An Existence Study for a Multiplied System with p-Laplacian Involving φ-Hilfer Derivatives

Abstract

In this paper, we study the existence of solutions for a multiplied system of fractional differential equations with nonlocal integro multi-point boundary conditions by using the p-Laplacian operator and the $\phi$-Hilfer derivatives. The presented results are obtained by the fixed point theorems of Krasnoselskii. An illustrative example is presented at the end to show the applicability of the obtained results. To the best of our knowledge, this is the first time where such a problem is considered.

1. Introduction

As cited in [1,2,3,4,5,6,7], fractional calculus has recently become more significant in a variety of scientific disciplines. Fractional integrals and derivatives are currently defined in a variety of ways, ranging from the most well-known Riemann–Liouville and Caputo fractional derivatives to less well-known approaches. In [8], R. Hilfer introduced the fractional Hilfer derivative, which is a generalization of the Riemann–Liouville and Caputo derivatives.

Recently, new fractional differential results have been published, see [5,9]. In most of these published articles, Schauder, Krasnoselskii, Darbo, or Monch theories have been used to prove the existence of solutions to nonlinear fractional equations.

In [10,11] and the references within, certain features and uses of the Helfer derivative are discussed. Several authors have looked into prime value difficulties involving fractional Hilfer derivatives (see [9,12,13]). However, there are few publications in the literature on the boundary value problems of fractional Hilfer derivatives. The authors defined nonlocal value problems for derivatives of Helfer’s fractions in [14]. We refer to the publications in [15,16,17] for some recent work on boundary value problems using fractional Hilfer derivatives. Some writers have worked on fractional issue solutions using p-Laplacian operators. For example, [18,19,20,21,22,23] where the solutions to a nonlinear fractional equation with the p-Laplacian operator were examined. We will mention some more studies here for the reader’s convenience. We begin with the work of A. Devi, A. Kumar, D. Baleanu, and A. Khan [19], who used Caputo derivatives of distinct orders and the ${\psi }_p$ Laplacian operator to obtain stability conclusions for the following nonlinear FDEs:

$$\left\{\begin{array}{c}{}^c{\mathcal{D}}^{r_1}{\psi }_p\left[{}^c{\mathcal{D}}^{r_2}\left(u\left(t\right)-\sum _{i=1}^m{v}_i\left(t\right)\right)\right]=-w(t,u\left(t\right)),t\in \left(0,1\right]\\ {\left.{\psi }_p\left[{}^c{\mathcal{D}}^{r_2}\left(u\left(t\right)-\sum _{i=1}^m{v}_i\left(t\right)\right)\right]\right|}_{t=0}=0,\\ u\left(0\right)=\sum _{i=1}^m{v}_i\left(0\right),\\ u^{\prime }\left(1\right)=\sum _{i=1}^m{v}_i^{\prime }\left(1\right),\\ u^j\left(0\right)=\sum _{i=1}^m{v}_i^j\left(0\right),\mathrm{for}j=2,3,\dots ,n-1\end{array}\right.$$

where $0<r_1\le 1,n-1<r_2\le n,n\ge 4$, and $v_i,w$ are continuous functions. ${}^c{\mathcal{D}}^{r_1}$ and ${}^c{\mathcal{D}}^{r_2}$, in Caputo’s interpretation, represent the derivative of fractional order $r_1$ and $r_2$, respectively, and ${\psi }_p\left(z\right)={\left|z\right|}^{p-2}z$ denotes the p-Laplacian operator, which is equal to $\frac{1}{p}+\frac{1}{q}=1,{\left({\psi }_p\right)}^{-1}={\psi }_q.$

In the present research, we study the existence of solutions for the following multiplied system of the $\phi$-Hilfer-type fractional differential equations order, with a p-Laplacian operator of the form:

$$\left\{\begin{array}{c}{}^H{\mathcal{D}}_{a^{+}}^{{\alpha }_{k1},{\beta }_{k1};\phi }{\psi }_{p_k}\left({}^H{\mathcal{D}}_{a^{+}}^{{\alpha }_{k2},{\beta }_{k2};\phi }{u}_k\right)\left(t\right)=F_k(t,u_1\left(t\right),\dots ,u_n\left(t\right)),t\in \left[a,b\right],k=\overline{1,n},\hfill \\ u_k\left(a\right)=0,u_k\left(b\right)={\displaystyle \sum _{i=1}^n}{\lambda }_i{u}_i\left({\zeta }_{ik}\right),a<{\zeta }_k<b,{\lambda }_i\in \left]0,+\infty \right[\hfill \\ {\psi }_{p_k}\left({}^H{\mathcal{D}}_{a^{+}}^{{\alpha }_{k2},{\beta }_{k2};\phi }{u}_k\right)\left(a\right)=0,{\psi }_{p_k}\left({}^H{\mathcal{D}}_{a^{+}}^{{\alpha }_{k2},{\beta }_{k2};\phi }{u}_k\right)\left(b\right)={\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right),a<\zeta <b,\hfill \end{array}\right.$$

(1)

Here, we take ${}^H{\mathcal{D}}_{a^{+}}^{{\alpha }_{k1},{\beta }_{k1};\phi }$ and ${}^H{\mathcal{D}}_{a^{+}}^{{\alpha }_{k2},{\beta }_{k2};\phi }$$\left(k=\overline{1,n},n\in \mathbb{N},n\ge 2\right)$, which are the $\phi$-Hilfer fractional derivative of orders ${\alpha }_{k1}$ and ${\alpha }_{k2}$ with $1<{\alpha }_{k1},{\alpha }_{k2}<2$ and ${\beta }_{k1},{\beta }_{k2}$ 2 parameters $0\le {\beta }_{k1},{\beta }_{k2}\le 1,$ and ${\mathcal{I}}_{0^{+}}^{\rho ;\phi }$ the left-handed $\phi$-Riemann–Liouville fractional integral of order $\rho$ with $\rho >0$, and ${\psi }_{p_k}\left(z\right)={\left|z\right|}^{p_k-2}z$ denotes the $p_k$-Laplacian operator, which is equal to $\frac{1}{p_k}+\frac{1}{q_k}=1,{\left({\psi }_{p_k}\right)}^{-1}={\psi }_{q_k},\left(q_k\ge 2\right)$ m, and $\phi :\left(a,b\right]\to \mathbb{R}$ is an increasing function such that ${\phi }^{\prime }\left(t\right)\ne 0$, for all $t\in \left[a,b\right]$, and $F_k:\left[a,b\right]\times {\mathbb{R}}^n\to \mathbb{R}$, $(k=\overline{1,n})$ are given functions that will be “well-defined” later.

2. Preliminaries

In this part, we introduce various notations and terminology for the $\phi$-percent Hilfer Derivatives Calculus, as well as preliminary results that will be used in our proofs later on; see [24,25,26] for more information.

Let $\phi :\left[a,b\right]\to \mathbb{R}$ be an increasing function with ${\phi }^{\prime }\left(t\right)\ne 0$, for all $t\in J$, and let $C(\left[a,b\right],\mathbb{R})$ be the Banach space.

For all $\upsilon >-1$ and $s,t\in [0,\infty ),\left(t\ge s\right)$, we denote ${\phi }_{\upsilon }(t,s)={(\phi \left(t\right)-\phi \left(s\right))}^{\upsilon }.$

Definition 1.

Let $(a,b)$, $\left(-\infty \le a<b\le \infty \right)$ percent be a finite or infinite interval of the half-axis $\left(0,\infty \right)$ and $\alpha >0$. Furthermore, consider $\phi \left(t\right)$ a positive rising function on $\left(a,b\right]$, with a continuous derivative ${\phi }^{\prime }\left(t\right)$ on $(a,b)$. The φ-Riemann–Liouville fractional integral of a function u with respect to another function φ on $\left[a,b\right]$ is defined by

$${\mathcal{I}}_{a^{+}}^{\alpha ;\phi }u\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle \underset{a}{\stackrel{t}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{\alpha -1}(t,s)u\left(s\right)ds,$$

(2)

where $\Gamma \left(.\right)$ is the Gamma function.

Definition 2.

Let $n\in \mathbb{N}$ and $\phi ,u\in C^n\left(J\right)$ percent be two functions such that φ is increasing and ${\phi }^{\prime }\left(t\right)\ne 0$, for all $t\in \left(a,b\right]$. The left-sided φ-Riemann–Liouville fractional derivative of a function u of order α is defined by

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{a^{+}}^{\alpha ;\phi }u\left(t\right)& =& {\left(\frac{1}{{\phi }^{\prime }\left(t\right)}\frac{d}{dt}\right)}^n{\mathcal{I}}_{a^{+}}^{n-\alpha ;\phi }u\left(t\right)\hfill \\ & =& \frac{1}{\Gamma (n-\alpha )}{\left(\frac{1}{{\phi }^{\prime }\left(t\right)}\frac{d}{dt}\right)}^n{\displaystyle \underset{a}{\stackrel{t}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{n-\alpha -1}(t,s)u\left(s\right)ds,\hfill \end{array}$$

where $n=\left[\alpha \right]+1,$$\left[\alpha \right]$ represents the integer part of the real number $\alpha .$

Definition 3.

Let $n-1<\alpha <n$ with $n\in \mathbb{N}$, $\left[a,b\right]$ be the interval such that $-\infty \le a<b\le \infty$ and $\phi ,u\in C^n\left(\left[a,b\right],\mathbb{R}\right)$ percent are two functions such that φ is increasing and ${\phi }^{\prime }\left(t\right)\ne 0$, for all $t\in \left[a,b\right]$. The φ-Hilfer fractional derivative of a function u of order a and type $0\le \beta \le 1$ is defined by

$${}^H{\mathcal{D}}_{a^{+}}^{\alpha ,\beta ;\phi }u\left(t\right)={\mathcal{I}}_{a^{+}}^{\beta (n-\alpha );\phi }{\left(\frac{1}{{\phi }^{\prime }\left(t\right)}\frac{d}{dt}\right)}^n{\mathcal{I}}_{a^{+}}^{(1-\beta )(n-\alpha );\phi }u\left(t\right)={\mathcal{I}}_{a^{+}}^{\gamma -\alpha ;\phi }{\mathcal{D}}_{a^{+}}^{\gamma ;\phi }u\left(t\right),$$

where $n=\left[\alpha \right]+1$, $\gamma -\alpha =\beta \left(n-\alpha \right).$

Auxiliary Lemma

Lemma 1.

Let $\alpha ,\rho >0$. Then, we have the following semigroup property given by

$${\mathcal{I}}_{a^{+}}^{\alpha ;\phi }{\mathcal{I}}_{a^{+}}^{\rho ;\phi }u\left(t\right)={\mathcal{I}}_{a^{+}}^{\alpha +\rho ;\phi }u\left(t\right),t>a.$$

Next, we present the φ-fractional integral and derivatives of a power function.

Proposition 1.

Let $\alpha \ge 0,\sigma >0$ and $t>a.$ Then, φ-fractional integral and derivative of a power function are given by

1.

${\mathcal{I}}_{a^{+}}^{\alpha ,\phi }{\phi }_{\sigma -1}(t,a)\left(t\right)=\frac{\Gamma \left(\sigma \right)}{\Gamma (\alpha +\sigma )}{\phi }_{\sigma +\alpha -1}(t,a).$

2.

${}^H{\mathcal{D}}_{a^{+}}^{\alpha ,\beta ;\phi }{\phi }_{\sigma -1}(t,a)\left(t\right)=\frac{\Gamma \left(\sigma \right)}{\Gamma (\sigma -\alpha )}{\phi }_{\sigma -\alpha -1}(t,a),n-1<\alpha <n,\sigma >n.$

Lemma 2.

If $u\in C^n([a,b],\mathbb{R}),n-1<\alpha <n,0\le \beta \le 1$ and $\gamma =\alpha +\beta (n-\alpha )$. Then

$${\mathcal{I}}_{a^{+}}^{\alpha ,\phi }{(}^H{\mathcal{D}}_{a^{+}}^{\alpha ,\beta ;\phi }u)\left(t\right)=u\left(t\right)-\sum _{k=1}^{k=n}\frac{{\phi }_{\gamma -k}(t,a)}{\Gamma (\gamma -k+1)}{\nabla }_{\phi }^{[n-k]}{\mathcal{I}}_{a^{+}}^{(1-\beta )(n-\alpha );\phi }u\left(a\right),t\in [a,b],$$

where ${\nabla }_{\phi }^{\left[n\right]}u\left(t\right):={\left(\frac{1}{{\psi }^{\prime }\left(t\right)}\frac{d}{dt}\right)}^{n}u\left(t\right).$

Lemma 3.

Let $u\in C^n\left[a,b\right]$ and $0<q<1,$ we have

$$\left|{\mathcal{I}}_{a^{+}}^{q;\phi }u\left(t_2\right)-{\mathcal{I}}_{a^{+}}^{q;\phi }u\left(t_1\right)\right|\le \frac{2∥u∥}{\Gamma \left(q+1\right)}{\phi }_q(t_2,t_1).$$

Lemma 4

([27]). For the p-Laplacian operator ${\psi }_p$, the following conditions hold true:

(1) 

If $\left|{\delta }_1\right|,\left|{\delta }_2\right|\ge \rho >0,$$1<p\le 2,{\delta }_1{\delta }_2>0,$ then

$$\left|{\psi }_p\left({\delta }_1\right)-{\psi }_p\left({\delta }_2\right)\right|\le \left(p-1\right){\rho }^{p-2}\left|{\delta }_1-{\delta }_2\right|.$$

(2) 

If $p>2,$$\left|{\delta }_1\right|,\left|{\delta }_2\right|\le {\rho }_{*}>0,$ then

$$\left|{\psi }_p\left({\delta }_1\right)-{\psi }_p\left({\delta }_2\right)\right|\le \left(p-1\right){\rho }_{*}^{p-2}\left|{\delta }_1-{\delta }_2\right|.$$

Lemma 5

([9]). For nonnegative $a_i,i=1,\cdots ,k$,

$${\left({\displaystyle \underset{i=1}{\sum ^k}}{a}_i\right)}^q\le k^{q-1}\left({\displaystyle \underset{i=1}{\sum ^k}}{a}_i^q\right),q\ge 1$$

Lemma 6

([28], Krasnosel’skii fixed point theorem). Let Δ be a closed, bounded, convex and nonempty subset of a Banach space X. Let Γ, Π be operators such that:

(i) $\Gamma x+\Pi y\in \Delta ,$ $x,y\in \Delta .$

(ii) $\Gamma$ is compact and continuous.

(iii) $\Pi$ is a contraction mapping.

Then there exists $\xi \in \Delta$ such that $\xi =\Gamma \xi +\Pi \xi$.

In this subsection, we consider now the multiplied system:

$$\left\{\begin{array}{c}{}^H{\mathcal{D}}_{a^{+}}^{{\alpha }_{k1},{\beta }_{k1};\phi }{\psi }_{p_k}\left({}^H{\mathcal{D}}_{a^{+}}^{{\alpha }_{k2},{\beta }_{k2};\phi }{u}_k\right)\left(t\right)=f_k\left(t\right),t\in \left[a,b\right],k=\overline{1,n}\hfill \\ u_k\left(a\right)=0,u_k\left(b\right)={\displaystyle \underset{i=1}{\sum ^n}}{\lambda }_i{u}_i\left({\zeta }_{ik}\right),a<{\zeta }_{ik}<b,{\lambda }_i\in \left]0,+\infty \right[\hfill \\ {\psi }_{p_k}\left({}^H{\mathcal{D}}_{a^{+}}^{{\alpha }_{k2},{\beta }_{k2};\phi }{u}_k\right)\left(a\right)=0,{\psi }_{p_k}\left({}^H{\mathcal{D}}_{a^{+}}^{{\alpha }_{k2},{\beta }_{k2};\phi }{u}_k\right)\left(b\right)={\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right),a<\zeta <b,\hfill \end{array}\right.$$

(3)

where $f_k:\left(a,b\right]\to \mathbb{R}$ are continuous functions, and

$$\left\{\begin{array}{c}a\ge 0,1<{\alpha }_{k1},{\alpha }_{k2}<2,0\le {\beta }_{k1},{\beta }_{k2}\le 1,k=\overline{1,n}\hfill \\ \mathrm{and}\hfill \\ {\gamma }_{k1}={\alpha }_{k1}+{\beta }_{k1}(2-{\alpha }_{k1}),{\gamma }_{k2}={\alpha }_{k2}+{\beta }_{k2}(2-{\alpha }_{k2}).\hfill \end{array}\right.$$

(4)

Lemma 7.

Let $f_k\in C\left(\left[a,b\right]\right)(k=\overline{1,n})$, the existence solution of the multiplied system (3) is given by:

$$\begin{array}{ccc}& & u_k\left(t\right)\hfill \\ & =& \frac{{\phi }_{{\gamma }_{k2}-1}\left(t,a\right)}{{\phi }_{{\gamma }_{k2}-1}\left(b,a\right)}{\displaystyle \underset{i=1}{\sum ^n}}{\lambda }_i{u}_i\left({\zeta }_{ik}\right)+\frac{1}{\Gamma \left({\alpha }_{k2}\right)}{\displaystyle \underset{a}{\stackrel{t}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k2}-1}(t,s)\hfill \\ & & \times \left\{{\psi }_{q_k}\left[\frac{1}{\Gamma \left({\alpha }_{k1}\right)}{\displaystyle \underset{a}{\stackrel{s}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k1}-1}(s,x)f_k\left(z\right)dz+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_1;\phi }{f}_k\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(s,a\right)\right]\right\}ds\hfill \\ & & -\frac{{\phi }_{{\gamma }_{k2}-1}\left(t,a\right)}{\Gamma \left({\alpha }_{k2}\right){\phi }_{{\gamma }_{k2}-1}\left(b,a\right)}{\displaystyle \underset{a}{\stackrel{b}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k2}-1}(b,y)\hfill \\ & & \times \left\{{\psi }_{q_k}\left[\frac{1}{\Gamma \left({\alpha }_{k1}\right)}{\displaystyle \underset{a}{\stackrel{y}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k1}-1}(y,z)f_k\left(z\right)dz+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_1;\phi }{f}_k\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(y,a\right)\right]\right\}dy.\hfill \end{array}$$

Proof. 

Assume u and v satisfies the system (3), and considering the first equation in the system (3). By applying the fractional integral operators ${\mathcal{I}}_{a^{+}}^{{\alpha }_1;\phi }$ to (3) and using Lemma 2, we conclude that:

$${\psi }_{p_k}\left({}^H{\mathcal{D}}_{a^{+}}^{{\alpha }_{k2},{\beta }_{k2};\phi }{u}_k\right)\left(t\right)={\mathcal{I}}_{a^{+}}^{{\alpha }_{k1};\phi }{f}_k\left(t\right)+\frac{c_1}{\Gamma \left({\gamma }_{k1}-1\right)}{\phi }_{{\gamma }_{k1}-2}\left(t,a\right)+\frac{c_2}{\Gamma \left({\gamma }_{k1}\right)}{\phi }_{{\gamma }_{k1}-1}\left(t,a\right),$$

(5)

for some real constantes $c_1$ and $c_2$. Now, using the first boundary condition (4)

$${\psi }_{p_k}\left({}^H{\mathcal{D}}_{a^{+}}^{{\alpha }_{k2},{\beta }_{k2};\phi }{u}_k\right)\left(a\right)=0,$$

in (3), we obtain

$$\frac{c_1}{\Gamma \left({\gamma }_{k1}-1\right)}{\phi }_{{\gamma }_{k1}-2}\left(a,a\right)+\frac{c_2}{\Gamma \left({\gamma }_1\right)}{\phi }_{{\gamma }_{k1}-1}\left(a,a\right)=0,$$

where

$${\phi }_{{\gamma }_{k1}-1}\left(a,a\right)=0\mathrm{and}{\phi }_{{\gamma }_{k1}-2}\left(t,a\right)\to \infty ,t\to a,$$

then $c_1=0.$ Using the second boundary condition

$${\psi }_{p_k}\left({}^H{\mathcal{D}}_{a^{+}}^{{\alpha }_{k2},{\beta }_{k2};\phi }{u}_k\right)\left(b\right)={\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)$$

in (3), we have

$${\mathcal{I}}_{a^{+}}^{{\alpha }_{k1};\phi }{f}_k\left(b\right)+\frac{c_2}{\Gamma \left({\gamma }_{k1}\right)}{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)={\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right),$$

these imply that

$$c_2=\frac{\Gamma \left({\gamma }_{k1}\right)}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_{k1};\phi }{f}_k\left(b\right)).$$

Substituting the value of $c_2$ in (5), we obtain

$${\psi }_{p_k}\left({}^H{\mathcal{D}}_{a^{+}}^{{\alpha }_{k2},{\beta }_{k2};\phi }{u}_k\right)\left(t\right)={\mathcal{I}}_{a^{+}}^{{\alpha }_{k1};\phi }{f}_k\left(t\right)+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_{k1};\phi }{f}_k\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(t,a\right),$$

thus, we have

$${}^H{\mathcal{D}}_{a^{+}}^{{\alpha }_{k2},{\beta }_{k2};\phi }{u}_k\left(t\right)={\psi }_{q_k}\left[{\mathcal{I}}_{a^{+}}^{{\alpha }_{k1};\phi }{f}_k\left(t\right)+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_{k1};\phi }{f}_k\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(t,a\right)\right],$$

(6)

by applying the fractional integral operators ${\mathcal{I}}_{a^{+}}^{{\alpha }_{k2};\phi }$ to (6) and using Lemma 2, we obtain

$$\begin{array}{ccc}\hfill u_k\left(t\right)& =& {\mathcal{I}}_{a^{+}}^{{\alpha }_{k2};\phi }\left\{{\psi }_{q_k}\left[{\mathcal{I}}_{a^{+}}^{{\alpha }_1;\phi }{f}_k\left(t\right)+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_{k1};\phi }{f}_k\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(t,a\right)\right]\right\}\hfill \\ & & +\frac{c_3}{\Gamma \left({\gamma }_{k2}-1\right)}{\phi }_{{\gamma }_{k2}-2}\left(t,a\right)+\frac{c_4}{\Gamma \left({\gamma }_{k2}\right)}{\phi }_{{\gamma }_{k2}-1}\left(t,a\right)\hfill \end{array}$$

(7)

for some real constants $c_3$ and $c_4.$ Using the second boundary condition $u\left(a\right)=0$ in (3), we obtain

$$\frac{c_3}{\Gamma \left({\gamma }_{k2}-1\right)}{\phi }_{{\gamma }_{k2}-2}\left(a,a\right)+\frac{c_4}{\Gamma \left({\gamma }_{k2}\right)}{\phi }_{{\gamma }_2-1}\left(a,a\right)=0,$$

and we have

$${\phi }_{{\gamma }_{k2}-1}\left(a,a\right)=0,{\phi }_{{\gamma }_{k2}-2}\left(t,a\right)\to \infty ,t\to a,$$

which implies that $c_3=0.$ Now, substituting the value of $c_3$ in (7), we obtain

$$\begin{array}{ccc}\hfill u_k\left(t\right)& =& {\mathcal{I}}_{a^{+}}^{{\alpha }_{k2};\phi }\left\{{\psi }_{q_k}\left({\mathcal{I}}_{a^{+}}^{{\alpha }_{k1};\phi }{f}_k\left(t\right)+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_{k1};\phi }{f}_k\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(t,a\right)\right)\right\}\hfill \\ & & +\frac{c_4}{\Gamma \left({\gamma }_{k2}\right)}{\phi }_{{\gamma }_{k2}-1}\left(t,a\right).\hfill \end{array}$$

(8)

In view of condition $u_k\left(b\right)={\displaystyle \underset{i=1}{\sum ^n}}{\lambda }_i{u}_i\left({\zeta }_{ik}\right)$ in (3), we obtain

$$\begin{array}{ccc}& & {\displaystyle \underset{i=1}{\sum ^n}}{\lambda }_i{u}_i\left({\zeta }_{ik}\right)\hfill \\ & =& {\mathcal{I}}_{a^{+}}^{{\alpha }_2;\phi }{\left\{{\psi }_{q_k}\left({\mathcal{I}}_{a^{+}}^{{\alpha }_{k1};\phi }f\left(t\right)+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }u\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_{k1};\phi }f\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(t,a\right)\right)\right\}}_{t\to b}\hfill \\ & & +\frac{c_4}{\Gamma \left({\gamma }_{k2}\right)}{\phi }_{{\gamma }_{k2}-1}\left(b,a\right),\hfill \end{array}$$

then

$$\begin{array}{ccc}\hfill c_4& =& \frac{\Gamma \left({\gamma }_{k2}\right)}{{\phi }_{{\gamma }_{k2}-1}\left(b,a\right)}{\displaystyle \underset{i=1}{\sum ^n}}{\lambda }_i{u}_i\left({\zeta }_{ik}\right)\hfill \\ & & -\frac{\Gamma \left({\gamma }_{k2}\right)}{{\phi }_{{\gamma }_2-1}\left(b,a\right)}{\mathcal{I}}_{a^{+}}^{{\alpha }_{k2};\phi }\left\{{\psi }_{q_k}{\left[{\mathcal{I}}_{a^{+}}^{{\alpha }_{k1};\phi }{f}_k\left(t\right)+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_{k1};\phi }{f}_k\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(t,a\right)\right]}_{t=b}\right\},\hfill \end{array}$$

substituting the value of $c_4$ in (8), we obtain

$$\begin{array}{ccc}\hfill u_k\left(t\right)& =& {\mathcal{I}}_{a^{+}}^{{\alpha }_{k2};\phi }\left\{{\psi }_{q_k}\left[{\mathcal{I}}_{a^{+}}^{{\alpha }_{k1};\phi }{f}_k\left(t\right)+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_{k1};\phi }{f}_k\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(t,a\right)\right]\right\}\hfill \\ & & -\frac{{\phi }_{{\gamma }_{k2}-1}\left(t,a\right)}{{\phi }_{{\gamma }_{k2}-1}\left(b,a\right)}{\mathcal{I}}_{a^{+}}^{{\alpha }_{k2};\phi }\left\{{\psi }_{q_k}{\left[{\mathcal{I}}_{a^{+}}^{{\alpha }_1;\phi }{f}_k\left(b\right)+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_{k1};\phi }{f}_k\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)\right]}_{t=b}\right\}\hfill \\ & & +\frac{{\phi }_{{\gamma }_{k2}-1}\left(t,a\right)}{{\phi }_{{\gamma }_{k2}-1}\left(b,a\right)}{\displaystyle \underset{i=1}{\sum ^n}}{\lambda }_i{u}_i\left({\zeta }_{ik}\right).\hfill \end{array}$$

The proof is finished. □

3. Main Results

We offer our primary results on the presence and stability of the above problem in this part. Let $\mathcal{C}=C(\left[a,b\right],\mathbb{R})$ be the Banach space of all continuous mappings from $\left[a,b\right]$ to $\mathbb{R}$ endowed with ${∥u∥}_{\mathcal{C}}={\displaystyle \underset{t\in \left[0,1\right]}{sup}}u\left(t\right)$ is a Banach space. It is clear that the space $\mathcal{X}=$${\mathcal{C}}^n$ is endowed with the norm

$${∥\left(u_1,\dots ,u_n\right)∥}_{\mathcal{X}}={\displaystyle \underset{i=1}{\sum ^n}}{∥u_i∥}_{\mathcal{C}}.$$

3.1. The Existence Result

Now, we list the hypotheses:

$\left({\mathcal{A}}_1\right):$ There exist positive real constants $N_k>0$, $k=1,n$, such that for all $t\in \left[a,b\right]$ and $u_k,v_k\in \mathbb{R}(k=1,n),$ we have

$$\left|F_k(t,u_1,\dots ,u_n)-F_k(t,v_1,\dots ,v_n)\right|\le N_k{\displaystyle \underset{i=1}{\sum ^n}}\left|u_i-v_i\right|.$$

$\left({\mathcal{A}}_2\right):$ There exist positive real constants ${{\rm Y}}_{ik}>0$, $i,k=1,n$, such that, for all $t\in \left[a,b\right],$ and $u_k\in \mathbb{R}(k=1,n),$ we have

$$\left|F_k(t,u_1,\dots ,u_n)\right|\le \sum _{i=1}^n{{\rm Y}}_{ik}∥u_i∥.$$

Now, we define the following quantities:

• $\begin{array}{ccc}\hfill K& =& \phi \left(b\right)-\phi \left(a\right)\hfill \\ \hfill {\Delta }_k& \ge & \frac{2K^{{\alpha }_{k1}}{N}_k}{\Gamma (1+{\alpha }_{k1})}+\frac{K^{\rho }}{\Gamma (1+\rho )},k=1,n,\hfill \\ \hfill {\mathcal{B}}_{k1}& =& \frac{3^{q_k-2}{K}^{{\alpha }_{k2}}}{\Gamma (1+{\alpha }_{k2})}\left(2{\left(\frac{K^{{\alpha }_{k1}}\left({\displaystyle \underset{i=1}{\sum ^n}}{{\rm Y}}_{ik}\right)}{\Gamma (1+{\alpha }_{k1})}\right)}^{q_k-1}+{\left(\frac{K^{\rho }}{\Gamma \left(\rho +1\right)}\right)}^{q_k-1}\right),k=1,n,\hfill \\ \hfill {\mathcal{B}}_{k2}& =& \frac{2\times 3^{q_k-2}}{\Gamma (1+{\alpha }_{k2})}\left(2{\left(\frac{K^{{\alpha }_{k1}}\left({\displaystyle \underset{i=1}{\sum ^n}}{{\rm Y}}_{ik}\right)}{\Gamma (1+{\alpha }_{k1})}\right)}^{q_k-1}+{\left(\frac{K^{\rho }}{\Gamma \left(\rho +1\right)}\right)}^{q_k-1}\right),k=1,n,\hfill \\ \hfill {\mathcal{B}}_{k3}& =& \frac{\left(q_k-1\right){\Delta }_k^{q_k-2}{K}^{{\alpha }_{k2}}}{\Gamma (1+{\alpha }_{k2})}\left[\frac{2K^{{\alpha }_{k1}}{N}_k}{\Gamma (1+{\alpha }_{k1})}+\frac{K^{\rho }}{\Gamma (1+\rho )}\right]+{\displaystyle \underset{i=1}{\sum ^n}}{\lambda }_i,k=1,n.\hfill \end{array}$

Based on the above hypotheses, we present to the reader the following result. Now, consider the following operator $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ by:

$$\mathcal{T}(u_1,\dots ,u_n)\left(t\right)=({\mathcal{T}}_1(u_1,\dots ,u_n)\left(t\right),{\mathcal{T}}_2(u_1,\dots ,u_n)\left(t\right),\dots ,{\mathcal{T}}_n(u_1,\dots ,u_n)\left(t\right)),$$

(9)

where

$$\begin{array}{ccc}& & {\mathcal{T}}_k(u_1,\dots ,u_n)\left(t\right)\hfill \\ & =& \frac{1}{\Gamma \left({\alpha }_{k2}\right)}{\displaystyle \underset{a}{\stackrel{t}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k2}-1}(t,s)\hfill \\ & & \times \left\{{\psi }_{q_k}\left[\frac{1}{\Gamma \left({\alpha }_{k1}\right)}{\displaystyle \underset{a}{\stackrel{s}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k1}-1}(s,z){\tilde{F}}_{u_k}\left(z\right)dz+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_1;\phi }{\tilde{F}}_{u_k}\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(s,a\right)\right]\right\}ds\hfill \\ & & +\frac{{\phi }_{{\gamma }_{k2}-1}\left(t,a\right)}{{\phi }_{{\gamma }_{k2}-1}\left(b,a\right)}{\displaystyle \underset{i=1}{\sum ^n}}{\lambda }_i{u}_i\left({\zeta }_{ik}\right)-\frac{{\phi }_{{\gamma }_{k2}-1}\left(t,a\right)}{\Gamma \left({\alpha }_{k2}\right){\phi }_{{\gamma }_{k2}-1}\left(b,a\right)}{\displaystyle \underset{a}{\stackrel{b}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k2}-1}(b,y)\hfill \\ & & \times \left\{{\psi }_{q_k}\left[\frac{1}{\Gamma \left({\alpha }_{k1}\right)}{\displaystyle \underset{a}{\stackrel{y}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k1}-1}(y,z){\tilde{F}}_{u_k}\left(z\right)dz+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_1;\phi }{\tilde{F}}_{u_k}\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(y,a\right)\right]\right\}dy.\hfill \end{array}$$

where

$${\tilde{F}}_{u_k}\left(t\right)=F_k(t,u_1,\dots ,u_n).$$

Here, we divide the operator $\mathcal{T}(u_1,\dots ,u_n)\left(t\right)$ as follows:

$$\mathcal{T}(u_1,\dots ,u_n)\left(t\right)={\mathcal{P}}_{1k}(u_1,\dots ,u_n)\left(t\right)+{\mathcal{P}}_{2k}(u_1,\dots ,u_n)\left(t\right),\left(k=\overline{1,n}\right).$$

(10)

where ($k=\overline{1,n}$)

$$\begin{array}{ccc}& & {\mathcal{P}}_{1k}(u_1,\dots ,u_n)\left(t\right)\hfill \\ & =& \frac{1}{\Gamma \left({\alpha }_{k2}\right)}{\displaystyle \underset{a}{\stackrel{t}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k2}-1}(t,s)\hfill \\ & & \times \left\{{\psi }_{q_k}\left[\frac{1}{\Gamma \left({\alpha }_{k1}\right)}{\displaystyle \underset{a}{\stackrel{s}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k1}-1}(s,z){\tilde{F}}_{u_k}\left(z\right)dz+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_1;\phi }{\tilde{F}}_{u_k}\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(s,a\right)\right]\right\}ds\hfill \end{array}$$

and

$$\begin{array}{ccc}& & {\mathcal{P}}_{2k}(u_1,\dots ,u_n)\left(t\right)\hfill \\ & =& \frac{{\phi }_{{\gamma }_{k2}-1}\left(t,a\right)}{{\phi }_{{\gamma }_{k2}-1}\left(b,a\right)}{\displaystyle \underset{i=1}{\sum ^n}}{\lambda }_i{u}_i\left({\zeta }_{ik}\right)\hfill \\ & & -\frac{{\phi }_{{\gamma }_{k2}-1}\left(t,a\right)}{\Gamma \left({\alpha }_{k2}\right){\phi }_{{\gamma }_{k2}-1}\left(b,a\right)}{\displaystyle \underset{a}{\stackrel{b}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k2}-1}(b,y)\hfill \\ & & \times \left\{{\psi }_{q_k}\left[\frac{1}{\Gamma \left({\alpha }_{k1}\right)}{\displaystyle \underset{a}{\stackrel{y}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k1}-1}(y,z){\tilde{F}}_{u_k}\left(z\right)dz+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_1;\phi }{\tilde{F}}_{u_k}\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(y,a\right)\right]\right\}dy.\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill \mathcal{T}& =& \left({\mathcal{P}}_{11},{\mathcal{P}}_{12},\dots ,{\mathcal{P}}_{1n}\right)+\left({\mathcal{P}}_{21},{\mathcal{P}}_{22},\dots ,{\mathcal{P}}_{2n}\right)\hfill \\ & =& {\mathcal{P}}_1+{\mathcal{P}}_2.\hfill \end{array}$$

Our first result concerning the existence of solutions of the problem (1) for which we have used the fixed point theorem of Krasnoselskii’s is as follows

Theorem 1.

Let $F_k:[a,b]\times {\mathbb{R}}^n\to \mathbb{R}$ be continuous functions that satisfy conditions (${\mathbb{A}}_1$) and (${\mathbb{A}}_2$). If

$$\left({\displaystyle \underset{i=1}{\sum ^n}}{\lambda }_i\right)<\frac{1}{n(n+1)},\mathit{and}\left({\displaystyle \underset{k=1}{\sum ^n}}{\mathcal{B}}_{k3}\right)<1.$$

Then, problem (1) admits at least one solution on $\left[a,b\right]$

Proof. 

The proof will be given in several steps. Let ${\mathbb{U}}_r=\{(u_1,\dots ,u_n)\in \mathcal{X},∥(u_1,\dots ,u_n)∥\le r\},$ so that

$${\displaystyle \underset{k=1,n}{max}}\left\{\frac{1}{{\left(2(n+1){\mathcal{B}}_{k1}\right)}^{q_k-2}}\right\}\le r$$

First step: We prove that

$${∥\mathcal{T}(u_1,\dots ,u_n)\left(t\right)∥}_{\mathcal{X}}\le r.$$

From (9), we obtain

$$\begin{array}{ccc}\hfill {∥\mathcal{T}(u_1,\dots ,u_n)\left(t\right)∥}_{\mathcal{X}}& =& {\displaystyle \underset{i=1}{\sum ^n}}{∥{\mathcal{T}}_i(u_1,\dots ,u_n)\left(t\right)∥}_{\mathcal{C}},\hfill \\ & & \mathrm{and}\hfill \\ \hfill {∥{\mathcal{T}}_i(u_1,\dots ,u_n)\left(t\right)∥}_{\mathcal{C}}& \le & {\displaystyle \underset{t\in \left[a,b\right]}{sup}}\left|{\mathcal{P}}_{1i}(u_1,\dots ,u_n)\left(t\right)\right|+{\displaystyle \underset{t\in \left[a,b\right]}{sup}}\left|{\mathcal{P}}_{2i}(u_1,\dots ,u_n)\left(t\right)\right|,i=\overline{1,n}.\hfill \end{array}$$

(11)

Let $(u_1,\dots ,u_n)\in {\mathbb{U}}_r$. by ${\psi }_{q_k}\left(z\right)=z{\left|z\right|}^{q_k-2},$ and for Lemma 5, we have

$$\begin{array}{ccc}& & \left|{\mathcal{P}}_{1k}(u_1,\dots ,u_n)\left(t\right)\right|\hfill \\ & \le & \frac{1}{\Gamma \left({\alpha }_{k2}\right)}\left|{\displaystyle \underset{a}{\stackrel{t}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k2}-1}(t,s)\right.\hfill \\ & & \times \left.\left\{{\psi }_{q_k}\left[\frac{1}{\Gamma \left({\alpha }_{k1}\right)}{\displaystyle \underset{a}{\stackrel{s}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k2}-1}(s,z){\tilde{F}}_{u_k}\left(z\right)dz+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_1;\phi }{\tilde{F}}_{u_k}\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(s,a\right)\right]\right\}ds\right|\hfill \end{array}$$

(12)

$$\begin{array}{ccc}& \le & \frac{K^{{\alpha }_{k2}}}{\Gamma (1+{\alpha }_{k2})}{\left|\frac{1}{\Gamma \left({\alpha }_{k1}\right)}{\displaystyle \underset{a}{\stackrel{s}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k2}-1}(s,z){\tilde{F}}_{u_k}\left(z\right)dz+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_1;\phi }{\tilde{F}}_{u_k}\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(s,a\right)\right|}^{q_k-1}\hfill \\ & \le & \frac{3^{q_k-2}{K}^{{\alpha }_{k2}}}{\Gamma (1+{\alpha }_{k2})}\left({\left|\frac{1}{\Gamma \left({\alpha }_{k1}\right)}{\displaystyle \underset{a}{\stackrel{s}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k2}-1}(s,z){\tilde{F}}_{u_k}\left(z\right)dz\right|}^{q_{k-1}}+{\left|{\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)\right|}^{q_{k-1}}+{\left|{\mathcal{I}}_{a^{+}}^{{\alpha }_1;\phi }{\tilde{F}}_{u_k}\left(b\right)\right|}^{q_{k-1}}\right)\hfill \\ & \le & \frac{3^{q_k-2}{K}^{{\alpha }_{k2}}}{\Gamma (1+{\alpha }_{k2})}\left(2{\left|\frac{K^{{\alpha }_{k1}}\left({\displaystyle \underset{i=1}{\sum ^n}}{{\rm Y}}_{ik}\right)\delta }{\Gamma (1+{\alpha }_{k1})}\right|}^{q_{k-1}}+{\left|\frac{K^{\rho }\delta }{\Gamma \left(\rho +1\right)}\right|}^{q_{k-1}}\right)\hfill \\ & \le & \frac{3^{q_k-2}{K}^{{\alpha }_{k2}}}{\Gamma (1+{\alpha }_{k2})}\left(2{\left(\frac{K^{{\alpha }_{k1}}\left({\displaystyle \underset{i=1}{\sum ^n}}{{\rm Y}}_{ik}\right)}{\Gamma (1+{\alpha }_{k1})}\right)}^{q_{k-1}}+{\left(\frac{K^{\rho }}{\Gamma \left(\rho +1\right)}\right)}^{q_{k-1}}\right){\delta }^{q_{k-1}}\hfill \\ & \le & {\mathcal{B}}_{k1}{r}^{q_{k-1}},\hfill \end{array}$$

(13)

and

$$\begin{array}{ccc}& & \left|{\mathcal{P}}_{2k}(u_1,\dots ,u_n)\left(t\right)\right|\hfill \\ & \hfill \le & \left|\frac{{\phi }_{{\gamma }_{k2}-1}\left(t,a\right)}{{\phi }_{{\gamma }_{k2}-1}\left(b,a\right)}{\displaystyle \underset{i=1}{\sum ^n}}{\lambda }_i{u}_i\left({\zeta }_{ik}\right)\right|\hfill \\ & & +\left|\frac{{\phi }_{{\gamma }_{k2}-1}\left(t,a\right)}{\Gamma \left({\alpha }_{k2}\right){\phi }_{{\gamma }_{k2}-1}\left(b,a\right)}{\displaystyle \underset{a}{\stackrel{b}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k2}-1}(b,y)\right.\hfill \\ & & \times \left.\left\{{\psi }_{q_k}\left[\frac{1}{\Gamma \left({\alpha }_{k1}\right)}{\displaystyle \underset{a}{\stackrel{y}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k1}-1}(y,z){\tilde{F}}_{u_k}\left(z\right)dz+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_1;\phi }{\tilde{F}}_{u_k}\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(y,a\right)\right]\right\}dy\right|\hfill \\ & \hfill \le & \left({\displaystyle \underset{i=1}{\sum ^n}}{\lambda }_i\right)r+{\mathcal{B}}_{k1}{r}^{q_{k-1}},\hfill \end{array}$$

(14)

by (11), (12), and (14), we obtain

$${∥\mathcal{T}(u_1,\dots ,u_n)\left(t\right)∥}_{\mathcal{X}}\le n\left({\displaystyle \underset{i=1}{\sum ^n}}{\lambda }_i\right)r+2{\displaystyle \underset{k=1}{\sum ^n}}{\mathcal{B}}_{k1}{r}^{q_{k-1}}\le r.$$

Second Step: ${\mathcal{P}}_2$ is a contraction. $\left({\mathcal{P}}_2={\displaystyle \underset{k=1}{\sum ^n}}{\mathcal{P}}_{2k}\right)$

Let $(u_1,\dots ,u_n),(v_1,\dots ,v_n)$$\in {\mathbb{U}}_r$, and we have the following estimate

$$\begin{array}{ccc}& & \left|{\mathcal{P}}_{2k}(u_1,\dots ,u_n)\left(t\right)-{\mathcal{P}}_{2k}(v_1,\dots ,v_n)\left(t\right)\right|\hfill \\ & \le & {\displaystyle \underset{i=1}{\sum ^n}}\left({\lambda }_i\left|u_i\left({\zeta }_{ik}\right)-v_i\left({\zeta }_{ik}\right)\right|\right)\hfill \\ & & +\frac{K^{{\alpha }_{k2}}}{\Gamma (1+{\alpha }_{k2})}\hfill \\ & & \times \left|{\psi }_{q_k}\left[\frac{1}{\Gamma \left({\alpha }_{k1}\right)}{\displaystyle \underset{a}{\stackrel{y}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k1}-1}(y,z){\tilde{F}}_{u_k}\left(z\right)dz+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_1;\phi }{\tilde{F}}_{u_k}\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(y,a\right)\right]\right.\hfill \\ & & -\left.{\psi }_{q_k}\left[\frac{1}{\Gamma \left({\alpha }_{k1}\right)}{\displaystyle \underset{a}{\stackrel{y}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k1}-1}(y,z){\tilde{F}}_{v_k}\left(z\right)dz+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{v}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_1;\phi }{\tilde{F}}_{v_k}\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(y,a\right)\right]\right|,\hfill \end{array}$$

by Lemma 4, we obtain

$$\begin{array}{ccc}& & \left|{\mathcal{P}}_{2k}(u_1,\dots ,u_n)\left(t\right)-{\mathcal{P}}_{2k}(v_1,\dots ,v_n)\left(t\right)\right|\hfill \\ & \le & {\displaystyle \underset{i=1}{\sum ^n}}\left({\lambda }_i\left|u_i\left({\zeta }_{ik}\right)-v_i\left({\zeta }_{ik}\right)\right|\right)\hfill \\ & & +\frac{\left(q_k-1\right){\Delta }^{q_k-2}{K}^{{\alpha }_{k2}}}{\Gamma (1+{\alpha }_{k2})}\hfill \\ & & \times \left[\frac{2K^{{\alpha }_{k1}}}{\Gamma (1+{\alpha }_{k1})}\left|{\tilde{F}}_{u_k}\left(b\right)-{\tilde{F}}_{v_k}\left(b\right)\right|+\left(\left|{\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{\rho ;\phi }{v}_k\left(\zeta \right)\right|\right)\right]\hfill \\ & \le & \left({\displaystyle \underset{i=1}{\sum ^n}}{\lambda }_i∥u_i-v_i∥\right)\hfill \\ & & +\frac{\left(q_k-1\right){\Delta }^{q_k-2}{K}^{{\alpha }_{k2}}}{\Gamma (1+{\alpha }_{k2})}\left[\frac{2K^{{\alpha }_{k1}}{N}_k}{\Gamma (1+{\alpha }_{k1})}+\frac{K^{\rho }}{\Gamma (1+\rho )}\right]{\displaystyle \underset{i=1}{\sum ^n}}∥u_i-v_i∥\hfill \\ & \le & \left(\frac{\left(q_k-1\right){\Delta }_k^{q_k-2}{K}^{{\alpha }_{k2}}}{\Gamma (1+{\alpha }_{k2})}\left[\frac{2K^{{\alpha }_{k1}}{N}_k}{\Gamma (1+{\alpha }_{k1})}+\frac{K^{\rho }}{\Gamma (1+\rho )}\right]+{\displaystyle \underset{i=1}{\sum ^n}}{\lambda }_i\right){\displaystyle \underset{i=1}{\sum ^n}}∥u_i-v_i∥\hfill \\ & \le & {\mathcal{B}}_{k3}{\displaystyle \underset{i=1}{\sum ^n}}∥u_i-v_i∥.\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}& & \left|({\mathcal{P}}_2(u_1,\dots ,u_n)\left(t\right)-({\mathcal{P}}_2(v_1,\dots ,v_n)\left(t\right)\right|\hfill \\ & \le & \left({\displaystyle \underset{k=1}{\sum ^n}}{\mathcal{B}}_{k3}\right){\displaystyle \underset{i=1}{\sum ^n}}∥u_i-v_i∥\le {\displaystyle \underset{i=1}{\sum ^n}}∥u_i-v_i∥.\hfill \end{array}$$

Since $\left({\displaystyle \underset{k=1}{\sum ^n}}{\mathcal{B}}_{k3}\right)<1,$ the operator ${\mathcal{P}}_2$ is a contraction.

Third Step: ${\mathcal{P}}_1$ is compact and continuous.$\left({\mathcal{P}}_1={\displaystyle \underset{k=1}{\sum ^n}}{\mathcal{P}}_{1k}\right)$

Since $F_k$ are a continuous functions, this implies that the operator ${\mathcal{P}}_1$ is continuous on ${\mathbb{U}}_r$ and by (14), we have

$$\left|{\mathcal{P}}_1(u_1,\dots ,u_n)\left(t\right)\right|\le {\displaystyle \underset{k=1}{\sum ^n}}{\mathcal{B}}_{k1}{r}^{q_k}.$$

(15)

Moreover, ${\mathcal{P}}_1(u_1,\dots ,u_n)$ is uniformly bounded by (15).

Next, we show equicontinuity. Let $(u_1,\dots ,u_n)\in {\mathbb{U}}_r$. Let $t_1,t_2\in [a,b]$ such that, $t_1<t_2$ we have

$$\begin{array}{ccc}& & \left|{\mathcal{P}}_{1k}(u_1,\dots ,u_n)\left(t_2\right)-{\mathcal{P}}_{1k}(u_1,\dots ,u_n)\left(t_1\right)\right|\hfill \\ & =& \frac{1}{\Gamma \left({\alpha }_{k2}\right)}\left|{\displaystyle \underset{a}{\stackrel{t_2}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k2}-1}(t_2,s)\right.\hfill \\ & & \times \left\{{\psi }_{q_k}\left[\frac{1}{\Gamma \left({\alpha }_{k1}\right)}{\displaystyle \underset{a}{\stackrel{s}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k1}-1}(s,x){\tilde{F}}_{u_k}\left(z\right)dz+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_1;\phi }{\tilde{F}}_{u_k}\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(s,a\right)\right]\right\}ds\hfill \\ & & -{\displaystyle \underset{a}{\stackrel{t_1}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k2}-1}(t_1,s)\hfill \\ & & \left.\times \left\{{\psi }_{q_k}\left[\frac{1}{\Gamma \left({\alpha }_{k1}\right)}{\displaystyle \underset{a}{\stackrel{s}{\int }}}{\phi }^{\prime }\left(s\right){\phi }_{{\alpha }_{k1}-1}(s,x){\tilde{F}}_{u_k}\left(z\right)dz+\frac{({\mathcal{I}}_{a^{+}}^{\rho ;\phi }{u}_k\left(\zeta \right)-{\mathcal{I}}_{a^{+}}^{{\alpha }_1;\phi }{\tilde{F}}_{u_k}\left(b\right))}{{\phi }_{{\gamma }_{k1}-1}\left(b,a\right)}{\phi }_{{\gamma }_{k1}-1}\left(s,a\right)\right]\right\}ds\right|\hfill \\ & \le & \frac{2\times 3^{q_k-2}}{\Gamma (1+{\alpha }_{k2})}\left(2{\left(\frac{K^{{\alpha }_{k1}}\left({\displaystyle \underset{i=1}{\sum ^n}}{{\rm Y}}_{ik}\right)}{\Gamma (1+{\alpha }_{k1})}\right)}^{q_k-1}+{\left(\frac{K^{\rho }}{\Gamma \left(\rho +1\right)}\right)}^{q_{k-1}}\right)r^{q_k-1}{\left(\phi \left(t_2\right)-\phi \left(t_2\right)\right)}^{{\alpha }_{k2}}\hfill \\ & \le & {\mathcal{B}}_{k2}\times r^{q_k-1}{\left(\phi \left(t_2\right)-\phi \left(t_2\right)\right)}^{{\alpha }_{k2}}.\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}& & \left|{\mathcal{P}}_1(u_1,\dots ,u_n)\left(t_2\right)-{\mathcal{P}}_1(u_1,\dots ,u_n)\left(t_1\right)\right|\hfill \\ & \le & {\displaystyle \underset{k=1}{\sum ^n}}{\mathcal{B}}_{k2}\times r^{q_k}{\left(\phi \left(t_2\right)-\phi \left(t_2\right)\right)}^{{\alpha }_{k2}}.\hfill \end{array}$$

Consequently,

$$\left|{\mathcal{P}}_1(u_1,\dots ,u_n)\left(t_2\right)-{\mathcal{P}}_1(u_1,\dots ,u_n)\left(t_1\right)\right|\to 0,\mathrm{as}{t}_1\to t_2.$$

This shows that ${\mathcal{P}}_1{\mathbb{U}}_r$ is equicontinuous. Hence, by the Arzelià–Ascoli theorem, ${\mathcal{P}}_1$ is completely continuous on ${\mathbb{U}}_r$. As a consequence of Krasnoselskii’s fixed point theorem, we conclude that it has a fixed point, which is a solution of (1). The proof of Theorem 1 is, thus, completely achieved. □

3.2. An Illustrative Example

Example 1.

Consider the coupled system for all $t\in \left(0,1\right](n=2)$:

$$\left\{\begin{array}{c}{}^c{\mathcal{D}}_{0^{+}}^{\frac{1}{2};t^2}{\psi }_2\left[{}^c{\mathcal{D}}_{0^{+}}^{\frac{1}{4};t^2}\left(u\left(t\right)\right)\right]=\frac{1}{1+t^2}\left(\frac{u\left(t\right)}{1+{\left(u\left(t\right)\right)}^2}\right),\\ {}^c{\mathcal{D}}_{0^{+}}^{\frac{1}{4};t^2}{\psi }_2\left[{}^c{\mathcal{D}}_{0^{+}}^{\frac{1}{2};t^2}\left(v\left(t\right)\right)\right]=\frac{t}{1+e^t}\left(\frac{v\left(t\right)}{1+{\left(v\left(t\right)\right)}^2}\right),\\ u\left(0\right)=v\left(0\right)=0,\\ {\left.{\psi }_2\left[{}^c{\mathcal{D}}_{0^{+}}^{\frac{1}{4};t^2}\left(u\left(t\right)\right)\right]\right|}_{t=0}={\left.{\psi }_p\left[{}^c{\mathcal{D}}_{0^{+}}^{\frac{1}{2};t^2}\left(v\left(t\right)\right)\right]\right|}_{t=0}=0,\\ u\left(1\right)={\displaystyle \underset{i=1}{\sum ^n}}\frac{1}{7(i!)}u\left({\zeta }_i\right),v\left(1\right)={\displaystyle \underset{i=1}{\sum ^n}}\frac{1}{9(i!)}v\left({\zeta }_i\right),{\zeta }_i\in \left(0,1\right],\\ {\left.{\psi }_2\left[{}^c{\mathcal{D}}_{0^{+}}^{\frac{1}{4};t^2}\left(u\left(t\right)\right)\right]\right|}_{t=1}={\left.{\psi }_p\left[{}^c{\mathcal{D}}_{0^{+}}^{\frac{1}{2};t^2}\left(v\left(t\right)\right)\right]\right|}_{t=1}=0\end{array}\right.$$

(16)

therefore,

$$\begin{array}{ccc}\hfill K& =& 1,\hfill \\ \hfill N_1& =& {{\rm Y}}_{11}=\frac{1}{2},\mathrm{and}{N}_2={{\rm Y}}_{12}=\frac{1}{4}.\hfill \end{array}$$

Thus, the assumptions (${\mathbb{A}}_1-{\mathbb{A}}_2$) are satisfied, and Theorem 1 implies that (16) has a solution on $[0,1].$

4. Conclusions

In this study, we have demonstrated the existence of solutions of a multiplied system of fractional differential equations with nonlocal integro multi-point boundary conditions by using the p-Laplacian operator and the $\phi$-Hilfer derivatives. Using the fixed point theorem of Krasnoselskii, we have completed our work, with an example presented at the end to show the applicability of the obtained results.