−</sup> <sup>1</sup>)*ηδ*1−2(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)*δ*1−*α*0−1*ϕ*<sup>1</sup> <sup>1</sup> 0 G1(*ζ*, *ϑ*)*f*(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I σ*1 <sup>0</sup>+*u*(*ϑ*), *I σ*2 <sup>0</sup>+*v*(*ϑ*)) *dϑ dζ* <sup>+</sup>(*δ*<sup>1</sup> <sup>−</sup> <sup>1</sup>)*ηδ*1−<sup>2</sup> a2 <sup>1</sup> 0 *n* ∑ *i*=1 <sup>1</sup> 0 g2*i*(*τ*, *ζ*) *d*H*i*(*τ*) *ϕ*<sup>1</sup> <sup>1</sup> 0 G1(*ζ*, *ϑ*)*f*(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I σ*1 <sup>0</sup>+*u*(*ϑ*), *I σ*2 <sup>0</sup>+*v*(*ϑ*)) *dϑ dζ*.

So for any *η* ∈ (0, 1) we find


Therefore, for *η* ∈ (0, 1) we obtain

$$\begin{split} |(\mathcal{A}\_{1}(\boldsymbol{u},\boldsymbol{v}))'(\boldsymbol{\eta})| &\leq \Sigma\_{2}^{\varrho\_{1}-1}M\_{1}^{\varrho\_{1}-1} \left[ \frac{1}{\Gamma(\delta\_{1}-1)} \left( \frac{\eta^{\delta\_{1}-2}}{\delta\_{1}-\mathsf{a}\_{0}} + \frac{\eta^{\delta\_{1}-1}}{\delta\_{1}-1} \right) \right. \\ &+ \frac{(\delta\_{1}-1)\eta^{\delta\_{1}-2}}{\mathsf{a}\_{2}} \int\_{0}^{1} \sum\_{i=1}^{n} \left( \int\_{0}^{1} \frac{1}{\Gamma(\delta\_{1}-\mathsf{a}\_{i})} (1-\xi)^{\delta\_{1}-\mathsf{a}\_{0}-1} \, d\zeta \right) \tau^{\delta\_{1}-\mathsf{a}\_{i}-1} \, d\mathcal{H}\_{i}(\boldsymbol{\tau}) \right] \\ &= \Sigma\_{2}^{\varrho\_{1}-1}M\_{1}^{\varrho\_{1}-1} \left[ \frac{1}{\Gamma(\delta\_{1}-1)} \left( \frac{\eta^{\delta\_{1}-2}}{\delta\_{1}-\mathsf{a}\_{0}} + \frac{\eta^{\delta\_{1}-1}}{\delta\_{1}-1} \right) + \frac{(\delta\_{1}-1)\eta^{\delta\_{1}-2}}{\mathsf{a}\_{2}(\delta\_{1}-\mathsf{a}\_{0})} \sum\_{i=1}^{n} \frac{1}{\Gamma(\delta\_{1}-\mathsf{a}\_{i})} \\ &\times \int\_{0}^{1} \tau^{\delta\_{1}-\mathsf{a}\_{i}-1} \, d\mathcal{H}\_{i}(\boldsymbol{\tau}) \right]. \end{split} \tag{23}$$

We denote by

$$\begin{split} \Theta\_{0}(\eta) &= \frac{1}{\Gamma(\delta\_{1}-1)} \left( \frac{\eta^{\delta\_{1}-2}}{\delta\_{1}-\alpha\_{0}} + \frac{\eta^{\delta\_{1}-1}}{\delta\_{1}-1} \right) \\ &+ \frac{(\delta\_{1}-1)\eta^{\delta\_{1}-2}}{\alpha\_{2}(\delta\_{1}-\alpha\_{0})} \sum\_{i=1}^{n} \frac{1}{\Gamma(\delta\_{1}-\alpha\_{i})} \int\_{0}^{1} \tau^{\delta\_{1}-\alpha\_{i}-1} d\mathcal{H}\_{i}(\tau), \ \eta \in (0,1). \end{split}$$

This function <sup>Θ</sup><sup>0</sup> <sup>∈</sup> *<sup>L</sup>*1(0, 1), because

$$\begin{split} \int\_{0}^{1} \Theta\_{0}(\eta) \, d\eta &= \frac{1}{\Gamma(\delta\_{1})} \left( \frac{1}{\delta\_{1} - a\_{0}} + \frac{1}{\delta\_{1}} \right) + \frac{1}{a\_{2}(\delta\_{1} - a\_{0})} \\ &\times \sum\_{i=1}^{n} \frac{1}{\Gamma(\delta\_{1} - a\_{i})} \int\_{0}^{1} \tau^{\delta\_{1} - a\_{i} - 1} \, d\mathcal{H}\_{i}(\tau) < \infty. \end{split} \tag{24}$$

Then for any *s*1,*s*<sup>2</sup> ∈ [0, 1] with *s*<sup>1</sup> < *s*<sup>2</sup> and (*u*, *v*) ∈ E, by (23) and (24) we conclude

$$\left| \mathcal{A}\_1(u,v)(s\_1) - \mathcal{A}\_1(u,v)(s\_2) \right| = \left| \int\_{s\_1}^{s\_2} (\mathcal{A}\_1(u,v))'(\tau) \, d\tau \right| \le \Xi\_2^{\varrho\_1 - 1} M\_1^{\varrho\_1 - 1} \int\_{s\_1}^{s\_2} \Theta\_0(\tau) \, d\tau. \tag{25}$$

By (24) and (25), we deduce that A1(E) is equicontinuous. By a similar method, we find that A2(E) is also equicontinuous, and then A(E) is equicontinuous too. Using the Arzela–Ascoli theorem, we conclude that A1(E) and A2(E) are relatively compact sets, and so A(E) is also relatively compact. In addition, we can show that A1, A<sup>2</sup> and A are continuous on P (see Lemma 1.4.1 from [14]). Hence, A is a completely continuous operator on P.

We define now the cone

$$\mathcal{P}\_0 = \{ (u, v) \in \mathcal{P}\_{\prime} \colon u(\eta) \ge \eta^{\delta\_1 - 1} ||u||\_{\prime} \ v(\eta) \ge \eta^{\delta\_2 - 1} ||v||\_{\prime} \ \eta \in [0, 1] \}.$$

Under the assumptions (*I*1) and (*I*2), by using Lemma 4, we deduce that A(P) ⊂ P0, and so A|P<sup>0</sup> : P<sup>0</sup> → P<sup>0</sup> (denoted again by A) is also a completely continuous operator. For *θ* > 0 we denote by *B<sup>θ</sup>* the open ball centered at zero of radius *θ*, and by *B<sup>θ</sup>* and *∂B<sup>θ</sup>* its closure and its boundary, respectively.

We also denote by *<sup>M</sup>*<sup>1</sup> =% <sup>1</sup> <sup>0</sup> <sup>J</sup>1(*τ*)*ψ*1(*τ*)*dτ*, *<sup>M</sup>*<sup>2</sup> <sup>=</sup>% <sup>1</sup> <sup>0</sup> <sup>J</sup>3(*τ*)*ψ*2(*τ*)*dτ*, *<sup>M</sup>*<sup>3</sup> <sup>=</sup>% <sup>1</sup> <sup>0</sup> J2(*τ*)*dτ*, *<sup>M</sup>*<sup>4</sup> = % <sup>1</sup> <sup>0</sup> <sup>J</sup>4(*τ*)*dτ*, and for *<sup>θ</sup>*1, *<sup>θ</sup>*<sup>2</sup> <sup>∈</sup> (0, 1), *<sup>θ</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>θ</sup>*2, *<sup>M</sup>*<sup>5</sup> <sup>=</sup> % *<sup>θ</sup>*<sup>2</sup> *<sup>θ</sup>*<sup>1</sup> J2(*ζ*) % *<sup>ζ</sup> <sup>θ</sup>*<sup>1</sup> G1(*ζ*, *<sup>τ</sup>*) *<sup>d</sup><sup>τ</sup>* 1−1 *dζ*, *<sup>M</sup>*<sup>6</sup> = % *<sup>θ</sup>*<sup>2</sup> *<sup>θ</sup>*<sup>1</sup> J4(*ζ*) % *<sup>ζ</sup> <sup>θ</sup>*<sup>1</sup> G3(*ζ*, *<sup>τ</sup>*) *<sup>d</sup><sup>τ</sup>* 2−1 *dζ*.

**Theorem 2.** *We suppose that assumptions* (*I*1)*,* (*I*2)*,*

(*I*3) *There exist ci* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, ... , 4 *with* <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *ci* <sup>&</sup>gt; <sup>0</sup>*, di* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, ... , 4 *with* <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *di* > 0*, and μ*<sup>1</sup> ≥ 1, *μ*<sup>2</sup> ≥ 1 *such that*

$$\chi\_{10} = \limsup\_{\sum\_{i=1}^4 c\_i z\_i \to 0} \max\_{\eta \in [0,1]} \frac{\chi\_1(\eta, z\_1, z\_2, z\_3, z\_4)}{\varphi\_{r\_1}((c\_1 z\_1 + c\_2 z\_2 + c\_3 z\_3 + c\_4 z\_4)^{\mu\_1})} < l\_1 \lambda$$

*and*

$$\chi\_{20} = \limsup\_{\sum\_{i=1}^4 d\_i z\_i \to 0} \max\_{\eta \in [0,1]} \frac{\chi\_2(\eta, z\_1, z\_2, z\_3, z\_4)}{\varphi\_{r\_2}((d\_1 z\_1 + d\_2 z\_2 + d\_3 z\_3 + d\_4 z\_4)^{\mu\_2})} < l\_2 \lambda$$

*where <sup>l</sup>*<sup>1</sup> = (2*r*1−1*M*1*Mr*1−<sup>1</sup> <sup>3</sup> *ρ μ*1(*r*1−1) <sup>1</sup> )−1*, <sup>l</sup>*<sup>2</sup> = (2*r*2−1*M*2*Mr*2−<sup>1</sup> <sup>4</sup> *ρ μ*2(*r*2−1) <sup>2</sup> )−1*, with <sup>ρ</sup>*<sup>1</sup> = 2 max *c*1, *c*2, *<sup>c</sup>*<sup>3</sup> <sup>Γ</sup>(*σ*1+1), *<sup>c</sup>*<sup>4</sup> Γ(*σ*2+1) *, <sup>ρ</sup>*<sup>2</sup> <sup>=</sup> 2 max *d*1, *d*2, *<sup>d</sup>*<sup>3</sup> <sup>Γ</sup>(*ς*1+1), *<sup>d</sup>*<sup>4</sup> Γ(*ς*2+1) *;*

(*I*4) *There exist pi* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, ... , 4 *with* <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *pi* <sup>&</sup>gt; <sup>0</sup>*, qi* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, ... , 4 *with* <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *qi* > 0*, θ*1, *θ*<sup>2</sup> ∈ (0, 1)*, θ*<sup>1</sup> < *θ*<sup>2</sup> *and λ*<sup>1</sup> > 1, *λ*<sup>2</sup> > 1 *such that*

$$f\_{\infty} = \liminf\_{\sum\_{i=1}^{4} p\_i z\_i \to \infty} \min\_{\eta \in [\theta\_1 \theta\_2]} \frac{f(\eta\_1 z\_1, z\_2, z\_3, z\_4)}{\overline{\rho\_{r\_1}(p\_1 z\_1 + p\_2 z\_2 + p\_3 z\_3 + p\_4 z\_4)}} > l\_3 \lambda$$

*or*

$$g\_{\infty} = \liminf\_{\sum\_{i=1}^{4} q\_i z\_i \to \infty} \min\_{\eta \in [\theta\_1, \theta\_2]} \frac{g(\eta\_1 z\_1, z\_2, z\_3, z\_4)}{\overline{\rho\_{r\_2} (q\_1 z\_1 + q\_2 z\_2 + q\_3 z\_3 + q\_4 z\_4)}} > l\_4 \lambda$$

$$\begin{split} & where \; l\_{3} = \lambda\_{1} (2\rho\_{3}M\_{5}\theta\_{1}^{\delta\_{1}-1})^{1-r\_{1}}, \; l\_{4} = \lambda\_{2} (2\rho\_{4}M\_{6}\theta\_{1}^{\delta\_{2}-1})^{1-r\_{2}} \; with \; \rho\_{3} = \min\left\{p\_{1}\theta\_{1}^{\delta\_{1}-1}\right\}, \\ & \; p\_{2}\theta\_{1}^{\delta\_{2}-1}, \; \frac{p\_{3}\theta\_{1}^{r\_{1}+\delta\_{1}-1}\Gamma(\delta\_{1})}{\Gamma(\delta\_{1}+\sigma\_{1})}, \; \frac{p\_{4}\theta\_{1}^{\sigma\_{2}+\delta\_{2}-1}\Gamma(\delta\_{2})}{\Gamma(\delta\_{2}+\sigma\_{2})} \right\}, \; \rho\_{4} = \min\left\{q\_{1}\theta\_{1}^{\delta\_{1}-1}, q\_{2}\theta\_{1}^{\delta\_{2}-1}, \; \frac{q\_{3}\theta\_{1}^{\delta\_{1}+\delta\_{1}-1}\Gamma(\delta\_{1})}{\Gamma(\delta\_{1}+\varepsilon\_{1})}, \; \rho\_{4} = \lambda\_{2} (2\rho\_{4}\theta\_{1}^{\delta\_{2}-1}, \; \rho\_{4} = \lambda\_{2}), \; \rho\_{4} = \lambda\_{2} (2\rho\_{4}\theta\_{1}^{\delta\_{2}}, \; \rho\_{4} = \lambda\_{2}), \\ & \; \frac{q\_{4}\theta\_{1}^{\delta\_{2}+\delta\_{2}-1}\Gamma(\delta\_{2})}{\Gamma(\delta\_{2}+\varepsilon\_{2})} \; \}, \; \end{split}$$

*hold. Then there exists a positive solution* (*u*(*τ*), *v*(*τ*)), *τ* ∈ [0, 1] *of problems (1) and (2).*

**Proof.** By (*I*3) there exists *R* ∈ (0, 1) such that

$$\begin{array}{l} \chi\_1(\eta\_\prime z\_1, z\_2, z\_3, z\_4) \le l\_1 \varrho\_{r\_1}((\mathfrak{c}\_1 z\_1 + \mathfrak{c}\_2 z\_2 + \mathfrak{c}\_3 z\_3 + \mathfrak{c}\_4 z\_4)^{\mu\_1}),\\ \chi\_2(\eta\_\prime z\_1, z\_2, z\_3, z\_4) \le l\_2 \varrho\_{r\_2}((d\_1 z\_1 + d\_2 z\_2 + d\_3 z\_3 + d\_4 z\_4)^{\mu\_2}),\end{array} \tag{26}$$

for all *<sup>η</sup>* <sup>∈</sup> [0, 1], *zi* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, ... , 4 with <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *cizi* <sup>≤</sup> *<sup>R</sup>* and <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *dizi* ≤ *R*. We define *R*<sup>1</sup> ≤ min{*R*/*ρ*1, *R*/*ρ*2, *R*}. For any (*u*, *v*) ∈ *BR*<sup>1</sup> ∩ P and *ζ* ∈ [0, 1] we have

$$\begin{cases} c\_1 u(\zeta) + c\_2 v(\zeta) + c\_3 t\_{0+}^{\sigma\_1} u(\zeta) + c\_4 I\_{0+}^{\sigma\_2} v(\zeta) \\ \quad \le 2 \max\left\{ c\_1, c\_2, \frac{c\_3}{\Gamma(\sigma\_1 + 1)'}, \frac{c\_4}{\Gamma(\sigma\_2 + 1)} \right\} \|(u, v)\|\|\_{\mathcal{V}} = \rho\_1 \|(u, v)\|\|\_{\mathcal{V}} \le \rho\_1 R\_1 \le R\_1 \\\ d\_1 u(\zeta) + d\_2 v(\zeta) + d\_3 t\_{0+}^{\xi\_1} u(\zeta) + d\_4 I\_{0+}^{\xi\_2} v(\zeta) \\\ \quad \le 2 \max\left\{ d\_1, d\_2, \frac{d\_3}{\Gamma(\zeta\_1 + 1)'}, \frac{d\_4}{\Gamma(\zeta\_2 + 1)} \right\} \|(u, v)\|\|\_{\mathcal{V}} = \rho\_2 \|(u, v)\|\|\_{\mathcal{V}} \le \rho\_2 R\_1 \le R. \end{cases}$$

Then by (26) and Lemma 3, for any (*u*, *v*) ∈ *∂BR*<sup>1</sup> ∩ P<sup>0</sup> and *η* ∈ [0, 1], we deduce

(A1(*u*, *v*))(*η*) ≤ <sup>1</sup> 0 J2(*ζ*)*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ϑ*)*f*(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I σ*1 <sup>0</sup>+*u*(*ϑ*), *I σ*2 <sup>0</sup>+*v*(*ϑ*)) *dϑ dζ* = *M*3*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ϑ*)*f*(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I σ*1 <sup>0</sup>+*u*(*ϑ*), *I σ*2 <sup>0</sup>+*v*(*ϑ*)) *dϑ* ≤ *M*3*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ϑ*)*ψ*1(*ϑ*)*χ*1(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I σ*1 <sup>0</sup>+*u*(*ϑ*), *I σ*2 <sup>0</sup>+*v*(*ϑ*)) *dϑ* ≤ *M*3*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ϑ*)*ψ*1(*ϑ*)*l*1*ϕr*<sup>1</sup> (*c*1*u*(*ϑ*) + *c*2*v*(*ϑ*) + *c*<sup>3</sup> *I σ*1 <sup>0</sup>+*u*(*ϑ*) + *c*<sup>4</sup> *I σ*2 0+*v*(*ϑ*))*μ*<sup>1</sup> *dϑ* ≤ *M*3*ϕ*<sup>1</sup> (*ϕr*<sup>1</sup> ((*ρ*<sup>1</sup> (*u*, *v*) <sup>Y</sup> )*μ*<sup>1</sup> ))*ϕ*<sup>1</sup> (*l*1)*ϕ*<sup>1</sup> (*M*1) <sup>=</sup> *<sup>M</sup>*3*M*1−<sup>1</sup> <sup>1</sup> *l* 1−1 <sup>1</sup> *ρ μ*1 1 (*u*, *v*) *μ*1 <sup>Y</sup> <sup>≤</sup> *<sup>M</sup>*3*M*1−<sup>1</sup> <sup>1</sup> *l* 1−1 <sup>1</sup> *ρ μ*1 1 (*u*, *v*) <sup>Y</sup> <sup>=</sup> <sup>1</sup> 2 (*u*, *v*) Y, (A2(*u*, *v*))(*η*) ≤ <sup>1</sup> 0 J4(*ζ*)*ϕ*<sup>2</sup> <sup>1</sup> 0 J3(*ϑ*)*g*(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I ς*1 <sup>0</sup>+*u*(*ϑ*), *I ς*2 <sup>0</sup>+*v*(*ϑ*)) *dϑ dζ* = *M*4*ϕ*<sup>2</sup> <sup>1</sup> 0 J3(*ϑ*)*g*(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I ς*1 <sup>0</sup>+*u*(*ϑ*), *I ς*2 <sup>0</sup>+*v*(*ϑ*)) *dϑ* ≤ *M*4*ϕ*<sup>2</sup> <sup>1</sup> 0 J3(*ϑ*)*ψ*2(*ϑ*)*χ*2(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I ς*1 <sup>0</sup>+*u*(*ϑ*), *I ς*2 <sup>0</sup>+*v*(*ϑ*)) *dϑ* ≤ *M*4*ϕ*<sup>2</sup> <sup>1</sup> 0 J3(*ϑ*)*ψ*2(*ϑ*)*l*2*ϕr*<sup>2</sup> (*d*1*u*(*ϑ*) + *d*2*v*(*ϑ*) + *d*<sup>3</sup> *I ς*1 <sup>0</sup>+*u*(*ϑ*) + *d*<sup>4</sup> *I ς*2 0+*v*(*ϑ*))*μ*<sup>2</sup> *dϑ* ≤ *M*4*ϕ*<sup>2</sup> (*ϕr*<sup>2</sup> ((*ρ*<sup>2</sup> (*u*, *v*) <sup>Y</sup> )*μ*<sup>2</sup> ))*ϕ*<sup>2</sup> (*l*2)*ϕ*<sup>2</sup> (*M*2) <sup>=</sup> *<sup>M</sup>*4*M*2−<sup>1</sup> <sup>2</sup> *l* 2−1 <sup>2</sup> *ρ μ*2 2 (*u*, *v*) *μ*2 <sup>Y</sup> <sup>≤</sup> *<sup>M</sup>*4*M*2−<sup>1</sup> <sup>2</sup> *l* 2−1 <sup>2</sup> *ρ μ*2 2 (*u*, *v*) <sup>Y</sup> <sup>=</sup> <sup>1</sup> 2 (*u*, *v*) Y.

Then we conclude that

$$\|\mathcal{A}(\boldsymbol{u},\boldsymbol{v})\|\_{\mathcal{Y}} = \|\mathcal{A}\_1(\boldsymbol{u},\boldsymbol{v})\| + \|\mathcal{A}\_2(\boldsymbol{u},\boldsymbol{v})\| \le \|(\boldsymbol{u},\boldsymbol{v})\|\_{\mathcal{Y}}, \; \forall (\boldsymbol{u},\boldsymbol{v}) \in \partial B\_{R\_1} \cap \mathcal{P}\_0. \tag{27}$$

Now we suppose in (*I*4) that *f*<sup>∞</sup> > *l*<sup>3</sup> (in a similar manner we study the case *g*<sup>∞</sup> > *l*4). Then there exists *C*<sup>1</sup> > 0 such that

$$f(\eta, z\_1, z\_2, z\_3, z\_4) \ge l\_3 \wp\_{r\_1}(p\_1 z\_1 + p\_2 z\_2 + p\_3 z\_3 + p\_4 z\_4) - \mathcal{C}\_1. \tag{28}$$

for all *η* ∈ [*θ*1, *θ*2] and *zi* ≥ 0, *i* = 1, ... , 4. By definition of *I σ*1 <sup>0</sup>+, for any (*u*, *v*) ∈ P<sup>0</sup> and *η* ∈ [0, 1] we have

$$\begin{array}{lcl} I\_{0+}^{\sigma\_{1}}u(\eta) = \frac{1}{\Gamma(\sigma\_{1})} \int\_{0}^{\eta} (\eta - \zeta)^{\sigma\_{1} - 1} u(\zeta) \, d\zeta \geq \frac{1}{\Gamma(\sigma\_{1})} \int\_{0}^{\eta} (\eta - \zeta)^{\sigma\_{1} - 1} \zeta^{\delta\_{1} - 1} ||u|| \, d\zeta\\ \overset{\circ}{=} \frac{||u||}{\Gamma(\sigma\_{1})} \int\_{0}^{1} (\eta - \eta y)^{\sigma\_{1} - 1} \eta^{\delta\_{1} - 1} y^{\delta\_{1} - 1} \eta \, dy = \frac{||u||}{\Gamma(\sigma\_{1})} \eta^{\sigma\_{1} + \delta\_{1} - 1} \int\_{0}^{1} y^{\delta\_{1} - 1} (1 - y)^{\sigma\_{1} - 1} \, dy\\ = \frac{||u||}{\Gamma(\sigma\_{1})} \eta^{\sigma\_{1} + \delta\_{1} - 1} B(\delta\_{1}, \sigma\_{1}) = \frac{||u|| \eta^{\sigma\_{1} + \delta\_{1} - 1} \Gamma(\delta\_{1})}{\Gamma(\delta\_{1} + \sigma\_{1})} \end{array}$$

and in a similar way

$$I\_{0+}^{\sigma\_2}v(\eta) \ge \frac{||v||\eta^{\sigma\_2+\delta\_2-1}\Gamma(\delta\_2)}{\Gamma(\delta\_2+\sigma\_2)}\nu$$

where *B*(*p*, *q*) is the first Euler function. Then by using (28) and (29), for any (*u*, *v*) ∈ P<sup>0</sup> and *η* ∈ [*θ*1, *θ*2] we obtain

(A1(*u*, *v*))(*η*) ≥ *<sup>θ</sup>*<sup>2</sup> *θ*1 G2(*η*, *ζ*)*ϕ*<sup>1</sup> *<sup>ζ</sup> θ*1 G1(*ζ*, *ϑ*)*f*(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I σ*1 <sup>0</sup>+*u*(*ϑ*), *I σ*2 <sup>0</sup>+*v*(*ϑ*))*dϑ dζ* ≥ *θ δ*1−1 1 *<sup>θ</sup>*<sup>2</sup> *θ*1 J2(*ζ*) *<sup>ζ</sup> θ*1 G1(*ζ*, *ϑ*) *l*3(*p*1*u*(*ϑ*) + *p*2*v*(*ϑ*) + *p*<sup>3</sup> *I σ*1 <sup>0</sup>+*u*(*ϑ*) + *p*<sup>4</sup> *I σ*2 0+*v*(*ϑ*))*r*1−<sup>1</sup> −*C*1]*dϑ*) 1−1 *dζ* ≥ *θ δ*1−1 1 *<sup>θ</sup>*<sup>2</sup> *θ*1 J2(*ζ*) *<sup>ζ</sup> θ*1 G1(*ζ*, *ϑ*) *l*3 *p*1*θ δ*1−1 1 *u* + *<sup>p</sup>*2*θδ*2−<sup>1</sup> 1 *v* +*p*<sup>3</sup> *θ σ*1+*δ*1−1 <sup>1</sup> Γ(*δ*1) <sup>Γ</sup>(*δ*<sup>1</sup> <sup>+</sup> *<sup>σ</sup>*1) *u* + *p*<sup>4</sup> *θ σ*2+*δ*2−1 <sup>1</sup> Γ(*δ*2) <sup>Γ</sup>(*δ*<sup>2</sup> <sup>+</sup> *<sup>σ</sup>*2) *v* -*r*1−<sup>1</sup> − *C*<sup>1</sup> ⎤ <sup>⎦</sup>*d<sup>ϑ</sup>* ⎞ ⎠ 1−1 *dζ* ≥ *θ δ*1−1 1 *<sup>θ</sup>*<sup>2</sup> *θ*1 J2(*ζ*) , *<sup>ζ</sup> θ*1 G1(*ζ*, *ϑ*) ! *l*3 , min *p*1*θ δ*1−1 <sup>1</sup> , *p*2*θ δ*2−1 <sup>1</sup> , *p*<sup>3</sup> *θ σ*1+*δ*1−1 <sup>1</sup> Γ(*δ*1) <sup>Γ</sup>(*δ*<sup>1</sup> <sup>+</sup> *<sup>σ</sup>*1) , *<sup>p</sup>*4*θσ*2+*δ*2−<sup>1</sup> <sup>1</sup> Γ(*δ*2) Γ(*δ*<sup>2</sup> + *σ*2) # 2 (*u*, *v*) Y -*r*1−<sup>1</sup> − *C*<sup>1</sup> ⎤ <sup>⎦</sup>*d<sup>ϑ</sup>* ⎞ ⎠ 1−1 *dζ* = *θ δ*1−1 1 *<sup>θ</sup>*<sup>2</sup> *θ*1 J2(*ζ*) *<sup>ζ</sup> θ*1 G1(*ζ*, *ϑ*) *l*3(2*ρ*3 (*u*, *v*) Y ) *<sup>r</sup>*1−<sup>1</sup> <sup>−</sup> *<sup>C</sup>*<sup>1</sup> *dϑ* 1−<sup>1</sup> *dζ* = *M*5*θ δ*1−1 1 *l*3(2*ρ*3 (*u*, *v*) Y ) *<sup>r</sup>*1−<sup>1</sup> <sup>−</sup> *<sup>C</sup>*<sup>1</sup> 1−<sup>1</sup> = *Mr*1−<sup>1</sup> <sup>5</sup> *θ* (*δ*1−1)(*r*1−1) <sup>1</sup> *<sup>l</sup>*32*<sup>r</sup>*1−1*ρr*1−<sup>1</sup> 3 (*u*, *v*) *r*1−1 <sup>Y</sup> <sup>−</sup> *<sup>M</sup>r*1−<sup>1</sup> <sup>5</sup> *θ* (*δ*1−1)(*r*1−1) <sup>1</sup> *C*<sup>1</sup> 1−1 = *λ*1 (*u*, *v*) *r*1−1 <sup>Y</sup> <sup>−</sup> *<sup>C</sup>*<sup>2</sup> 1−1 , *<sup>C</sup>*<sup>2</sup> = *<sup>M</sup>r*1−<sup>1</sup> <sup>5</sup> *θ* (*δ*1−1)(*r*1−1) <sup>1</sup> *C*1.

Then we deduce

$$\|\mathcal{A}(\boldsymbol{u},\boldsymbol{v})\|\|\_{\mathcal{Y}} \ge \|\mathcal{A}\_{1}(\boldsymbol{u},\boldsymbol{v})\| \ge |\mathcal{A}\_{1}(\boldsymbol{u},\boldsymbol{v})(\theta\_{1})| \ge \left(\lambda\_{1} \|(\boldsymbol{u},\boldsymbol{v})\|\|\_{\mathcal{Y}}^{r\_{1}-1} - \mathbb{C}\_{2}\right)^{\varrho\_{1}-1}, \ \forall \,(\boldsymbol{u},\boldsymbol{v}) \in \mathcal{P}\_{0}.$$

$$\text{We choose } \mathcal{R}\_{2} \ge \max\left\{1, \mathcal{C}\_{2}^{\varrho\_{1}-1}/(\lambda\_{1}-1)^{\varrho\_{1}-1}\right\} \text{ and we obtain}$$

$$\|\mathcal{A}(\boldsymbol{u},\boldsymbol{v})\|\|\boldsymbol{y} \ge \|(\boldsymbol{u},\boldsymbol{v})\|\|\boldsymbol{y}, \ \forall \,(\boldsymbol{u},\boldsymbol{v}) \in \partial B\_{\mathcal{R}\_{2}} \cap \mathcal{P}\_{0}.\tag{30}$$

By Lemma 5, (27), (30) and Theorem 1 (i), we conclude that A has a fixed point (*u*, *v*) ∈ (*BR*<sup>2</sup> \ *BR*<sup>1</sup> ) ∩ P0, so *R*<sup>1</sup> ≤ (*u*, *v*) <sup>Y</sup> <sup>≤</sup> *<sup>R</sup>*2, and *<sup>u</sup>*(*τ*) <sup>≥</sup> *<sup>τ</sup>δ*1−<sup>1</sup> *u* and *<sup>v</sup>*(*τ*) <sup>≥</sup> *<sup>τ</sup>δ*2−<sup>1</sup> *v* for all *τ* ∈ [0, 1]. Then *u* > 0 or *v* > 0, that is *u*(*τ*) > 0 for all *τ* ∈ (0, 1] or *v*(*τ*) > 0 for all *τ* ∈ (0, 1]. Hence (*u*(*τ*), *v*(*τ*)), *τ* ∈ [0, 1] is a positive solution of problem (1) and (2).

**Theorem 3.** *We suppose that assumptions* (*I*1)*,* (*I*2)*,*

(29)

(*I*5) *There exist ei* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, ... , 4 *with* <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *ei* <sup>&</sup>gt; <sup>0</sup>*, ki* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, ... , 4 *with* <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *ki* > 0 *such that*

$$\chi\_{1\infty} = \limsup\_{\sum\_{i=1}^{4} \varepsilon\_{i} z\_{i} \to \infty} \max\_{\eta \in [0,1]} \frac{\chi\_{1}(\eta, z\_{1}, z\_{2}, z\_{3}, z\_{4})}{\overline{\rho\_{r\_{1}}(e\_{1}z\_{1} + e\_{2}z\_{2} + e\_{3}z\_{3} + e\_{4}z\_{4})} < m\_{1}\lambda$$

*and*

$$\chi\_{2\infty} = \limsup\_{\sum\_{i=1}^4 k\_i z\_i \to \infty} \max\_{\eta \in [0,1]} \frac{\chi\_2(\eta, z\_1, z\_2, z\_3, z\_4)}{\overline{\rho\_{r\_2}(k\_1 z\_1 + k\_2 z\_2 + k\_3 z\_3 + k\_4 z\_4)}} < m\_{2\lambda}$$

*where m*<sup>1</sup> <sup>&</sup>lt; min{1/(2*M*1(*ξ*1*M*3)*r*1−1), 1/(*M*1(2*ξ*1*M*3)*r*1−1)}*, <sup>m</sup>*<sup>2</sup> <sup>&</sup>lt; min{1/(2*M*2(*ξ*2*M*4)*r*2−1), 1/(*M*2(2*ξ*2*M*4)*r*2−1)} *with <sup>ξ</sup>*<sup>1</sup> <sup>=</sup> 2 max *e*1,*e*2, *<sup>e</sup>*<sup>3</sup> <sup>Γ</sup>(*σ*1+1), *<sup>e</sup>*<sup>4</sup> Γ(*σ*2+1) *, <sup>ξ</sup>*<sup>2</sup> <sup>=</sup> 2 max *k*1, *k*2, *<sup>k</sup>*<sup>3</sup> <sup>Γ</sup>(*ς*1+1), *<sup>k</sup>*<sup>4</sup> Γ(*ς*2+1) *;* (*I*6) *There exist si* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, ... , 4 *with* <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *si* <sup>&</sup>gt; <sup>0</sup>*, ti* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, ... , 4 *with* <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *ti* > 0*,*

 $\text{10) 1 \text{ttr} \text{ } \text{error} \text{s}\_1 \le \text{\textquotedblleft}\_1 \text{ } -1 \text{ } -1 \text{ } -1 \text{ } -1 \text{ } -1 \text{ } \text{\textquotedblright} \text{ } \text{\textquotedblleft}\_1 \text{\textquotedblright}\_1 \le \text{\textquotedblleft}\_1 \text{\textquotedblright}\_1 \le \text{\textquotedblleft}\_1 \text{\textquotedblright}\_1 \le \text{\textquotedblleft}\_1 \text{\textquotedblright}\_1 \le \text{\textquotedblleft}\_1 \text{\textquotedblright}\_1 \text{\textquotedblright}\_1 \le \text{\textquotedblleft}\_1 \text{\textquotedblleft}\_1 \text{\textquotedblright}\_1 \le \text{\textquotedblleft}\_1 \text{\textquotedblleft}\_1 \text{\textquotedblleft}\_1 \text{\textquotedbl}\_1 \le \text{\textquotedblleft}\_1 \text{\textquotedbl}\_1 \text{\textquotedbl}\_1 \text{\textquotedbl}\_1 \le \text{\textquotedblleft}\_1 \text{\textquotedbl}\_1 \text{\textquotedbl}\_1 \le \text{\textquotedbl}\_1 \text{\textquotedbl}\_1 \text{\textquotedbl}\_1 \le \text{\textquotedbl}\_1 \text{\textquotedbl}\_1 \text{\textquotedbl}\_1 \le \text{\textquotedbl}\_1 \text{\textquotedbl}\_1 \text{\textquotedbl}\_1 \le \text{\textquotedbl}\_1 \text{\textquotedbl}\_1 \text{\textquotedbl}\_1 \le \text{\textquotedbl}\_1 \text{\textquotedbl}\_1 \text{\textquotedbl}\_1 \le \text{\textquotedbl}\_1 \text{\textquotedbl}\_1 \text{\textquotedbl}\_1 \le \text{\textquotedbl}\_1 \text{\textquotedbl}\_1 \text{\textquotedbl}\_1 \le \text{\textquotedbl}\_1 \text{\textquotedbl}\_1$ 

$$f\_0 = \liminf\_{\sum\_{i=1}^4 s\_i z\_i \to 0} \min\_{\eta \in [\theta\_1 \theta\_2]} \frac{f(\eta, z\_1, z\_2, z\_3, z\_4)}{\varrho\_{r\_1} ((s\_1 z\_1 + s\_2 z\_2 + s\_3 z\_3 + s\_4 z\_4)^{\nu\_1})} > m\_3 \nu$$

*or*

$$g\_0 = \liminf\_{\sum\_{i=1}^4 t\_i z\_i \to 0} \min\_{\eta \in [\theta\_1, \theta\_2]} \frac{g(\eta, z\_1, z\_2, z\_3, z\_4)}{\varphi\_{r\_2}((t\_1 z\_1 + t\_2 z\_2 + t\_3 z\_3 + t\_4 z\_4)^{\nu\_2})} > m\_{4\nu}$$

$$\begin{split} \text{where } m\_{\mathfrak{3}} &= \lambda\_{\mathfrak{3}}^{r\_{1}-1} (M\_{5} 2^{\nu\_{1}} \xi\_{\mathfrak{3}}^{\nu\_{1}} \theta\_{1}^{\delta\_{1}-1})^{1-r\_{1}}, m\_{4} = \lambda\_{\mathfrak{4}}^{r\_{2}-1} (M\_{6} 2^{\nu\_{2}} \xi\_{\mathfrak{4}}^{\nu\_{2}} \theta\_{1}^{\delta\_{2}-1})^{1-r\_{2}}, \text{with } \xi\_{\mathfrak{3}} = \mathfrak{4}, \\ \min \left\{ s\_{1} \theta\_{1}^{\delta\_{1}-1}, s\_{2} \theta\_{1}^{\delta\_{2}-1}, \frac{s\_{3} \theta\_{1}^{\nu\_{1}+\delta\_{1}-1} \Gamma(\delta\_{1})}{\Gamma(\delta\_{1}+\sigma\_{1})}, \frac{s\_{4} \theta\_{1}^{\nu\_{2}+\delta\_{2}-1} \Gamma(\delta\_{2})}{\Gamma(\delta\_{2}+\sigma\_{2})} \right\}, \xi\_{4} = \min \left\{ t\_{1} \theta\_{1}^{\delta\_{1}-1}, t\_{2} \theta\_{1}^{\delta\_{2}-1} \right\} \\ \frac{t\_{3} \theta\_{1}^{\delta\_{1}+\delta\_{1}-1} \Gamma(\delta\_{1})}{\Gamma(\delta\_{1}+\varepsilon\_{1})}, \frac{t\_{4} \theta\_{1}^{\delta\_{2}+\delta\_{2}-1} \Gamma(\delta\_{2})}{\Gamma(\delta\_{2}+\varepsilon\_{2})} \end{split}$$

*hold. Then there exists a positive solution* (*u*(*τ*), *v*(*τ*)), *τ* ∈ [0, 1] *of problem (1) and (2).*

**Proof.** From (*I*5) there exist *C*<sup>3</sup> > 0, *C*<sup>4</sup> > 0 such that

$$\begin{array}{c} \chi\_1(\eta, z\_1, z\_2, z\_3, z\_4) \le m\_1 \varrho\_{r\_1} (e\_1 z\_1 + e\_2 z\_2 + e\_3 z\_3 + e\_4 z\_4) + \mathcal{C}\_3 \\ \chi\_2(\eta, z\_1, z\_2, z\_3, z\_4) \le m\_2 \varrho\_{r\_2} (k\_1 z\_1 + k\_2 z\_2 + k\_3 z\_3 + k\_4 z\_4) + \mathcal{C}\_4 \end{array} \tag{31}$$

for any *η* ∈ [0, 1] and *zi* ≥ 0, *i* = 1, ... , 4. By using (*I*2) and (31) for any (*u*, *v*) ∈ P<sup>0</sup> and *η* ∈ [0, 1] we find

A1(*u*, *v*)(*η*) ≤ <sup>1</sup> 0 J2(*ζ*)*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ϑ*)*f*(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I σ*1 <sup>0</sup>+*u*(*ϑ*), *I σ*2 <sup>0</sup>+*v*(*ϑ*)) *dϑ dζ* ≤ *M*3*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ϑ*)*ψ*1(*ϑ*)*χ*1(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I σ*1 <sup>0</sup>+*u*(*ϑ*), *I σ*2 <sup>0</sup>+*v*(*ϑ*)) *dϑ* ≤ *M*3*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ϑ*)*ψ*1(*ϑ*) 0 *m*1*ϕr*<sup>1</sup> *e*1*u*(*ϑ*) + *e*2*v*(*ϑ*) + *e*<sup>3</sup> *I σ*1 <sup>0</sup>+*u*(*ϑ*) + *e*<sup>4</sup> *I σ*2 <sup>0</sup>+*v*(*ϑ*) + *C*<sup>3</sup> 1 *dϑ* ≤ *M*3*ϕ*<sup>1</sup> , <sup>1</sup> 0 J1(*ϑ*)*ψ*1(*ϑ*) ! *m*<sup>1</sup> *e*1 *u* + *e*<sup>2</sup> *v* + *e*3 *u* <sup>Γ</sup>(*σ*<sup>1</sup> <sup>+</sup> <sup>1</sup>) <sup>+</sup> *e*4 *v* Γ(*σ*<sup>2</sup> + 1) *r*1−<sup>1</sup> + *C*<sup>3</sup> " *dϑ* - ≤ *M*3*ϕ*<sup>1</sup> \* *m*<sup>1</sup> max *e*1,*e*2, *<sup>e</sup>*<sup>3</sup> <sup>Γ</sup>(*σ*1+1), *<sup>e</sup>*<sup>4</sup> Γ(*σ*2+1) 2 (*u*, *v*) Y *r*1−1 + *C*<sup>3</sup> + × <sup>1</sup> 0 J1(*ϑ*)*ψ*1(*ϑ*) *dϑ* 1−<sup>1</sup> = *M*1−<sup>1</sup> <sup>1</sup> *M*<sup>3</sup> *<sup>m</sup>*1*ξr*1−<sup>1</sup> 1 (*u*, *v*) *r*1−1 <sup>Y</sup> <sup>+</sup> *<sup>C</sup>*<sup>3</sup> 1−1 ,

and

A2(*u*, *v*)(*η*) ≤ <sup>1</sup> 0 J4(*ζ*)*ϕ*<sup>2</sup> <sup>1</sup> 0 J3(*ϑ*)*g*(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I ς*1 <sup>0</sup>+*u*(*ϑ*), *I ς*2 <sup>0</sup>+*v*(*ϑ*)) *dϑ dζ* ≤ *M*4*ϕ*<sup>2</sup> <sup>1</sup> 0 J3(*ϑ*)*ψ*2(*ϑ*)*χ*2(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I ς*1 <sup>0</sup>+*u*(*ϑ*), *I ς*2 <sup>0</sup>+*v*(*ϑ*)) *dϑ* ≤ *M*4*ϕ*<sup>2</sup> <sup>1</sup> 0 J3(*ϑ*)*ψ*2(*ϑ*) 0 *m*2*ϕr*<sup>2</sup> *k*1*u*(*ϑ*) + *k*2*v*(*ϑ*) + *k*<sup>3</sup> *I ς*1 <sup>0</sup>+*u*(*ϑ*) + *k*<sup>4</sup> *I ς*2 <sup>0</sup>+*v*(*ϑ*) + *C*<sup>4</sup> 1 *dϑ* ≤ *M*4*ϕ*<sup>2</sup> , <sup>1</sup> 0 J3(*ϑ*)*ψ*2(*ϑ*) ! *m*<sup>2</sup> *k*1 *u* + *k*<sup>2</sup> *v* + *k*3 *u* <sup>Γ</sup>(*ς*<sup>1</sup> <sup>+</sup> <sup>1</sup>) <sup>+</sup> *k*4 *v* Γ(*ς*<sup>2</sup> + 1) *r*2−<sup>1</sup> + *C*<sup>4</sup> " *dϑ* - ≤ *M*4*ϕ*<sup>2</sup> \* *m*<sup>2</sup> max *k*1, *k*2, *<sup>k</sup>*<sup>3</sup> <sup>Γ</sup>(*ς*1+1), *<sup>k</sup>*<sup>4</sup> Γ(*ς*2+1) 2 (*u*, *v*) Y *r*2−1 + *C*<sup>4</sup> + × <sup>1</sup> 0 J3(*ϑ*)*ψ*2(*ϑ*) *dϑ* 2−<sup>1</sup> = *M*2−<sup>1</sup> <sup>2</sup> *M*<sup>4</sup> *<sup>m</sup>*2*ξr*2−<sup>1</sup> 2 (*u*, *v*) *r*2−1 <sup>Y</sup> <sup>+</sup> *<sup>C</sup>*<sup>4</sup> 2−1 .

Then we obtain

$$\begin{aligned} \left||\mathcal{A}\_1(\boldsymbol{u},\boldsymbol{v})\right|| &\leq M\_1^{\varrho\_1 - 1} M\_3 \left(m\_1 \mathfrak{f}\_1^{r\_1 - 1} ||(\boldsymbol{u},\boldsymbol{v})||\_{\mathcal{Y}}^{r\_1 - 1} + \mathcal{C}\_3\right)^{\varrho\_1 - 1} \\ \left||\mathcal{A}\_2(\boldsymbol{u},\boldsymbol{v})\right|| &\leq M\_2^{\varrho\_2 - 1} M\_4 \left(m\_2 \mathfrak{f}\_2^{r\_2 - 1} ||(\boldsymbol{u},\boldsymbol{v})||\_{\mathcal{Y}}^{r\_2 - 1} + \mathcal{C}\_4\right)^{\varrho\_2 - 1} \end{aligned}$$

and so

$$\begin{aligned} \|\mathcal{A}(\boldsymbol{u},\boldsymbol{v})\|\_{\mathcal{Y}} &\leq M\_{1}^{\varrho\_{1}-1} M\_{3} \Big( m\_{1} \mathfrak{E}\_{1}^{r\_{1}-1} \|(\boldsymbol{u},\boldsymbol{v})\|\|\_{\mathcal{Y}}^{r\_{1}-1} + \mathsf{C}\_{3} \Big)^{\varrho\_{1}-1} \\ &+ M\_{2}^{\varrho\_{2}-1} M\_{4} \Big( m\_{2} \mathfrak{E}\_{2}^{r\_{2}-1} \|(\boldsymbol{u},\boldsymbol{v})\|\|\_{\mathcal{Y}}^{r\_{2}-1} + \mathsf{C}\_{4} \Big)^{\varrho\_{2}-1} \end{aligned}$$

for all (*u*, *v*) ∈ P0. We choose

*<sup>R</sup>*<sup>3</sup> <sup>≥</sup> max 1, *M*1−<sup>1</sup> <sup>1</sup> *<sup>M</sup>*321−2*C*1−<sup>1</sup> <sup>3</sup> <sup>+</sup> *<sup>M</sup>*2−<sup>1</sup> <sup>2</sup> *<sup>M</sup>*422−2*C*2−<sup>1</sup> 4 <sup>1</sup> <sup>−</sup> (*M*1−<sup>1</sup> <sup>1</sup> *<sup>M</sup>*321−2*m*1−<sup>1</sup> <sup>1</sup> *<sup>ξ</sup>*<sup>1</sup> <sup>+</sup> *<sup>M</sup>*2−<sup>1</sup> <sup>2</sup> *<sup>M</sup>*422−2*m*2−<sup>1</sup> <sup>2</sup> *ξ*2) , *M*1−<sup>1</sup> <sup>1</sup> *<sup>M</sup>*3*C*1−<sup>1</sup> <sup>3</sup> <sup>+</sup> *<sup>M</sup>*2−<sup>1</sup> <sup>2</sup> *<sup>M</sup>*4*C*2−<sup>1</sup> 4 <sup>1</sup> <sup>−</sup> (*M*1−<sup>1</sup> <sup>1</sup> *<sup>M</sup>*3*m*1−<sup>1</sup> <sup>1</sup> *<sup>ξ</sup>*<sup>1</sup> <sup>+</sup> *<sup>M</sup>*2−<sup>1</sup> <sup>2</sup> *<sup>M</sup>*4*m*2−<sup>1</sup> <sup>2</sup> *ξ*2) , *M*1−<sup>1</sup> <sup>1</sup> *<sup>M</sup>*3*C*1−<sup>1</sup> <sup>3</sup> <sup>+</sup> *<sup>M</sup>*2−<sup>1</sup> <sup>2</sup> *<sup>M</sup>*422−2*C*2−<sup>1</sup> 4 <sup>1</sup> <sup>−</sup> (*M*1−<sup>1</sup> <sup>1</sup> *<sup>M</sup>*3*m*1−<sup>1</sup> <sup>1</sup> *<sup>ξ</sup>*<sup>1</sup> <sup>+</sup> *<sup>M</sup>*2−<sup>1</sup> <sup>2</sup> *<sup>M</sup>*422−2*m*2−<sup>1</sup> <sup>2</sup> *ξ*2) , (32) *M*1−<sup>1</sup> <sup>1</sup> *<sup>M</sup>*321−2*C*1−<sup>1</sup> <sup>3</sup> <sup>+</sup> *<sup>M</sup>*2−<sup>1</sup> <sup>2</sup> *<sup>M</sup>*4*C*2−<sup>1</sup> 4 <sup>1</sup> <sup>−</sup> (*M*1−<sup>1</sup> <sup>1</sup> *<sup>M</sup>*321−2*m*1−<sup>1</sup> <sup>1</sup> *<sup>ξ</sup>*<sup>1</sup> <sup>+</sup> *<sup>M</sup>*2−<sup>1</sup> <sup>2</sup> *<sup>M</sup>*4*m*2−<sup>1</sup> <sup>2</sup> *ξ*2) # ,

and then we conclude

$$\|\mathcal{A}(\mu, v)\|\_{\mathcal{Y}} \le \|(\mu, v)\|\_{\mathcal{Y}'} \,\,\forall (\mu, v) \in \partial B\_{R\_3} \cap \mathcal{P}\_0. \tag{33}$$

The above number *<sup>R</sup>*<sup>3</sup> was chosen based on the inequalities (*<sup>x</sup>* <sup>+</sup> *<sup>y</sup>*) <sup>≤</sup> <sup>2</sup><sup>−</sup>1(*<sup>x</sup>* <sup>+</sup> *<sup>y</sup>*) for <sup>≥</sup> 1 and *<sup>x</sup>*, *<sup>y</sup>* <sup>≥</sup> 0, and (*<sup>x</sup>* <sup>+</sup> *<sup>y</sup>*) <sup>≤</sup> *<sup>x</sup>* <sup>+</sup> *<sup>y</sup>* for <sup>∈</sup> (0, 1] and *<sup>x</sup>*, *<sup>y</sup>* <sup>≥</sup> 0. Here = <sup>1</sup> − 1 or <sup>2</sup> − 1. We prove the inequality (33) in one case, namely <sup>1</sup> ∈ [2, ∞) and <sup>2</sup> <sup>∈</sup> [2, <sup>∞</sup>). In this case, by using (32) and the relations *<sup>M</sup>*1−<sup>1</sup> <sup>1</sup> *<sup>M</sup>*321−2*m*1−<sup>1</sup> <sup>1</sup> *ξ*<sup>1</sup> < 1/2 and

*M*2−<sup>1</sup> <sup>2</sup> *<sup>M</sup>*422−2*m*2−<sup>1</sup> <sup>2</sup> *ξ*<sup>2</sup> < 1/2 (from the inequalities for *m*<sup>1</sup> and *m*<sup>2</sup> in (*I*5)) we have the inequalities

$$\begin{array}{l} \mathcal{M}\_{1}^{\varrho\_{1}-1}M\_{3}(m\_{1}\mathfrak{z}\_{1}^{r\_{1}-1}\mathcal{R}\_{3}^{r\_{1}-1}+\mathcal{C}\_{3})^{\varrho\_{1}-1}+\mathcal{M}\_{2}^{\varrho\_{2}-1}M\_{4}(m\_{2}\mathfrak{z}\_{2}^{r\_{2}-1}\mathcal{R}\_{3}^{r\_{2}-1}+\mathcal{C}\_{4})^{\varrho\_{2}-1} \\ \leq \mathcal{M}\_{1}^{\varrho\_{1}-1}M\_{3}\mathfrak{2}^{\varrho\_{1}-2}(m\_{1}^{\varrho\_{1}-1}\tilde{\xi}\_{1}\mathcal{R}\_{3}+\mathcal{C}\_{3}^{\varrho\_{1}-1})+M\_{2}^{\varrho\_{2}-1}M\_{4}\mathfrak{2}^{\varrho\_{2}-2}(m\_{2}^{\varrho\_{2}-1}\tilde{\xi}\_{2}\mathcal{R}\_{3}+\mathcal{C}\_{4}^{\varrho\_{2}-1}) \\ = (\mathcal{M}\_{1}^{\varrho\_{1}-1}M\_{3}\mathfrak{2}^{\varrho\_{1}-2}m\_{1}^{\varrho\_{1}-1}\tilde{\xi}\_{1}+M\_{2}^{\varrho\_{2}-1}M\_{4}\mathfrak{2}^{\varrho\_{2}-2}m\_{2}^{\varrho\_{2}-1}\tilde{\xi}\_{2})\mathcal{R}\_{3} \\ + (M\_{1}^{\varrho\_{1}-1}M\_{3}\mathfrak{2}^{\varrho\_{1}-2}C\_{3}^{\varrho\_{1}-1}+M\_{2}^{\varrho\_{2}-1}M\_{4}\mathfrak{2}^{\varrho\_{2}-2}C\_{4}^{\varrho\_{2}-1}) \le \mathcal{R}\_{3}. \end{array}$$

In a similar manner we consider the cases <sup>1</sup> ∈ (1, 2] and <sup>2</sup> ∈ (1, 2]; <sup>1</sup> ∈ [2, ∞) and <sup>2</sup> ∈ (1, 2]; <sup>1</sup> ∈ (1, 2] and <sup>2</sup> ∈ [2, ∞).

In (*I*6), we suppose that *g*<sup>0</sup> > *m*<sup>4</sup> (in a similar manner we can study the case *f*<sup>0</sup> > *m*3). We deduce that there exists *<sup>R</sup>*<sup>4</sup> <sup>∈</sup> (0, 1] such that

$$g(\eta, z\_1, z\_2, z\_3, z\_4) \ge m\_4 \wp\_2((t\_1 z\_1 + t\_2 z\_2 + t\_3 z\_3 + t\_4 z\_4)^{\nu\_2}),\tag{34}$$

for all *<sup>η</sup>* <sup>∈</sup> [*θ*1, *<sup>θ</sup>*2], *zi* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, ... , 4, <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *tizi* <sup>≤</sup> *<sup>R</sup>*4. We take *<sup>R</sup>*<sup>4</sup> <sup>≤</sup> min{*<sup>R</sup>*4/*<sup>ξ</sup>* 4, *<sup>R</sup>*<sup>4</sup>}, where *ξ* <sup>4</sup> <sup>=</sup> 2 max *t*1, *t*2, *<sup>t</sup>*<sup>3</sup> <sup>Γ</sup>(*ς*1+1), *<sup>t</sup>*<sup>4</sup> Γ(*ς*2+1) . Then for any (*u*, *v*) ∈ *BR*<sup>4</sup> ∩ P and *η* ∈ [0, 1] we have

$$\begin{aligned} &t\_1 u(\zeta) + t\_2 v(\zeta) + t\_3 l\_{0+}^\xi u(\zeta) + t\_4 l\_{0+}^\xi v(\zeta) \leq t\_1 \|u\| + t\_2 \|v\| + \frac{t\_3 \|u\|}{\Gamma(\zeta\_1 + 1)} + \frac{t\_4 \|v\|}{\Gamma(\zeta\_2 + 1)} \\ &\leq \max\left\{ t\_1, t\_2, \frac{t\_3}{\Gamma(\zeta\_1 + 1)}, \frac{t\_4}{\Gamma(\zeta\_2 + 1)} \right\} 2 \|(u, v)\|\_{\mathcal{V}} = \tilde{\xi}\_4 \|(u, v)\|\_{\mathcal{V}} \leq \tilde{\xi}\_4 R\_4 \leq \tilde{R}\_4. \end{aligned}$$

Therefore by using (34) and (29), we obtain for any (*u*, *v*) ∈ *∂BR*<sup>4</sup> ∩ P<sup>0</sup> and *η* ∈ [*θ*1, *θ*2]

A2(*u*, *v*)(*η*) ≥ *<sup>θ</sup>*<sup>2</sup> *θ*1 G4(*η*, *ζ*)*ϕ*<sup>2</sup> *<sup>ζ</sup> θ*1 G3(*ζ*, *ϑ*)*g*(*ϑ*, *u*(*ϑ*), *v*(*ϑ*), *I ς*1 <sup>0</sup>+*u*(*ϑ*), *I ς*2 <sup>0</sup>+*v*(*ϑ*))*dϑ dζ* ≥ *θ δ*2−1 1 *<sup>θ</sup>*<sup>2</sup> *θ*1 J4(*ζ*)*ϕ*<sup>2</sup> *<sup>ζ</sup> θ*1 G3(*ζ*, *ϑ*) 0 *m*4*ϕr*<sup>2</sup> (*t*1*u*(*ϑ*) + *t*2*v*(*ϑ*) + *t*<sup>3</sup> *I ς*1 <sup>0</sup>+*u*(*ϑ*) +*t*<sup>4</sup> *I ς*2 <sup>0</sup>+*v*(*ϑ*) *ν*2 *dϑ dζ* <sup>≥</sup> *<sup>θ</sup>δ*2−<sup>1</sup> 1 *<sup>θ</sup>*<sup>2</sup> *θ*1 J4(*ζ*)*ϕ*<sup>2</sup> *<sup>ζ</sup> θ*1 G3(*ζ*, *ϑ*) *m*<sup>4</sup> *t*1*θ δ*1−1 1 *u* + *<sup>t</sup>*2*θδ*2−<sup>1</sup> 1 *v* + +*t*<sup>3</sup> *θ ς*1+*δ*1−1 <sup>1</sup> Γ(*δ*1) <sup>Γ</sup>(*δ*<sup>1</sup> <sup>+</sup> *<sup>ς</sup>*1) *u* + *t*<sup>4</sup> *θ ς*2+*δ*2−1 <sup>1</sup> Γ(*δ*2) <sup>Γ</sup>(*δ*<sup>2</sup> <sup>+</sup> *<sup>ς</sup>*2) *v* -*ν*2(*r*2−1) ⎤ <sup>⎦</sup>*d<sup>ϑ</sup>* ⎞ <sup>⎠</sup>*d<sup>ζ</sup>* ≥ *θ δ*2−1 1 *<sup>θ</sup>*<sup>2</sup> *θ*1 J4(*ζ*) *<sup>ζ</sup> θ*1 G3(*ζ*, *ϑ*)*m*4(2*ξ*<sup>4</sup> (*u*, *v*) Y ) *ν*2(*r*2−1) *dϑ* 2−<sup>1</sup> *dζ* = *θ δ*2−1 <sup>1</sup> *<sup>m</sup>*2−<sup>1</sup> <sup>4</sup> (2*ξ*4)*ν*2(2−1)(*r*2−1) (*u*, *v*) *ν*2 Y , *<sup>θ</sup>*<sup>2</sup> *θ*1 J4(*ζ*) *<sup>ζ</sup> θ*1 G3(*ζ*, *ϑ*) *dϑ* 2−<sup>1</sup> *dζ* - = *M*6*θ δ*2−1 <sup>1</sup> *<sup>m</sup>*2−<sup>1</sup> <sup>4</sup> <sup>2</sup>*ν*<sup>2</sup> *<sup>ξ</sup>ν*<sup>2</sup> 4 (*u*, *v*) *ν*2 <sup>Y</sup> <sup>=</sup> *<sup>λ</sup>*<sup>4</sup> (*u*, *v*) *ν*2 <sup>Y</sup> ≥ (*u*, *v*) *ν*2 <sup>Y</sup> ≥ (*u*, *v*) Y.

Then we deduce A2(*u*, *v*) ≥ (*u*, *v*) <sup>Y</sup> and then

$$\|\mathcal{A}(\mu, v)\|\_{\mathcal{Y}} \ge \|(\mu, v)\|\_{\mathcal{Y}'} \,\,\forall (\mu, v) \in \partial B\_{R\_4} \cap \mathcal{P}\_0. \tag{35}$$

From Lemma 5, (33), (35) and Theorem 1 (ii), we conclude that A has a fixed point (*u*, *v*) ∈ (*BR*<sup>3</sup> \ *BR*<sup>4</sup> ) ∩ P0, so *R*<sup>4</sup> ≤ (*u*, *v*) <sup>Y</sup> ≤ *R*3, which is a positive solution of problem (1) and (2).

**Theorem 4.** *We suppose that assumptions* (*I*1)*,* (*I*2)*,* (*I*4) *and* (*I*6) *hold. In addition, the functions ψ<sup>i</sup> and χi*, *i* = 1, 2 *satisfy the condition*

$$(I\mathcal{T})\ \ M\_3 M\_1^{\varrho\_1 - 1} D\_0^{\varrho\_1 - 1} < \frac{1}{2}\\_\.\ M\_2 M\_2^{\varrho\_2 - 1} D\_0^{\varrho\_2 - 1} < \frac{1}{2}\\_\. where$$

$$D\_0 = \max\left\{\max\_{\eta \in [0, 1], z\_i \in [0, \omega\_0], i = 1, \dots, 4} \chi\_1(\eta, z\_1, z\_2, z\_3, z\_4), \right\},$$

$$\max\_{\eta \in [0, 1], z\_i \in [0, \omega\_0], i = 1, \dots, 4} \chi\_2(\eta, z\_1, z\_2, z\_3, z\_4)\right\},$$

$$with\ \omega\_0 = \max\left\{1, \frac{1}{\Gamma(\sigma\_1 + 1)}, \frac{1}{\Gamma(\sigma\_2 + 1)}, \frac{1}{\Gamma(\sigma\_1 + 1)}, \frac{1}{\Gamma(\varsigma\_2 + 1)}\right\}.$$

*Then there exist two positive solutions* (*u*1(*τ*), *v*1(*τ*)), (*u*2(*τ*), *v*2(*τ*)), *τ* ∈ [0, 1] *of problem (1) and (2).*

**Proof.** Under assumptions (*I*1), (*I*2) and (*I*4), Theorem 2 gives us the existence of *R*<sup>2</sup> > 1 such that

$$\|\mathcal{A}(\boldsymbol{u},\boldsymbol{v})\|\boldsymbol{\mathcal{y}} \ge \|(\boldsymbol{u},\boldsymbol{v})\|\boldsymbol{\mathcal{y}}, \; \forall (\boldsymbol{u},\boldsymbol{v}) \in \partial B\_{R\_2} \cap \mathcal{P}\_0. \tag{36}$$

Under assumptions (*I*1), (*I*2) and (*I*6), Theorem 3 gives us the existence of *R*<sup>4</sup> < 1 such that

$$\|\mathcal{A}(\mu, v)\|\_{\mathcal{Y}} \ge \|(\mu, v)\|\_{\mathcal{Y}'} \,\,\forall (\mu, v) \in \partial B\_{R\_4} \cap \mathcal{P}\_0. \tag{37}$$

Now we consider the set *B*<sup>1</sup> = {(*u*, *v*) ∈ Y, (*u*, *v*) <sup>Y</sup> < 1}. By (*I*7), for any (*u*, *<sup>v</sup>*) ∈ *∂B*<sup>1</sup> ∩ P<sup>0</sup> and *η* ∈ [0, 1], we obtain

$$\begin{split} \mathcal{A}\_{1}(u,v)(\eta) &\leq \int\_{0}^{1} \mathcal{J}\_{2}(\xi) \, \uprho\_{\theta 1} \Big( \int\_{0}^{1} \mathcal{J}\_{1}(\theta) \, \uprho\_{1}(\theta) \chi\_{1}(\theta), v(\theta), J\_{0+}^{\mathcal{O}\_{1}} u(\theta), J\_{0+}^{\mathcal{O}\_{1}} v(\theta) \, d\theta \Big) d\tilde{\xi} \\ &\leq D\_{0}^{\varrho\_{1}-1} \Big( \int\_{0}^{1} \mathcal{J}\_{2}(\xi) \, d\xi \Big) \Big( \int\_{0}^{1} \mathcal{J}\_{1}(\theta) \, \uprho\_{1}(\theta) \, d\theta \Big)^{\varrho\_{1}-1} = M\_{3} D\_{0}^{\varrho\_{1}-1} M\_{1}^{\varrho\_{1}-1} < \frac{1}{2} \\ \mathcal{A}\_{2}(u,v)(\eta) &\leq \int\_{0}^{1} \mathcal{J}\_{4}(\xi) \, \uprho\_{\theta 2} \Big( \int\_{0}^{1} \mathcal{J}\_{3}(\theta) \, \uprho\_{2}(\theta) \chi\_{2}(\theta, u(\theta), v(\theta), J\_{0+}^{\mathbb{C}1} u(\theta), J\_{0+}^{\mathbb{C}2} v(\theta) \, d\theta \Big) d\xi \\ &\leq D\_{0}^{\varrho\_{2}-1} \Big( \int\_{0}^{1} \mathcal{J}\_{4}(\xi) \, d\xi \Big) \Big( \int\_{0}^{1} \mathcal{J}\_{3}(\theta) \, \uprho\_{2}(\theta) \, d\theta \Big)^{\varrho\_{2}-1} = M\_{4} D\_{0}^{\varrho\_{2}-1} M\_{2}^{\varrho\_{2}-1} < \frac{1}{2}. \end{split}$$

Then A*i*(*u*, *v*) < 1/2 for all (*u*, *v*) ∈ *∂B*<sup>1</sup> ∩ P0, *i* = 1, 2. Hence

$$\|\mathcal{A}(\mathfrak{u},\boldsymbol{\upsilon})\|\_{\mathcal{Y}} = \|\mathcal{A}\_{1}(\mathfrak{u},\boldsymbol{\upsilon})\| + \|\mathcal{A}\_{2}(\mathfrak{u},\boldsymbol{\upsilon})\| < 1 = \|(\mathfrak{u},\boldsymbol{\upsilon})\|\_{\mathcal{Y}}, \ \forall \,(\mathfrak{u},\boldsymbol{\upsilon}) \in \partial B\_{1} \cap \mathcal{P}\_{0}.\tag{38}$$

So from (36), (38) and Theorem 1, we deduce that problem (1) and (2) has one positive solution (*u*1, *v*1) ∈ P<sup>0</sup> with 1 < (*u*1, *v*1) <sup>Y</sup> ≤ *R*2. From (37) and (38) and the Guo– Krasnosel'skii fixed point theorem, we conclude that problem (1) and (2) have another positive solution (*u*2, *v*2) ∈ P<sup>0</sup> with *R*<sup>4</sup> ≤ (*u*2, *v*2) <sup>Y</sup> < 1. Then problem (1) and (2) have at least two positive solutions (*u*1(*τ*), *v*1(*τ*)), (*u*2(*τ*), *v*2(*τ*)), *τ* ∈ [0, 1].

#### **4. Examples**

Let *γ*<sup>1</sup> = 3/2, *γ*<sup>2</sup> = 7/6, *p* = 4, *q* = 3, *δ*<sup>1</sup> = 10/3, *δ*<sup>2</sup> = 12/5, *σ*<sup>1</sup> = 2/5, *σ*<sup>2</sup> = 29/7, *ς*<sup>1</sup> = 11/9, *ς*<sup>2</sup> = 21/4, *n* = 2, *m* = 1, *α*<sup>0</sup> = 13/8, *α*<sup>1</sup> = 5/7, *α*<sup>2</sup> = 3/4, *β*<sup>0</sup> = 10/9, *β*<sup>1</sup> = 7/8, *r*<sup>1</sup> = 17/4, *r*<sup>2</sup> = 25/8, <sup>1</sup> = 17/13, <sup>2</sup> = 25/17, H0(*t*) = {2/7, *t* ∈ [0, 3/4); 11/4, *t* ∈ [3/4, 1]}, H1(*t*) = *t*/2, *t* ∈ [0, 1], H2(*t*) = {1/2, *t* ∈ [0, 1/2); 13/10, *t* ∈ [1/2, 1]}, K0(*t*) = 4*t*/9, *t* ∈ [0, 1], K1(*t*) = {1/4, *t* ∈ [0, 1/3); 29/20, *t* ∈ [1/3, 1]}.

We consider the system of fractional differential equations

$$\begin{cases} D\_{0+}^{3/2} \left( \boldsymbol{\varrho}\_{17/4} \left( D\_{0+}^{10/3} \boldsymbol{u}(\tau) \right) \right) = f(\boldsymbol{\tau}, \boldsymbol{u}(\tau), \boldsymbol{v}(\tau), I\_{0+}^{2/5} \boldsymbol{u}(\tau), I\_{0+}^{29/7} \boldsymbol{v}(\tau)), & \boldsymbol{\tau} \in (0, 1), \\\ D\_{0+}^{7/6} \left( \boldsymbol{\varrho}\_{25/8} \left( D\_{0+}^{12/5} \boldsymbol{v}(\tau) \right) \right) = \boldsymbol{g}(\boldsymbol{\tau}, \boldsymbol{u}(\tau), \boldsymbol{v}(\tau), I\_{0+}^{11/9} \boldsymbol{u}(\tau), I\_{0+}^{21/4} \boldsymbol{v}(\tau)), & \boldsymbol{\tau} \in (0, 1), \end{cases} \tag{39}$$

with the boundary conditions

⎧ ⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

$$\begin{array}{ll} u(0) = u'(0) = u''(0) = 0, \ D\_{0+}^{10/3} u(0) = 0, \ D\_{0+}^{10/3} u(1) = \frac{1}{2^{4/3}} D\_{0+}^{10/3} u(\frac{3}{4}), \\\ D\_{0+}^{13/8} u(1) = \frac{1}{2} \int\_0^1 D\_{0+}^{5/7} u(\eta) \, d\eta + \frac{4}{5} D\_{0+}^{3/4} u\left(\frac{1}{2}\right), \\\ v(0) = v'(0) = 0, \ D\_{0+}^{12/5} v(0) = 0, \ \ \ \ \ \ \ \end{array} \begin{array}{ll} \rho\_{0+}^{1/3} v(\frac{3}{4}), \\\ \ \ \ \ \ \end{array} \tag{40}$$
  $v(0) = v'(0) = 0, \ D\_{0+}^{12/5} v(0) = 0, \ \ \ \end{array} \begin{array}{ll} \rho\_{25/8}^{12/5} v(1) \ = \frac{4}{5} \int\_0^1 \rho\_{25/8} \left( D\_{0+}^{12/5} v(\eta) \right) d\eta, \\\ \ \ \ \ D\_{0+}^{10/9} v(1) = \frac{4}{5} D\_{0+}^{7/8} v\left(\frac{1}{3}\right). \end{array}$ 

We have here a<sup>1</sup> ≈ 0.56698729 > 0, a<sup>2</sup> ≈ 2.16111947 > 0, b<sup>1</sup> ≈ 0.61904762 > 0, b<sup>2</sup> ≈ 0.43774133 > 0. So, assumption (*I*1) is satisfied. We also obtain

<sup>g</sup>1(*τ*, *<sup>η</sup>*) = <sup>1</sup> Γ(3/2) - *<sup>τ</sup>*1/2(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)1/2 <sup>−</sup> (*<sup>τ</sup>* <sup>−</sup> *<sup>η</sup>*)1/2, 0 <sup>≤</sup> *<sup>η</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, *<sup>τ</sup>*1/2(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)1/2, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>η</sup>* <sup>≤</sup> 1, <sup>g</sup>2(*τ*, *<sup>η</sup>*) = <sup>1</sup> Γ(10/3) - *<sup>τ</sup>*7/3(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)17/24 <sup>−</sup> (*<sup>τ</sup>* <sup>−</sup> *<sup>η</sup>*)7/3, 0 <sup>≤</sup> *<sup>η</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, *<sup>τ</sup>*7/3(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)17/24, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>η</sup>* <sup>≤</sup> 1, <sup>g</sup>21(*τ*, *<sup>η</sup>*) = <sup>1</sup> Γ(55/21) - *<sup>τ</sup>*34/21(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)17/24 <sup>−</sup> (*<sup>τ</sup>* <sup>−</sup> *<sup>η</sup>*)34/21, 0 <sup>≤</sup> *<sup>η</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, *<sup>τ</sup>*34/21(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)17/24, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>η</sup>* <sup>≤</sup> 1, <sup>g</sup>22(*τ*, *<sup>η</sup>*) = <sup>1</sup> Γ(31/12) - *<sup>τ</sup>*19/12(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)17/24 <sup>−</sup> (*<sup>τ</sup>* <sup>−</sup> *<sup>η</sup>*)19/12, 0 <sup>≤</sup> *<sup>η</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, *<sup>τ</sup>*19/12(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)17/24, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>η</sup>* <sup>≤</sup> 1, <sup>g</sup>3(*τ*, *<sup>η</sup>*) = <sup>1</sup> Γ(7/6) - *<sup>τ</sup>*1/6(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)1/6 <sup>−</sup> (*<sup>τ</sup>* <sup>−</sup> *<sup>η</sup>*)1/6, 0 <sup>≤</sup> *<sup>η</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, *<sup>τ</sup>*1/6(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)1/6, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>η</sup>* <sup>≤</sup> 1, <sup>g</sup>4(*τ*, *<sup>η</sup>*) = <sup>1</sup> Γ(12/5) - *<sup>τ</sup>*7/5(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)13/45 <sup>−</sup> (*<sup>τ</sup>* <sup>−</sup> *<sup>η</sup>*)7/5, 0 <sup>≤</sup> *<sup>η</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, *<sup>τ</sup>*7/5(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)13/45, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>η</sup>* <sup>≤</sup> 1, <sup>g</sup>41(*τ*, *<sup>η</sup>*) = <sup>1</sup> Γ(61/40) - *<sup>τ</sup>*21/40(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)13/45 <sup>−</sup> (*<sup>τ</sup>* <sup>−</sup> *<sup>η</sup>*)21/40, 0 <sup>≤</sup> *<sup>η</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, *<sup>τ</sup>*21/40(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)13/45, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>η</sup>* <sup>≤</sup> 1, <sup>G</sup>1(*τ*, *<sup>η</sup>*) = <sup>g</sup>1(*τ*, *<sup>η</sup>*) + *<sup>τ</sup>*1/2 2a<sup>1</sup> g1 3 4 , *η* , (*τ*, *η*) ∈ [0, 1] × [0, 1], <sup>G</sup>2(*τ*, *<sup>η</sup>*) = <sup>g</sup>2(*τ*, *<sup>η</sup>*) + *<sup>τ</sup>*7/3 a2 1 2 <sup>1</sup> 0 g21(*ϑ*, *η*) *dϑ* + 4 5 <sup>g</sup>22 <sup>1</sup> 2 , *η* , (*τ*, *<sup>η</sup>*) <sup>∈</sup> [0, 1] <sup>×</sup> [0, 1], <sup>G</sup>3(*τ*, *<sup>η</sup>*) = <sup>g</sup>3(*τ*, *<sup>η</sup>*) + <sup>4</sup>*τ*1/6 9b<sup>1</sup> <sup>1</sup> 0 g3(*ϑ*, *η*) *dϑ*, (*τ*, *η*) ∈ [0, 1] × [0, 1], <sup>G</sup>4(*τ*, *<sup>η</sup>*) = <sup>g</sup>4(*τ*, *<sup>η</sup>*) + <sup>6</sup>*τ*7/5 5b<sup>2</sup> <sup>g</sup>41 <sup>1</sup> 3 , *η* , (*τ*, *η*) ∈ [0, 1] × [0, 1], <sup>h</sup>1(*η*) = <sup>1</sup> Γ(3/2) (<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)1/2, <sup>h</sup>2(*η*) = <sup>1</sup> Γ(10/3) (<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)17/24(<sup>1</sup> <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)13/8), *<sup>η</sup>* <sup>∈</sup> [0, 1], <sup>h</sup>3(*η*) = <sup>1</sup> Γ(7/6) (<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)1/6, <sup>h</sup>4(*η*) = <sup>1</sup> Γ(12/5) (<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)13/45(<sup>1</sup> <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)10/9), *<sup>η</sup>* <sup>∈</sup> [0, 1].

Besides we deduce

J1(*ζ*) = ⎧ ⎨ ⎩ h1(*ζ*) + <sup>1</sup> <sup>2</sup>a1Γ(3/2) <sup>3</sup> 4 1/2(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)1/2 <sup>−</sup> <sup>3</sup> <sup>4</sup> − *ζ* 1/2 , 0 <sup>≤</sup> *<sup>ζ</sup>* <sup>≤</sup> <sup>3</sup> 4 , h1(*ζ*) + <sup>1</sup> <sup>2</sup>a1Γ(3/2) 3 4 1/2(<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)1/2, <sup>3</sup> <sup>4</sup> < *ζ* ≤ 1, J2(*ζ*) = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎩ h2(*ζ*) + <sup>1</sup> a2 1 2Γ(76/21) (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)17/24 <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)55/21 + <sup>4</sup> 5Γ(31/12) \* <sup>1</sup> 2 19/12 (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)17/24 <sup>−</sup> 1 <sup>2</sup> − *ζ* 19/12+), 0 <sup>≤</sup> *<sup>ζ</sup>* <sup>≤</sup> <sup>1</sup> 2 , h2(*ζ*) + <sup>1</sup> a2 1 2Γ(76/21) (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)17/24 <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)55/21 + <sup>4</sup> 5Γ(31/12) 1 2 19/12 (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)17/24) , <sup>1</sup> <sup>2</sup> < *ζ* ≤ 1, <sup>J</sup>3(*ζ*) = <sup>h</sup>3(*ζ*) + <sup>4</sup> <sup>9</sup>b1Γ(13/6) (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)1/6 <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)7/6 , *ζ* ∈ [0, 1], J4(*ζ*) = ⎧ ⎪⎨ ⎪⎩ h4(*ζ*) + <sup>6</sup> <sup>5</sup>b2Γ(61/40) \* <sup>1</sup> 3 21/40 (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)13/45 <sup>−</sup> 1 <sup>3</sup> − *ζ* 21/40+ , 0 <sup>≤</sup> *<sup>ζ</sup>* <sup>≤</sup> <sup>1</sup> 3 , h4(*ζ*) + <sup>6</sup> <sup>5</sup>b2Γ(61/40) 1 3 21/40 (<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*)13/45, <sup>1</sup> <sup>3</sup> < *ζ* ≤ 1.

**Example 1.** *We consider the functions*

$$f(\eta, z\_1, z\_2, z\_3, z\_4) = \frac{(2z\_1 + z\_2 + 5z\_3 + 7z\_4)^{13a/4}}{\eta^{\frac{3}{4}}(1 - \eta)^{\frac{3}{2}}}, \text{ } g(\eta, z\_1, z\_2, z\_3, z\_4) = \frac{(3z\_1 + 8z\_2 + 2z\_3 + 9z\_4)^{17b/8}}{\eta^{\frac{6}{3}}(1 - \eta)^{\frac{3}{4}}}, \text{ } (41)$$

*for η* ∈ (0, 1)*, zi* ≥ 0, *i* = 1, ... , 4*, where a* > 1*, b* > 1*, κ*<sup>1</sup> ∈ (0, 1)*, κ*<sup>2</sup> ∈ (0, 3/2)*, <sup>κ</sup>*<sup>3</sup> <sup>∈</sup> (0, 1)*, <sup>κ</sup>*<sup>4</sup> <sup>∈</sup> (0, 7/6)*. Here <sup>ψ</sup>*1(*η*) = <sup>1</sup> *<sup>η</sup>κ*<sup>1</sup> (1−*η*)*κ*<sup>2</sup> *, <sup>ψ</sup>*2(*η*) = <sup>1</sup> *<sup>η</sup>κ*<sup>3</sup> (1−*η*)*κ*<sup>4</sup> *for <sup>η</sup>* <sup>∈</sup> (0, 1)*, χ*1(*η*, *z*1, *z*2, *z*3, *z*4)=(2*z*<sup>1</sup> + *z*<sup>2</sup> + 5*z*<sup>3</sup> + 7*z*4)13*a*/4 *and χ*2(*η*, *z*1, *z*2, *z*3, *z*4)=(3*z*<sup>1</sup> + 8*z*<sup>2</sup> + 2*z*<sup>3</sup> + <sup>9</sup>*z*4)17*b*/8 *for <sup>η</sup>* <sup>∈</sup> [0, 1]*, zi* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, ... , 4. *We also find* <sup>Λ</sup><sup>1</sup> <sup>=</sup> % <sup>1</sup> <sup>0</sup> (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)1/2*ψ*1(*τ*) *<sup>d</sup><sup>τ</sup>* <sup>=</sup> *B* <sup>1</sup> <sup>−</sup> *<sup>κ</sup>*1, <sup>3</sup> <sup>2</sup> − *κ*<sup>2</sup> <sup>∈</sup> (0, <sup>∞</sup>)*,* <sup>Λ</sup><sup>2</sup> <sup>=</sup> % <sup>1</sup> <sup>0</sup> (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)1/6*ψ*2(*τ*) *<sup>d</sup><sup>τ</sup>* <sup>=</sup> *<sup>B</sup>*(<sup>1</sup> <sup>−</sup> *<sup>κ</sup>*3, <sup>7</sup> <sup>6</sup> − *κ*<sup>4</sup>  ∈ (0, ∞)*. Then assumption* (*I*2) *is also satisfied. Moreover, in* (*I*3)*, for c*<sup>1</sup> = 2*, c*<sup>2</sup> = 1*, c*<sup>3</sup> = 5*, c*<sup>4</sup> = 7*, μ*<sup>1</sup> = 1*, d*<sup>1</sup> = 3*, d*<sup>2</sup> = 8*, d*<sup>3</sup> = 2*, d*<sup>4</sup> = 9*, μ*<sup>2</sup> = 1*, we obtain χ*<sup>10</sup> = 0*, χ*<sup>20</sup> = 0*. In* (*I*4)*, for* [*θ*1, *θ*2] ⊂ (0, 1)*, p*<sup>1</sup> = 2*, p*<sup>2</sup> = 1*, p*<sup>3</sup> = 5*, p*<sup>4</sup> = 7*, we have f*<sup>∞</sup> = ∞*. By Theorem 2, we deduce that there exists a positive solution* (*u*(*τ*), *v*(*τ*)), *τ* ∈ [0, 1] *of problems (39) and (40) with the nonlinearities (41).*

#### **Example 2.** *We consider the functions*

$$\begin{cases} f(\eta, z\_1, z\_2, z\_3, z\_4) = \frac{s\_0(\eta+2)}{(\eta^2+6)\sqrt[3]{\eta^2}} \Big[ \left(\frac{1}{4}z\_1 + \frac{1}{3}z\_2 + z\_3 + \frac{1}{2}z\_4\right)^{\omega\_1} \\ \quad + \left(\frac{1}{4}z\_1 + \frac{1}{3}z\_2 + z\_3 + \frac{1}{2}z\_4\right)^{\omega\_2}, \ \eta \in (0, 1], \ z\_i \ge 0, \ i = 1, \ldots, 4, \\ g(\eta, z\_1, z\_2, z\_3, z\_4) = \frac{t\_0(3 + \sin\eta)}{(\eta+2)^4 \sqrt[3]{(1-\eta)^3}} \left(e^{z\_1} + \ln(z\_2 + z\_3 + 1) + z\_4^{\omega\_3}\right), \\ \quad \eta \in [0, 1), \ z\_i \ge 0, \ i = 1, \ldots, 4, \end{cases} \tag{42}$$

*where s*<sup>0</sup> > 0*, t*<sup>0</sup> > 0*, ω*<sup>1</sup> > <sup>13</sup> <sup>4</sup> *, ω*<sup>2</sup> ∈ 0, <sup>13</sup> 4 *, ω*<sup>3</sup> > 0*. Here, we have ψ*1(*η*) = <sup>1</sup> <sup>√</sup><sup>3</sup> *<sup>η</sup>*<sup>2</sup> *, <sup>η</sup>* <sup>∈</sup> (0, 1]*, <sup>χ</sup>*1(*η*, *<sup>z</sup>*1, *<sup>z</sup>*2, *<sup>z</sup>*3, *<sup>z</sup>*4) = *<sup>s</sup>*0(*η*+2) (*η*2+6) 1 <sup>4</sup> *<sup>z</sup>*<sup>1</sup> <sup>+</sup> <sup>1</sup> <sup>3</sup> *<sup>z</sup>*<sup>2</sup> <sup>+</sup> *<sup>z</sup>*<sup>3</sup> <sup>+</sup> <sup>1</sup> <sup>2</sup> *z*<sup>4</sup> *ω*1 + 1 <sup>4</sup> *<sup>z</sup>*<sup>1</sup> <sup>+</sup> <sup>1</sup> <sup>3</sup> *z*<sup>2</sup> + *z*3+ 1 <sup>2</sup> *z*<sup>4</sup> *<sup>ω</sup>*<sup>2</sup> *, <sup>η</sup>* <sup>∈</sup> [0, 1]*, zi* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, ... , 4*, <sup>ψ</sup>*2(*η*) = <sup>1</sup> <sup>√</sup><sup>5</sup> (1−*η*)<sup>3</sup> *, <sup>η</sup>* <sup>∈</sup> [0, 1)*, <sup>χ</sup>*2(*η*, *<sup>z</sup>*1, *<sup>z</sup>*2, *<sup>z</sup>*3, *<sup>z</sup>*4) = *t*0(3+sin *η*) (*η*+2)<sup>4</sup> *ez*<sup>1</sup> + ln(*z*<sup>2</sup> + *z*<sup>3</sup> + 1) + *z ω*<sup>3</sup> 4 *, η* ∈ [0, 1]*, zi* ≥ 0*, i* = 1, ... , 4*. We find* Λ<sup>1</sup> = % 1 <sup>0</sup> (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)1/2 <sup>1</sup> <sup>√</sup><sup>3</sup> *<sup>τ</sup>*<sup>2</sup> *<sup>d</sup><sup>τ</sup>* <sup>=</sup> *<sup>B</sup>* 1 3 , 3 2 <sup>∈</sup> (0, <sup>∞</sup>)*,* <sup>Λ</sup><sup>2</sup> <sup>=</sup> % <sup>1</sup> <sup>0</sup> (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)1/6 <sup>1</sup> <sup>√</sup><sup>5</sup> (1−*τ*)<sup>3</sup> *<sup>d</sup><sup>τ</sup>* <sup>=</sup> <sup>30</sup> <sup>17</sup> ∈ (0, ∞)*. Then assumption* (*I*2) *is satisfied. For* [*θ*1, *θ*2] ⊂ (0, 1)*, p*<sup>1</sup> = 1/4*, p*<sup>2</sup> = 1/3*, p*<sup>3</sup> = 1*, p*<sup>4</sup> = 1/2*, we obtain f*<sup>∞</sup> = ∞*, and for s*<sup>1</sup> = 1/4*, s*<sup>2</sup> = 1/3*, s*<sup>3</sup> = 1*, s*<sup>4</sup> = 1/2 *and ν*<sup>1</sup> ∈ 4*ω*<sup>2</sup> <sup>13</sup> , 1 *, we have f*<sup>0</sup> = ∞*. So assumptions* (*I*4) *and* (*I*6) *are satisfied. Then after some computations, we deduce <sup>M</sup>*<sup>1</sup> = % <sup>1</sup> <sup>0</sup> <sup>J</sup>1(*τ*)*ψ*1(*τ*) *<sup>d</sup><sup>τ</sup>* <sup>≈</sup> 3.04682891*, <sup>M</sup>*<sup>2</sup> <sup>=</sup> % <sup>1</sup> <sup>0</sup> J3(*τ*)*ψ*2(*τ*) *dτ* ≈ 2.64937892*, <sup>M</sup>*<sup>3</sup> = % <sup>1</sup> <sup>0</sup> <sup>J</sup>2(*τ*) *<sup>d</sup><sup>τ</sup>* <sup>≈</sup> 0.15582207*, <sup>M</sup>*<sup>4</sup> <sup>=</sup> % <sup>1</sup> <sup>0</sup> J4(*τ*) *dτ* ≈ 1.25629509*. In addition, we obtain that ω*<sup>0</sup> = <sup>1</sup> <sup>Γ</sup>(7/5) <sup>≈</sup> 1.12706049*, <sup>D</sup>*<sup>0</sup> <sup>=</sup> max '<sup>3</sup>*s*<sup>0</sup> 7 <sup>25</sup> <sup>12</sup>*ω*<sup>0</sup> *<sup>ω</sup>*<sup>1</sup> + <sup>25</sup> <sup>12</sup>*ω*<sup>0</sup> *<sup>ω</sup>*<sup>2</sup> , *t*0*m*0[*eω*<sup>0</sup> + ln(2*ω*<sup>0</sup> + <sup>1</sup>)+*ωω*<sup>3</sup> 0 ] ( *, with m*<sup>0</sup> = max*η*∈[0,1] 3+sin *η* (*η*+2)<sup>4</sup> ≈ 3.0123699*. If*

$$\begin{cases} s\_0 < \min\left\{\frac{7}{3(2M\_3)^{13/4}M\_1[(25\omega\_0/12)^{\omega\_1}+(25\omega\_0/12)^{\omega\_2}]} \mid \frac{7}{3(2M\_4)^{17/8}M\_2[(25\omega\_0/12)^{\omega\_1}+(25\omega\_0/12)^{\omega\_2}]}\right\} \\ t\_0 < \min\left\{\frac{1}{(2M\_3)^{13/4}M\_1m\_0[\epsilon^{\omega\_0}+\ln(2\omega\_0+1)+\omega\_0^{\omega\_3}]} \mid \frac{1}{(2M\_4)^{17/8}M\_2m\_0[\epsilon^{\omega\_0}+\ln(2\omega\_0+1)+\omega\_0^{\omega\_3}]}\right\} \end{cases}$$

*then the inequalities M*3*M*4/13 <sup>1</sup> *<sup>D</sup>*4/13 <sup>0</sup> <sup>&</sup>lt; <sup>1</sup> <sup>2</sup> *, <sup>M</sup>*4*M*8/17 <sup>2</sup> *<sup>D</sup>*8/17 <sup>0</sup> <sup>&</sup>lt; <sup>1</sup> <sup>2</sup> *are satisfied (that is, assumption* (*I*7) *is satisfied). For example, if ω*<sup>1</sup> = 4*, ω*<sup>2</sup> = 2*, ω*<sup>3</sup> = 3*, and s*<sup>0</sup> ≤ 0.0034 *and t*<sup>0</sup> ≤ 0.0031*, then the above inequalities are satisfied. By Theorem 4, we conclude that problem (39) and (40) with the nonlinearities (42) has at least two positive solutions* (*u*1(*τ*), *v*1(*τ*)), (*u*2(*τ*), *v*2(*τ*)), *τ* ∈ [0, 1]*.*

#### **5. Conclusions**

In this paper we investigate the system of Riemann–Liouville fractional differential Equations (1) with *r*1-Laplacian and *r*2-Laplacian operators and fractional integral terms,

subject to the uncoupled boundary conditions (2) which contain Riemann–Stieltjes integrals and fractional derivatives of various orders. The nonlinearities *f* and *g* from the system are nonnegative functions and they may be singular at *τ* = 0 and/or *τ* = 1. First we present the Green functions associated to our problem (1) and (2) and some of their properties. Then we give various conditions for the functions *f* and *g* such that (1) and (2) has at least one or two positive solutions. In the proof of our main results we use the Guo–Krasnosel'skii fixed point theorem of cone expansion and compression of norm type. We finally present two examples for illustrating the obtained existence theorems.

**Author Contributions:** Conceptualization, R.L.; formal analysis, A.T. and R.L.; methodology, A.T. and R.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors thank the referees for their valuable comments and suggestions.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Systems of Riemann–Liouville Fractional Differential Equations with** *ρ***-Laplacian Operators and Nonlocal Coupled Boundary Conditions**

**Alexandru Tudorache <sup>1</sup> and Rodica Luca 2,\***


**Abstract:** In this paper, we study the existence of positive solutions for a system of fractional differential equations with *ρ*-Laplacian operators, Riemann–Liouville derivatives of diverse orders and general nonlinearities which depend on several fractional integrals of differing orders, supplemented with nonlocal coupled boundary conditions containing Riemann–Stieltjes integrals and varied fractional derivatives. The nonlinearities from the system are continuous nonnegative functions and they can be singular in the time variable. We write equivalently this problem as a system of integral equations, and then we associate an operator for which we are looking for its fixed points. The main results are based on the Guo–Krasnosel'skii fixed point theorem of cone expansion and compression of norm type.

**Keywords:** Riemann–Liouville fractional differential equations; nonlocal coupled boundary conditions; singular functions; positive solutions; multiplicity

**MSC:** 34A08; 34B10; 34B16; 34B18

#### **1. Introduction**

We consider the system of Riemann–Liouville fractional differential equations with *ρ*1-Laplacian and *ρ*2-Laplacian operators

$$\begin{cases} D\_{0+}^{\delta\_1}(\boldsymbol{\varrho}\_{\boldsymbol{\rho}\_1}(D\_{0+}^{\gamma\_1}\mathbf{x}(t))) = \mathfrak{f}(t, \mathbf{x}(t), \boldsymbol{y}(t), I\_{0+}^{\mu\_1}\mathbf{x}(t), I\_{0+}^{\mu\_2}\boldsymbol{y}(t)), & t \in (0, 1), \\\ D\_{0+}^{\delta\_2}(\boldsymbol{\varrho}\_{\boldsymbol{\rho}\_2}(D\_{0+}^{\gamma\_2}\mathbf{y}(t))) = \mathfrak{g}(t, \mathbf{x}(t), \boldsymbol{y}(t), I\_{0+}^{\nu\_1}\mathbf{x}(t), I\_{0+}^{\nu\_2}\boldsymbol{y}(t)), & t \in (0, 1), \end{cases} \tag{1}$$

subject to the nonlocal coupled boundary conditions

$$\begin{cases} \begin{aligned} \mathbf{x}^{(j)}(0) = 0, \; j = 0, \ldots, p-2, \; D\_{0+}^{\gamma\_1} \mathbf{x}(0) = 0, \\ \quad \mathbf{q}\_{\rho\_1}(D\_{0+}^{\gamma\_1} \mathbf{x}(1)) = \int\_0^1 \mathbf{q}\_{\rho\_1}(D\_{0+}^{\gamma\_1} \mathbf{x}(\tau)) \, d\mathfrak{M}\_0(\tau), \; D\_{0+}^{a\_0} \mathbf{x}(1) = \sum\_{k=1}^n \int\_0^1 D\_{0+}^{a\_k} y(\tau) \, d\mathfrak{M}\_k(\tau), \\ \quad y^{(j)}(0) = 0, \; j = 0, \ldots, q-2, \; D\_{0+}^{\gamma\_2} y(0) = 0, \\ \quad \mathbf{q}\_{\rho\_2}(D\_{0+}^{\gamma\_2} y(1)) = \int\_0^1 \mathbf{q}\_{\rho\_2}(D\_{0+}^{\gamma\_2} y(\tau)) \, d\mathfrak{M}\_0(\tau), \; D\_{0+}^{\beta\_0} y(1) = \sum\_{k=1}^m \int\_0^1 D\_{0+}^{\beta\_k} x(\tau) \, d\mathfrak{M}\_k(\tau), \end{aligned} \end{cases} \tag{2}$$

where *<sup>δ</sup>*1, *<sup>δ</sup>*<sup>2</sup> <sup>∈</sup> (1, 2], *<sup>γ</sup>*<sup>1</sup> <sup>∈</sup> (*<sup>p</sup>* <sup>−</sup> 1, *<sup>p</sup>*], *<sup>p</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>p</sup>* <sup>≥</sup> 3, *<sup>γ</sup>*<sup>2</sup> <sup>∈</sup> (*<sup>q</sup>* <sup>−</sup> 1, *<sup>q</sup>*], *<sup>q</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>q</sup>* <sup>≥</sup> 3, *<sup>n</sup>*, *<sup>m</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>μ</sup>*1, *<sup>μ</sup>*2, *<sup>ν</sup>*1, *<sup>ν</sup>*<sup>2</sup> <sup>&</sup>gt; 0, *<sup>α</sup><sup>k</sup>* <sup>∈</sup> <sup>R</sup>, *<sup>k</sup>* <sup>=</sup> 0, ... , *<sup>n</sup>*, 0 <sup>≤</sup> *<sup>α</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>α</sup>*<sup>2</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *<sup>α</sup><sup>n</sup>* <sup>≤</sup> *<sup>β</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>γ</sup>*<sup>2</sup> <sup>−</sup> 1, *<sup>β</sup>*<sup>0</sup> <sup>≥</sup> 1, *<sup>β</sup><sup>k</sup>* <sup>∈</sup> <sup>R</sup>, *<sup>k</sup>* <sup>=</sup> 0, ... , *<sup>m</sup>*, 0 <sup>≤</sup> *<sup>β</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>β</sup>*<sup>2</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *<sup>β</sup><sup>m</sup>* <sup>≤</sup> *<sup>α</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>γ</sup>*<sup>1</sup> <sup>−</sup> 1, *<sup>α</sup>*<sup>0</sup> <sup>≥</sup> 1, *ϕρ<sup>i</sup>* (*s*) = |*s*| *<sup>ρ</sup>i*−2*s*, *ϕ*−<sup>1</sup> *<sup>ρ</sup><sup>i</sup>* = *ϕ<sup>i</sup>* , *<sup>i</sup>* = *<sup>ρ</sup><sup>i</sup> <sup>ρ</sup>i*−<sup>1</sup> , *<sup>i</sup>* <sup>=</sup> 1, 2, *<sup>ρ</sup><sup>i</sup>* <sup>&</sup>gt; 1, *<sup>i</sup>* <sup>=</sup> 1, 2, <sup>f</sup>, <sup>g</sup> : (0, 1) <sup>×</sup> <sup>R</sup><sup>4</sup> <sup>+</sup> <sup>→</sup> <sup>R</sup><sup>+</sup> are continuous functions, singular at *t* = 0 and/or *t* = 1, (R<sup>+</sup> = [0, ∞)), *I<sup>θ</sup>* <sup>0</sup><sup>+</sup> is the Riemann–Liouville fractional integral of order *θ* (for *θ* = *μ*1, *μ*2, *ν*1, *ν*2), *D<sup>θ</sup>* <sup>0</sup><sup>+</sup> is the Riemann–Liouville fractional derivative of order *θ* (for *θ* = *δ*1, *γ*1, *δ*2, *γ*2, *α*0, ... , *αn*, *β*0, ... , *βm*), and the integrals from the

**Citation:** Tudorache, A.; Luca, R. Systems of Riemann–Liouville Fractional Differential Equations with *ρ*-Laplacian Operators and Nonlocal Coupled Boundary Conditions. *Fractal Fract.* **2022**, *6*, 610. https:// doi.org/10.3390/fractalfract6100610

Academic Editor: Maria Rosaria Lancia

Received: 25 September 2022 Accepted: 14 October 2022 Published: 19 October 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

boundary conditions (2) are Riemann–Stieltjes integrals with <sup>M</sup>*<sup>i</sup>* : [0, 1] <sup>→</sup> <sup>R</sup>, *<sup>i</sup>* <sup>=</sup> 0, ... , *<sup>n</sup>* and <sup>N</sup>*<sup>j</sup>* : [0, 1] <sup>→</sup> <sup>R</sup>, *<sup>j</sup>* <sup>=</sup> 0, ... , *<sup>m</sup>* functions of bounded variation. The present work was motivated by the applications of *ρ*-Laplacian operators in various fields such as nonlinear elasticity, glaciology, nonlinear electrorheological fluids, fluid flows through porous media, etc. see for details the paper [1] and its references.

In this paper, we present varied conditions for the functions f and g such that problem (1), (2) has a positive solution, and then it has two positive solutions. A positive solution of (1), (2) is a pair of functions (*x*, *<sup>y</sup>*) <sup>∈</sup> (*C*([0, 1], <sup>R</sup>+))<sup>2</sup> satisfying the system (1) and the boundary conditions (2), with *x*(*s*) > 0 for all *s* ∈ (0, 1] or *y*(*s*) > 0 for all *s* ∈ (0, 1]. We apply the Guo–Krasnosel'skii fixed point theorem of cone expansion and compression of norm type (see [2]) in the proof of our main results. Connected to our problem, we mention the following papers. In [3], the authors studied the existence of multiple positive solutions of the system of nonlinear fractional differential equations with *p*1-Laplacian and *p*2-Laplacian operators

$$\begin{cases} D\_{0+}^{\mathcal{B}\_1}(\mathcal{q}\_{p\_1}(D\_{0+}^{a\_1}\mathfrak{x}(\mathbf{s}))) = \mathfrak{f}(\mathbf{s}, \mathfrak{x}(\mathbf{s}), \mathfrak{y}(\mathbf{s})), & \mathbf{s} \in (0, 1), \\\ D\_{0+}^{\mathcal{B}\_2}(\mathcal{q}\_{p\_2}(D\_{0+}^{a\_2}\mathfrak{y}(\mathbf{s}))) = \mathfrak{g}(\mathbf{s}, \mathfrak{x}(\mathbf{s}), \mathfrak{y}(\mathbf{s})), & \mathbf{s} \in (0, 1). \end{cases}$$

supplemented with the nonlocal uncoupled boundary conditions

$$\begin{cases} \begin{aligned} \varkappa(0) = 0, \; \; D\_{0+}^{\gamma\_1} \varkappa(1) = \sum\_{k=1}^{m-2} \zeta\_{1k} D\_{0+}^{\gamma\_1} \varkappa(\eta\_{1k}), \\ \; \; D\_{0+}^{\kappa\_1} \varkappa(0) = 0, \; \; \; \; \varphi\_{p\_1}(D\_{0+}^{\kappa\_1} \varkappa(1)) = \sum\_{k=1}^{m-2} \zeta\_{1k} \varphi\_{p\_1}(D\_{0+}^{\kappa\_1} \varkappa(\eta\_{1k})), \\ \; \; y(0) = 0, \; \; D\_{0+}^{\gamma\_2} y(1) = \sum\_{k=1}^{m} \zeta\_{2k} D\_{0+}^{\gamma\_2} y(\eta\_{2k}), \\ \; \; \; D\_{0+}^{\kappa\_2} y(0) = 0, \; \; \; \; \; \; \; \; \; \varphi\_{p\_2}(D\_{0+}^{\kappa\_2} y(1)) = \sum\_{k=1}^{m-2} \zeta\_{2k} \varphi\_{p\_2}(D\_{0+}^{\kappa\_2} y(\eta\_{2k})). \end{aligned} \end{cases}$$

where *αi*, *β<sup>i</sup>* ∈ (1, 2], *γ<sup>i</sup>* ∈ (0, 1], *α<sup>i</sup>* + *β<sup>i</sup>* ∈ (3, 4], *α<sup>i</sup>* > *γ<sup>i</sup>* + 1, *i* = 1, 2, *ξ*1*k*, *η*1*k*, *ζ*1*k*, *ξ*2*k*, *η*2*k*, *ζ*2*<sup>k</sup>* ∈ (0, 1) for *k* = 1, ... , *m* − 2, *p*1, *p*<sup>2</sup> > 1, and f and g are nonnegative and nonsingular functions. They applied the Leray-Schauder alternative theorem, the Leggett-Williams fixed point theorem and the Avery-Henderson fixed point theorem in the proof of the existence results. In [4], the authors studied the existence and nonexistence of positive solutions for the system of Riemann–Liouville fractional differential equations with 1-Laplacian and 2-Laplacian operators

$$\begin{cases} D\_{0+}^{\gamma\_1}(\boldsymbol{\varrho}\_{\boldsymbol{\varrho}\_1}(D\_{0+}^{\delta\_1}\boldsymbol{\omega}(\boldsymbol{s}))) + \lambda \mathfrak{f}(\boldsymbol{s}, \boldsymbol{\varkappa}(\boldsymbol{s}), \boldsymbol{y}(\boldsymbol{s})) = \boldsymbol{0}, & \boldsymbol{s} \in (0, 1), \\\ D\_{0+}^{\gamma\_2}(\boldsymbol{\varrho}\_{\boldsymbol{\varrho}\_2}(D\_{0+}^{\delta\_2}\boldsymbol{\varrho}(\boldsymbol{s}))) + \mu \mathfrak{g}(\boldsymbol{s}, \boldsymbol{\varkappa}(\boldsymbol{s}), \boldsymbol{y}(\boldsymbol{s})) = \boldsymbol{0}, & \boldsymbol{s} \in (0, 1), \end{cases} \tag{3}$$

subject to the coupled nonlocal boundary conditions

$$\begin{cases} \begin{aligned} \text{x}^{(j)}(0) = 0, \; j = 0, \dots, p-2; \; D\_{0+}^{\delta\_1} \text{x}(0) = 0, \; D\_{0+}^{a\_0} \text{x}(1) = \sum\_{k=1}^{n} \int\_{0}^{1} D\_{0+}^{a\_k} y(\zeta) \, d\mathfrak{M}\_k(\zeta), \\\ y^{(j)}(0) = 0, \; j = 0, \dots, q-2; \; D\_{0+}^{\delta\_2} y(0) = 0, \; D\_{0+}^{\delta\_0} y(1) = \sum\_{k=1}^{m} \int\_{0}^{1} D\_{0+}^{\delta\_k} \text{x}(\zeta) \, d\mathfrak{M}\_k(\zeta), \end{aligned} \end{cases} \tag{4}$$

where *λ* and *μ* are positive parameters, *γ*1, *γ*<sup>2</sup> ∈ (0, 1], *δ*<sup>1</sup> ∈ (*p* − 1, *p*], *δ*<sup>2</sup> ∈ (*q* − 1, *q*], *p*, *q* ∈ <sup>N</sup>, *<sup>p</sup>*, *<sup>q</sup>* <sup>≥</sup> 3, *<sup>n</sup>*, *<sup>m</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>α</sup><sup>k</sup>* <sup>∈</sup> <sup>R</sup> for all *<sup>k</sup>* <sup>=</sup> 0, ... , *<sup>n</sup>*, 0 <sup>≤</sup> *<sup>α</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>α</sup>*<sup>2</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *<sup>α</sup><sup>n</sup>* <sup>≤</sup> *<sup>β</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>δ</sup>*<sup>2</sup> <sup>−</sup> 1, *<sup>β</sup>*<sup>0</sup> <sup>≥</sup> 1, *<sup>β</sup><sup>k</sup>* <sup>∈</sup> <sup>R</sup> for all *<sup>k</sup>* <sup>=</sup> 0, ... , *<sup>m</sup>*, 0 <sup>≤</sup> *<sup>β</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>β</sup>*<sup>2</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *<sup>β</sup><sup>m</sup>* <sup>≤</sup> *<sup>α</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>δ</sup>*<sup>1</sup> <sup>−</sup> 1, *<sup>α</sup>*<sup>0</sup> <sup>≥</sup> 1, 1, <sup>2</sup> <sup>&</sup>gt; 1, the functions <sup>f</sup>, <sup>g</sup> <sup>∈</sup> *<sup>C</sup>*([0, 1] <sup>×</sup> <sup>R</sup><sup>+</sup> <sup>×</sup> <sup>R</sup>+, <sup>R</sup>+), and the functions <sup>M</sup>*j*, *<sup>j</sup>* <sup>=</sup> 1, ... , *<sup>n</sup>* and N*k*, *k* = 1, ... , *m* are bounded variation functions. They presented sufficient conditions on the functions f and g, and intervals for the parameters *λ* and *μ* such that problem (3), (4) has positive solutions. In [5], the authors investigated the existence and multiplicity of

positive solutions for the system (3) with *λ* = *μ* = 1, supplemented with the uncoupled nonlocal boundary conditions

$$\begin{cases} \mathbf{x}^{(j)}(0) = 0, \; j = 0, \ldots, p - 2; \; D\_{0+}^{\delta\_1} \mathbf{x}(0) = 0, \; D\_{0+}^{\kappa\_0} \mathbf{x}(1) = \sum\_{k=1}^{n} \int\_0^1 D\_{0+}^{\kappa\_k} \mathbf{x}(\zeta) \, d\mathfrak{M}\_k(\zeta), \\\ y^{(j)}(0) = 0, \; j = 0, \ldots, q - 2; \; D\_{0+}^{\delta\_2} y(0) = 0, \; D\_{0+}^{\delta\_0} y(1) = \sum\_{k=1}^{m} \int\_0^1 D\_{0+}^{\delta\_k} y(\zeta) \, d\mathfrak{M}\_k(\zeta), \end{cases}$$

where *<sup>n</sup>*, *<sup>m</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>α</sup><sup>k</sup>* <sup>∈</sup> <sup>R</sup> for all *<sup>k</sup>* <sup>=</sup> 0, 1, ... , *<sup>n</sup>*, 0 <sup>≤</sup> *<sup>α</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>α</sup>*<sup>2</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *<sup>α</sup><sup>n</sup>* <sup>≤</sup> *<sup>α</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>δ</sup>*<sup>1</sup> <sup>−</sup> 1, *<sup>α</sup>*<sup>0</sup> <sup>≥</sup> 1, *<sup>β</sup><sup>k</sup>* <sup>∈</sup> <sup>R</sup> for all *<sup>k</sup>* <sup>=</sup> 0, 1, ... , *<sup>m</sup>*, 0 <sup>≤</sup> *<sup>β</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>β</sup>*<sup>2</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *<sup>β</sup><sup>m</sup>* <sup>≤</sup> *<sup>β</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>δ</sup>*<sup>2</sup> <sup>−</sup> 1, *<sup>β</sup>*<sup>0</sup> <sup>≥</sup> 1, the functions f and g from system (3) are nonnegative and continuous, and they may be singular at *s* = 0 and/or *s* = 1, and M*j*, *j* = 1, ... , *n* and N*k*, *k* = 1, ... , *m* are functions of bounded variation. They applied the Guo–Krasnosel'skii fixed point theorem in the proof of the main existence results. In [6] the authors studied the existence and multiplicity of positive solutions for the system (1) subject to general uncoupled boundary conditions in the point *t* = 1. We mention that our problem (1), (2) is different than the problems from papers [4,6]. Indeed the orders of the first fractional derivatives in the system (3) (from [4]) are positive numbers less than or equal to 1, and in our system (1) the first fractional derivatives are numbers greater than 1 and less than or equal to 2. This difference conducts to the consideration of different boundary conditions (more precisely, for our problem, we have a bigger number of such boundary conditions)—see (2) and (4). Another differences are the presence of the parameters in system (3)—here, we do not have any parameters, and also the nonlinearities f and g from (3) which are nonsingular functions, as opposed to our problem in which the functions f and g are singular; so here is a more difficult case to study. On the other hand, the essential difference between the present problem (1), (2) and the problem studied in [6], is given by the boundary conditions. In [6] the last boundary conditions for the unknown functions are uncoupled in the point 1, and here in (2), the last boundary conditions for the unknown functions *x* and *y* are coupled in the point 1; that is, the fractional derivative of order *α*<sup>0</sup> of function *x* in the point 1 is dependent of varied fractional derivatives of function *y*, and the fractional derivative of order *β*<sup>0</sup> of function *y* in 1 is dependent of various fractional derivatives of function *x*. Hence the novelty of our problem (1), (2) is represented by a combination between the existence of *ρ*-Laplacian operators in system (1), the dependence of the nonlinearities in (1) on diverse fractional integrals, and the nature of the last boundary conditions in the point 1 which are coupled here. We also mention the recent papers [7–12] in which the authors study fractional differential equations and systems with *ρ*-Laplacian operators, and some recent monographs devoted to the investigation of boundary value problems for fractional differential equations and systems, namely [13–17].

The paper is organized in the following way. In Section 2, some auxiliary results which include the properties of the Green functions associated to our problem (1), (2) are given. In Section 3 we present the system of integral equations corresponding to our problem, and the main existence and multiplicity theorems for positive solutions of (1), (2), and Section 4 contains their proofs. Finally, two examples which illustrate our obtained results are presented in Section 5, and the conclusions are given in Section 6.

#### **2. Auxiliary Results**

In this section, we consider the system of fractional differential equations

$$\begin{cases} D\_{0+}^{\delta\_1} (\varrho\_{\rho\_1} (D\_{0+}^{\gamma\_1} \mathfrak{x}(t))) = u(t), \ t \in (0,1), \\\ D\_{0+}^{\delta\_2} (\varrho\_{\rho\_2} (D\_{0+}^{\gamma\_2} \mathfrak{y}(t))) = v(t), \ t \in (0,1), \end{cases} \tag{5}$$

with the coupled boundary conditions (2), where *<sup>u</sup>*, *<sup>v</sup>* <sup>∈</sup> *<sup>C</sup>*(0, 1) <sup>∩</sup> *<sup>L</sup>*1(0, 1).

We denote *ϕρ*<sup>1</sup> (*Dγ*<sup>1</sup> <sup>0</sup>+*x*(*t*)) = *<sup>h</sup>*(*t*), *ϕρ*<sup>2</sup> (*Dγ*<sup>2</sup> <sup>0</sup>+*y*(*t*)) = *k*(*t*). Then problem (2), (5) is equivalent to the following three problems

$$\begin{cases} \, \, \_D^{\delta\_1} h(t) = u(t), \,\, t \in (0,1),\\\, h(0) = 0, \,\, h(1) = \int\_0^1 h(\tau) \, d\mathfrak{M}\_0(\tau), \end{cases} \tag{6}$$

$$\begin{cases} \, \, \_{0+}^{\delta\_2} k(t) = v(t), \,\, t \in (0,1),\\\, k(0) = 0, \,\, k(1) = \int\_0^1 k(\tau) \, d\mathfrak{N}\_0(\tau) \end{cases} \tag{7}$$

and

$$\begin{cases} D\_{0+}^{\gamma\_1} \mathfrak{x}(t) = \mathfrak{q}\_{\mathbb{\ell}\_1}(h(t)), & t \in (0,1), \\\ D\_{0+}^{\gamma\_2} \mathfrak{y}(t) = \mathfrak{q}\_{\mathbb{\ell}\_2}(k(t)), & t \in (0,1), \end{cases} \tag{8}$$

with the boundary conditions

$$\begin{cases} \begin{array}{c} \mathbf{x}^{(j)}(0) = 0, \ j = 0, \ldots, p - 2, \ D\_{0+}^{a\_0} \mathbf{x}(1) = \sum\_{k=1}^{n} \int\_{0}^{1} D\_{0+}^{a\_k} y(\tau) \, d\mathfrak{M}\_k(\tau),\\ \mathbf{y}^{(j)}(0) = 0, \ j = 0, \ldots, q - 2, \ D\_{0+}^{\mathcal{B}\_0} y(1) = \sum\_{k=1}^{m} \int\_{0}^{1} D\_{0+}^{\mathcal{B}\_k} \mathbf{x}(\tau) \, d\mathfrak{M}\_k(\tau). \end{array} \end{cases} \tag{9}$$

By Lemma 4.1.5 from [16], the unique solution *h* ∈ *C*[0, 1] of problem (6) is

$$h(t) = -\int\_0^1 \mathfrak{G}\_1(t, \tau) u(\tau) \, d\tau, \; t \in [0, 1], \tag{10}$$

where

$$\begin{split} \mathfrak{G}\_{1}(t,\tau) &= \mathfrak{g}\_{1}(t,\tau) + \frac{t^{\delta\_{1}-1}}{\mathfrak{a}\_{1}} \int\_{0}^{1} \mathfrak{g}\_{1}(\zeta,\tau) \, d\mathfrak{M}\_{0}(\zeta),\\ \mathfrak{g}\_{1}(t,\tau) &= \frac{1}{\Gamma(\delta\_{1})} \begin{cases} \begin{array}{l} t^{\delta\_{1}-1}(1-\tau)^{\delta\_{1}-1} - (t-\tau)^{\delta\_{1}-1}, \ 0 \le \tau \le t \le 1, \\\ t^{\delta\_{1}-1}(1-\tau)^{\delta\_{1}-1}, \ 0 \le t \le \tau \le 1, \end{cases} \end{split}$$

for (*t*, *<sup>τ</sup>*) <sup>∈</sup> [0, 1] <sup>×</sup> [0, 1], with <sup>a</sup><sup>1</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> % <sup>1</sup> <sup>0</sup> *<sup>ζ</sup>δ*1−<sup>1</sup> *<sup>d</sup>*M0(*ζ*) <sup>=</sup> 0.

By the same lemma (Lemma 4.1.5 from [16]), the unique solution *k* ∈ *C*[0, 1] of problem (7) is

$$k(t) = -\int\_0^1 \mathfrak{G}\_2(t, \tau) v(\tau) \, d\tau, \; t \in [0, 1], \tag{11}$$

where

$$\begin{split} \mathfrak{G}\_{2}(t,\tau) &= \mathfrak{g}\_{2}(t,\tau) + \frac{t^{\delta\_{2}-1}}{\mathfrak{a}\_{2}} \int\_{0}^{1} \mathfrak{g}\_{2}(\zeta,\tau) \, d\mathfrak{N}\_{0}(\zeta),\\ \mathfrak{g}\_{2}(t,\tau) &= \frac{1}{\Gamma(\delta\_{2})} \begin{cases} t^{\delta\_{2}-1}(1-\tau)^{\delta\_{2}-1} - (t-\tau)^{\delta\_{2}-1}, & 0 \le \tau \le t \le 1, \\\ t^{\delta\_{2}-1}(1-\tau)^{\delta\_{2}-1}, & 0 \le t \le \tau \le 1, \end{cases} \end{split}$$

for (*t*, *<sup>τ</sup>*) <sup>∈</sup> [0, 1] <sup>×</sup> [0, 1], with <sup>a</sup><sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> % <sup>1</sup> <sup>0</sup> *<sup>ζ</sup>δ*2−<sup>1</sup> *<sup>d</sup>*N0(*ζ*) <sup>=</sup> 0. By Lemma 2.2 from [4], the unique solution (*x*, *<sup>y</sup>*) <sup>∈</sup> (*C*[0, 1])<sup>2</sup> of problem (8), (9) is

$$\begin{cases} \begin{aligned} x(t) &= -\int\_0^1 \mathfrak{G}\_3(t,\tau) \, \boldsymbol{\varrho}\_{\boldsymbol{\varrho}\_1}(h(\tau)) \, d\tau - \int\_0^1 \mathfrak{G}\_4(t,\tau) \, \boldsymbol{\varrho}\_{\boldsymbol{\varrho}\_2}(k(\tau)) \, d\tau, \; t \in [0,1], \\\ y(t) &= -\int\_0^1 \mathfrak{G}\_5(t,\tau) \, \boldsymbol{\varrho}\_{\boldsymbol{\varrho}\_1}(h(\tau)) \, d\tau - \int\_0^1 \mathfrak{G}\_6(t,\tau) \, \boldsymbol{\varrho}\_{\boldsymbol{\varrho}\_2}(k(\tau)) \, d\tau, \; t \in [0,1]. \end{aligned} \end{cases} \tag{12}$$

where

<sup>G</sup>3(*t*, *<sup>τ</sup>*) = <sup>g</sup>3(*t*, *<sup>τ</sup>*) + *<sup>t</sup> <sup>γ</sup>*1−1b<sup>1</sup> b , *m* ∑ *i*=1 <sup>1</sup> 0 g3*i*(*ϑ*, *τ*) *d*N*i*(*ϑ*) - , <sup>G</sup>4(*t*, *<sup>τ</sup>*) = *<sup>t</sup> <sup>γ</sup>*1−1Γ(*γ*2) bΓ(*γ*<sup>2</sup> − *β*0) *n* ∑ *i*=1 <sup>1</sup> 0 g4*i*(*ϑ*, *τ*) *d*M*i*(*ϑ*), <sup>G</sup>5(*t*, *<sup>τ</sup>*) = *<sup>t</sup> <sup>γ</sup>*2−1Γ(*γ*1) bΓ(*γ*<sup>1</sup> − *α*0) *m* ∑ *i*=1 <sup>1</sup> 0 g3*i*(*ϑ*, *τ*) *d*N*i*(*ϑ*), <sup>G</sup>6(*t*, *<sup>τ</sup>*) = <sup>g</sup>4(*t*, *<sup>τ</sup>*) + *<sup>t</sup> <sup>γ</sup>*2−1b<sup>2</sup> b , *n* ∑ *i*=1 <sup>1</sup> 0 g4*i*(*ϑ*, *τ*) *d*M*i*(*ϑ*) - , <sup>g</sup>3(*t*, *<sup>τ</sup>*) = <sup>1</sup> Γ(*γ*1) - *t <sup>γ</sup>*1−1(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*γ*1−*α*0−<sup>1</sup> <sup>−</sup> (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)*γ*1<sup>−</sup>1, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> 1, *t <sup>γ</sup>*1−1(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*γ*1−*α*0<sup>−</sup>1, 0 <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, <sup>g</sup>3*i*(*ϑ*, *<sup>τ</sup>*) = <sup>1</sup> Γ(*γ*<sup>1</sup> − *βi*) - *<sup>ϑ</sup>γ*1−*βi*−1(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*γ*1−*α*0−<sup>1</sup> <sup>−</sup> (*<sup>ϑ</sup>* <sup>−</sup> *<sup>τ</sup>*)*γ*1−*βi*<sup>−</sup>1, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>ϑ</sup>* <sup>≤</sup> 1, *<sup>ϑ</sup>γ*1−*βi*−1(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*γ*1−*α*0<sup>−</sup>1, 0 <sup>≤</sup> *<sup>ϑ</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, <sup>g</sup>4(*t*, *<sup>τ</sup>*) = <sup>1</sup> Γ(*γ*2) - *t <sup>γ</sup>*2−1(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*γ*2−*β*0−<sup>1</sup> <sup>−</sup> (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)*γ*2<sup>−</sup>1, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> 1, *t <sup>γ</sup>*2−1(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*γ*2−*β*0<sup>−</sup>1, 0 <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, <sup>g</sup>4*j*(*ϑ*, *<sup>τ</sup>*) = <sup>1</sup> Γ(*γ*<sup>2</sup> − *αj*) *<sup>ϑ</sup>γ*2−*αj*−1(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*γ*2−*β*0−<sup>1</sup> <sup>−</sup> (*<sup>ϑ</sup>* <sup>−</sup> *<sup>τ</sup>*) *γ*2−*αj*−1 , 0 ≤ *τ* ≤ *ϑ* ≤ 1, *<sup>ϑ</sup>γ*2−*αj*−1(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*γ*2−*β*0<sup>−</sup>1, 0 <sup>≤</sup> *<sup>ϑ</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1,

for all *<sup>t</sup>*, *<sup>τ</sup>*, *<sup>ϑ</sup>* <sup>∈</sup> [0, 1], *<sup>i</sup>* <sup>=</sup> 1, ... , *<sup>m</sup>*, *<sup>j</sup>* <sup>=</sup> 1, ... , *<sup>n</sup>*, and <sup>b</sup><sup>1</sup> <sup>=</sup> <sup>∑</sup>*<sup>n</sup> i*=1 Γ(*γ*2) Γ(*γ*2−*αi*) % 1 <sup>0</sup> *<sup>ζ</sup>γ*2−*αi*−<sup>1</sup> *<sup>d</sup>*M*i*(*ζ*), b<sup>2</sup> = ∑*<sup>m</sup> i*=1 Γ(*γ*1) Γ(*γ*1−*βi*) % 1 <sup>0</sup> *<sup>ζ</sup>γ*1−*βi*−<sup>1</sup> *<sup>d</sup>*N*i*(*ζ*), and <sup>b</sup> <sup>=</sup> <sup>Γ</sup>(*γ*1)Γ(*γ*2) <sup>Γ</sup>(*γ*1−*α*0)Γ(*γ*2−*β*0) <sup>−</sup> <sup>b</sup>1b<sup>2</sup> <sup>=</sup> 0.

Combining the above Formulas (10)–(12) for *h*(*t*), *k*(*t*), *x*(*t*), *y*(*t*), *t* ∈ [0, 1], we obtain the following result.

**Lemma 1.** *If* <sup>a</sup><sup>1</sup> <sup>=</sup> <sup>0</sup>*,* <sup>a</sup><sup>2</sup> <sup>=</sup> <sup>0</sup> *and* <sup>b</sup> <sup>=</sup> <sup>0</sup>*, then the unique solution* (*x*, *<sup>y</sup>*) <sup>∈</sup> (*C*[0, 1])<sup>2</sup> *of problem (5), (2) is given by*

$$\begin{split} \mathfrak{x}(t) &= \int\_{0}^{1} \mathfrak{G}\_{3}(t,\tau) \varrho\_{\varrho\_{1}} \left( \int\_{0}^{1} \mathfrak{G}\_{1}(\tau,\zeta) u(\zeta) \,d\zeta \right) d\tau \\ &+ \int\_{0}^{1} \mathfrak{G}\_{4}(t,\tau) \varrho\_{\varrho\_{2}} \left( \int\_{0}^{1} \mathfrak{G}\_{2}(\tau,\zeta) v(\zeta) \,d\zeta \right) d\tau, \; \forall t \in [0,1], \\ \mathfrak{y}(t) &= \int\_{0}^{1} \mathfrak{G}\_{5}(t,\tau) \varrho\_{\varrho\_{1}} \left( \int\_{0}^{1} \mathfrak{G}\_{1}(\tau,\zeta) u(\zeta) \,d\zeta \right) d\tau \\ &+ \int\_{0}^{1} \mathfrak{G}\_{6}(t,\tau) \varrho\_{\varrho\_{2}} \left( \int\_{0}^{1} \mathfrak{G}\_{2}(\tau,\zeta) v(\zeta) \,d\zeta \right) d\tau, \; \forall t \in [0,1]. \end{split}$$

Now by using the properties of functions g1, g2, g3, g3*i*, *i* = 1, ... , *m*, g4, g4*j*, *j* = 1, . . . , *n* (see [14,16]), we deduce the following properties of the functions G*i*, *i* = 1, . . . , 6.

**Lemma 2.** *We suppose that* a<sup>1</sup> > 0*,* a<sup>2</sup> > 0 *and* b > 0*,* M*i*, *i* = 1, ... , *n and* N*j*, *j* = 0, ... , *m are nondecreasing functions. Then the functions* G*i*, *i* = 1, . . . , 6 *have the properties:*


$$\mathfrak{J}\_1(\tau) = \mathfrak{h}\_1(\tau) + \frac{1}{\mathfrak{a}\_1} \int\_0^1 \mathfrak{g}\_1(\zeta, \tau) \, d\mathfrak{M}\_0(\zeta), \,\,\,\forall \,\, \tau \in [0, 1]\_\tau$$

*with* h1(*τ*) = <sup>1</sup> <sup>Γ</sup>(*δ*1)(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*δ*1<sup>−</sup>1, *<sup>τ</sup>* <sup>∈</sup> [0, 1]*. (c)* G2(*t*, *τ*) ≤ J2(*τ*), *for all* (*t*, *τ*) ∈ [0, 1] × [0, 1]*, where*

$$\mathfrak{J}\_2(\tau) = \mathfrak{h}\_2(\tau) + \frac{1}{\mathfrak{a}\_2} \int\_0^1 \mathfrak{g}\_2(\zeta, \tau) \, d\mathfrak{N}\_0(\zeta), \,\,\,\forall \,\, \tau \in [0, 1].$$

$$\begin{aligned} \text{with } \mathfrak{h}\_2(\mathfrak{r}) &= \frac{1}{\Gamma(\delta\_2)} (1 - \mathfrak{r})^{\delta\_2 - 1}, \ \mathfrak{r} \in [0, 1]. \\ (d) \quad \mathfrak{G}\_3(t, \mathfrak{r}) &\le \mathfrak{J}\_3(\mathfrak{r}), \text{for all } (t, \mathfrak{r}) \in [0, 1] \times [0, 1], \text{ where} \end{aligned}$$

$$\mathfrak{J}\_3(\boldsymbol{\pi}) = \mathfrak{h}\_3(\boldsymbol{\pi}) + \frac{\mathfrak{b}\_1}{\mathfrak{b}} \left( \sum\_{i=1}^m \int\_0^1 \mathfrak{g}\_{3i}(\boldsymbol{\theta}, \boldsymbol{\pi}) \, d\mathfrak{N}\_i(\boldsymbol{\theta}) \right) , \,\,\forall \,\boldsymbol{\pi} \in [0, 1]\_\sigma$$

$$with \, \mathfrak{h}\_{\mathfrak{J}}(\tau) = \frac{1}{\Gamma(\gamma\_1)} (1 - \tau)^{\gamma\_1 - \mathfrak{a}\_0 - 1} (1 - (1 - \tau)^{a\_0}), \,\,\, \tau \in [0, 1].$$

$$\mathfrak{S}\_{\mathbf{i}}(\mathfrak{e}) \quad \mathfrak{G}\_{\mathfrak{Z}}(\mathfrak{t}, \mathfrak{r}) \ge t^{\gamma\_1 - 1} \mathfrak{J}\_{\mathfrak{Z}}(\mathfrak{r})\_{\prime} \check{f} \text{ for all } (\mathfrak{t}, \mathfrak{r}) \in [0, 1] \times [0, 1].$$

*(f)* G4(*t*, *τ*) ≤ J4(*τ*)*, for all* (*t*, *τ*) ∈ [0, 1] × [0, 1]*, where*

$$\mathfrak{J}\_4(\tau) = \frac{\Gamma(\gamma\_2)}{\mathfrak{b}\Gamma(\gamma\_2 - \beta\_0)} \sum\_{i=1}^n \int\_0^1 \mathfrak{g}\_{4i}(\theta, \tau) \, d\mathfrak{M}\_i(\theta), \,\,\,\forall \,\,\tau \in [0, 1].$$

*(g)* G4(*t*, *τ*) = *t <sup>γ</sup>*1−1J4(*τ*)*, for all* (*t*, *<sup>τ</sup>*) <sup>∈</sup> [0, 1] <sup>×</sup> [0, 1]*.*

$$\mathfrak{a}(\mathfrak{h}) \quad \mathfrak{G}\_5(t,\tau) \le \mathfrak{J}\_5(\tau), for all \ (t,\tau) \in [0,1] \times [0,1], where$$

$$\mathfrak{J}\_5(\tau) = \frac{\Gamma(\gamma\_1)}{\mathfrak{b}\Gamma(\gamma\_1 - \mathfrak{a}\_0)} \sum\_{i=1}^m \int\_0^1 \mathfrak{g}\_{3i}(\theta, \tau) \, d\mathfrak{N}\_i(\theta), \,\,\,\forall \,\tau \in [0, 1].$$

$$\mathfrak{a}(\mathfrak{i}) \quad \mathfrak{G}\_5(\mathfrak{t}, \mathfrak{r}) = t^{\gamma\_2 - 1} \mathfrak{J}\_5(\mathfrak{r})\_\prime \not\mathfrak{f} \not\mathfrak{r} \, all \, (\mathfrak{t}, \mathfrak{r}) \in [0, 1] \times [0, 1] \,\omega$$

*(j)* G6(*t*, *τ*) ≤ J6(*τ*)*, for all* (*t*, *τ*) ∈ [0, 1] × [0, 1]*, where*

$$\mathfrak{J}\_6(\tau) = \mathfrak{h}\_4(\tau) + \frac{\mathfrak{b}\_2}{\mathfrak{b}} \left( \sum\_{i=1}^n \int\_0^1 \mathfrak{g}\_{4i}(\vartheta, \tau) \, d\mathfrak{M}\_i(\vartheta) \right), \,\,\forall \,\tau \in [0, 1].$$

$$\begin{aligned} \text{with } \mathfrak{h}\_{4}(\tau) &= \frac{1}{\Gamma(\gamma\_{2})} (1 - \tau)^{\gamma\_{2} - \beta\_{0} - 1} (1 - (1 - \tau)^{\beta\_{0}}), \ \ \ \tau \in [0, 1]. \\ \text{(k)} \quad \mathfrak{G}\_{6}(t, \tau) &\geq t^{\gamma\_{2} - 1} \mathfrak{J}\_{6}(\tau), \text{for all } (t, \tau) \in [0, 1] \times [0, 1]. \end{aligned}$$

Under the assumptions of Lemma 2, we find that J*i*(*τ*) ≥ 0 for all *τ* ∈ [0, 1] and *i* = 1, ... , 6, and J1, J2, J3, J<sup>6</sup> ≡ 0. In addition, J<sup>4</sup> ≡ 0 if all the functions M*i*, *i* = 1, ... , *n* are constant, and J<sup>5</sup> ≡ 0 if all the functions N*j*, *j* = 1, . . . , *m* are constant.

We also deduce easily the next lemma.

**Lemma 3.** *We suppose that* a<sup>1</sup> > 0*,* a<sup>2</sup> > 0 *and* b > 0*,* M*i*, *i* = 1, ... , *n and* N*j*, *j* = 0, ... , *m are nondecreasing functions, <sup>u</sup>*, *<sup>v</sup>* <sup>∈</sup> *<sup>C</sup>*(0, 1) <sup>∩</sup> *<sup>L</sup>*1(0, 1) *with <sup>u</sup>*(*s*) <sup>≥</sup> <sup>0</sup>*, <sup>v</sup>*(*s*) <sup>≥</sup> <sup>0</sup> *for all <sup>s</sup>* <sup>∈</sup> (0, 1)*. Then the solution* (*x*, *y*) *of problem (5), (2) satisfies the inequalities x*(*s*) ≥ 0*, y*(*s*) ≥ 0 *for all <sup>s</sup>* <sup>∈</sup> [0, 1]*, and x*(*s*) <sup>≥</sup> *<sup>s</sup>γ*1−1*x*(*τ*) *and y*(*s*) <sup>≥</sup> *<sup>s</sup>γ*2−1*y*(*τ*) *for all s*, *<sup>τ</sup>* <sup>∈</sup> [0, 1]*.*

#### **3. Main Theorems**

By using Lemma 1, the pair of functions (*x*, *y*) is a solution of problem (1), (2) if and only if (*x*, *y*) is a solution of the system

$$\begin{split} \mathfrak{x}(t) &= \int\_{0}^{1} \mathfrak{G}\_{3}(t,\tau) \mathfrak{p}\_{\varrho\_{1}} \Big( \int\_{0}^{1} \mathfrak{G}\_{1}(\tau,\zeta) \mathfrak{f}(\zeta,\mathtt{x}(\zeta),\mathtt{y}(\zeta),\mathtt{l}\_{0+}^{\mu\_{1}}\mathtt{x}(\zeta),\mathtt{l}\_{0+}^{\mu\_{2}}\mathtt{y}(\zeta)) \, d\zeta \Big) d\tau \\ &+ \int\_{0}^{1} \mathfrak{G}\_{4}(t,\tau) \mathfrak{p}\_{\varrho\_{2}} \Big( \int\_{0}^{1} \mathfrak{G}\_{2}(\tau,\zeta) \mathfrak{g}(\zeta,\mathtt{x}(\zeta),\mathtt{y}(\zeta),\mathtt{l}\_{0+}^{\nu\_{1}}\mathtt{x}(\zeta),\mathtt{l}\_{0+}^{\nu\_{2}}\mathtt{y}(\zeta)) \, d\zeta \Big) d\tau, \\ \mathfrak{y}(t) &= \int\_{0}^{1} \mathfrak{G}\_{5}(t,\tau) \mathfrak{g}\_{\varrho\_{1}} \Big( \int\_{0}^{1} \mathfrak{G}\_{1}(\tau,\zeta) \mathfrak{f}(\zeta,\mathtt{x}(\zeta),\mathtt{y}(\zeta),\mathtt{l}\_{0+}^{\mu\_{1}}\mathtt{x}(\zeta),\mathtt{l}\_{0+}^{\mu\_{2}}\mathtt{y}(\zeta)) \, d\zeta \Big) d\tau \\ &+ \int\_{0}^{1} \mathfrak{G}\_{6}(t,\tau) \mathfrak{g}\_{\varrho\_{2}} \Big( \int\_{0}^{1} \mathfrak{G}\_{2}(\tau,\zeta) \mathfrak{g}(\zeta,\mathtt{x}(\zeta),\mathtt{y}(\zeta),\mathtt{l}\_{0+}^{\nu\_{1}}\mathtt{x}(\zeta),\mathtt{l}\_{0+}^{\nu\_{2}}\mathtt{y}(\zeta)) \, d\zeta \Big) d\tau, \end{split}$$

for all *t* ∈ [0, 1]. We introduce the Banach space U = *C*[0, 1] with supremum norm *x* <sup>=</sup> sup*s*∈[0,1] <sup>|</sup>*x*(*s*)|, and the Banach space <sup>V</sup> <sup>=</sup> <sup>U</sup> <sup>×</sup> <sup>U</sup> with the norm (*x*, *y*) V <sup>=</sup> *x* + *y* . We define the cone

$$\mathfrak{U} = \{(\mathfrak{x}, \mathfrak{y}) \in \mathfrak{V}, \ \mathfrak{x}(\mathfrak{s}) \ge 0, \ \mathfrak{y}(\mathfrak{s}) \ge 0, \ \forall \, \mathfrak{s} \in [0, 1] \}.$$

We also define the operators E1, E<sup>2</sup> : V → U and E : V → V by

<sup>E</sup>1(*x*, *<sup>y</sup>*)(*t*) = <sup>1</sup> 0 G3(*t*, *τ*)*ϕ*<sup>1</sup> <sup>1</sup> 0 G1(*τ*, *ζ*)f(*ζ*, *x*(*ζ*), *y*(*ζ*), *I μ*1 <sup>0</sup>+*x*(*ζ*), *I μ*2 <sup>0</sup>+*y*(*ζ*)) *dζ dτ* + <sup>1</sup> 0 G4(*t*, *τ*)*ϕ*<sup>2</sup> <sup>1</sup> 0 G2(*τ*, *ζ*)g(*ζ*, *x*(*ζ*), *y*(*ζ*), *I ν*1 <sup>0</sup>+*x*(*ζ*), *I ν*2 <sup>0</sup>+*y*(*ζ*)) *dζ dτ*, <sup>E</sup>2(*x*, *<sup>y</sup>*)(*t*) = <sup>1</sup> 0 G5(*t*, *τ*)*ϕ*<sup>1</sup> <sup>1</sup> 0 G1(*τ*, *ζ*)f(*ζ*, *x*(*ζ*), *y*(*ζ*), *I μ*1 <sup>0</sup>+*x*(*ζ*), *I μ*2 <sup>0</sup>+*y*(*ζ*)) *dζ dτ* + <sup>1</sup> 0 G6(*t*, *τ*)*ϕ*<sup>2</sup> <sup>1</sup> 0 G2(*τ*, *ζ*)g(*ζ*, *x*(*ζ*), *y*(*ζ*), *I ν*1 <sup>0</sup>+*x*(*ζ*), *I ν*2 <sup>0</sup>+*y*(*ζ*)) *dζ dτ*,

for all *t* ∈ [0, 1] and (*x*, *y*) ∈ V, and E(*x*, *y*)=(E1(*x*, *y*),E2(*x*, *y*)), (*x*, *y*) ∈ V. We remark that (*x*, *y*) is a solution of problem (1), (2) if and only if (*x*, *y*) is a fixed point of operator E.

We define the constants: <sup>Ξ</sup>*<sup>i</sup>* <sup>=</sup> % <sup>1</sup> <sup>0</sup> <sup>J</sup>*i*(*τ*)*ξi*(*τ*) *<sup>d</sup>τ*, *<sup>i</sup>* <sup>=</sup> 1, 2, <sup>Ξ</sup>*<sup>j</sup>* <sup>=</sup> % <sup>1</sup> <sup>0</sup> J*j*(*τ*) *dτ*, *j* = 3, . . . , 6, and for *<sup>σ</sup>*1, *<sup>σ</sup>*<sup>2</sup> <sup>∈</sup> (0, 1), *<sup>σ</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>σ</sup>*2, <sup>Ξ</sup><sup>7</sup> <sup>=</sup> % *<sup>σ</sup>*<sup>2</sup> *<sup>σ</sup>*<sup>1</sup> <sup>J</sup>3(*τ*) % *<sup>τ</sup> <sup>σ</sup>*<sup>1</sup> <sup>G</sup>1(*τ*, *<sup>ζ</sup>*) *<sup>d</sup><sup>ζ</sup>* 1−1 *<sup>d</sup>τ*, <sup>Ξ</sup><sup>8</sup> = % *<sup>σ</sup>*<sup>2</sup> *<sup>σ</sup>*<sup>1</sup> <sup>J</sup>6(*τ*) % *<sup>τ</sup> <sup>σ</sup>*<sup>1</sup> <sup>G</sup>2(*τ*, *<sup>ζ</sup>*) *<sup>d</sup><sup>ζ</sup>* 2−1 *dτ*.

We now present the assumptions that we will use in our theorems.


$$f(t, w\_1, w\_2, w\_3, w\_4) \le \xi\_1(t)\psi\_1(t, w\_1, w\_2, w\_3, w\_4), \\ g(t, w\_1, w\_2, w\_3, w\_4) \le \xi\_2(t)\psi\_2(t, w\_1, w\_2, w\_3, w\_4).$$

for any *<sup>t</sup>* <sup>∈</sup> (0, 1), *wi* <sup>∈</sup> <sup>R</sup>+, *<sup>i</sup>* <sup>=</sup> 1, . . . , 4.

(*H*3) There exist *li* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, ... , 4 with <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *li* <sup>&</sup>gt; 0, *mi* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, ... , 4 with <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *mi* > 0, and *θ*<sup>1</sup> ≥ 1, *θ*<sup>2</sup> ≥ 1 such that

$$\begin{split} \psi\_{10} &= \limsup\_{\sum\_{i=1}^{4} l\_{i}w\_{i} \to 0} \max\_{t \in [0,1]} \frac{\psi\_{1}(t, w\_{1}, w\_{2}, w\_{3}, w\_{4})}{\varphi\_{\rho\_{1}}((l\_{1}w\_{1} + l\_{2}w\_{2} + l\_{3}w\_{3} + l\_{4}w\_{4})^{\theta\_{1}})} < c\_{1}, \\ \text{and} \ \psi\_{20} &= \limsup\_{\sum\_{i=1}^{4} m\_{i}w\_{i} \to 0} \max\_{t \in [0,1]} \frac{\psi\_{2}(t, w\_{1}, w\_{2}, w\_{3}, w\_{4})}{\varphi\_{\rho\_{2}}((m\_{1}w\_{1} + m\_{2}w\_{2} + m\_{3}w\_{3} + m\_{4}w\_{4})^{\theta\_{2}})} < c\_{2}. \end{split}$$

where

*c*<sup>1</sup> = - min- <sup>4</sup>*ρ*1−1Ξ1Ξ*ρ*1−<sup>1</sup> <sup>3</sup> *d θ*1(*ρ*1−1) 1 −1 , <sup>4</sup>*ρ*1−1Ξ1Ξ*ρ*1−<sup>1</sup> <sup>5</sup> *d θ*1(*ρ*1−1) 1 −1 ) , if <sup>Ξ</sup><sup>5</sup> <sup>=</sup> 0; <sup>4</sup>*ρ*1−1Ξ1Ξ*ρ*1−<sup>1</sup> <sup>3</sup> *d θ*1(*ρ*1−1) 1 −1 , if Ξ<sup>5</sup> = 0 ) , *c*<sup>2</sup> = - min- <sup>4</sup>*ρ*2−1Ξ2Ξ*ρ*2−<sup>1</sup> <sup>4</sup> *d θ*2(*ρ*2−1) 2 −1 , <sup>4</sup>*ρ*2−1Ξ2Ξ*ρ*2−<sup>1</sup> <sup>6</sup> *d θ*2(*ρ*2−1) 2 −1 ) , if <sup>Ξ</sup><sup>4</sup> <sup>=</sup> 0; <sup>4</sup>*ρ*2−1Ξ2Ξ*ρ*2−<sup>1</sup> <sup>6</sup> *d θ*2(*ρ*2−1) 2 −1 , if Ξ<sup>4</sup> = 0 ) , with *<sup>d</sup>*<sup>1</sup> <sup>=</sup> <sup>2</sup> max *l*1, *l*2, *<sup>l</sup>*<sup>3</sup> <sup>Γ</sup>(*μ*1+1), *<sup>l</sup>*<sup>4</sup> Γ(*μ*2+1) , *<sup>d</sup>*<sup>2</sup> <sup>=</sup> 2 max *m*1, *m*2, *<sup>m</sup>*<sup>3</sup> <sup>Γ</sup>(*ν*1+1), *<sup>m</sup>*<sup>4</sup> Γ(*ν*2+1) .

(*H*4)There exist *si* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, ... , 4 with <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *si* <sup>&</sup>gt; 0, *ti* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, ... , 4 with <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *ti* > 0, *σ*1, *σ*<sup>2</sup> ∈ (0, 1), *σ*<sup>1</sup> < *σ*<sup>2</sup> and *η*<sup>1</sup> > 1, *η*<sup>2</sup> > 1 such that

$$\mathfrak{f}\_{\infty} = \liminf\_{\sum\_{i=1}^{4} s\_i w\_i \to \infty} \min\_{t \in [\sigma\_1 \sigma\_2]} \frac{\mathfrak{f}(t, w\_1, w\_2, w\_3, w\_4)}{\mathfrak{q}\_{\rho\_1}(s\_1 w\_1 + s\_2 w\_2 + s\_3 w\_3 + s\_4 w\_4)} > c\_3.$$

$$\text{or} \quad \mathfrak{g}\_{\infty} = \liminf\_{\sum\_{i=1}^{4} t\_i w\_i \to \infty} \min\_{t \in [\sigma\_1 \sigma\_2]} \frac{\mathfrak{g}(t, w\_1, w\_2, w\_3, w\_4)}{\mathfrak{q}\_{\rho\_2}(t\_1 w\_1 + t\_2 w\_2 + t\_3 w\_3 + t\_4 w\_4)} > c\_{4\prime}.$$

where

$$c\_{3} = \eta\_{1} \left( 2d\_{3} \overline{\omega}\_{T} \sigma\_{1}^{\gamma\_{1} - 1} \right)^{1 - \rho\_{1}}, c\_{4} = \eta\_{2} \left( 2d\_{4} \overline{\omega}\_{\sigma} \sigma\_{1}^{\gamma\_{2} - 1} \right)^{1 - \rho\_{2}} \text{ with } d\_{3} = \min \left\{ s\_{1} \sigma\_{1}^{\gamma\_{1} - 1}, s\_{2} \sigma\_{1}^{\gamma\_{2} - 1}, s\_{3} \frac{s\_{1}^{\gamma\_{1} + \gamma\_{1} - 1} \Gamma(\gamma\_{1})}{\Gamma(\gamma\_{1} + \rho\_{1})}, s\_{4} \frac{s\_{1}^{\gamma\_{2} + \gamma\_{2}} \Gamma(\gamma\_{2})}{\Gamma(\gamma\_{2} + \rho\_{2})} \right\},$$

$$d\_{4} = \min \left\{ t\_{1} \sigma\_{1}^{\gamma\_{1} - 1}, t\_{2} \sigma\_{1}^{\gamma\_{2} - 1}, t\_{3} \frac{s\_{1}^{\gamma\_{1} + \gamma\_{1} - 1} \Gamma(\gamma\_{1})}{\Gamma(\gamma\_{1} + \gamma\_{1})}, t\_{4} \frac{s\_{1}^{\gamma\_{2} + \gamma\_{2} - 1} \Gamma(\gamma\_{2})}{\Gamma(\gamma\_{2} + \gamma\_{2})} \right\}.$$

$$(H5) \text{There exist } u\_{i} \ge 0, \ i = 1, \ldots, 4 \text{ with } \sum\_{i=1}^{4} u\_{i} > 0, v\_{i} \ge 0, \ i = 1, \ldots, 4 \text{ with } \sum\_{i=1}^{4} v\_{i} > 0, \ i = 1, \ldots, 4 \text{ with } \sum\_{i=1}^{4} v\_{i} > 0, \zeta($$

$$\begin{aligned} \psi\_{1\infty} &= \limsup\_{\sum\_{i=1}^4 u\_i w\_i \to \infty} \max\_{t \in [0,1]} \frac{\psi\_1(t, w\_1, w\_2, w\_3, w\_4)}{\overline{\varphi\_{\rho\_1}(u\_1 w\_1 + u\_2 w\_2 + u\_3 w\_3 + u\_4 w\_4)}} < e\_{1\prime}, \\ \text{and } \psi\_{2\infty} &= \limsup\_{\sum\_{i=1}^4 v\_i w\_i \to \infty} \max\_{t \in [0,1]} \frac{\psi\_2(t, w\_1, w\_2, w\_3, w\_4)}{\overline{\varphi\_{\rho\_2}(v\_1 w\_1 + v\_2 w\_2 + v\_3 w\_3 + v\_4 w\_4)}} < e\_{2\prime}. \end{aligned}$$

$$\begin{array}{c} \text{where} \\ e\_1 < \left[ 2\Xi\_1^{q-1} (\Xi\_3 + \Xi\_5) \Lambda\_1 k\_1 \right]^{1-\rho\_1}, e\_2 < \left[ 2\Xi\_2^{q-1} (\Xi\_4 + \Xi\_6) \Lambda\_2 k\_2 \right]^{1-\rho\_2}, \text{with } \Lambda\_1 = \max\{ 2^{q\_i - 2}, 1 \}, \text{ } \Lambda\_2 = \max\{ 2^{q\_i - 2}, 1 \}, \\ k\_1 = 2 \max\{ u\_1, u\_2, \frac{v\_3}{\Gamma(\mu\_1 + 1)}, \frac{v\_4}{\Gamma(\mu\_2 + 1)} \}, k\_2 = 2 \max\left\{ v\_1, v\_2, \frac{v\_3}{\Gamma(\mu\_1 + 1)}, \frac{v\_4}{\Gamma(\mu\_2 + 1)} \right\}. \\ (H6) \text{ There exist } p\_i \ge 0, \ i = 1, \dots, 4 \text{ with } \sum\_{i=1}^4 p\_i > 0, q\_i \ge 0, \ i = 1, \dots, 4 \text{ with } \sum\_{i=1}^4 q\_i > 0, \\ \sigma\_1, \sigma\_2 \in (0, 1), \sigma\_1 < \sigma\_2 \text{ and } \xi\_1 \in (0, 1], \zeta\_2 \in (0, 1], \eta\_3 \ge 1, \ \eta\_4 \ge 1 \text{ such that} \end{array}$$

$$\mathfrak{f}\_0 = \liminf\_{\sum\_{i=1}^4 p\_i w\_i \to 0} \min\_{t \in [\sigma\_1 \sigma\_2]} \frac{\mathfrak{f}(t, w\_1, w\_2, w\_3, w\_4)}{\mathfrak{g}\_{\rho\_1}((p\_1 w\_1 + p\_2 w\_2 + p\_3 w\_3 + p\_4 w\_4)^{\xi\_1})} > e\_{3\prime}$$

$$\text{for} \quad \mathfrak{g}\_0 = \liminf\_{\sum\_{i=1}^4 q\_i w\_i \to 0} \min\_{t \in [\sigma\_1, \sigma\_2]} \frac{\mathfrak{g}(t, w\_1, w\_2, w\_3, w\_4)}{\overline{\mathfrak{g}\_{\rho\_2}((q\_1 w\_1 + q\_2 w\_2 + q\_3 w\_3 + q\_4 w\_4)^{\varsigma\_2})}} > e\_{4\prime}$$

where *e*<sup>3</sup> = *σγ*1−<sup>1</sup> <sup>1</sup> <sup>2</sup>*ς*<sup>1</sup> *<sup>k</sup> ς*1 <sup>3</sup> Ξ<sup>7</sup> 1−*ρ*<sup>1</sup> , *e*<sup>4</sup> = *σγ*2−<sup>1</sup> <sup>1</sup> <sup>2</sup>*ς*<sup>2</sup> *<sup>k</sup> ς*2 <sup>4</sup> Ξ<sup>8</sup> 1−*ρ*<sup>2</sup> , with *<sup>k</sup>*<sup>3</sup> <sup>=</sup> min *<sup>p</sup>*1*σγ*1−<sup>1</sup> <sup>1</sup> , *<sup>p</sup>*2*σγ*2−<sup>1</sup> <sup>1</sup> , *p*<sup>3</sup> *σ μ*1+*γ*1−1 <sup>1</sup> Γ(*γ*1) <sup>Γ</sup>(*γ*1+*μ*1) , *<sup>p</sup>*<sup>4</sup> *σ μ*2+*γ*2−1 <sup>1</sup> Γ(*γ*2) Γ(*γ*2+*μ*2) ) , *<sup>k</sup>*<sup>4</sup> <sup>=</sup> min *<sup>q</sup>*1*σγ*1−<sup>1</sup> <sup>1</sup> , *<sup>q</sup>*2*σγ*2−<sup>1</sup> <sup>1</sup> , *q*3 *σ ν*1+*γ*1−1 <sup>1</sup> Γ(*γ*1) <sup>Γ</sup>(*γ*1+*ν*1) , *<sup>q</sup>*<sup>4</sup> *σ ν*2+*γ*2−1 <sup>1</sup> Γ(*γ*2) Γ(*γ*2+*ν*2) ) . (*H*7) *A*1−<sup>1</sup> <sup>0</sup> <sup>Ξ</sup>3Ξ1−<sup>1</sup> <sup>1</sup> <sup>&</sup>lt; <sup>1</sup> <sup>4</sup> , *<sup>A</sup>*2−<sup>1</sup> <sup>0</sup> <sup>Ξ</sup>4Ξ2−<sup>1</sup> <sup>2</sup> <sup>&</sup>lt; <sup>1</sup> <sup>4</sup> , *<sup>A</sup>*1−<sup>1</sup> <sup>0</sup> <sup>Ξ</sup>5Ξ1−<sup>1</sup> <sup>1</sup> <sup>&</sup>lt; <sup>1</sup> <sup>4</sup> , *<sup>A</sup>*2−<sup>1</sup> <sup>0</sup> <sup>Ξ</sup>6Ξ2−<sup>1</sup> <sup>2</sup> <sup>&</sup>lt; <sup>1</sup> 4 , where

*<sup>A</sup>*<sup>0</sup> <sup>=</sup> max max*t*∈[0,1], *wi*∈[0,], *<sup>i</sup>*=1,...,4 *<sup>ψ</sup>*1(*t*, *<sup>w</sup>*1, *<sup>w</sup>*2, *<sup>w</sup>*3, *<sup>w</sup>*4), max*t*∈[0,1], *wi*∈[0,], *<sup>i</sup>*=1,...,4 *<sup>ψ</sup>*2(*t*, *<sup>w</sup>*1, *<sup>w</sup>*2, *<sup>w</sup>*3, *<sup>w</sup>*4)}, with = max 1, <sup>1</sup> <sup>Γ</sup>(*μ*1+1), <sup>1</sup> <sup>Γ</sup>(*μ*2+1), <sup>1</sup> <sup>Γ</sup>(*ν*1+1), <sup>1</sup> Γ(*ν*2+1) .

> **Lemma 4.** *We suppose that* (*H*1) *and* (*H*2) *hold. Then* E : Q → Q *is a completely continuous operator.*

We introduce now the cone

$$\mathfrak{U}\_0 = \{ (\mathfrak{x}, y) \in \mathfrak{Q}\_{\prime} \: \ x(\mathfrak{x}) \ge \tau^{\gamma\_1 - 1} \|\mathfrak{x}\|\_{\prime} \ y(\mathfrak{x}) \ge \tau^{\gamma\_2 - 1} \|y\|\_{\prime} \ \forall \, \tau \in [0, 1] \}.$$

If (*H*1) and(*H*2) are satisfied, then by Lemma 3 we obtain E(Q) ⊂ Q<sup>0</sup> and then the operator <sup>E</sup>|Q<sup>0</sup> : <sup>Q</sup><sup>0</sup> <sup>→</sup> <sup>Q</sup><sup>0</sup> (which we will denote again by <sup>E</sup>) is completely continuous. For *κ* > 0 we denote by *B<sup>κ</sup>* the open ball centered at zero of radius *κ*, and by *B<sup>κ</sup>* and *∂B<sup>κ</sup>* its closure and its boundary, respectively.

Our main existence results are the following theorems.

**Theorem 1.** *We suppose that assumptions* (*H*1)*–*(*H*4) *hold. Then there exists a positive solution* (*x*(*t*), *y*(*t*)), *t* ∈ [0, 1] *of problem (1), (2).*

**Theorem 2.** *We suppose that assumptions* (*H*1)*,* (*H*2)*,* (*H*5)*,* (*H*6) *hold. Then there exists a positive solution* (*x*(*t*), *y*(*t*)), *t* ∈ [0, 1] *of problem (1), (2).*

**Theorem 3.** *We suppose that assumptions* (*H*1)*,* (*H*2)*,* (*H*4)*,* (*H*6) *and* (*H*7) *hold. Then there exist two positive solutions* (*x*1(*t*), *y*1(*t*)), (*x*2(*t*), *y*2(*t*)), *t* ∈ [0, 1] *of problem (1), (2).*

#### **4. Proofs of the Results**

**Proof of Lemma 4.** By (*H*2), we have <sup>Ξ</sup><sup>1</sup> = % <sup>1</sup> <sup>0</sup> <sup>J</sup>1(*τ*)*ξ*1(*τ*) *<sup>d</sup><sup>τ</sup>* <sup>&</sup>gt; 0 and <sup>Ξ</sup><sup>2</sup> <sup>=</sup> % <sup>1</sup> <sup>0</sup> J2(*τ*)*ξ*2(*τ*) *dτ* > 0. In addition, by using Lemma 2.2 we find

$$\begin{array}{l} \Xi\_{1} \leq \frac{M\_{1}}{\Gamma(\delta\_{1})} \left[ 1 + \frac{1}{\mathfrak{a}\_{1}} \left( \int\_{0}^{1} \zeta^{\delta\_{1}-1} \, d\mathfrak{M}\_{0}(\zeta) \right) \right] < \infty, \\\Xi\_{2} \leq \frac{M\_{2}}{\Gamma(\delta\_{2})} \left[ 1 + \frac{1}{\mathfrak{a}\_{2}} \left( \int\_{0}^{1} \zeta^{\delta\_{2}-1} \, d\mathfrak{M}\_{0}(\zeta) \right) \right] < \infty. \end{array}$$

Using now Lemma 3, we deduce that the operator E maps Q into Q.

Next, we will show that E transforms the bounded sets into relatively compact sets. Let S ⊂ Q be a bounded set. So there exists *L*<sup>1</sup> > 0 such that (*x*, *y*) <sup>V</sup> <sup>≤</sup> *<sup>L</sup>*<sup>1</sup> for all (*x*, *y*) ∈ S. Because *ψ*<sup>1</sup> and *ψ*<sup>2</sup> are continuous functions, we find that there exists *L*<sup>2</sup> > 0 such that *<sup>L</sup>*<sup>2</sup> <sup>=</sup> max sup*τ*∈[0,1], *wi*∈[0,Λ], *<sup>i</sup>*=1,...,4 *<sup>ψ</sup>*1(*τ*, *<sup>w</sup>*1, *<sup>w</sup>*2, *<sup>w</sup>*3, *<sup>w</sup>*4), sup*τ*∈[0,1], *wi*∈[0,Λ], *<sup>i</sup>*=1,...,4 *<sup>ψ</sup>*2(*τ*, *<sup>w</sup>*1, *<sup>w</sup>*2, *<sup>w</sup>*3, *<sup>w</sup>*4)}, where <sup>Λ</sup> <sup>=</sup> *<sup>L</sup>*<sup>1</sup> max 1, <sup>1</sup> <sup>Γ</sup>(*μ*1+1), <sup>1</sup> <sup>Γ</sup>(*μ*2+1), <sup>1</sup> <sup>Γ</sup>(*ν*1+1), <sup>1</sup> Γ(*ν*2+1) . Because |*Iω* <sup>0</sup>+*z*(*t*)| ≤ *z* <sup>Γ</sup>(*ω*+1) for *ω* > 0 and *z* ∈ *C*[0, 1], by Lemma 2 we obtain that for any (*x*, *y*) ∈ S and *t* ∈ [0, 1]

E1(*x*, *y*)(*t*) ≤ <sup>1</sup> 0 J3(*τ*)*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ζ*)*ξ*1(*ζ*)*ψ*1(*ζ*, *x*(*ζ*), *y*(*ζ*), *I μ*1 <sup>0</sup>+*x*(*ζ*), *I μ*2 <sup>0</sup>+*y*(*ζ*)) *dζ dτ* + <sup>1</sup> 0 J4(*τ*)*ϕ*<sup>2</sup> <sup>1</sup> 0 J2(*ζ*)*ξ*2(*ζ*)*ψ*2(*ζ*, *x*(*ζ*), *y*(*ζ*), *I ν*1 <sup>0</sup>+*x*(*ζ*), *I ν*2 <sup>0</sup>+*y*(*ζ*)) *dζ dτ* <sup>≤</sup> *<sup>L</sup>*1−<sup>1</sup> <sup>2</sup> *ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ζ*)*ξ*1(*ζ*) *dζ* <sup>1</sup> 0 J3(*τ*) *dτ* +*L*2−<sup>1</sup> <sup>2</sup> *ϕ*<sup>2</sup> <sup>1</sup> 0 J2(*ζ*)*ξ*2(*ζ*) *dζ* <sup>1</sup> 0 J4(*τ*) *dτ* = *L*1−<sup>1</sup> <sup>2</sup> <sup>Ξ</sup>1−<sup>1</sup> <sup>1</sup> <sup>Ξ</sup><sup>3</sup> <sup>+</sup> *<sup>L</sup>*2−<sup>1</sup> <sup>2</sup> <sup>Ξ</sup>2−<sup>1</sup> <sup>2</sup> Ξ4.

In a similar way we have

$$\mathfrak{E}\_2(\mathfrak{x}, y)(t) \le L\_2^{\varrho\_1 - 1} \Xi\_1^{\varrho\_1 - 1} \Xi\_5 + L\_2^{\varrho\_2 - 1} \Xi\_2^{\varrho\_2 - 1} \Xi\_6.$$

Therefore

$$\begin{array}{c} \|\mathfrak{E}\_{1}(\boldsymbol{x},\boldsymbol{y})\| \leq L\_{2}^{\varrho\_{1}-1}\Xi\_{1}^{\varrho\_{1}-1}\Xi\_{3} + L\_{2}^{\varrho\_{2}-1}\Xi\_{2}^{\varrho\_{2}-1}\Xi\_{4},\\ \|\mathfrak{E}\_{2}(\boldsymbol{x},\boldsymbol{y})\| \leq L\_{2}^{\varrho\_{1}-1}\Xi\_{1}^{\varrho\_{1}-1}\Xi\_{5} + L\_{2}^{\varrho\_{2}-1}\Xi\_{2}^{\varrho\_{2}-1}\Xi\_{6}. \end{array}$$

for all (*x*, *y*) ∈ S, and then E1(S), E2(S) and E(S) are bounded.

In what follows, we prove that E(S) is equicontinuous. By Lemma 1, for (*x*, *y*) ∈ S and *t* ∈ [0, 1] we find

<sup>E</sup>1(*x*, *<sup>y</sup>*)(*t*) = <sup>1</sup> 0 ! <sup>g</sup>3(*t*, *<sup>τ</sup>*) + *<sup>t</sup> <sup>γ</sup>*1−1b<sup>1</sup> b , *m* ∑ *i*=1 <sup>1</sup> 0 g3*i*(*ϑ*, *τ*) *d*N*i*(*ϑ*) -" ×*ϕ*<sup>1</sup> <sup>1</sup> 0 G1(*τ*, *ζ*)f *ζ*, *x*(*ζ*), *y*(*ζ*), *I μ*1 <sup>0</sup>+*x*(*ζ*), *I μ*2 <sup>0</sup>+*y*(*ζ*) *dζ dτ* + <sup>1</sup> 0 *t <sup>γ</sup>*1−1Γ(*γ*2) bΓ(*γ*<sup>2</sup> − *β*0) , *n* ∑ *i*=1 <sup>1</sup> 0 g4*i*(*ϑ*, *τ*) *d*M*i*(*ϑ*) - ×*ϕ*<sup>2</sup> <sup>1</sup> 0 G2(*τ*, *ζ*)g *ζ*, *x*(*ζ*), *y*(*ζ*), *I ν*1 <sup>0</sup>+*x*(*ζ*), *I ν*2 <sup>0</sup>+*y*(*ζ*) *dζ dτ* = *<sup>t</sup>* 0 1 Γ(*γ*1) *t <sup>γ</sup>*1−1(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*γ*1−*α*0−<sup>1</sup> <sup>−</sup> (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)*γ*1−<sup>1</sup> ×*ϕ*<sup>1</sup> <sup>1</sup> 0 G1(*τ*, *ζ*)f *ζ*, *x*(*ζ*), *y*(*ζ*), *I μ*1 <sup>0</sup>+*x*(*ζ*), *I μ*2 <sup>0</sup>+*y*(*ζ*) *dζ dτ* + <sup>1</sup> *t* 1 Γ(*γ*1) *t <sup>γ</sup>*1−1(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*γ*1−*α*0−<sup>1</sup> ×*ϕ*<sup>1</sup> <sup>1</sup> 0 G1(*τ*, *ζ*)f(*ζ*, *x*(*ζ*), *y*(*ζ*), *I μ*1 <sup>0</sup>+*x*(*ζ*), *I μ*2 <sup>0</sup>+*y*(*ζ*)) *dζ dτ* + *t <sup>γ</sup>*1−1b<sup>1</sup> b <sup>1</sup> 0 , *m* ∑ *i*=1 <sup>1</sup> 0 g3*i*(*ϑ*, *τ*) *d*N*i*(*ϑ*) - ×*ϕ*<sup>1</sup> <sup>1</sup> 0 G1(*τ*, *ζ*)f(*ζ*, *x*(*ζ*), *y*(*ζ*), *I μ*1 <sup>0</sup>+*x*(*ζ*), *I μ*2 <sup>0</sup>+*y*(*ζ*)) *dζ dτ* + *t <sup>γ</sup>*1−1Γ(*γ*2) bΓ(*γ*<sup>2</sup> − *β*0) <sup>1</sup> 0 , *n* ∑ *i*=1 <sup>1</sup> 0 g4*i*(*ϑ*, *τ*) *d*M*i*(*ϑ*) - ×*ϕ*<sup>2</sup> <sup>1</sup> 0 G2(*τ*, *ϑ*)g(*ζ*, *x*(*ζ*), *y*(*ζ*), *I ν*1 <sup>0</sup>+*x*(*ζ*), *I ν*2 <sup>0</sup>+*y*(*ζ*)) *dζ dτ*.

Then for any *t* ∈ (0, 1), we obtain

(E1(*x*, *y*)) (*t*) = *<sup>t</sup>* 0 1 Γ(*γ*1) (*γ*<sup>1</sup> − 1)*t <sup>γ</sup>*1−2(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*γ*1−*α*0−<sup>1</sup> <sup>−</sup> (*γ*<sup>1</sup> <sup>−</sup> <sup>1</sup>)(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)*γ*1−<sup>2</sup> ×*ϕ*<sup>1</sup> <sup>1</sup> 0 G1(*τ*, *ζ*)f(*ζ*, *x*(*ζ*), *y*(*ζ*), *I μ*1 <sup>0</sup>+*x*(*ζ*), *I μ*2 <sup>0</sup>+*y*(*ζ*))*dζ dτ* + <sup>1</sup> *t* 1 Γ(*γ*1) (*γ*<sup>1</sup> − 1)*t <sup>γ</sup>*1−2(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)*γ*1−*α*0−<sup>1</sup> ×*ϕ*<sup>1</sup> <sup>1</sup> 0 G1(*τ*, *ζ*)f(*ζ*, *x*(*ζ*), *y*(*ζ*), *I μ*1 <sup>0</sup>+*x*(*ζ*), *I μ*2 <sup>0</sup>+*y*(*ζ*)) *dζ dτ* <sup>+</sup>(*γ*<sup>1</sup> <sup>−</sup> <sup>1</sup>)*<sup>t</sup> <sup>γ</sup>*1−2b<sup>1</sup> b <sup>1</sup> 0 , *m* ∑ *i*=1 <sup>1</sup> 0 g3*i*(*ϑ*, *τ*) *d*N*i*(*ϑ*) - ×*ϕ*<sup>1</sup> <sup>1</sup> 0 G1(*τ*, *ζ*)f(*ζ*, *x*(*ζ*), *y*(*ζ*), *I μ*1 <sup>0</sup>+*x*(*ζ*), *I μ*2 <sup>0</sup>+*y*(*ζ*)) *dζ dτ* <sup>+</sup>(*γ*<sup>1</sup> <sup>−</sup> <sup>1</sup>)*<sup>t</sup> <sup>γ</sup>*1−2Γ(*γ*2) bΓ(*γ*<sup>2</sup> − *β*0) <sup>1</sup> 0 , *n* ∑ *i*=1 <sup>1</sup> 0 g4*i*(*ϑ*, *τ*) *d*M*i*(*ϑ*) - ×*ϕ*<sup>2</sup> <sup>1</sup> 0 G2(*τ*, *ζ*)g(*ζ*, *x*(*ζ*), *y*(*ζ*), *I ν*1 <sup>0</sup>+*x*(*ζ*), *I ν*2 <sup>0</sup>+*y*(*ζ*)) *dζ dτ*.

So for any *t* ∈ (0, 1) we deduce


Hence for any *t* ∈ (0, 1) we find


We denote by

$$\Theta\_1(t) = \frac{(\mathfrak{b} + \mathfrak{b}\_1 \mathfrak{b}\_2) t^{\gamma\_1 - 2}}{\mathfrak{b}(\gamma\_1 - \mathfrak{a}\_0) \Gamma(\gamma\_1 - 1)} + \frac{t^{\gamma\_1 - 1}}{\Gamma(\gamma\_1)}, \ \Theta\_2(t) = \frac{(\gamma\_1 - 1) t^{\gamma\_1 - 2} \mathfrak{b}\_1}{\mathfrak{b} \Gamma(\gamma\_2 - \beta\_0 + 1)}, \ t \in (0, 1).$$

Then for any *t*1, *t*<sup>2</sup> ∈ [0, 1] with *t*<sup>1</sup> < *t*<sup>2</sup> and (*x*, *y*) ∈ S, we deduce

$$\begin{split} & \left| \mathfrak{E}\_{1}(\boldsymbol{\chi}, \boldsymbol{\upmu})(t\_{1}) - \mathfrak{E}\_{1}(\boldsymbol{\upmu}, \boldsymbol{\upmu})(t\_{2}) \right| = \left| \int\_{t\_{1}}^{t\_{2}} (\mathfrak{E}\_{1}(\boldsymbol{\upmu}, \boldsymbol{\upmu}))'(\boldsymbol{\uppi}) \, d\boldsymbol{\uppi} \right| \\ & \leq L\_{2}^{\varrho\_{1} - 1} \Xi\_{1}^{\varrho\_{1} - 1} \int\_{t\_{1}}^{t\_{2}} \Theta\_{1}(\boldsymbol{\uppi}) \, d\boldsymbol{\uppi} + L\_{2}^{\varrho\_{2} - 1} \Xi\_{2}^{\varrho\_{2} - 1} \int\_{t\_{1}}^{t\_{2}} \Theta\_{2}(\boldsymbol{\uppi}) \, d\boldsymbol{\uppi}. \end{split} \tag{13}$$

Because <sup>Θ</sup>1, <sup>Θ</sup><sup>2</sup> <sup>∈</sup> *<sup>L</sup>*1(0, 1), by (13), we conclude that <sup>E</sup>1(S) is equicontinuous. By using a similar technique, we deduce that E2(S) is also equicontinuous, and so E(S) is equicontinuous. We apply now the Arzela-Ascoli theorem and we obtain that E1(S) and E2(S) are relatively compact sets, and then E(S) is relatively compact, too. In addition, we can prove that E1, E<sup>2</sup> and E are continuous operators on Q (see Lemma 1.4.1 from [16]). Therefore, the operator E is completely continuous on Q.

**Proof of Theorem 1.** From (*H*3) we deduce that there exists *r* ∈ (0, 1) such that

$$\begin{array}{l} \psi\_{1}(t, w\_{1}, w\_{2}, w\_{3}, w\_{4}) \leq c\_{1} \varphi\_{\rho\_{1}}((l\_{1}w\_{1} + l\_{2}w\_{2} + l\_{3}w\_{3} + l\_{4}w\_{4})^{\theta\_{1}}),\\ \psi\_{2}(t, w\_{1}, w\_{2}, w\_{3}, w\_{4}) \leq c\_{2} \varphi\_{\rho\_{2}}((m\_{1}w\_{1} + m\_{2}w\_{2} + m\_{3}w\_{3} + m\_{4}w\_{4})^{\theta\_{2}}),\end{array} \tag{14}$$

for all *<sup>t</sup>* <sup>∈</sup> [0, 1], *wi* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, ... , 4 with <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *liwi* <sup>≤</sup> *<sup>r</sup>* and <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *miwi* ≤ *r*. We consider firstly the case Ξ<sup>4</sup> = 0 and Ξ<sup>5</sup> = 0. We define *r*<sup>1</sup> ≤ min{*r*/*d*1,*r*/*d*2,*r*}. For any (*x*, *y*) ∈ *Br*<sup>1</sup> ∩ Q and *τ* ∈ [0, 1] we find

$$\begin{array}{l} l\_{1}\mathbf{x}(\tau) + l\_{2}\mathbf{y}(\tau) + l\_{3}I\_{0+}^{\mu\_{1}}\mathbf{x}(\tau) + l\_{4}I\_{0+}^{\mu\_{2}}\mathbf{y}(\tau) \\ \leq 2\max\left\{l\_{1}, l\_{2}, \frac{l\_{3}}{\Gamma(\mu\_{1}+1)'}\frac{l\_{4}}{\Gamma(\mu\_{2}+1)}\right\} \|(\mathbf{x}, \mathbf{y})\|\_{\mathfrak{W}} = d\_{1} \|(\mathbf{x}, \mathbf{y})\|\_{\mathfrak{W}} \leq d\_{1}r\_{1} \leq r\_{1} \\ m\_{1}\mathbf{x}(\tau) + m\_{2}\mathbf{y}(\tau) + m\_{3}I\_{0+}^{\mu\_{1}}\mathbf{x}(\tau) + m\_{4}I\_{0+}^{\mu\_{2}}\mathbf{y}(\tau) \\ \leq 2\max\left\{m\_{1}, m\_{2}, \frac{m\_{3}}{\Gamma(\nu\_{1}+1)'}\frac{m\_{4}}{\Gamma(\nu\_{2}+1)}\right\} \|(\mathbf{x}, \mathbf{y})\|\_{\mathfrak{W}} = d\_{2} \|(\mathbf{x}, \mathbf{y})\|\_{\mathfrak{W}} \leq d\_{2}r\_{1} \leq r. \end{array}$$

Therefore by (14) and Lemma 2, for any (*x*, *y*) ∈ *∂Br*<sup>1</sup> ∩ Q<sup>0</sup> and *t* ∈ [0, 1] we deduce

E1(*x*, *y*)(*t*) ≤ <sup>1</sup> 0 J3(*τ*)*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ζ*)f(*ζ*, *x*(*ζ*), *y*(*ζ*), *I μ*1 <sup>0</sup>+*x*(*ζ*), *I μ*2 <sup>0</sup>+*y*(*ζ*)) *dζ dτ* + <sup>1</sup> 0 J4(*τ*)*ϕ*<sup>2</sup> <sup>1</sup> 0 J2(*ζ*)g(*ζ*, *x*(*ζ*), *y*(*ζ*), *I ν*1 <sup>0</sup>+*x*(*ζ*), *I ν*2 <sup>0</sup>+*y*(*ζ*)) *dζ dτ* = Ξ3*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ζ*)f *ζ*, *x*(*ζ*), *y*(*ζ*), *I μ*1 <sup>0</sup>+*x*(*ζ*), *I μ*2 <sup>0</sup>+*y*(*ζ*) *dζ* +Ξ4*ϕ*<sup>2</sup> <sup>1</sup> 0 J2(*ζ*)g *ζ*, *x*(*ζ*), *y*(*ζ*), *I ν*1 <sup>0</sup>+*x*(*ζ*), *I ν*2 <sup>0</sup>+*y*(*ζ*) *dζ* ≤ Ξ3*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ζ*)*ξ*1(*ζ*)*ψ*<sup>1</sup> *ζ*, *x*(*ζ*), *y*(*ζ*), *I μ*1 <sup>0</sup>+*x*(*ζ*), *I μ*2 <sup>0</sup>+*y*(*ζ*) *dζ* +Ξ4*ϕ*<sup>2</sup> <sup>1</sup> 0 J2(*ζ*)*ξ*2(*ζ*)*ψ*<sup>2</sup> *ζ*, *x*(*ζ*), *y*(*ζ*), *I ν*1 <sup>0</sup>+*x*(*ζ*), *I ν*2 <sup>0</sup>+*y*(*ζ*) *dζ* ≤ Ξ3*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ζ*)*ξ*1(*ζ*)*c*1*ϕρ*<sup>1</sup> *l*1*x*(*ζ*) + *l*2*y*(*ζ*) + *l*<sup>3</sup> *I μ*1 <sup>0</sup>+*x*(*ζ*) + *l*<sup>4</sup> *I μ*2 <sup>0</sup>+*y*(*ζ*) *θ*1 *dζ* +Ξ4*ϕ*<sup>2</sup> <sup>1</sup> 0 J2(*ζ*)*ξ*2(*ζ*)*c*2*ϕρ*<sup>2</sup> *m*1*x*(*ζ*) + *m*2*y*(*ζ*) + *m*<sup>3</sup> *I ν*1 <sup>0</sup>+*x*(*ζ*) + *m*<sup>4</sup> *I ν*2 <sup>0</sup>+*y*(*ζ*) *θ*2 *dζ* ≤ Ξ3*ϕ*<sup>1</sup> *ϕρ*<sup>1</sup> (*d*1 (*x*, *y*) V) *θ*1 *ϕ*<sup>1</sup> (*c*1)*ϕ*<sup>1</sup> (Ξ1) +Ξ4*ϕ*<sup>2</sup> *ϕρ*<sup>2</sup> (*d*2 (*x*, *y*) V) *θ*2 *ϕ*<sup>2</sup> (*c*2)*ϕ*<sup>2</sup> (Ξ2) <sup>=</sup> <sup>Ξ</sup>3Ξ1−<sup>1</sup> <sup>1</sup> *c* 1−1 <sup>1</sup> *<sup>d</sup>θ*<sup>1</sup> 1 (*x*, *y*) *θ*1 V <sup>+</sup> <sup>Ξ</sup>4Ξ2−<sup>1</sup> <sup>2</sup> *c* 2−1 <sup>2</sup> *<sup>d</sup>θ*<sup>2</sup> 2 (*x*, *y*) *θ*2 V <sup>≤</sup> <sup>Ξ</sup>3Ξ1−<sup>1</sup> <sup>1</sup> *c* 1−1 <sup>1</sup> *<sup>d</sup>θ*<sup>1</sup> 1 (*x*, *y*) V <sup>+</sup> <sup>Ξ</sup>4Ξ2−<sup>1</sup> <sup>2</sup> *c* 2−1 <sup>2</sup> *<sup>d</sup>θ*<sup>2</sup> 2 (*x*, *y*) V <sup>≤</sup> <sup>1</sup> 4 (*x*, *y*) V <sup>+</sup> <sup>1</sup> 4 (*x*, *y*) V <sup>=</sup> <sup>1</sup> 2 (*x*, *y*) V.

In a similar manner we obtain

$$\begin{array}{l} \mathfrak{C}\_{2}(\mathfrak{x},\mathfrak{y})(t) \leq \Xi\_{5} \Xi\_{1}^{\varrho\_{1}-1} c\_{1}^{\varrho\_{1}-1} d\_{1}^{\theta\_{1}} ||(\mathfrak{x},\mathfrak{y})||\_{\mathfrak{W}} + \Xi\_{6} \Xi\_{2}^{\varrho\_{2}-1} c\_{2}^{\varrho\_{2}-1} d\_{2}^{\theta\_{2}} ||(\mathfrak{x},\mathfrak{y})||\_{\mathfrak{W}} \\ \leq \frac{1}{4} ||(\mathfrak{x},\mathfrak{y})||\_{\mathfrak{W}} + \frac{1}{4} ||(\mathfrak{x},\mathfrak{y})||\_{\mathfrak{W}} = \frac{1}{2} ||(\mathfrak{x},\mathfrak{y})||\_{\mathfrak{W}}. \end{array}$$

Then we conclude

$$\|\|\mathfrak{E}(\mathbf{x},\mathbf{y})\|\|\_{\mathfrak{V}} = \|\|\mathfrak{E}\_1(\mathbf{x},\mathbf{y})\| + \|\|\mathfrak{E}\_2(\mathbf{x},\mathbf{y})\| \le \|(\mathbf{x},\mathbf{y})\|\_{\mathfrak{V}\mathfrak{V}} \,\,\,\forall (\mathbf{x},\mathbf{y}) \in \partial B\_{\mathfrak{V}\_1} \cap \mathfrak{Q}\_0. \tag{15}$$

If Ξ<sup>4</sup> = 0 or Ξ<sup>5</sup> = 0 we also find in a similar manner inequality (15).

In what follows, in (*H*4) we assume that g<sup>∞</sup> > *c*<sup>4</sup> (in a similar manner we study the case f<sup>∞</sup> > *c*3). Then there exists a positive constant *C*<sup>1</sup> > 0 such that

$$\mathfrak{g}(t, w\_1, w\_2, w\_3, w\_4) \ge c\_4 \mathfrak{p}\_{\rho\_2}(t\_1 w\_1 + t\_2 w\_2 + t\_3 w\_3 + t\_4 w\_4) - \mathbb{C}\_1. \tag{16}$$

for all *t* ∈ [*σ*1, *σ*2] and *wi* ≥ 0, *i* = 1, ... , 4. From the definition of *I ν*1 <sup>0</sup>+, for any (*x*, *y*) ∈ Q<sup>0</sup> and *τ* ∈ [0, 1], we find

$$\begin{split} I\_{0+}^{\boldsymbol{\nu}\_{1}}\mathbf{x}(\tau) &= \frac{1}{\Gamma(\nu\_{1})} \int\_{0}^{\tau} (\tau - \boldsymbol{\zeta})^{\nu\_{1} - 1} \mathbf{x}(\boldsymbol{\zeta}) \, d\boldsymbol{\zeta} \geq \frac{1}{\Gamma(\nu\_{1})} \int\_{0}^{\tau} (\tau - \boldsymbol{\zeta})^{\nu\_{1} - 1} \boldsymbol{\zeta}^{\gamma\_{1} - 1} \|\mathbf{x}\| \, d\boldsymbol{\zeta} \\ &\overset{\scriptstyle \zeta = \tau \boldsymbol{z}}{=} \frac{\|\mathbf{x}\|}{\Gamma(\nu\_{1})} \int\_{0}^{1} (\tau - \tau \boldsymbol{z})^{\nu\_{1} - 1} \tau^{\gamma\_{1} - 1} \boldsymbol{z}^{\gamma\_{1} - 1} \tau \, d\boldsymbol{z} = \frac{\|\mathbf{x}\|}{\Gamma(\nu\_{1})} \tau^{\nu\_{1} + \gamma\_{1} - 1} \int\_{0}^{1} \boldsymbol{z}^{\gamma\_{1} - 1} (1 - \boldsymbol{z})^{\nu\_{1} - 1} \, d\boldsymbol{z} \\ &= \frac{\|\mathbf{x}\|}{\Gamma(\nu\_{1})} \tau^{\nu\_{1} + \gamma\_{1} - 1} B(\gamma\_{1}, \nu\_{1}) = \frac{\|\mathbf{x}\| \|\boldsymbol{\tau}^{\nu\_{1} + \gamma\_{1} - 1} \Gamma(\gamma\_{1})}{\Gamma(\gamma\_{1} + \nu\_{1})}, \end{split} \tag{17}$$

and similarly

$$I\_{0+}^{\nu\_2}y(\tau) \ge \frac{||y||\pi^{\nu\_2+\gamma\_2-1}\Gamma(\gamma\_2)}{\Gamma(\gamma\_2+\nu\_2)}\tau$$

where *<sup>B</sup>*(*z*1, *<sup>z</sup>*2) is the first Euler function defined by *<sup>B</sup>*(*z*1, *<sup>z</sup>*2) = % <sup>1</sup> 0 *t <sup>z</sup>*1−1(<sup>1</sup> <sup>−</sup> *<sup>t</sup>*)*z*2−<sup>1</sup> *dt*, *z*1, *z*<sup>2</sup> > 0. Then by using (16) and (17), for any (*x*, *y*) ∈ Q<sup>0</sup> and *t* ∈ [*σ*1, *σ*2] we obtain

E2(*x*, *y*)(*t*) ≥ *<sup>σ</sup>*<sup>2</sup> *σ*1 G6(*t*, *τ*)*ϕ*<sup>2</sup> *<sup>τ</sup> σ*1 G2(*τ*, *ζ*)g *ζ*, *x*(*ζ*), *y*(*ζ*), *I ν*1 <sup>0</sup>+*x*(*ζ*), *I ν*2 <sup>0</sup>+*y*(*ζ*) *dζ dτ* <sup>≥</sup> *<sup>σ</sup>γ*2−<sup>1</sup> 1 *<sup>σ</sup>*<sup>2</sup> *σ*1 J6(*τ*) *<sup>τ</sup> σ*1 G2(*τ*, *ζ*) *c*4 *t*1*x*(*ζ*) + *t*2*y*(*ζ*) + *t*<sup>3</sup> *I ν*1 <sup>0</sup>+*x*(*ζ*) + *t*<sup>4</sup> *I ν*2 <sup>0</sup>+*y*(*ζ*) *<sup>ρ</sup>*2−<sup>1</sup> −*C*1]*dζ*) 2−1 *dτ* <sup>≥</sup> *<sup>σ</sup>γ*2−<sup>1</sup> 1 *<sup>σ</sup>*<sup>2</sup> *σ*1 J6(*τ*) *<sup>τ</sup> σ*1 G2(*τ*, *ζ*) *c*4 *<sup>t</sup>*1*σγ*1−<sup>1</sup> 1 *x* + *<sup>t</sup>*2*σγ*2−<sup>1</sup> 1 *y* +*t*<sup>3</sup> *σν*1+*γ*1−<sup>1</sup> <sup>1</sup> Γ(*γ*1) <sup>Γ</sup>(*γ*<sup>1</sup> <sup>+</sup> *<sup>ν</sup>*1) *x* + *t*<sup>4</sup> *σν*2+*γ*2−<sup>1</sup> <sup>1</sup> Γ(*γ*2) <sup>Γ</sup>(*γ*<sup>2</sup> <sup>+</sup> *<sup>ν</sup>*2) *y* -*ρ*2−<sup>1</sup> − *C*<sup>1</sup> ⎤ <sup>⎦</sup>*d<sup>ζ</sup>* ⎞ ⎠ 2−1 *dτ* <sup>≥</sup> *<sup>σ</sup>γ*2−<sup>1</sup> 1 *<sup>σ</sup>*<sup>2</sup> *σ*1 J6(*τ*) , *<sup>τ</sup> σ*1 G2(*τ*, *ζ*) ! *c*4 , min *<sup>t</sup>*1*σγ*1−<sup>1</sup> <sup>1</sup> , *<sup>t</sup>*2*σγ*2−<sup>1</sup> <sup>1</sup> , *t*<sup>3</sup> *σν*1+*γ*1−<sup>1</sup> <sup>1</sup> Γ(*γ*1) <sup>Γ</sup>(*γ*<sup>1</sup> <sup>+</sup> *<sup>ν</sup>*1) , *t*4 *σν*2+*γ*2−<sup>1</sup> <sup>1</sup> Γ(*γ*2) Γ(*γ*<sup>2</sup> + *ν*2) # 2 (*x*, *y*) V -*ρ*2−<sup>1</sup> − *C*<sup>1</sup> ⎤ <sup>⎦</sup> *<sup>d</sup><sup>ζ</sup>* ⎞ ⎠ 2−1 *dτ* = *σγ*2−<sup>1</sup> 1 *<sup>σ</sup>*<sup>2</sup> *σ*1 J6(*τ*) *<sup>τ</sup> σ*1 G2(*τ*, *ζ*) *c*4(2*d*4 (*x*, *y*) V) *<sup>ρ</sup>*2−<sup>1</sup> <sup>−</sup> *<sup>C</sup>*<sup>1</sup> *dζ* 2−<sup>1</sup> *dτ* = <sup>Ξ</sup>8*σγ*2−<sup>1</sup> 1 *c*4(2*d*4 (*x*, *y*) V) *<sup>ρ</sup>*2−<sup>1</sup> <sup>−</sup> *<sup>C</sup>*<sup>1</sup> 2−<sup>1</sup> = Ξ*ρ*2−<sup>1</sup> <sup>8</sup> *<sup>σ</sup>*(*γ*2−1)(*ρ*2−1) <sup>1</sup> *<sup>c</sup>*42*<sup>ρ</sup>*2−1*<sup>d</sup> ρ*2−1 4 (*x*, *y*) *ρ*2−1 V <sup>−</sup> <sup>Ξ</sup>*ρ*2−<sup>1</sup> <sup>8</sup> *<sup>σ</sup>*(*γ*2−1)(*ρ*2−1) <sup>1</sup> *C*<sup>1</sup> 2−1 = *η*2 (*x*, *y*) *ρ*2−1 V <sup>−</sup> *<sup>C</sup>*<sup>2</sup> 2−1 , *<sup>C</sup>*<sup>2</sup> <sup>=</sup> <sup>Ξ</sup>*ρ*2−<sup>1</sup> <sup>8</sup> *<sup>σ</sup>*(*γ*1−1)(*ρ*2−1) <sup>1</sup> *C*1.

So we find

$$\|\|\mathfrak{E}(\mathbf{x},\mathbf{y})\|\|\_{\mathfrak{W}} \ge \|\|\mathfrak{E}\_2(\mathbf{x},\mathbf{y})\|\| \ge \mathfrak{E}\_2(\mathbf{x},\mathbf{y})(\sigma\_1) \ge \left(\eta\_2 \|(\mathbf{x},\mathbf{y})\|\|\_{\mathfrak{W}}^{\varrho\_2 - 1} - \mathbf{C}\_2\right)^{\varrho\_2 - 1}, \forall \,(\mathbf{x},\mathbf{y}) \in \mathfrak{Q}\_0.$$

We choose *<sup>r</sup>*<sup>2</sup> <sup>≥</sup> max 1, *C*2−<sup>1</sup> <sup>2</sup> /(*η*<sup>2</sup> <sup>−</sup> <sup>1</sup>)2−<sup>1</sup> and we deduce

$$\|\|\mathfrak{E}(\mathbf{x},\mathbf{y})\|\|\_{\mathfrak{W}} \ge \|(\mathbf{x},\mathbf{y})\|\|\_{\mathfrak{W}} \,\,\forall \,(\mathbf{x},\mathbf{y}) \in \partial B\_{r\_2} \cap \mathfrak{Q}\_0. \tag{18}$$

Now based on Lemma 4, the relations (15), (18) and the Guo–Krasnosel'skii fixed point theorem we conclude that the operator E has a fixed point (*x*, *y*) ∈ (*Br*<sup>2</sup> \ *Br*<sup>1</sup> ) ∩ Q<sup>0</sup> with *r*<sup>1</sup> ≤ (*x*, *y*) V <sup>≤</sup> *<sup>r</sup>*<sup>2</sup> and *<sup>x</sup>*(*s*) <sup>≥</sup> *<sup>s</sup>γ*1−<sup>1</sup> *x* , *<sup>y</sup>*(*s*) <sup>≥</sup> *<sup>s</sup>γ*2−<sup>1</sup> *y* for all *s* ∈ [0, 1]. So *x* > 0 or *y* > 0, that is *x*(*s*) > 0 for all *s* ∈ (0, 1] or *y*(*s*) > 0 for all *s* ∈ (0, 1]. Therefore, (*x*(*t*), *y*(*t*)), *t* ∈ [0, 1] is a positive solution of problem (1), (2).

**Proof of Theorem 2.** From assumption (*H*5) we deduce that there exist *C*<sup>3</sup> > 0, *C*<sup>4</sup> > 0 such that

$$\begin{array}{c} \psi\_1(t, w\_1, w\_2, w\_3, w\_4) \le e\_1 \wp\_{\rho\_1}(\mathfrak{u}\_1 w\_1 + \mathfrak{u}\_2 w\_2 + \mathfrak{u}\_3 w\_3 + \mathfrak{u}\_4 w\_4) + \mathcal{C}\_3\\ \psi\_2(t, w\_1, w\_2, w\_3, w\_4) \le e\_2 \wp\_{\rho\_2}(\mathfrak{u}\_1 w\_1 + \mathfrak{u}\_2 w\_2 + \mathfrak{u}\_3 w\_3 + \mathfrak{u}\_4 w\_4) + \mathcal{C}\_4 \end{array} \tag{19}$$

for any *t* ∈ [0, 1] and *wi* ≥ 0, *i* = 1, ... , 4. By using (*H*2) and (19), for any (*x*, *y*) ∈ Q<sup>0</sup> and *t* ∈ [0, 1] we obtain

E1(*x*, *y*)(*t*) ≤ <sup>1</sup> 0 J3(*τ*)*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ζ*)f(*ζ*, *x*(*ζ*), *y*(*ζ*), *I μ*1 <sup>0</sup>+*x*(*ζ*), *I μ*2 <sup>0</sup>+*y*(*ζ*)) *dζ dτ* + <sup>1</sup> 0 J4(*τ*)*ϕ*<sup>2</sup> <sup>1</sup> 0 J2(*ζ*)g(*ζ*, *x*(*ζ*), *y*(*ζ*), *I ν*1 <sup>0</sup>+*x*(*ζ*), *I ν*2 <sup>0</sup>+*y*(*ζ*)) *dζ dτ* ≤ Ξ3*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ζ*)*ξ*1(*ζ*)*ψ*1(*ζ*, *x*(*ζ*), *y*(*ζ*), *I μ*1 <sup>0</sup>+*x*(*ζ*), *I μ*2 <sup>0</sup>+*y*(*ζ*)) *dζ* +Ξ4*ϕ*<sup>2</sup> <sup>1</sup> 0 J2(*ζ*)*ξ*2(*ζ*)*ψ*2(*ζ*, *x*(*ζ*), *y*(*ζ*), *I ν*1 <sup>0</sup>+*x*(*ζ*), *I ν*2 <sup>0</sup>+*y*(*ζ*)) *dζ* ≤ Ξ3*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ζ*)*ξ*1(*ζ*) *e*1*ϕρ*<sup>1</sup> *u*1*x*(*ζ*) + *u*2*y*(*ζ*) + *u*<sup>3</sup> *I μ*1 <sup>0</sup>+*x*(*ζ*) + *u*<sup>4</sup> *I μ*2 <sup>0</sup>+*y*(*ζ*) + *C*<sup>3</sup> *dζ* +Ξ4*ϕ*<sup>2</sup> <sup>1</sup> 0 J2(*ζ*)*ξ*2(*ζ*) 0 *e*2*ϕρ*<sup>2</sup> *v*1*x*(*ζ*) + *v*2*y*(*ζ*) + *v*<sup>3</sup> *I ν*1 <sup>0</sup>+*x*(*ζ*) + *v*<sup>4</sup> *I ν*2 <sup>0</sup>+*y*(*ζ*) + *C*<sup>4</sup> 1 *dζ* ≤ Ξ3*ϕ*<sup>1</sup> , <sup>1</sup> 0 J1(*ζ*)*ξ*1(*ζ*) ! *e*1 *u*1 *x* + *u*<sup>2</sup> *y* + *u*3 *x* <sup>Γ</sup>(*μ*<sup>1</sup> <sup>+</sup> <sup>1</sup>) <sup>+</sup> *u*4 *y* Γ(*μ*<sup>2</sup> + 1) *ρ*1−<sup>1</sup> + *C*<sup>3</sup> " *dζ* - +Ξ4*ϕ*<sup>2</sup> , <sup>1</sup> 0 J2(*ζ*)*ξ*2(*ζ*) ! *e*2 *v*1 *x* + *v*<sup>2</sup> *y* + *v*3 *x* <sup>Γ</sup>(*ν*<sup>1</sup> <sup>+</sup> <sup>1</sup>) <sup>+</sup> *v*4 *y* Γ(*ν*<sup>2</sup> + 1) *ρ*2−<sup>1</sup> + *C*<sup>4</sup> " *dζ* - ≤ Ξ3*ϕ*<sup>1</sup> \* *e*1 max *u*1, *u*2, *<sup>u</sup>*<sup>3</sup> <sup>Γ</sup>(*μ*1+1), *<sup>u</sup>*<sup>4</sup> Γ(*μ*2+1) 2 (*x*, *y*) V *ρ*1−1 + *C*<sup>3</sup> + × <sup>1</sup> 0 J1(*ζ*)*ξ*1(*ζ*) *dζ* 1−<sup>1</sup> +Ξ4*ϕ*<sup>2</sup> \* *e*2 max *v*1, *v*2, *<sup>v</sup>*<sup>3</sup> <sup>Γ</sup>(*ν*1+1), *<sup>v</sup>*<sup>4</sup> Γ(*ν*2+1) 2 (*x*, *y*) V *ρ*2−1 + *C*<sup>4</sup> + × <sup>1</sup> 0 J2(*ζ*)*ξ*2(*ζ*) *dζ* 2−<sup>1</sup> = Ξ1−<sup>1</sup> <sup>1</sup> Ξ<sup>3</sup> *e*1*k ρ*1−1 1 (*x*, *y*) *ρ*1−1 V <sup>+</sup> *<sup>C</sup>*<sup>3</sup> 1−1 + Ξ2−<sup>1</sup> <sup>2</sup> Ξ<sup>4</sup> *e*2*k ρ*2−1 2 (*x*, *y*) *ρ*2−1 V <sup>+</sup> *<sup>C</sup>*<sup>4</sup> 2−1 .

In a similar way we find

$$\begin{split} \mathfrak{E}\_{2}(\boldsymbol{\chi},\boldsymbol{y})(t) &\leq \Xi\_{1}^{\varrho\_{1}-1} \Xi\_{5} \Big( e\_{1} k\_{1}^{\rho\_{1}-1} || (\boldsymbol{\chi},\boldsymbol{y}) ||\_{\mathfrak{V}}^{\rho\_{1}-1} + \mathsf{C}\_{3} \Big)^{\varrho\_{1}-1} \\ &+ \Xi\_{2}^{\varrho\_{2}-1} \Xi\_{6} \Big( e\_{2} k\_{2}^{\rho\_{2}-1} || (\boldsymbol{\chi},\boldsymbol{y}) ||\_{\mathfrak{V}}^{\rho\_{2}-1} + \mathsf{C}\_{4} \Big)^{\varrho\_{2}-1} . \end{split}$$

Hence we conclude

$$\begin{split} \|\mathfrak{E}\_{1}(\boldsymbol{x},\boldsymbol{y})\| &\leq \Xi\_{1}^{\varrho\_{1}-1}\Xi\_{3}\Big(e\_{1}k\_{1}^{\rho\_{1}-1}||(\boldsymbol{x},\boldsymbol{y})||\_{\mathfrak{V}}^{\rho\_{1}-1}+\mathsf{C}\_{3}\Big)^{\varrho\_{1}-1} \\ &+\Xi\_{2}^{\varrho\_{2}-1}\Xi\_{4}\Big(e\_{2}k\_{2}^{\rho\_{2}-1}||(\boldsymbol{x},\boldsymbol{y})||\_{\mathfrak{V}}^{\rho\_{2}-1}+\mathsf{C}\_{4}\Big)^{\varrho\_{2}-1} \\ \|\|\mathfrak{E}\_{2}(\boldsymbol{x},\boldsymbol{y})\|\|\leq\Xi\_{1}^{\varrho\_{1}-1}\Xi\_{5}\Big(e\_{1}k\_{1}^{\rho\_{1}-1}||(\boldsymbol{x},\boldsymbol{y})||\_{\mathfrak{V}}^{\rho\_{1}-1}+\mathsf{C}\_{3}\Big)^{\varrho\_{1}-1} \\ &+\Xi\_{2}^{\varrho\_{2}-1}\Xi\_{6}\Big(e\_{2}k\_{2}^{\rho\_{2}-1}||(\boldsymbol{x},\boldsymbol{y})||\_{\mathfrak{V}}^{\rho\_{2}-1}+\mathsf{C}\_{4}\Big)^{\varrho\_{2}-1} \end{split}$$

and then

$$\begin{split} \|\mathfrak{E}(\mathbf{x},\boldsymbol{y})\|\_{\mathfrak{V}} &\leq \Xi\_{1}^{\varrho\_{1}-1} (\Xi\_{3} + \Xi\_{5}) \Big( e\_{1} k\_{1}^{\varrho\_{1}-1} \|(\mathbf{x},\boldsymbol{y})\|\_{\mathfrak{V}}^{\varrho\_{1}-1} + \mathsf{C}\_{3} \Big)^{\varrho\_{1}-1} \\ &+ \Xi\_{2}^{\varrho\_{2}-1} (\Xi\_{4} + \Xi\_{6}) \Big( e\_{2} k\_{2}^{\varrho\_{2}-1} \|(\mathbf{x},\boldsymbol{y})\|\_{\mathfrak{V}}^{\varrho\_{2}-1} + \mathsf{C}\_{4} \Big)^{\varrho\_{2}-1} \end{split} \tag{20}$$

for all (*x*, *y*) ∈ Q0. We choose

$$r\_3 \ge \max\left\{1, \frac{\Xi\_1^{\varrho\_1 - 1} (\Xi\_3 + \Xi\_5) \Lambda\_1 \mathfrak{C}\_3^{\varrho\_1 - 1} + \Xi\_2^{\varrho\_2 - 1} (\Xi\_4 + \Xi\_6) \Lambda\_2 \mathfrak{C}\_4^{\varrho\_2 - 1}}{1 - \left[\Xi\_1^{\varrho\_1 - 1} (\Xi\_3 + \Xi\_5) \Lambda\_1 \mathfrak{C}\_1^{\varrho\_1 - 1} k\_1 + \Xi\_2^{\varrho\_2 - 1} (\Xi\_4 + \Xi\_6) \Lambda\_2 \mathfrak{C}\_2^{\varrho\_2 - 1} k\_2\right]}\right\}.$$

Then by (20) and the inequalities (*<sup>a</sup>* <sup>+</sup> *<sup>b</sup>*)*i*−<sup>1</sup> <sup>≤</sup> <sup>Λ</sup>*i*(*ai*−<sup>1</sup> <sup>+</sup> *<sup>b</sup>i*−1), for *<sup>a</sup>*, *<sup>b</sup>* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, 2 we deduce

$$\|\|\mathfrak{E}(\mathbf{x},\mathbf{y})\|\|\_{\mathfrak{D}} \le \|(\mathbf{x},\mathbf{y})\|\|\_{\mathfrak{D}'} \; \forall \,(\mathbf{x},\mathbf{y}) \in \partial B\_{r\_3} \cap \mathfrak{D}\_0. \tag{21}$$

Now, in (*H*6) we assume that f<sup>0</sup> > *e*<sup>3</sup> (the case g<sup>0</sup> > *e*<sup>4</sup> is treated in a similar way). So there exists *<sup>r</sup>*<sup>4</sup> <sup>∈</sup> (0, 1] such that

$$f(t, w\_1, w\_2, w\_3, w\_4) \ge e\_4 \rho\_{\rho\_1} ((p\_1 w\_1 + p\_2 w\_2 + p\_3 w\_3 + p\_4 w\_4)^{\zeta\_1}),\tag{22}$$

for all *<sup>t</sup>* <sup>∈</sup> [*σ*1, *<sup>σ</sup>*2], *wi* <sup>≥</sup> 0, *<sup>i</sup>* <sup>=</sup> 1, ... , 4, <sup>∑</sup><sup>4</sup> *<sup>i</sup>*=<sup>1</sup> *piwi* <sup>≤</sup> *<sup>r</sup>*4. We define *<sup>r</sup>*<sup>4</sup> <sup>≤</sup> min{*<sup>r</sup>*4/*<sup>k</sup>*3,*<sup>r</sup>*4}, where *<sup>k</sup>*<sup>3</sup> <sup>=</sup> <sup>2</sup> max *<sup>p</sup>*1, *<sup>p</sup>*2, *<sup>p</sup>*<sup>3</sup> <sup>Γ</sup>(*μ*1+1), *<sup>p</sup>*<sup>4</sup> Γ(*μ*2+1) . Hence for any (*x*, *y*) ∈ *Br*<sup>4</sup> ∩ Q and *t* ∈ [0, 1] we find *p*1*x*(*τ*) + *p*2*y*(*τ*) + *p*<sup>3</sup> *I μ*1 *μ*2

$$\begin{cases} p\_1 \ge (\tau) + p\_2 y(\tau) + p\_3 I\_{0+}^{\mu\_1} \mathfrak{x}(\tau) + p\_4 I\_{0+}^{\mu\_2} y(\tau) \\ \le 2 \max \left\{ p\_1, p\_2, \frac{p\_3}{\Gamma(\mu\_1 + 1)}, \frac{p\_4}{\Gamma(\mu\_2 + 1)} \right\} || (\mathfrak{x}, y) ||\_{\mathfrak{W}} = \tilde{k}\_3 r\_4 \le \tilde{r}\_4. \end{cases}$$

Therefore, by using (22) and the inequalities *I μ*1 <sup>0</sup>+*x*(*τ*) ≥ *x <sup>τ</sup>μ*1+*γ*1−1Γ(*γ*1) <sup>Γ</sup>(*γ*1+*μ*1) and *<sup>I</sup> μ*2 <sup>0</sup>+*y*(*τ*) ≥ *y <sup>τ</sup>μ*2+*γ*2−1Γ(*γ*2) <sup>Γ</sup>(*γ*2+*μ*2) , for all *<sup>τ</sup>* <sup>∈</sup> [0, 1] and (*x*, *<sup>y</sup>*) <sup>∈</sup> <sup>Q</sup>0, we obtain for any (*x*, *<sup>y</sup>*) <sup>∈</sup> *Br*<sup>4</sup> <sup>∩</sup> <sup>Q</sup><sup>0</sup> and *t* ∈ [*σ*1, *σ*2]

E1(*x*, *y*)(*t*) ≥ *<sup>σ</sup>*<sup>2</sup> *σ*1 G3(*t*, *τ*)*ϕ*<sup>1</sup> *<sup>τ</sup> σ*1 G1(*τ*, *ζ*)f(*ζ*, *x*(*ζ*), *y*(*ζ*), *I μ*1 <sup>0</sup>+*x*(*ζ*), *I μ*2 <sup>0</sup>+*y*(*ζ*)) *dζ dτ* <sup>≥</sup> *<sup>σ</sup>γ*1−<sup>1</sup> 1 *<sup>σ</sup>*<sup>2</sup> *σ*1 J3(*τ*) *<sup>τ</sup> σ*1 G1(*τ*, *ζ*)*e*<sup>3</sup> *p*1*x*(*ζ*) + *p*2*y*(*ζ*) + *p*<sup>3</sup> *I μ*1 <sup>0</sup>+*x*(*ζ*) +*p*<sup>4</sup> *I μ*2 <sup>0</sup>+*y*(*ζ*) *ς*1(*ρ*1−1) *dζ* 1−<sup>1</sup> *dτ* <sup>≥</sup> *<sup>σ</sup>γ*1−<sup>1</sup> 1 *<sup>σ</sup>*<sup>2</sup> *σ*1 J3(*τ*) *<sup>τ</sup> σ*1 G1(*τ*, *ζ*)*e*<sup>3</sup> *<sup>p</sup>*1*σγ*1−<sup>1</sup> 1 *x* + *<sup>p</sup>*2*σγ*2−<sup>1</sup> 1 *y* + *p*<sup>3</sup> *σμ*1+*γ*1−<sup>1</sup> <sup>1</sup> Γ(*γ*1) <sup>Γ</sup>(*γ*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*1) *x* + *p*<sup>4</sup> *σμ*2+*γ*2−<sup>1</sup> <sup>1</sup> Γ(*γ*2) <sup>Γ</sup>(*γ*<sup>2</sup> <sup>+</sup> *<sup>μ</sup>*2) *y* -*ς*1(*ρ*1−1) *dζ* ⎞ ⎠ 1−1 *dτ*

<sup>≥</sup> *<sup>σ</sup>γ*1−<sup>1</sup> 1 *<sup>σ</sup>*<sup>2</sup> *σ*1 J3(*τ*) , *<sup>τ</sup> σ*1 G1(*τ*, *ζ*)*e*<sup>3</sup> , min *<sup>p</sup>*1*σγ*1−<sup>1</sup> <sup>1</sup> , *<sup>p</sup>*2*σγ*2−<sup>1</sup> <sup>1</sup> , *p*<sup>3</sup> *σμ*1+*γ*1−<sup>1</sup> <sup>1</sup> Γ(*γ*1) <sup>Γ</sup>(*γ*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*1) , *p*4 *σμ*2+*γ*2−<sup>1</sup> <sup>1</sup> Γ(*γ*2) Γ(*γ*<sup>2</sup> + *μ*2) # 2 (*x*, *y*) V -*ς*1(*ρ*1−1) *dζ* ⎞ ⎠ 1−1 *dτ* = *σγ*1−<sup>1</sup> 1 *<sup>σ</sup>*<sup>2</sup> *σ*1 J3(*τ*) *<sup>τ</sup> σ*1 G1(*τ*, *ζ*)*e*3(2*k*<sup>3</sup> (*x*, *y*) V) *ς*1(*ρ*1−1) *dζ* 1−<sup>1</sup> *dτ* = *σγ*1−<sup>1</sup> <sup>1</sup> *e* 1−1 <sup>3</sup> <sup>2</sup>*ς*<sup>1</sup> *<sup>k</sup> ς*1 3 (*x*, *y*) *ς*1 V *<sup>σ</sup>*<sup>2</sup> *σ*1 J3(*τ*) *<sup>τ</sup> σ*1 G1(*τ*, *ζ*) *dζ* 1−<sup>1</sup> *dτ* = *σγ*1−<sup>1</sup> <sup>1</sup> *e* 1−1 <sup>3</sup> <sup>2</sup>*ς*<sup>1</sup> *<sup>k</sup> ς*1 <sup>3</sup> Ξ<sup>7</sup> (*x*, *y*) *ς*1 V <sup>≥</sup> *<sup>σ</sup>γ*1−<sup>1</sup> <sup>1</sup> *e* 1−1 <sup>3</sup> <sup>2</sup>*ς*<sup>1</sup> *<sup>k</sup> ς*1 <sup>3</sup> Ξ<sup>7</sup> (*x*, *y*) V <sup>=</sup> (*x*, *y*) V.

Then we deduce

$$\|\mathfrak{E}(\mathbf{x},\mathbf{y})\|\_{\mathfrak{V}} \ge \|\mathfrak{E}\_1(\mathbf{x},\mathbf{y})\| \ge \mathfrak{E}\_1(\mathbf{x},\mathbf{y})(\sigma\_1) \ge \|(\mathbf{x},\mathbf{y})\|\_{\mathfrak{V}}, \ \forall (\mathbf{x},\mathbf{y}) \in \partial B\_{r\_4} \cap \mathfrak{Q}\_0. \tag{23}$$

By Lemma 4, (21), (23) and the Guo–Krasnosel'skii fixed point theorem, we conclude that E has a fixed point (*x*, *y*) ∈ (*Br*<sup>3</sup> \ *Br*<sup>4</sup> ) ∩ Q0, so *r*<sup>4</sup> ≤ (*x*, *y*) V <sup>≤</sup> *<sup>r</sup>*3, and *<sup>x</sup>*(*s*) <sup>≥</sup> *<sup>s</sup>γ*1−<sup>1</sup> *x* , *<sup>y</sup>*(*s*) <sup>≥</sup> *<sup>s</sup>γ*2−<sup>1</sup> *y* for all *s* ∈ [0, 1], which is a positive solution of problem (1), (2).

**Proof of Theorem 3.** Because assumptions (*H*1), (*H*2) and (*H*4) hold, then by Theorem 1 we deduce that there exists *r*<sup>2</sup> > 1 such that

$$\|\|\mathfrak{E}(\mathbf{x},\mathbf{y})\|\|\_{\mathfrak{V}} \ge \|\|(\mathbf{x},\mathbf{y})\|\|\_{\mathfrak{V}} \,\,\, \forall \,(\mathbf{x},\mathbf{y}) \in \partial B\_{r\_2} \cap \mathfrak{Q}\_0. \tag{24}$$

Next because assumptions (*H*1), (*H*2) and (*H*6) hold, then by Theorem 2 we conclude that there exists *r*<sup>4</sup> < 1 such that

$$\|\|\mathfrak{E}(\mathbf{x},\mathbf{y})\|\|\_{\mathfrak{D}} \ge \|(\mathbf{x},\mathbf{y})\|\|\_{\mathfrak{D}'} \,\,\forall \,(\mathbf{x},\mathbf{y}) \in \partial B\_{r\_4} \cap \mathfrak{Q}\_0. \tag{25}$$

Now, consider the set *B*<sup>1</sup> = {(*x*, *y*) ∈ V, (*x*, *y*) V <sup>&</sup>lt; <sup>1</sup>}. By assumption (*H*7) for any (*x*, *y*) ∈ *∂B*<sup>1</sup> ∩ Q<sup>0</sup> and *t* ∈ [0, 1] we find

E1(*x*, *y*)(*t*) ≤ <sup>1</sup> 0 J3(*τ*)*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ζ*)*ξ*1(*ζ*)*ψ*<sup>1</sup> *ζ*, *x*(*ζ*), *y*(*ζ*), *I μ*1 <sup>0</sup>+*x*(*ζ*), *I μ*2 <sup>0</sup>+*y*(*ζ*) *dζ dτ* + <sup>1</sup> 0 J4(*τ*)*ϕ*<sup>2</sup> <sup>1</sup> 0 J2(*ζ*)*ξ*2(*ζ*)*ψ*<sup>2</sup> *ζ*, *x*(*ζ*), *y*(*ζ*), *I ν*1 <sup>0</sup>+*x*(*ζ*), *I ν*2 <sup>0</sup>+*y*(*ζ*) *dζ dτ* <sup>≤</sup> *<sup>A</sup>*1−<sup>1</sup> 0 <sup>1</sup> 0 J3(*τ*)*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ζ*)*ξ*1(*ζ*) *dζ dτ* + *A*2−<sup>1</sup> 0 <sup>1</sup> 0 J4(*τ*)*ϕ*<sup>2</sup> <sup>1</sup> 0 J2(*ζ*)*ξ*2(*ζ*) *dζ dτ* = *A*1−<sup>1</sup> 0 <sup>1</sup> 0 J3(*τ*) *dτ* <sup>1</sup> 0 J1(*ζ*)*ξ*1(*ζ*) *dζ* 1−<sup>1</sup> +*A*2−<sup>1</sup> 0 <sup>1</sup> 0 J4(*τ*) *dτ* <sup>1</sup> 0 J2(*ζ*)*ξ*2(*ζ*) *dζ* 2−<sup>1</sup> = *A*1−<sup>1</sup> <sup>0</sup> <sup>Ξ</sup>3Ξ1−<sup>1</sup> <sup>1</sup> <sup>+</sup> *<sup>A</sup>*2−<sup>1</sup> <sup>0</sup> <sup>Ξ</sup>4Ξ2−<sup>1</sup> <sup>2</sup> <sup>&</sup>lt; <sup>1</sup> <sup>4</sup> <sup>+</sup> <sup>1</sup> <sup>4</sup> <sup>=</sup> <sup>1</sup> 2 , E2(*x*, *y*)(*t*) ≤ <sup>1</sup> 0 J5(*τ*)*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ζ*)*ξ*1(*ζ*)*ψ*<sup>1</sup> *ζ*, *x*(*ζ*), *y*(*ζ*), *I μ*1 <sup>0</sup>+*x*(*ζ*), *I μ*2 <sup>0</sup>+*y*(*ζ*) *dζ dτ* + <sup>1</sup> 0 J6(*τ*)*ϕ*<sup>2</sup> <sup>1</sup> 0 J2(*ζ*)*ξ*2(*ζ*)*ψ*<sup>2</sup> *ζ*, *x*(*ζ*), *y*(*ζ*), *I ν*1 <sup>0</sup>+*x*(*ζ*), *I ν*2 <sup>0</sup>+*y*(*ζ*) *dζ dτ* <sup>≤</sup> *<sup>A</sup>*1−<sup>1</sup> 0 <sup>1</sup> 0 J5(*τ*)*ϕ*<sup>1</sup> <sup>1</sup> 0 J1(*ζ*)*ξ*1(*ζ*) *dζ dτ* + *A*2−<sup>1</sup> 0 <sup>1</sup> 0 J6(*τ*)*ϕ*<sup>2</sup> <sup>1</sup> 0 J2(*ζ*)*ξ*2(*ζ*) *dζ dτ* = *A*1−<sup>1</sup> 0 <sup>1</sup> 0 J5(*τ*) *dτ* <sup>1</sup> 0 J1(*ζ*)*ξ*1(*ζ*) *dζ* 1−<sup>1</sup> +*A*2−<sup>1</sup> 0 <sup>1</sup> 0 J6(*τ*) *dτ* <sup>1</sup> 0 J2(*ζ*)*ξ*2(*ζ*) *dζ* 2−<sup>1</sup> = *A*1−<sup>1</sup> <sup>0</sup> <sup>Ξ</sup>5Ξ1−<sup>1</sup> <sup>1</sup> <sup>+</sup> *<sup>A</sup>*2−<sup>1</sup> <sup>0</sup> <sup>Ξ</sup>6Ξ2−<sup>1</sup> <sup>2</sup> <sup>&</sup>lt; <sup>1</sup> <sup>4</sup> <sup>+</sup> <sup>1</sup> <sup>4</sup> <sup>=</sup> <sup>1</sup> 2 .

Therefore we deduce E1(*x*, *y*) < <sup>1</sup> <sup>2</sup> , E2(*x*, *y*) < <sup>1</sup> <sup>2</sup> for all (*x*, *y*) ∈ *∂B*<sup>1</sup> ∩ Q0. So we obtain

$$\|\|\mathfrak{E}(\mathbf{x},\mathbf{y})\|\|\_{\mathfrak{W}} = \|\|\mathfrak{E}\_1(\mathbf{x},\mathbf{y})\| + \|\|\mathfrak{E}\_2(\mathbf{x},\mathbf{y})\|\| < 1 = \|(\mathbf{x},\mathbf{y})\|\|\_{\mathfrak{W}} \; \forall \,(\mathbf{x},\mathbf{y}) \in \partial B\_1 \cap \mathfrak{Q}\_0. \tag{26}$$

Then, by (24) and (26) we conclude that there exists a positive solution (*x*1, *y*1) ∈ Q<sup>0</sup> with 1 < (*x*1, *y*1) V <sup>≤</sup> *<sup>r</sup>*<sup>2</sup> for problem (1), (2). By (25) and (26) we deduce that there exists another positive solution (*x*2, *y*2) ∈ Q<sup>0</sup> with *r*<sup>4</sup> ≤ (*x*2, *y*2) V <sup>&</sup>lt; 1 for problem (1), (2). Hence problem (1), (2) has at least two positive solutions (*x*1(*t*), *y*1(*t*)), (*x*2(*t*), *y*2(*t*)), *t* ∈ [0, 1].

#### **5. Examples**

Let *δ*<sup>1</sup> = <sup>7</sup> <sup>4</sup> , *<sup>δ</sup>*<sup>2</sup> <sup>=</sup> <sup>5</sup> <sup>3</sup> , *<sup>p</sup>* <sup>=</sup> 3, *<sup>q</sup>* <sup>=</sup> 4, *<sup>γ</sup>*<sup>1</sup> <sup>=</sup> <sup>5</sup> <sup>2</sup> , *<sup>γ</sup>*<sup>2</sup> <sup>=</sup> <sup>17</sup> <sup>5</sup> , *<sup>n</sup>* <sup>=</sup> 1, *<sup>m</sup>* <sup>=</sup> 2, *<sup>μ</sup>*<sup>1</sup> <sup>=</sup> <sup>23</sup> <sup>6</sup> , *<sup>μ</sup>*<sup>2</sup> <sup>=</sup> <sup>19</sup> 7 , *ν*<sup>1</sup> = <sup>47</sup> <sup>9</sup> , *<sup>ν</sup>*<sup>2</sup> <sup>=</sup> <sup>22</sup> <sup>3</sup> , *<sup>α</sup>*<sup>0</sup> <sup>=</sup> <sup>4</sup> <sup>3</sup> , *<sup>α</sup>*<sup>1</sup> <sup>=</sup> <sup>2</sup> <sup>3</sup> , *<sup>β</sup>*<sup>0</sup> <sup>=</sup> <sup>9</sup> <sup>4</sup> , *<sup>β</sup>*<sup>1</sup> <sup>=</sup> <sup>3</sup> <sup>4</sup> , *<sup>β</sup>*<sup>2</sup> <sup>=</sup> <sup>5</sup> <sup>6</sup> , *<sup>ρ</sup>*<sup>1</sup> <sup>=</sup> <sup>27</sup> <sup>8</sup> , *<sup>ρ</sup>*<sup>2</sup> <sup>=</sup> <sup>38</sup> <sup>9</sup> , <sup>1</sup> <sup>=</sup> <sup>27</sup> 19 , <sup>2</sup> = <sup>38</sup> <sup>29</sup> , <sup>M</sup>0(*τ*) = <sup>5</sup>*<sup>τ</sup>* <sup>7</sup> , *<sup>τ</sup>* <sup>∈</sup> [0, 1], <sup>N</sup>0(*τ*) = <sup>1</sup> <sup>2</sup> , *τ* ∈ 0, <sup>1</sup> 3 ; <sup>11</sup> <sup>10</sup> , *τ* ∈ 1 <sup>3</sup> , 1 , <sup>M</sup>1(*τ*) = 3 <sup>4</sup> , *τ* ∈ 0, <sup>1</sup> 2 ; <sup>93</sup> <sup>28</sup> , *τ* ∈ 1 <sup>2</sup> , 1 , <sup>N</sup>1(*τ*) = <sup>1</sup> <sup>3</sup> , *τ* ∈ 0, <sup>4</sup> 5 ; <sup>29</sup> <sup>24</sup> , *τ* ∈ 4 <sup>5</sup> , 1 , <sup>N</sup>2(*τ*) = <sup>3</sup>*<sup>τ</sup>* 2 , *τ* ∈ [0, 1].

We consider the system of fractional differential equations

$$\begin{cases} D\_{0+}^{7/4} \left( \varphi\_{27/8} \left( D\_{0+}^{5/2} \mathbf{x}(t) \right) \right) = \mathfrak{f} \left( t, \mathbf{x}(t), y(t), I\_{0+}^{23/6} \mathbf{x}(t), I\_{0+}^{19/7} y(t) \right), & t \in (0, 1), \\\ D\_{0+}^{5/3} \left( \varphi\_{38/9} \left( D\_{0+}^{17/5} y(t) \right) \right) = \mathfrak{g} \left( t, \mathbf{x}(t), y(t), I\_{0+}^{47/9} \mathbf{x}(t), I\_{0+}^{22/3} y(t) \right), & t \in (0, 1), \end{cases} \tag{27}$$

with the boundary conditions

$$\begin{cases} \begin{aligned} x(0) = \mathbf{x}'(0) = 0, \ D\_{0+}^{5/2} \mathbf{x}(0) = 0, \ \ \ \mathbf{q}\_{27/8} \left( D\_{0+}^{5/2} \mathbf{x}(1) \right) = \frac{5}{7} \int\_{0}^{1} \mathbf{q}\_{27/8} \left( D\_{0+}^{5/2} \mathbf{x}(\tau) \right) d\tau, \\\ D\_{0+}^{4/3} \mathbf{x}(1) = \frac{18}{7} D\_{0+}^{2/3} \mathcal{Y} \left( \frac{1}{2} \right), \\\ y(0) = y'(0) = y''(0) = 0, \ D\_{0+}^{37/5} y(0) = 0, \ D\_{0+}^{17/5} y(1) = \left( \frac{3}{5} \right)^{9/29} D\_{0+}^{17/5} y \left( \frac{1}{3} \right), \\\ D\_{0+}^{9/4} y(1) = \frac{7}{8} D\_{0+}^{3/4} \mathbf{x} \left( \frac{4}{5} \right) + \frac{3}{2} \int\_{0}^{1} D\_{0+}^{5/6} \mathbf{x}(\tau) \, d\tau. \end{aligned} \end{cases} \tag{28}$$

We obtain here a<sup>1</sup> ≈ 0.59183673 > 0, a<sup>2</sup> ≈ 0.71155008 > 0, b<sup>1</sup> ≈ 1.45311179, b<sup>2</sup> ≈ 2.39587178, b ≈ 1.09690108 > 0. Then assumption (*H*1) is satisfied. We also find

<sup>g</sup>1(*t*, *<sup>τ</sup>*) = <sup>1</sup> Γ(7/4) - *t* 3/4(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)3/4 <sup>−</sup> (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)3/4, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> 1, *t* 3/4(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)3/4, 0 <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, <sup>g</sup>2(*t*, *<sup>τ</sup>*) = <sup>1</sup> Γ(5/3) - *t* 2/3(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)2/3 <sup>−</sup> (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)2/3, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> 1, *t* 2/3(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)2/3, 0 <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, <sup>g</sup>3(*t*, *<sup>τ</sup>*) = <sup>1</sup> Γ(5/2) - *t* 3/2(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)1/6 <sup>−</sup> (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)3/2, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> 1, *t* 3/2(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)1/6, 0 <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, <sup>g</sup>31(*ϑ*, *<sup>τ</sup>*) = <sup>1</sup> Γ(7/4) - *<sup>ϑ</sup>*3/4(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)1/6 <sup>−</sup> (*<sup>ϑ</sup>* <sup>−</sup> *<sup>τ</sup>*)3/4, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>ϑ</sup>* <sup>≤</sup> 1, *<sup>ϑ</sup>*3/4(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)1/6, 0 <sup>≤</sup> *<sup>ϑ</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, <sup>g</sup>32(*t*, *<sup>τ</sup>*) = <sup>1</sup> Γ(5/3) - *<sup>ϑ</sup>*2/3(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)1/6 <sup>−</sup> (*<sup>ϑ</sup>* <sup>−</sup> *<sup>τ</sup>*)2/3, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>ϑ</sup>* <sup>≤</sup> 1, *<sup>ϑ</sup>*2/3(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)1/6, 0 <sup>≤</sup> *<sup>ϑ</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, <sup>g</sup>4(*t*, *<sup>τ</sup>*) = <sup>1</sup> Γ(17/5) - *t* 12/5(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)3/20 <sup>−</sup> (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)12/5, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> 1, *t* 12/5(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)3/20, 0 <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, <sup>g</sup>41(*ϑ*, *<sup>τ</sup>*) = <sup>1</sup> Γ(41/15) - *<sup>ϑ</sup>*26/15(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)3/20 <sup>−</sup> (*<sup>ϑ</sup>* <sup>−</sup> *<sup>τ</sup>*)26/15, 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>ϑ</sup>* <sup>≤</sup> 1, *<sup>ϑ</sup>*26/15(<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)3/20, 0 <sup>≤</sup> *<sup>ϑ</sup>* <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> 1, <sup>G</sup>1(*t*, *<sup>τ</sup>*) = <sup>g</sup>1(*t*, *<sup>τ</sup>*) + <sup>5</sup>*<sup>t</sup>* 3/4 7a<sup>1</sup> <sup>1</sup> 0 g1(*ζ*, *τ*) *dζ*, <sup>G</sup>2(*t*, *<sup>τ</sup>*) = <sup>g</sup>2(*t*, *<sup>τ</sup>*) + <sup>3</sup>*<sup>t</sup>* 2/3 5a<sup>2</sup> g2 1 3 , *τ* ,

$$\begin{split} & \mathfrak{G}\_{3}(t,\tau) = \mathfrak{g}\_{3}(t,\tau) + \frac{t^{3/2}\mathfrak{b}\_{1}}{\mathfrak{b}} \left[ \frac{7}{8} \mathfrak{g}\_{31} \left( \frac{7}{8},\tau \right) + \frac{3}{2} \int\_{0}^{1} \mathfrak{g}\_{32}(\theta,\tau) \, d\theta \right], \\ & \mathfrak{G}\_{4}(t,\tau) = \frac{18t^{3/2}\Gamma(17/5)}{7\mathfrak{b}\Gamma(23/20)} \mathfrak{g}\_{41} \left( \frac{1}{2},\tau \right), \\ & \mathfrak{G}\_{5}(t,\tau) = \frac{t^{12/5}\Gamma(5/2)}{\mathfrak{b}\Gamma(7/6)} \left[ \frac{7}{8} \mathfrak{g}\_{31} \left( \frac{4}{5},\tau \right) + \frac{3}{2} \int\_{0}^{1} \mathfrak{g}\_{32}(\theta,\tau) \, d\theta \right], \\ & \mathfrak{G}\_{6}(t,\tau) = \mathfrak{g}\_{4}(t,\tau) + \frac{18t^{12/5}\mathfrak{b}\_{2}}{7\mathfrak{b}} \mathfrak{g}\_{41} \left( \frac{1}{2},\tau \right), \\ & \mathfrak{h}\_{1}(\tau) = \frac{1}{\Gamma(7/4)} (1-\tau)^{3/4}, \ \mathfrak{b}\_{2}(\tau) = \frac{1}{\Gamma(5/3)} (1-\tau)^{2/3}, \\ & \mathfrak{h}\_{3}(\tau) = \frac{1}{\Gamma(5/2)} (1-\tau)^{1/6} (1-(1-\tau)^{4/3}), \\ & \mathfrak{h}\_{4}(\tau) = \frac{1}{\Gamma(7/5)} (1-\tau)^{3/20} (1-(1-\tau)^{9/4}), \end{split}$$

for all *t*, *τ*, *ϑ* ∈ [0, 1]. In addition we deduce

J1(*τ*) = h1(*τ*) + <sup>5</sup> <sup>7</sup>a1Γ(11/4) (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)3/4 <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)7/4 , *τ* ∈ [0, 1], J2(*τ*) = ⎧ ⎪⎨ ⎪⎩ h2(*τ*) + <sup>3</sup> <sup>5</sup>a2Γ(5/3) \* <sup>1</sup> 3 2/3 (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)2/3 <sup>−</sup> 1 <sup>3</sup> − *τ* 2/3+ , 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> <sup>1</sup> 3 , h2(*τ*) + <sup>3</sup> <sup>5</sup>a2Γ(5/3) 1 3 2/3 (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)2/3, <sup>1</sup> <sup>3</sup> < *τ* ≤ 1, J3(*τ*) = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎩ h3(*τ*) + <sup>b</sup><sup>1</sup> b - 7 8Γ(7/4) \* <sup>4</sup> 5 3/4 (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)1/6 <sup>−</sup> 4 <sup>5</sup> − *τ* 3/4+ + <sup>3</sup> 2Γ(8/3) (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)1/6 <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)5/3 , 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> <sup>4</sup> 5 , h3(*τ*) + <sup>b</sup><sup>1</sup> b - 7 8Γ(7/4) 4 5 3/4 (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)1/6 + <sup>3</sup> 2Γ(8/3) (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)1/6 <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)5/3 , <sup>4</sup> <sup>5</sup> < *τ* ≤ 1, J4(*τ*) = ⎧ ⎪⎨ ⎪⎩ 18Γ(17/5) <sup>7</sup>bΓ(23/20)Γ(41/15) \* <sup>1</sup> 2 26/15 (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)3/20 <sup>−</sup> 1 <sup>2</sup> − *τ* 26/15+ , 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> <sup>1</sup> 2 , 18Γ(17/5) <sup>7</sup>bΓ(23/20)Γ(41/15) 1 2 26/15 (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)3/20, <sup>1</sup> <sup>2</sup> < *τ* ≤ 1, J5(*τ*) = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎩ Γ(5/2) bΓ(7/6) - 7 8Γ(7/4) \* <sup>4</sup> 5 3/4 (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)1/6 <sup>−</sup> 4 <sup>5</sup> − *τ* 3/4+ + <sup>3</sup> 2Γ(8/3) (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)1/6 <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)5/3 , 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> <sup>4</sup> 5 , Γ(5/2) bΓ(7/6) - 7 8Γ(7/4) 4 5 3/4 (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)1/6 + <sup>3</sup> 2Γ(8/3) (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)1/6 <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)5/3 , <sup>4</sup> <sup>5</sup> < *τ* ≤ 1, J6(*τ*) = ⎧ ⎪⎨ ⎪⎩ h4(*τ*) + <sup>18</sup>b<sup>2</sup> <sup>7</sup>bΓ(41/15) \* <sup>1</sup> 2 26/15 (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)3/20 <sup>−</sup> 1 <sup>2</sup> − *τ* 26/15+ , 0 <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> <sup>1</sup> 2 , h4(*τ*) + <sup>18</sup>b<sup>2</sup> <sup>7</sup>bΓ(41/15) 1 2 26/15 (<sup>1</sup> <sup>−</sup> *<sup>τ</sup>*)3/20, <sup>1</sup> <sup>2</sup> < *τ* ≤ 1.

**Example 1.** *We introduce the functions*

$$\begin{split} \mathfrak{f}(t, w\_1, w\_2, w\_3, w\_4) &= \frac{(3w\_1 + 2w\_2 + w\_3 + 5w\_4)^{19a/8}}{t^{z\_1}(1-t)^{z\_2}},\\ \mathfrak{g}(t, w\_1, w\_2, w\_3, w\_4) &= \frac{(w\_1 + 7w\_2 + 4w\_3 + 2w\_4)^{29b/9}}{t^{z\_3}(1-t)^{z\_4}} \end{split} \tag{29}$$

*for t* ∈ (0, 1), *wi* ≥ 0, *i* = 1, ... , 4*, where a* > 1*, b* > 1*, z*<sup>1</sup> ∈ (0, 1)*, z*<sup>2</sup> ∈ 0, <sup>7</sup> 4 *, z*<sup>3</sup> ∈ (0, 1)*, z*<sup>4</sup> ∈ 0, <sup>5</sup> 3 *. Here ξ*1(*t*) = <sup>1</sup> *t <sup>z</sup>*<sup>1</sup> (1−*t*)*z*<sup>2</sup> *, <sup>ξ</sup>*2(*t*) = <sup>1</sup> *t <sup>z</sup>*<sup>3</sup> (1−*t*)*z*<sup>4</sup> *for <sup>t</sup>* <sup>∈</sup> (0, 1)*, <sup>ψ</sup>*1(*t*, *<sup>w</sup>*1, *<sup>w</sup>*2, *<sup>w</sup>*3, *<sup>w</sup>*4) = (3*w*<sup>1</sup> + 2*w*<sup>2</sup> + *w*<sup>3</sup> + 5*w*4)19*a*/8 *and ψ*2(*t*, *w*1, *w*2, *w*3, *w*4)=(*w*<sup>1</sup> + 7*w*<sup>2</sup> + 4*w*<sup>3</sup> + 2*w*4)29*b*/9 *for t* ∈ [0, 1]*, wi* ≥ 0*, i* = 1, ... , 4*. We also obtain M*<sup>1</sup> = *B*(1 − *z*1, 7/4 − *z*2) ∈ (0, ∞)*, M*<sup>2</sup> = *B*(1 − *z*3, 5/3 − *z*4) ∈ (0, ∞)*. Then assumption* (*H*2) *is satisfied. In addition, in* (*H*3)*, for l*<sup>1</sup> = 3*, l*<sup>2</sup> = 2*, l*<sup>3</sup> = 1*, l*<sup>4</sup> = 5*, θ*<sup>1</sup> = 1*, m*<sup>1</sup> = 1*, m*<sup>2</sup> = 7*, m*<sup>3</sup> = 4*, m*<sup>4</sup> = 2*, θ*<sup>2</sup> = 1*, we find ψ*<sup>10</sup> = 0 *and ψ*<sup>20</sup> = 0*. In* (*H*4)*, for* [*σ*1, *σ*2] ⊂ (0, 1)*, s*<sup>1</sup> = 3*, s*<sup>2</sup> = 2*, s*<sup>3</sup> = 1*, s*<sup>4</sup> = 5*, we obtain*

f<sup>∞</sup> = ∞*. Then by Theorem 1 we deduce that problem (27), (28) with the nonlinearities (29) has at least one solution* (*x*1(*t*), *y*1(*t*)), *t* ∈ [0, 1]*.*

**Example 2.** *We define the functions*

$$\begin{split} f(t, w\_1, w\_2, w\_3, w\_4) &= \frac{p\_0(t+3)}{(t^2+8)\sqrt[4]{t^3}} \left[ \left(\frac{1}{2}w\_1 + w\_2 + \frac{1}{4}w\_3 + \frac{1}{5}w\_4\right)^{v\_1} \\ &+ \left(\frac{1}{2}w\_1 + w\_2 + \frac{1}{4}w\_3 + \frac{1}{5}w\_4\right)^{v\_2} \Big|\_{,} \ t \in (0, 1], \ w\_i \ge 0, \ i = 1, \ldots, 4, \\ g(t, w\_1, w\_2, w\_3, w\_4) &= \frac{q\_0(2+\sin t)}{(t+6)^4\sqrt[4]{(1-t)^5}} \left(w\_1^{v\_3} + e^{w\_2} + \ln\left(w\_3 + w\_4 + 1\right)\right), \\ t \in [0, 1), \ w\_i \ge 0, \ i = 1, \ldots, 4, \end{split} \tag{30}$$

*where <sup>p</sup>*<sup>0</sup> <sup>&</sup>gt; <sup>0</sup>*, <sup>q</sup>*<sup>0</sup> <sup>&</sup>gt; <sup>0</sup>*, <sup>υ</sup>*<sup>1</sup> <sup>&</sup>gt; 19/8*, <sup>υ</sup>*<sup>2</sup> <sup>∈</sup> (0, 19/8)*, <sup>υ</sup>*<sup>3</sup> <sup>&</sup>gt; <sup>0</sup>*. Here we have <sup>ξ</sup>*1(*t*) = <sup>1</sup> <sup>√</sup><sup>4</sup> *<sup>t</sup>*<sup>3</sup> *, <sup>t</sup>* <sup>∈</sup> (0, 1]*, <sup>ψ</sup>*1(*t*, *<sup>w</sup>*1, *<sup>w</sup>*2, *<sup>w</sup>*3, *<sup>w</sup>*4) = *<sup>p</sup>*0(*t*+3) (*t*2+8) 1 <sup>2</sup>*w*<sup>1</sup> <sup>+</sup> *<sup>w</sup>*<sup>2</sup> <sup>+</sup> <sup>1</sup> <sup>4</sup>*w*<sup>3</sup> <sup>+</sup> <sup>1</sup> <sup>5</sup>*w*<sup>4</sup> *υ*1 + 1 <sup>2</sup>*w*<sup>1</sup> <sup>+</sup> *<sup>w</sup>*<sup>2</sup> <sup>+</sup> <sup>1</sup> <sup>4</sup>*w*3+ 1 <sup>5</sup>*w*<sup>4</sup> *υ*2 *, <sup>t</sup>* <sup>∈</sup> [0, 1]*, wi* <sup>≥</sup> <sup>0</sup>*, <sup>i</sup>* <sup>=</sup> 1, ... , 4*, <sup>ξ</sup>*2(*t*) = <sup>1</sup> <sup>√</sup><sup>7</sup> (1−*t*)<sup>5</sup> , *<sup>t</sup>* <sup>∈</sup> [0, 1)*, <sup>ψ</sup>*2(*t*, *<sup>w</sup>*1, *<sup>w</sup>*2, *<sup>w</sup>*3, *<sup>w</sup>*4) = *q*0(2+sin *t*) (*t*+6)<sup>4</sup> (*wυ*<sup>3</sup> <sup>1</sup> <sup>+</sup> *<sup>e</sup>w*<sup>2</sup> <sup>+</sup> ln(*w*<sup>3</sup> <sup>+</sup> *<sup>w</sup>*<sup>4</sup> <sup>+</sup> <sup>1</sup>))*, <sup>t</sup>* <sup>∈</sup> [0, 1]*, wi* <sup>≥</sup> <sup>0</sup>*, <sup>i</sup>* <sup>=</sup> 1, ... , 4*. We obtain <sup>M</sup>*<sup>1</sup> <sup>=</sup> *<sup>B</sup>*(1/4, 7/4) <sup>∈</sup> (0, <sup>∞</sup>)*, <sup>M</sup>*<sup>2</sup> <sup>=</sup> <sup>21</sup> <sup>20</sup> ∈ (0, ∞)*. Then assumption* (*H*2) *is satisfied. For* [*σ*1, *σ*2] ⊂ (0, 1)*, s*<sup>1</sup> = <sup>1</sup> <sup>2</sup> *, <sup>s</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup>*, <sup>s</sup>*<sup>3</sup> <sup>=</sup> <sup>1</sup> <sup>4</sup> *, <sup>s</sup>*<sup>4</sup> <sup>=</sup> <sup>1</sup> <sup>5</sup> *, we find* <sup>f</sup><sup>∞</sup> <sup>=</sup> <sup>∞</sup> *(in* (*H*4)*), and for <sup>p</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> *, p*<sup>2</sup> = 1*, p*<sup>3</sup> = <sup>1</sup> <sup>4</sup> *, <sup>p</sup>*<sup>4</sup> <sup>=</sup> <sup>1</sup> <sup>5</sup> *, ς*<sup>1</sup> ∈ 8*υ*<sup>2</sup> <sup>19</sup> , 1 *, we have* f<sup>0</sup> = ∞ *(in* (*H*6)*). So assumptions* (*H*4) *and* (*H*6) *are satisfied. Then after some computations we deduce* Ξ<sup>1</sup> ≈ 3.93816256*,* Ξ<sup>2</sup> ≈ 1.53523525*,* Ξ<sup>3</sup> ≈ 1.40740842*,* Ξ<sup>4</sup> ≈ 0.97489748*,* Ξ<sup>5</sup> ≈ 1.04873754*,* Ξ<sup>6</sup> ≈ 0.92404828*,* = 1*, and <sup>A</sup>*<sup>0</sup> <sup>=</sup> max<sup>4</sup>*p*<sup>0</sup> 9 <sup>39</sup> <sup>20</sup> *<sup>υ</sup>*<sup>1</sup> <sup>+</sup> <sup>39</sup> <sup>20</sup> *<sup>υ</sup>*<sup>2</sup> , *q*0*m*0(1 + *e* + ln 3) *, where <sup>m</sup>*<sup>0</sup> = max*t*∈[0,1] 2+sin *t* (*t*+6)<sup>4</sup> ≈ 2.00035047*. If*

$$\begin{aligned} p\_0 &< \frac{9}{\left(\frac{39}{20}\right)^{\upsilon\_1} + \left(\frac{39}{20}\right)^{\upsilon\_2}} \min\left\{ \frac{1}{4^{27/8}\Xi\_3^{19/8}\Xi\_1}, \frac{1}{4^{38/9}\Xi\_4^{29/9}\Xi\_2}, \frac{1}{4^{27/8}\Xi\_5^{19/8}\Xi\_1}, \frac{1}{4^{38/9}\Xi\_6^{29/9}\Xi\_2} \right\},\\ q\_0 &< \frac{1}{m\_0(1+e+\ln 3)} \min\left\{ \frac{1}{4^{19/8}\Xi\_3^{19/8}\Xi\_1}, \frac{1}{4^{29/9}\Xi\_4^{29/9}\Xi\_2}, \frac{1}{4^{19/8}\Xi\_5^{19/8}\Xi\_1}, \frac{1}{4^{19/8}\Xi\_5^{19/8}\Xi\_1}, \frac{1}{4^{29/9}\Xi\_6^{29/9}\Xi\_2} \right\}. \end{aligned}$$

*then the inequalities A*8/19 <sup>0</sup> <sup>Ξ</sup>3Ξ8/19 <sup>1</sup> <sup>&</sup>lt; <sup>1</sup> <sup>4</sup> *, <sup>A</sup>*9/29 <sup>0</sup> <sup>Ξ</sup>4Ξ9/29 <sup>2</sup> <sup>&</sup>lt; <sup>1</sup> <sup>4</sup> *, <sup>A</sup>*8/19 <sup>0</sup> <sup>Ξ</sup>5Ξ8/19 <sup>1</sup> <sup>&</sup>lt; <sup>1</sup> 4 *, A*9/29 <sup>0</sup> <sup>Ξ</sup>6Ξ9/29 <sup>2</sup> <sup>&</sup>lt; <sup>1</sup> <sup>4</sup> *are satisfied, (that is, assumption* (*H*7) *is satisfied). For example, if υ*<sup>1</sup> = 2*, υ*<sup>2</sup> = 3 *and p*<sup>0</sup> ≤ 0.0008*, q*<sup>0</sup> ≤ 0.0004*, then the above inequalities are verified. By Theorem 3, we conclude that problem (27), (28) with the nonlinearities (30) has at least two positive solutions* (*x*1(*t*), *y*1(*t*)), (*x*2(*t*), *y*2(*t*)), *t* ∈ [0, 1]*.*

#### **6. Conclusions**

In this paper we investigated the system of coupled fractional differential equations (1) with *ρ*-Laplacian operators and Riemann–Liouville fractional derivatives of varied orders, supplemented with general nonlocal boundary conditions (2) containing Riemann–Stieltjes integrals and fractional derivatives of differing orders. The nonlinearities from the system are dependent on various fractional integrals and they are nonnegative and singular in the points *t* = 0 and *t* = 1. The last boundary conditions for the unknown functions *x* and *y* are coupled in the point 1, in contrast to the boundary conditions from paper [6] in which they are uncoupled in the point 1. We presented diverse assumptions on the functions f and g so that problem (1), (2) has one positive solution (in Theorems 1 and 2), and two positive solutions (in Theorem 3). We also gave the corresponding Green functions and their properties used in the proof of the main results. We transformed our problem into a system of integral equations and we associated an operator E for which we looked for the fixed points by applying the Guo–Krasnosel'skii fixed point theorem of cone expansion and compression of norm type. We presented finally two examples for illustrating our main

theorems. For some future research directions we have in mind the study of some systems of fractional differential equations with other nonlocal coupled or uncoupled boundary conditions.

**Author Contributions:** Conceptualization, R.L.; Formal analysis, A.T. and R.L.; Methodology, A.T. and R.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **On a System of Riemann–Liouville Fractional Boundary Value Problems with -Laplacian Operators and Positive Parameters**

**Johnny Henderson <sup>1</sup> , Rodica Luca 2,\* and Alexandru Tudorache <sup>3</sup>**


**Abstract:** In this paper, we study the existence and nonexistence of positive solutions of a system of Riemann–Liouville fractional differential equations with -Laplacian operators, supplemented with coupled nonlocal boundary conditions containing Riemann–Stieltjes integrals, fractional derivatives of various orders, and positive parameters. We apply the Schauder fixed point theorem in the proof of the existence result.

**Keywords:** Riemann–Liouville fractional differential equations; nonlocal coupled boundary conditions; positive solutions; existence; nonexistence; positive parameters

**MSC:** 34A08; 34B10; 34B18

#### **1. Introduction**

We consider the system of fractional differential equations with 1-Laplacian and 2-Laplacian operators

$$\begin{cases} D\_{0+}^{\gamma\_1} (\varrho\_{\varrho\_1} (D\_{0+}^{\delta\_1} \mathbf{u}(t))) + \mathfrak{a}(t) \mathfrak{f}(\mathbf{v}(t)) = 0, & t \in (0,1), \\\ D\_{0+}^{\gamma\_2} (\varrho\_{\varrho\_2} (D\_{0+}^{\delta\_2} \mathbf{v}(t))) + \mathfrak{b}(t) \mathfrak{g}(\mathbf{u}(t)) = 0, & t \in (0,1), \end{cases} \tag{1}$$

subject to the coupled nonlocal boundary conditions

$$\begin{cases} \begin{aligned} \mathbf{u}^{(j)}(0) = 0, \; j = 0, \dots, p - 2; \; D\_{0+}^{\delta\_1} \mathbf{u}(0) = 0, \; D\_{0+}^{a\_0} \mathbf{u}(1) = \sum\_{j=1}^{n} \int\_0^1 D\_{0+}^{\delta\_j} \mathbf{v}(\tau) \, d\mathfrak{H}\_{\mathbf{j}}(\tau) + a\_0, \\\ \mathbf{v}^{(j)}(0) = 0, \; j = 0, \dots, q - 2; \; D\_{0+}^{\delta\_2} \mathbf{v}(0) = 0, \; D\_{0+}^{\delta\_0} \mathbf{v}(1) = \sum\_{j=1}^{m} \int\_0^1 D\_{0+}^{\delta\_j} \mathbf{u}(\tau) \, d\mathfrak{H}\_{\mathbf{j}}(\tau) + \mathfrak{h}\_0. \end{aligned} \end{cases} \tag{2}$$

where *<sup>γ</sup>*1, *<sup>γ</sup>*<sup>2</sup> <sup>∈</sup> (0, 1], *<sup>δ</sup>*<sup>1</sup> <sup>∈</sup> (*<sup>p</sup>* <sup>−</sup> 1, *<sup>p</sup>*], *<sup>δ</sup>*<sup>2</sup> <sup>∈</sup> (*<sup>q</sup>* <sup>−</sup> 1, *<sup>q</sup>*], *<sup>p</sup>*, *<sup>q</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>p</sup>*, *<sup>q</sup>* <sup>≥</sup> 3, *<sup>n</sup>*, *<sup>m</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>α</sup><sup>j</sup>* <sup>∈</sup> <sup>R</sup> for all *<sup>j</sup>* <sup>=</sup> 0, 1, ... , *<sup>n</sup>*, 0 <sup>≤</sup> *<sup>α</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>α</sup>*<sup>2</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *<sup>α</sup><sup>n</sup>* <sup>≤</sup> *<sup>β</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>δ</sup>*<sup>2</sup> <sup>−</sup> 1, *<sup>β</sup>*<sup>0</sup> <sup>≥</sup> 1, *<sup>β</sup><sup>j</sup>* <sup>∈</sup> <sup>R</sup> for all *j* = 0, 1, ... , *m*, 0 ≤ *β*<sup>1</sup> < *β*<sup>2</sup> < ··· < *β<sup>m</sup>* ≤ *α*<sup>0</sup> < *δ*<sup>1</sup> − 1, *α*<sup>0</sup> ≥ 1, the functions <sup>f</sup>, <sup>g</sup> : <sup>R</sup><sup>+</sup> <sup>→</sup> <sup>R</sup><sup>+</sup> and <sup>a</sup>, <sup>b</sup> : [0, 1] <sup>→</sup> <sup>R</sup><sup>+</sup> are continuous, (R<sup>+</sup> = [0, <sup>∞</sup>)), <sup>c</sup><sup>0</sup> and <sup>d</sup><sup>0</sup> are positive parameters, 1, <sup>2</sup> > 1, *ϕ<sup>i</sup>* (*ζ*) = |*ζ*| *i*−2*ζ*, *ϕ*−<sup>1</sup> *<sup>i</sup>* = *ϕρ<sup>i</sup>* , *<sup>ρ</sup><sup>i</sup>* = *<sup>i</sup> i*−<sup>1</sup> , and *<sup>i</sup>* <sup>=</sup> 1, 2. The integrals from the conditions (2) are Riemann–Stieltjes integrals with H*j*, *j* = 1, ... , *n* and K*i*, *i* = 1, ... , *m* functions of bounded variation, and *D<sup>k</sup>* <sup>0</sup><sup>+</sup> denotes the Riemann–Liouville derivative of order *k* (for *k* = *γ*1, *δ*1, *γ*2, *δ*2, *αj*; for *j* = 0, 1, ... , *n*, *βi*; and for *i* = 0, 1, ... , *m*).

We present in this paper sufficient conditions for the functions f and g, and intervals for the parameters c<sup>0</sup> and d<sup>0</sup> such that problem (1) and (2) has at least one positive solution, or it has no positive solutions. We apply the Schauder fixed point theorem in the proof of the main existence result. A positive solution of (1) and (2) is a pair of functions (u, v) <sup>∈</sup> (*C*([0, 1]; <sup>R</sup>+))<sup>2</sup> that satisfy the system (1) and the boundary conditions (2), with

**Citation:** Henderson, J.; Luca, R.; Tudorache, A. On a System of Riemann–Liouville Fractional Boundary Value Problems with -Laplacian Operators and Positive Parameters. *Fractal Fract.* **2022**, *6*, 299. https://doi.org/10.3390/ fractalfract6060299

Academic Editor: Maria Rosaria Lancia

Received: 22 April 2022 Accepted: 27 May 2022 Published: 29 May 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

u(*t*) > 0 and v(*t*) > 0 for all *t* ∈ (0, 1]. Now, we present some recent results related to our problem. By using the Guo–Krasnosel'skii fixed point theorem, in [1], the authors investigated the system of fractional differential equations

$$\begin{cases} D\_{0+}^{\gamma\_1} (\boldsymbol{\varrho}\_{\boldsymbol{\theta}\_1} (\boldsymbol{D}\_{0+}^{\delta\_1} \mathbf{u}(t))) + \lambda f(t, \mathbf{u}(t), \mathbf{v}(t)) = \mathbf{0}, & t \in (0, 1), \\\ D\_{0+}^{\gamma\_2} (\boldsymbol{\varrho}\_{\boldsymbol{\theta}\_2} (\boldsymbol{D}\_{0+}^{\delta\_2} \mathbf{v}(t))) + \mu \boldsymbol{g}(t, \mathbf{u}(t), \mathbf{v}(t)) = \mathbf{0}, & t \in (0, 1), \end{cases} \tag{3}$$

supplemented with the boundary conditions (2) with c<sup>0</sup> = d<sup>0</sup> = 0, where *λ* and *μ* are positive parameters, and *<sup>f</sup>* , *<sup>g</sup>* <sup>∈</sup> *<sup>C</sup>*([0, 1] <sup>×</sup> <sup>R</sup><sup>+</sup> <sup>×</sup> <sup>R</sup>+, <sup>R</sup>+). They presented various intervals for *λ* and *μ* such that problem (2) and (3) with c<sup>0</sup> = d<sup>0</sup> = 0 has at least one positive solution (u(*t*) > 0 for all *t* ∈ (0, 1], or v(*t*) > 0 for all *t* ∈ (0, 1]). They also studied the nonexistence of positive solutions. In [2], the author investigated the existence and nonexistence of positive solutions for the system (3) with the uncoupled boundary conditions

$$\begin{cases} \mathbf{u}^{(j)}(0) = 0, \; j = 0, \ldots, p - 2; \; D\_{0+}^{\delta\_1} \mathbf{u}(0) = 0, \; D\_{0+}^{\mathbf{u}\_0} \mathbf{u}(1) = \sum\_{j=1}^{n} \int\_0^1 D\_{0+}^{\delta\_j} \mathbf{u}(\tau) \, d\mathfrak{H}\_j(\tau), \\\ \mathbf{v}^{(j)}(0) = 0, \; j = 0, \ldots, q - 2; \; D\_{0+}^{\delta\_2} \mathbf{v}(0) = 0, \; D\_{0+}^{\delta\_0} \mathbf{v}(1) = \sum\_{j=1}^{m} \int\_0^1 D\_{0+}^{\delta\_j} \mathbf{v}(\tau) \, d\mathfrak{H}\_j(\tau). \end{cases}$$

where *<sup>α</sup><sup>j</sup>* <sup>∈</sup> <sup>R</sup> for all *<sup>j</sup>* <sup>=</sup> 0, 1, ... , *<sup>n</sup>*, 0 <sup>≤</sup> *<sup>α</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>α</sup>*<sup>2</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *<sup>α</sup><sup>n</sup>* <sup>≤</sup> *<sup>α</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>δ</sup>*<sup>1</sup> <sup>−</sup> 1, *<sup>α</sup>*<sup>0</sup> <sup>≥</sup> 1, *<sup>β</sup><sup>j</sup>* <sup>∈</sup> <sup>R</sup> for all *j* = 0, 1, ... , *m*, 0 ≤ *β*<sup>1</sup> < *β*<sup>2</sup> < ··· < *β<sup>m</sup>* ≤ *β*<sup>0</sup> < *δ*<sup>2</sup> − 1, *β*<sup>0</sup> ≥ 1, H*i*, *i* = 1, ... , *n*, and K*j*, *j* = 1, ··· , *m* are functions of bounded variation. In [3], the authors studied the positive solutions for the system of nonlinear fractional differential equations

$$\begin{cases} D\_{0+}^{\alpha} \mathbf{u}(t) + \mathfrak{a}(t) \mathfrak{f}(\mathbf{v}(t)) = 0, & t \in (0,1), \\\ D\_{0+}^{\beta} \mathbf{v}(t) + \mathfrak{b}(t) \mathfrak{g}(\mathbf{u}(t)) = 0, & t \in (0,1), \end{cases}$$

subject to the coupled integral boundary conditions

$$\begin{cases} \mathbf{u}(0) = \mathbf{u}'(0) = \dots = \mathbf{u}^{(n-2)}(0) = 0, \ \mathbf{u}(1) = \int\_0^1 \mathbf{v}(\tau) d\mathfrak{H}(\tau) + \mathfrak{e}\_{0\prime} \\\ \mathbf{v}(0) = \mathbf{v}'(0) = \dots = \mathbf{v}^{(m-2)}(0) = 0, \ \mathbf{v}(1) = \int\_0^1 \mathbf{u}(\tau) d\mathfrak{H}(\tau) + \mathfrak{e}\_{0\prime} \end{cases}$$

where *<sup>n</sup>* <sup>−</sup> <sup>1</sup> <sup>&</sup>lt; *<sup>α</sup>* <sup>≤</sup> *<sup>n</sup>*, *<sup>m</sup>* <sup>−</sup> <sup>1</sup> <sup>&</sup>lt; *<sup>β</sup>* <sup>≤</sup> *<sup>m</sup>*, *<sup>n</sup>*, *<sup>m</sup>* <sup>∈</sup> <sup>N</sup>, *<sup>n</sup>*, *<sup>m</sup>* <sup>≥</sup> 3, <sup>a</sup>, <sup>b</sup>, <sup>f</sup>, <sup>g</sup> are nonnegative continuous functions, c<sup>0</sup> and d<sup>0</sup> are positive parameters, and H and K are bounded variation functions. In [4], the authors investigated the existence and nonexistence of positive solutions for the system (1) with the nonlocal uncoupled boundary conditions with positive parameters

$$\begin{cases} \mathbf{u}^{(j)}(0) = 0, \; j = 0, \dots, p - 2; \; D\_{0+}^{\delta\_1} \mathbf{u}(0) = 0, \; D\_{0+}^{a\_0} \mathbf{u}(1) = \sum\_{j=1}^{n} \int\_0^1 D\_{0+}^{a\_j} \mathbf{u}(\tau) \, d\mathfrak{H}\_j(\tau) + \mathfrak{c}\_{0+}, \\\ \mathbf{v}^{(j)}(0) = 0, \; j = 0, \dots, q - 2; \; D\_{0+}^{\delta\_2} \mathbf{v}(0) = 0, \; D\_{0+}^{\delta\_0} \mathbf{v}(1) = \sum\_{j=1}^{m} \int\_0^1 D\_{0+}^{\delta\_j} \mathbf{v}(\tau) \, d\mathfrak{H}\_j(\tau) + \mathfrak{c}\_{0+}. \end{cases}$$

We note that our problem (1) and (2) is different than the problem studied in [4], because of the boundary conditions, which are coupled in (2) and uncoupled in [4]. Based on this difference, here, we will use, for problem (1) and (2), other Green functions, different systems of integral equations, and different operators than those in [4]. We would also like to mention the papers [5–10], and the monographs [11–13], which contain other recent results for fractional differential equations and systems of fractional differential equations with or without Laplacian operators, and for various applications. The novelties of our problem (1) and (2) with respect to the above papers consist in the consideration of positive parameters c<sup>0</sup> and d<sup>0</sup> in the coupled nonlocal boundary conditions (2) containing fractional

derivatives of various orders and Riemann–Stieltjes integrals, combined with the system of fractional differential Equation (1), which has -Laplacian operators.

The paper is structured as follows. In Section 2, we present some auxiliary results, which include the Green functions associated with our problem (1) and (2) and their properties. In Section 3, we give the main theorems for the existence and nonexistence of positive solutions for (1) and (2), and Section 4 contains an example illustrating our results. Finally, in Section 5, we present the conclusions of this work.

#### **2. Auxiliary Results**

In this section, we present some results from [1], which will be used in our main theorems in the next section.

We consider the system of fractional differential equations

$$\begin{cases} D\_{0+}^{\gamma\_1} (\boldsymbol{\varrho}\_{\boldsymbol{\varrho}\_1} (\boldsymbol{D}\_{0+}^{\delta\_1} \mathbf{u}(t))) + \tilde{\mathbf{h}}(t) = \mathbf{0}, & t \in (0, 1), \\\ D\_{0+}^{\gamma\_2} (\boldsymbol{\varrho}\_{\boldsymbol{\varrho}\_2} (\boldsymbol{D}\_{0+}^{\delta\_2} \mathbf{v}(t))) + \tilde{\mathbf{k}}(t) = \mathbf{0}, & t \in (0, 1), \end{cases} \tag{4}$$

with the coupled boundary conditions

$$\begin{cases} \begin{aligned} \mathbf{u}^{(j)}(0) = 0, \; j = 0, \dots, p - 2; \; D\_{0+}^{\delta\_1} \mathbf{u}(0) = 0, \; D\_{0+}^{a\_0} \mathbf{u}(1) = \sum\_{j=1}^{n} \int\_0^1 D\_{0+}^{a\_j} \mathbf{v}(\tau) \, d\mathfrak{H}\_{\mathbf{j}}(\tau), \\ \mathbf{v}^{(j)}(0) = 0, \; j = 0, \dots, q - 2; \; D\_{0+}^{\delta\_2} \mathbf{v}(0) = 0, \; D\_{0+}^{\delta\_0} \mathbf{v}(1) = \sum\_{j=1}^{m} \int\_0^1 D\_{0+}^{\delta\_j} \mathbf{u}(\tau) \, d\mathfrak{H}\_{\mathbf{j}}(\tau), \end{aligned} \end{cases} \tag{5}$$

where h, <sup>k</sup> <sup>∈</sup> *<sup>C</sup>*[0, 1]. We denote this by

$$\begin{split} \Delta\_{1} &= \sum\_{i=1}^{n} \frac{\Gamma(\delta\_{2})}{\Gamma(\delta\_{2} - \alpha\_{i})} \int\_{0}^{1} \tau^{\delta\_{2} - \alpha\_{i} - 1} d\mathfrak{H}\_{l}(\tau), \ \Delta\_{2} = \sum\_{i=1}^{m} \frac{\Gamma(\delta\_{1})}{\Gamma(\delta\_{1} - \beta\_{i})} \int\_{0}^{1} \tau^{\delta\_{1} - \beta\_{i} - 1} d\mathfrak{H}\_{l}(\tau), \\ \Delta &= \frac{\Gamma(\delta\_{1})\Gamma(\delta\_{2})}{\Gamma(\delta\_{1} - \alpha\_{0})\Gamma(\delta\_{2} - \beta\_{0})} - \Delta\_{1} \Delta\_{2}. \end{split}$$

**Lemma 1** ([1])**.** *If* <sup>Δ</sup> <sup>=</sup> <sup>0</sup>*, then the unique solution* (u, v) <sup>∈</sup> (*C*[0, 1])<sup>2</sup> *of problem (4) and (5) is given by*

$$\begin{cases} \begin{aligned} \mathbf{u}(t) &= \int\_{0}^{1} \mathfrak{G}\_{1}(t,\zeta) \, \rho\_{\rho\_{1}}(I\_{0+}^{\gamma\_{1}}\tilde{\mathbf{h}}(\zeta)) \, d\zeta + \int\_{0}^{1} \mathfrak{G}\_{2}(t,\zeta) \, \rho\_{\rho\_{2}}(I\_{0+}^{\gamma\_{2}}\tilde{\mathbf{k}}(\zeta)) \, d\zeta, \, \forall t \in [0,1], \\\ \mathbf{v}(t) &= \int\_{0}^{1} \mathfrak{G}\_{3}(t,\zeta) \, \rho\_{\rho\_{1}}(I\_{0+}^{\gamma\_{1}}\tilde{\mathbf{h}}(\zeta)) \, d\zeta + \int\_{0}^{1} \mathfrak{G}\_{4}(t,\zeta) \, \rho\_{\rho\_{2}}(I\_{0+}^{\gamma\_{2}}\tilde{\mathbf{k}}(\zeta)) \, d\zeta, \, \forall t \in [0,1], \end{aligned} \end{cases} \end{cases} \tag{6}$$

*where*

$$\begin{split} \mathfrak{G}\_{1}(t,\xi) &= \mathfrak{g}\_{1}(t,\xi) + \frac{t^{\delta\_{1}-1}\Delta\_{1}}{\Delta} \left( \sum\_{j=1}^{m} \int\_{0}^{1} \mathfrak{g}\_{1j}(\tau,\xi) \, d\mathfrak{H}\_{j}(\tau) \right), \\ \mathfrak{G}\_{2}(t,\xi) &= \frac{t^{\delta\_{1}-1}\Gamma(\delta\_{2})}{\Delta\Gamma(\delta\_{2}-\beta\_{0})} \sum\_{j=1}^{n} \int\_{0}^{1} \mathfrak{g}\_{2j}(\tau,\xi) \, d\mathfrak{H}\_{j}(\tau), \\ \mathfrak{G}\_{3}(t,\xi) &= \frac{t^{\delta\_{2}-1}\Gamma(\delta\_{1})}{\Delta\Gamma(\delta\_{1}-\alpha\_{0})} \sum\_{j=1}^{m} \int\_{0}^{1} \mathfrak{g}\_{1j}(\tau,\xi) \, d\mathfrak{H}\_{j}(\tau), \\ \mathfrak{G}\_{4}(t,\xi) &= \mathfrak{g}\_{2}(t,\xi) + \frac{t^{\delta\_{2}-1}\Delta\_{2}}{\Delta} \left( \sum\_{j=1}^{n} \int\_{0}^{1} \mathfrak{g}\_{2j}(\tau,\xi) \, d\mathfrak{H}\_{j}(\tau) \right), \end{split} \tag{7}$$

*for all* (*t*, *ζ*) ∈ [0, 1] × [0, 1] *and*

$$\begin{split} \mathfrak{g}\_{1}(t,\boldsymbol{\zeta}) &= \frac{1}{\Gamma(\delta\_{1})} \begin{cases} t^{\delta\_{1}-1} (1-\boldsymbol{\zeta})^{\delta\_{1}-a\_{0}-1} - (t-\boldsymbol{\zeta})^{\delta\_{1}-1}, & 0 \le \boldsymbol{\zeta} \le t \le 1, \\\ t^{\delta\_{1}-1} (1-\boldsymbol{\zeta})^{\delta\_{1}-a\_{0}-1}, & 0 \le t \le \boldsymbol{\zeta} \le 1, \\\ \mathfrak{g}\_{1\rangle}(\boldsymbol{\tau}, \boldsymbol{\zeta}) &= \frac{1}{\Gamma(\delta\_{1}-\beta\_{\boldsymbol{\zeta}})} \begin{cases} \tau^{\delta\_{1}-\beta\_{\boldsymbol{\zeta}}-1} (1-\boldsymbol{\zeta})^{\delta\_{1}-a\_{0}-1} - (\boldsymbol{\tau}-\boldsymbol{\zeta})^{\delta\_{1}-\beta\_{\boldsymbol{\zeta}}-1}, & 0 \le \boldsymbol{\zeta} \le \boldsymbol{\tau} \le 1, \\\ \tau^{\delta\_{1}-\beta\_{\boldsymbol{\zeta}}-1} (1-\boldsymbol{\zeta})^{\delta\_{1}-a\_{0}-1}, & 0 \le \boldsymbol{\tau} \le \boldsymbol{\zeta} \le 1, \end{cases} \end{split}$$

$$\begin{split} \mathfrak{g}\_{2}(t,\zeta) &= \frac{1}{\Gamma(\delta\_{2})} \begin{cases} t^{\delta\_{2}-1} (1-\zeta)^{\delta\_{2}-\beta\_{0}-1} - (t-\zeta)^{\delta\_{2}-1}, & 0 \le \zeta \le t \le 1, \\ t^{\delta\_{2}-1} (1-\zeta)^{\delta\_{2}-\beta\_{0}-1}, & 0 \le t \le \zeta \le 1, \\ \tau^{\delta\_{2}-a\_{k}} \left( \begin{array}{c} \tau^{\delta\_{2}-a\_{k}-1} (1-\zeta)^{\delta\_{2}-\beta\_{0}-1} - (\tau-\zeta)^{\delta\_{2}-a\_{k}-1}, \ 0 \le \zeta \le \tau \le 1, \\ \tau^{\delta\_{2}-a\_{k}-1} (1-\zeta)^{\delta\_{2}-\beta\_{0}-1}, & 0 \le \tau \le \zeta \le 1, \\ \tau^{\delta\_{2}-a\_{k}} \left( \end{array} \right) & \begin{array}{c} 0 \le \tau \le \zeta \le 1, \\ \end{array} \end{split}$$

*for all j* = 1, . . . , *m and k* = 1, . . . , *n.*

**Lemma 2** ([1])**.** *We suppose that* Δ > 0*,* H*j*, *j* = 1, ... , *n,* K*j*, *j* = 1, ... , *m are nondecreasing functions. Therefore, the functions* G*i*, *i* = 1, . . . , 4 *(given by (7)) have the following properties:*


$$\mathfrak{J}\_1(\zeta) = \mathfrak{h}\_1(\zeta) + \frac{\Delta\_1}{\Delta} \left( \sum\_{j=1}^m \int\_0^1 g\_{1j}(\tau, \zeta) d\mathfrak{K}\_j(\tau) \right) , \,\forall \, \zeta \in [0, 1].$$

$$\mathfrak{h}\_{\mathfrak{m}\_1(\zeta)} = \frac{1}{\Delta} \int\_{(1 \, \vert \, \tau)} \mathfrak{h}\_{\mathfrak{m}\_1 \mathfrak{K}\_1 \vert \, \tau} (1 - \mathfrak{a}\_{\mathfrak{m}\_1} \underline{\mathfrak{K}}\_{\mathfrak{K}\_1}(\underline{1} - \underline{\tau}) \mathfrak{K}\_{\mathfrak{K}}) , \, \mathfrak{a}\_{\mathfrak{m}\_1 \mathfrak{K}} \underline{\mathfrak{K}}\_{\mathfrak{K}}(\underline{1} - \underline{\mathfrak{K}}\_{\mathfrak{K}}(\underline{1})) $$

$$\text{and } \mathfrak{h}\_1(\mathbb{Q}) = \frac{1}{\Gamma(\delta\_1)} (1 - \zeta)^{\delta\_1 - \mathfrak{a}\_0 - 1} (1 - (1 - \zeta)^{\mathfrak{a}\_0}), \text{ for all } \zeta \in [0, 1].$$

*(3)* G1(*t*, *ζ*) ≥ *t <sup>δ</sup>*1−1J1(*ζ*) *for all* (*t*, *<sup>ζ</sup>*) <sup>∈</sup> [0, 1] <sup>×</sup> [0, 1]*; (4)* G2(*t*, *ζ*) ≤ J2(*ζ*), *for all* (*t*, *ζ*) ∈ [0, 1] × [0, 1]*, where*

$$\mathfrak{J}\_2(\zeta) = \frac{\Gamma(\delta\_2)}{\Delta \Gamma(\delta\_2 - \beta\_0)} \sum\_{j=1}^n \int\_0^1 g\_{2j}(\tau, \zeta) d\mathfrak{H}\_j(\tau), \ \forall \, \zeta \in [0, 1];$$

*(5)* G2(*t*, *ζ*) = *t <sup>δ</sup>*1−1J2(*ζ*) *for all* (*t*, *<sup>ζ</sup>*) <sup>∈</sup> [0, 1] <sup>×</sup> [0, 1]*;*

$$\begin{aligned} (6) \quad \mathfrak{G}\_3(t,\zeta) \le \mathfrak{J}\_3(\zeta) \text{ for all } (t,\zeta) \in [0,1] \times [0,1], where \\ \mathfrak{J}\_3(\zeta) = \frac{\Gamma(\delta\_1)}{\Delta \Gamma(\delta\_1 - \mathfrak{a}\_0)} \sum\_{j=1}^m \int\_0^1 \mathfrak{g}\_{1j}(\tau,\zeta) d\mathfrak{K}\_j(\tau), \ \forall \,\zeta \in [0,1]; \end{aligned}$$

$$(\top)\\_\mathcal{G}\_{\eth}(t,\zeta) = t^{\delta\_2 - 1}\Im\_{\eth}(\zeta)\not\!/\not\!/\not\!/\not\!/\not\!/\not\!\//\not\!\/=\not\!\!/\not\!\!/\not\!\!/\not\!\!/\not\!\!/\not\!\!/\not\!\!/\not\!\!/\not\!\!/\not\!\!/\not\!\!/\not\!\!/\not\!\!\times$$

$$(8)\quad \mathfrak{G}\_4(t,\zeta) \le \mathfrak{J}\_4(\zeta) \text{ for all } (t,\zeta) \in [0,1] \times [0,1], where$$

$$\begin{split} \mathfrak{J}\_{4}(\zeta) &= \mathfrak{h}\_{2}(\zeta) + \frac{\Delta\_{2}}{\Delta} \left( \sum\_{j=1}^{n} \int\_{0}^{1} \mathfrak{g}\_{2j}(\tau, \zeta) dH\_{j}(\tau) \right), \; \forall \, \zeta \in [0, 1], \\ \text{and } \mathfrak{h}\_{2}(\zeta) &= \frac{1}{\Gamma(\delta\_{2})} (1 - \zeta)^{\delta\_{2} - \beta\_{0} - 1} (1 - (1 - \zeta)^{\beta\_{0}}), \; for \; all \; \zeta \in [0, 1]. \\ \text{or } \sigma\_{1}(\nu, \tau) &> \iota \delta - 1 \gamma, \; \tau \ll \nu \text{ and } (\nu, \tau) \in [0, 1]. \end{split}$$

$$(\mathfrak{G}) \quad \mathfrak{G}\_4(t,\mathbb{Q}) \ge t^{\delta\_2 - 1} \mathfrak{J}\_4(\mathbb{Q}), for all \ (t,\mathbb{Q}) \in [0,1] \times [0,1].$$

**Lemma 3.** *We suppose that* Δ > 0*,* H*i*, *i* = 1, ... , *n,* K*j*, *j* = 1, ... , *m are nondecreasing functions, and* <sup>h</sup>, <sup>k</sup> <sup>∈</sup> *<sup>C</sup>*([0, 1]; <sup>R</sup>+)*. Therefore, the solution* (u(*t*), v(*t*)), *<sup>t</sup>* <sup>∈</sup> [0, 1] *of problem (4) and (5) (given by (6)) satisfies the inequalities* u(*t*) ≥ 0*,* v(*t*) ≥ 0*,* u(*t*) ≥ *t <sup>δ</sup>*1<sup>−</sup>1u(*ν*)*,* v(*t*) ≥ *t <sup>δ</sup>*2<sup>−</sup>1v(*ν*) *for all t*, *<sup>ν</sup>* <sup>∈</sup> [0, 1]*.*

**Proof.** Under the assumptions of this lemma, by using relations (6) and Lemma 2, we find that u(*t*) ≥ 0 and v(*t*) ≥ 0 for all *t* ∈ [0, 1]. In addition, for all *t*, *ν* ∈ [0, 1], we obtain the following inequalities:

u(*t*) ≥ *t δ*1−1 <sup>1</sup> 0 J1(*ζ*)*ϕρ*<sup>1</sup> (*I γ*1 <sup>0</sup>+<sup>h</sup>(*ζ*)) *<sup>d</sup><sup>ζ</sup>* <sup>+</sup> <sup>1</sup> 0 J2(*ζ*)*ϕρ*<sup>2</sup> (*I γ*2 <sup>0</sup>+<sup>k</sup>(*ζ*)) *<sup>d</sup><sup>ζ</sup>* ≥ *t δ*1−1 <sup>1</sup> 0 G1(*ν*, *ζ*)*ϕρ*<sup>1</sup> (*I γ*1 <sup>0</sup>+<sup>h</sup>(*ζ*)) *<sup>d</sup><sup>ζ</sup>* <sup>+</sup> <sup>1</sup> 0 G2(*ν*, *ζ*)*ϕρ*<sup>2</sup> (*I γ*2 <sup>0</sup>+<sup>k</sup>(*ζ*)) *<sup>d</sup><sup>ζ</sup>* = *t <sup>δ</sup>*1<sup>−</sup>1u(*ν*), v(*t*) ≥ *t δ*2−1 <sup>1</sup> 0 J3(*ζ*)*ϕρ*<sup>1</sup> (*I γ*1 <sup>0</sup>+<sup>h</sup>(*ζ*)) *<sup>d</sup><sup>ζ</sup>* <sup>+</sup> <sup>1</sup> 0 J4(*ζ*)*ϕρ*<sup>2</sup> (*I γ*2 <sup>0</sup>+<sup>k</sup>(*ζ*)) *<sup>d</sup><sup>ζ</sup>* ≥ *t δ*2−1 <sup>1</sup> 0 G3(*ν*, *ζ*)*ϕρ*<sup>1</sup> (*I γ*1 <sup>0</sup>+<sup>h</sup>(*ζ*)) *<sup>d</sup><sup>ζ</sup>* <sup>+</sup> <sup>1</sup> 0 G4(*ν*, *ζ*)*ϕρ*<sup>2</sup> (*I γ*2 <sup>0</sup>+<sup>k</sup>(*ζ*)) *<sup>d</sup><sup>ζ</sup>* = *t <sup>δ</sup>*2<sup>−</sup>1v(*ν*).

#### **3. Main Results**

In this section, we study the existence and nonexistence of positive solutions for problem (1) and (2) under some conditions on a, b, f, and g, when the positive parameters c<sup>0</sup> and d<sup>0</sup> belong to some intervals.

We now give the assumptions that we will use in the next part.


$$\begin{split} & \frac{\iota\_{L}}{L}, \mathfrak{g}(z) < \frac{\iota\_{0}}{L} \text{ for all } z \in [0, \mathfrak{e}\_{0}], \text{ where} \\ & L = \max \left\{ \frac{2^{\varrho\_{1}-1} \Sigma\_{1}}{\Gamma(\gamma\_{1}+1)} \left( \int\_{0}^{1} \mathfrak{J}\_{i}(\zeta) \zeta^{\gamma\_{1}(\rho\_{1}-1)} \, d\zeta \right)^{\varrho\_{1}-1}, i \in \{1,3\} \right\}, \\ & \frac{2^{\varrho\_{2}-1} \Sigma\_{2}}{\Gamma(\gamma\_{2}+1)} \left( \int\_{0}^{1} \mathfrak{J}\_{j}(\zeta) \zeta^{\gamma\_{2}(\rho\_{2}-1)} \, d\zeta \right)^{\varrho\_{2}-1}, j \in \{2,4\} \end{split}$$

0

with <sup>Ξ</sup><sup>1</sup> <sup>=</sup> sup*τ*∈[0,1] <sup>a</sup>(*τ*), <sup>Ξ</sup><sup>2</sup> <sup>=</sup> sup*τ*∈[0,1] <sup>b</sup>(*τ*).

(*K*4) The functions <sup>f</sup>, <sup>g</sup> : <sup>R</sup><sup>+</sup> <sup>→</sup> <sup>R</sup><sup>+</sup> are continuous and satisfy the conditions lim*w*→<sup>∞</sup> <sup>f</sup>(*w*) *w*1−<sup>1</sup> <sup>=</sup> <sup>∞</sup> and lim*w*→<sup>∞</sup> <sup>g</sup>(*w*) *<sup>w</sup>*2−<sup>1</sup> <sup>=</sup> <sup>∞</sup>.

By assumptions (*K*1) and (*K*2) and Lemma 2, we obtain that the constant *L* from assumption (*K*3) is positive.

Now, we consider the following system of fractional differential equations:

$$\begin{cases} \begin{array}{l} D\_{0+}^{\gamma\_1} \left( \varphi\_{\ell 1} (D\_{0+}^{\delta\_1} \mathfrak{x}(t)) \right) = 0, \ t \in (0,1), \\\ D\_{0+}^{\gamma\_2} \left( \varphi\_{\ell 2} (D\_{0+}^{\delta\_2} \mathfrak{y}(t)) \right) = 0, \ t \in (0,1), \end{array} \tag{8}$$
