Fractional-Order Integro-Differential Multivalued Problems with Fixed and Nonlocal Anti-Periodic Boundary Conditions

Abstract

This paper studies a new class of fractional differential inclusions involving two Caputo fractional derivatives of different orders and a Riemann–Liouville type integral nonlinearity, supplemented with a combination of fixed and nonlocal (dual) anti-periodic boundary conditions. The existence results for the given problem are obtained for convex and non-convex cases of the multi-valued map by applying the standard tools of the fixed point theory. Examples illustrating the obtained results are presented.

1. Introduction

The tools of fractional calculus significantly improved the mathematical modeling of many real world phenomena in viscoelastic materials [1], transport processes [2], economic processes [3,4], etc. Inspired by widespread applications of fractional calculus, many researchers turned to the area of fractional order boundary value problems, for example, see the monograph [5] and the articles [6,7,8,9,10,11,12]. Anti-periodic fractional order boundary value problems also received considerable attention, for instance, see a survey paper [13] and the references cited therein.

Differential inclusions play an important role in the study of dynamical systems and stochastic processes ([14,15,16]). One can find the useful application of this concept in queuing networks, finance, climate control, etc. [17]. Concerning the recent studies on differential inclusions of fractional order, we refer the reader to the articles [18,19,20,21,22,23,24,25,26,27].

Recently, in [28], the authors introduced and studied a new class of boundary value problems involving dual anti-periodic boundary conditions of the form:

$$\left\{\begin{array}{c}{}^c{D}^q\left[{(}^c{D}^p+\alpha )y\left(t\right)+\beta I^{\sigma }g(t,y\left(t\right))\right]=f(t,y\left(t\right)),1<p,q\le 2,t\in [x_1,x_2],\hfill \\ y\left(x_1\right)+y\left(x_2\right)=0,y^{\prime }\left(x_1\right)+y^{\prime }\left(x_2\right)=0,y\left(\xi \right)+y\left(\eta \right)=0,y^{\prime }\left(\xi \right)+y^{\prime }\left(\eta \right)=0,\hfill \end{array}\right.$$

(1)

where ${}^c{D}^{ϵ}$ denotes the Caputo fractional differential operator of order $ϵ(ϵ=p,q)$, $\alpha$ and $\beta$ are real numbers, $\sigma >0,$ $I^{\sigma }$ is the Riemann–Liouville fractional integral, $g,f:[x_1,x_2]\times \mathbb{R}⟶\mathbb{R}$ are given continuous functions, and $-\infty <x_1<\xi <\eta <x_2<\infty .$

The objective of this paper is to investigate the inclusions variant of the problem (1) given by

$$\begin{array}{ccc}& & {}^c{D}^q\left[{(}^c{D}^p+\alpha )y\left(t\right)+\beta I^{\sigma }g(t,y\left(t\right))\right]\in \mathcal{F}(t,y\left(t\right)),1<p,q\le 2,t\in [x_1,x_2],\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& & y\left(x_1\right)+y\left(x_2\right)=0,y^{\prime }\left(x_1\right)+y^{\prime }\left(x_2\right)=0,y\left(\xi \right)+y\left(\eta \right)=0,y^{\prime }\left(\xi \right)+y^{\prime }\left(\eta \right)=0,\hfill \end{array}$$

(3)

where $\mathcal{F}:[x_1,x_2]\times \mathbb{R}\to \mathcal{P}\left(\mathbb{R}\right)$ is a given multivalued map, ($\mathcal{P}\left(\mathbb{R}\right)$ is the family of all nonempty subsets of $\mathbb{R}$) and the other quantities are the same as defined in (1).

Notice that (2) deals with both single-valued map g and multi-valued map $\mathcal{F}.$ On the other hand the boundary conditions (3) describe the anti-periodicity of the unknown function (solution of the problem) at the end points $x_1$ and $x_2$ and at a pair of arbitrary interior points $\eta$ and $\xi$ in the interval $[x_1,x_2]$. These boundary conditions can be interpreted as a combination of fixed and nonlocal anti-periodic boundary conditions. In case of convex-valued multi-valued map $\mathcal{F},$ the existence result for the problem (2)–(3) is proved by applying the Leray–Schauder nonlinear alternative for multivalued maps, while the case of non-convex valued multi-valued map $\mathcal{F}$ is dealt with the aid of Covitz and Nadler fixed point theorem for contractive maps. Though we use the standard fixed point theorems to obtain the main results for the problem at hand, yet their exposition enrich the literature on anti-periodic fractional order nonlinear boundary value problems.

In the rest of the paper, we organize the material as follows. Section 2 contains some preliminary material for the problem (2)–(3). Our main work is presented in Section 3.

2. Basic Result

Before presenting an auxiliary lemma, let us recall some related definitions of fractional calculus [29].

Definition 1.

The Riemann–Liouville fractional integral $I_{x_1}^{\alpha }v$ of order $\alpha >0$ for a function $v\in L_1[a,b],-\infty <x_1<x_2<+\infty ,$ existing almost everywhere on $[x_1,x_2],$ is defined by

$$I_{x_1}^{\alpha }v\left(t\right)=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\underset{x_1}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}v\left(s\right)ds,$$

where Γ denotes the Euler gamma function.

Definition 2.

Let $v,v^{\left(m\right)}\in L_1[x_1,x_2].$ Then the Riemann–Liouville fractional derivative $D_{x_1}^{\alpha }v$ of order $\alpha \in (m-1,m],m\in \mathbb{N},$ existing almost everywhere on $[x_1,x_2],$ is defined as

$$D_{x_1}^{\alpha }v\left(t\right)=\frac{d^m}{d{t}^m}{I}_{x_1}^{m-\alpha }v\left(t\right)=\frac{1}{\mathsf{\Gamma }\left(m-\alpha \right)}\frac{d^m}{d{t}^m}\underset{x_1}{\overset{t}{\int }}{\left(t-s\right)}^{m-1-\alpha }v\left(s\right)ds.$$

The Caputo fractional derivative ${}^c{D}_a^{\alpha }v$ of order $\alpha \in (m-1,m],m\in N$ is defined as

$${}^c{D}_{x_1}^{\alpha }v\left(t\right)=D_{x_1}^{\alpha }\left[v\left(t\right)-v\left(a\right)-v^{\prime }\left(x_1\right)\frac{(t-x_1)}{1!}-\dots -v^{(m-1)}\left(x_1\right)\frac{{(t-x_1)}^{m-1}}{(m-1)!}\right].$$

Remark 1.

If $v\in A{C}^m[x_1,x_2],$ then the Caputo fractional derivative ${}^c{D}_{x_1}^{\alpha }v$ of order $\alpha \in (m-1,m],m\in N,$ existing almost everywhere on $[x_1,x_2],$ is defined as

$${}^c{D}_{x_1}^{\alpha }v\left(t\right)=I_{x_1}^{m-\alpha }{v}^{\left(m\right)}\left(t\right)=\frac{1}{\mathsf{\Gamma }\left(m-\alpha \right)}\underset{x_1}{\overset{t}{\int }}{\left(t-s\right)}^{m-1-\alpha }{v}^{\left(m\right)}\left(s\right)ds.$$

In passing we remark that the fractional integral and derivative operators $I_{x_1}^{\alpha }$ and ${}^c{D}_{x_1}^{\alpha }$ are respectively written as $I^{\alpha }$ and ${}^c{D}^{\alpha }$ for the sake of convenience.

Now we recall the following known lemma [28] that we need to study the problem (2)–(3).

Lemma 1.

For $H,F\in C([x_1,x_2],\mathbb{R})$, the linear fractional integro-differential equation

$${}^c{D}^q\left[{(}^c{D}^p+\alpha )y\left(t\right)+\beta I^{\sigma }H\left(t\right)\right]=F\left(t\right),1<p,q\le 2,t\in (x_1,x_2),$$

(4)

subject to the boundary conditions (3) is equivalent to the integral equation:

$$\begin{array}{ccc}\hfill y\left(t\right)& =& -\alpha I^{p}y\left(t\right)-\beta I^{\sigma +p}H\left(t\right)+I^{p+q}F\left(t\right)\hfill \\ & & +{\omega }_1\left(t\right)\left[\alpha I^{p}y\left(x_2\right)+\beta I^{\sigma +p}g(x_2,y\left(x_2\right))-I^{p+q}F\left(x_2\right)\right]\hfill \\ & & +{\omega }_2\left(t\right)\left[\alpha I^{p-1}y\left(x_2\right)+\beta I^{\sigma +p-1}g(x_2,y\left(x_2\right))-I^{p+q-1}F\left(x_2\right)\right]\hfill \\ & & -{\omega }_3\left(t\right)[\alpha \left(I^{p}y\left(\xi \right)+I^{p}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p}g(\xi ,y\left(\xi \right))+I^{\sigma +p}H\left(\eta \right)\right)\hfill \\ & & -\left(I^{p+q}F\left(\xi \right)+I^{p+q}F\left(\eta \right)\right)]\hfill \\ & & +{\omega }_4\left(t\right)[\alpha \left(I^{p-1}y\left(\xi \right)+I^{p-1}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p-1}g(\xi ,y\left(\xi \right))+I^{\sigma +p-1}H\left(\eta \right)\right)\hfill \\ & & -\left(I^{p+q-1}F\left(\xi \right)+I^{p+q-1}F\left(\eta \right)\right)],\hfill \end{array}$$

where

$$\begin{array}{c}\hfill \left\{{\displaystyle \begin{array}{ccc}\hfill {\omega }_1\left(t\right)& =& {\displaystyle {\nu }_1\frac{{(t-x_1)}^p}{\mathsf{\Gamma }(p+1)}+{\nu }_4\frac{{(t-x_1)}^{p+1}}{\mathsf{\Gamma }(p+2)}+{\nu }_7(t-x_1)+{\rho }_1,}\hfill \\ \hfill {\omega }_2\left(t\right)& =& {\displaystyle {\nu }_2\frac{{(t-x_1)}^p}{\mathsf{\Gamma }(p+1)}+{\nu }_5\frac{{(t-x_1)}^{p+1}}{\mathsf{\Gamma }(p+2)}+{\nu }_8(t-x_1)+{\rho }_2,}\hfill \\ \hfill {\omega }_3\left(t\right)& =& {\displaystyle {\nu }_1\frac{{(t-x_1)}^p}{\mathsf{\Gamma }(p+1)}+{\nu }_4\frac{{(t-x_1)}^{p+1}}{\mathsf{\Gamma }(p+2)}+{\nu }_7(t-x_1)-{\rho }_3,}\hfill \\ \hfill {\omega }_4\left(t\right)& =& {\displaystyle {\nu }_3\frac{{(t-x_1)}^p}{\mathsf{\Gamma }(p+1)}+{\nu }_6\frac{{(t-x_1)}^{p+1}}{\mathsf{\Gamma }(p+2)}+{\nu }_9(t-x_1)+{\rho }_4,}\hfill \end{array}}\right.\end{array}$$

(5)

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill {\nu }_1& =& {\displaystyle \frac{2{\gamma }_1-2{\gamma }_5}{{\delta }_1+{\delta }_2},{\nu }_2=\frac{{\gamma }_3{\gamma }_5-{\gamma }_5{\gamma }_7-2{\gamma }_2+2{\gamma }_6}{{\delta }_1+{\delta }_2},}\hfill \\ \hfill {\nu }_3& =& {\displaystyle \frac{{\gamma }_1{\gamma }_7-{\gamma }_1{\gamma }_3+2{\gamma }_2-2{\gamma }_6}{{\delta }_1+{\delta }_2},{\nu }_4=\frac{2{\gamma }_8-2{\gamma }_4}{{\delta }_1+{\delta }_2},}\hfill \\ \hfill {\nu }_5& =& {\displaystyle \frac{-{\gamma }_3{\gamma }_8+{\gamma }_7{\gamma }_8+2{\gamma }_1-2{\gamma }_5}{{\delta }_1+{\delta }_2},{\nu }_6=\frac{{\gamma }_3{\gamma }_4-{\gamma }_4{\gamma }_7-2{\gamma }_1+2{\gamma }_5}{{\delta }_1+{\delta }_2},}\hfill \\ \hfill {\nu }_7& =& {\displaystyle \frac{{\gamma }_4{\gamma }_5-{\gamma }_1{\gamma }_8}{{\delta }_1+{\delta }_2},{\nu }_8=\frac{{\gamma }_2{\gamma }_8-{\gamma }_1{\gamma }_5+{\gamma }_5^2-{\gamma }_6{\gamma }_8}{{\delta }_1+{\delta }_2},}\hfill \\ \hfill {\nu }_9& =& {\displaystyle \frac{{\gamma }_1^2-{\gamma }_1{\gamma }_5-{\gamma }_2{\gamma }_4+{\gamma }_4{\gamma }_6}{{\delta }_1+{\delta }_2},}\hfill \end{array}\right.\end{array}$$

(6)

$$\begin{array}{c}\hfill {\displaystyle \left\{\begin{array}{ccc}\hfill {\rho }_1& =& {\displaystyle \frac{{\gamma }_1{\gamma }_7{\gamma }_8-{\gamma }_4{\gamma }_5{\gamma }_7-2{\gamma }_1{\gamma }_5+2{\gamma }_4{\gamma }_6+2{\gamma }_5^2-2{\gamma }_6{\gamma }_8}{2({\delta }_1+{\delta }_2)},}\hfill \\ \hfill {\rho }_2& =& {\displaystyle \frac{{\gamma }_1{\gamma }_5{\gamma }_7-{\gamma }_2{\gamma }_7{\gamma }_8-{\gamma }_3{\gamma }_5^2+{\gamma }_3{\gamma }_6{\gamma }_8-2{\gamma }_1{\gamma }_6+2{\gamma }_2{\gamma }_5}{2({\delta }_1+{\delta }_2)},}\hfill \\ \hfill {\rho }_3& =& {\displaystyle \frac{-{\gamma }_1{\gamma }_3{\gamma }_8+{\gamma }_3{\gamma }_4{\gamma }_5+2{\gamma }_1^2-2{\gamma }_1{\gamma }_5-2{\gamma }_2{\gamma }_4+2{\gamma }_2{\gamma }_8}{2({\delta }_1+{\delta }_2)},}\hfill \\ \hfill {\rho }_4& =& {\displaystyle \frac{-{\gamma }_1^2{\gamma }_7+{\gamma }_1{\gamma }_3{\gamma }_5+{\gamma }_2{\gamma }_4{\gamma }_7-{\gamma }_3{\gamma }_4{\gamma }_6+2{\gamma }_1{\gamma }_6-2{\gamma }_2{\gamma }_5}{2({\delta }_1+{\delta }_2)},}\hfill \end{array}\right.}\end{array}$$

(7)

$$\begin{array}{c}\hfill {\displaystyle \left\{\begin{array}{ccc}\hfill {\gamma }_1& =& {\displaystyle \frac{{(x_2-x_1)}^p}{\mathsf{\Gamma }(p+1)},{\gamma }_2=\frac{{(x_2-x_1)}^{p+1}}{\mathsf{\Gamma }(p+2)},{\gamma }_3=(x_2-x_1),}\hfill \\ \hfill {\gamma }_4& =& {\displaystyle \frac{{(x_2-x_1)}^{p-1}}{\mathsf{\Gamma }\left(p\right)},{\gamma }_5=\frac{{(\xi -x_1)}^p+{(\eta -x_1)}^p}{\mathsf{\Gamma }(p+1)},}\hfill \\ \hfill {\gamma }_6& =& {\displaystyle \frac{{(\xi -x_1)}^{p+1}+{(\eta -x_1)}^{p+1}}{\mathsf{\Gamma }(p+2)},{\gamma }_7=(\xi -x_1)+(\eta -x_1),}\hfill \\ \hfill {\gamma }_8& =& {\displaystyle \frac{{(\xi -x_1)}^{p-1}+{(\eta -x_1)}^{p-1}}{\mathsf{\Gamma }\left(p\right)},}\hfill \end{array}\right.}\end{array}$$

(8)

and it is assumed that ${\delta }_1+{\delta }_2\ne 0$ with

$$\begin{array}{c}\hfill {\displaystyle \left\{\begin{array}{ccc}\hfill {\delta }_1& =& -{\gamma }_1{\gamma }_3{\gamma }_8+{\gamma }_1{\gamma }_7{\gamma }_8+{\gamma }_3{\gamma }_4{\gamma }_5-{\gamma }_4{\gamma }_5{\gamma }_7+2{\gamma }_1^2,\hfill \\ \hfill {\delta }_2& =& -4{\gamma }_1{\gamma }_5-2{\gamma }_2{\gamma }_4+2{\gamma }_2{\gamma }_8+2{\gamma }_4{\gamma }_6+2{\gamma }_5^2-2{\gamma }_6{\gamma }_8.\hfill \end{array}\right.}\end{array}$$

(9)

3. Existence Results

Let $C([x_1,x_2],\mathbb{R})$ denote the Banach space of all continuous functions from $[x_1,x_2]⟶\mathbb{R}$ endowed with sup-norm ${\displaystyle \parallel y\parallel =sup\left\{\right|y\left(t\right)|,t\in [x_1,x_2]\}}$.

Definition 3.

A function $y\in AC([x_1,x_2],\mathbb{R})$ is said to be a solution of the problem (2)–(3) if there exists a function $v\in L^1([x_1,x_2],\mathbb{R})$ with $v\left(t\right)\in \mathcal{F}(t,y)$ for a.e. $t\in [x_1,x_2]$ such that y satisfies the differential equation ${}^c{D}^q\left[{(}^c{D}^p+\alpha )y\left(t\right)+\beta I^{\sigma }g(t,y\left(t\right))\right]=v\left(t\right)$ on $[x_1,x_2]$ and the boundary conditions (3).

In the forthcoming analysis, we use the following notation:

$$\begin{array}{ccc}\hfill {\mathsf{\Phi }}_0& =& \{\frac{{(x_2-x_1)}^p}{\mathsf{\Gamma }(p+1)}+{\tilde{\omega }}_1\frac{{(x_2-x_1)}^p}{\mathsf{\Gamma }(p+1)}+{\tilde{\omega }}_2\frac{{(x_2-x_1)}^{p-1}}{\mathsf{\Gamma }\left(p\right)}+{\tilde{\omega }}_3\left(\frac{{(\xi -x_1)}^p+{(\eta -x_1)}^p}{\mathsf{\Gamma }(p+1)}\right)\hfill \\ \hfill & & +{\tilde{\omega }}_4\left(\frac{{(\xi -x_1)}^{p-1}+{(\eta -x_1)}^{p-1}}{\mathsf{\Gamma }\left(p\right)}\right)\},\hfill \\ \hfill {\mathsf{\Phi }}_1& =& \{\frac{{(x_2-x_1)}^{\sigma +p}}{\mathsf{\Gamma }(\sigma +p+1)}+{\tilde{\omega }}_1\frac{{(x_2-x_1)}^{\sigma +p}}{\mathsf{\Gamma }(\sigma +p+1)}+{\tilde{\omega }}_2\frac{{(x_2-x_1)}^{\sigma +p-1}}{\mathsf{\Gamma }(\sigma +p)}\hfill \\ \hfill & & +{\tilde{\omega }}_3\left(\frac{{(\xi -x_1)}^{\sigma +p}+{(\eta -x_1)}^{\sigma +p}}{\mathsf{\Gamma }(\sigma +p+1)}\right)+{\tilde{\omega }}_4\left(\frac{{(\xi -x_1)}^{\sigma +p-1}+{(\eta -x_1)}^{\sigma +p-1}}{\mathsf{\Gamma }(\sigma +p)}\right)\},\hfill \\ \hfill {\mathsf{\Phi }}_2& =& \{\frac{{(x_2-x_1)}^{p+q}}{\mathsf{\Gamma }(p+q+1)}+{\tilde{\omega }}_1\frac{{(x_2-x_1)}^{p+q}}{\mathsf{\Gamma }(p+q+1)}+{\tilde{\omega }}_2\frac{{(x_2-x_1)}^{p+q-1}}{\mathsf{\Gamma }(p+q)}\hfill \\ & & +{\tilde{\omega }}_3\left(\frac{{(\xi -x_1)}^{p+q}+{(\eta -x_1)}^{p+q}}{\mathsf{\Gamma }(p+q+1)}\right)+{\tilde{\omega }}_4\left(\frac{{(\xi -x_1)}^{p+q-1}+{(\eta -x_1)}^{p+q-1}}{\mathsf{\Gamma }(p+q)}\right)\},\hfill \end{array}$$

(10)

where ${\displaystyle {\tilde{\omega }}_1=\underset{t\in [x_1,x_2]}{sup}\left|{\omega }_1\left(t\right)\right|}$, ${\displaystyle {\tilde{\omega }}_2=\underset{t\in [x_1,x_2]}{sup}\left|{\omega }_2\left(t\right)\right|}$, ${\displaystyle {\tilde{\omega }}_3=\underset{t\in [x_1,x_2]}{sup}\left|{\omega }_3\left(t\right)\right|}$, ${\displaystyle {\tilde{\omega }}_4=\underset{t\in [x_1,x_2]}{sup}\left|{\omega }_4\left(t\right)\right|}$.

3.1. The Upper Semicontinuous Case

Let us begin this section by defining some spaces related to our work as follows: $\mathcal{P}\left(X\right)=\{Y\subset X:Y\ne \varnothing \};$${\mathcal{P}}_{cp}\left(X\right)=\{Y\in \mathcal{P}\left(X\right):Y\mathrm{is}\mathrm{compact}\};$${\mathcal{P}}_{c,cp}\left(X\right)=\{Y\in \mathcal{P}\left(X\right):Y\mathrm{is}\mathrm{convex}\mathrm{and}\mathrm{compact}\};$ ${\mathcal{P}}_{b,cl}\left(X\right)=\{Y\in \mathcal{P}\left(X\right):Y\mathrm{is}\mathrm{bounded}\mathrm{and}\mathrm{closed}\}$ and ${\mathcal{P}}_{cl}\left(X\right)=\{Y\in \mathcal{P}\left(X\right):Y\mathrm{is}\mathrm{closed}\}.$

Our first existence result for the problem (2)–(3) deals with the convex valued multi-valued map $\mathcal{F}$ and relies on the following Leray–Schauder nonlinear alternative.

Lemma 2

(Nonlinear alternative for Kakutani maps [30]). Let C be a closed convex subset of a Banach space E and U be an open subset of C with $0\in U.$ Suppose that $G:\overline{U}\to {\mathcal{P}}_{c,cp}\left(C\right)$ is an upper semicontinuous compact map. Then either G has a fixed point in $\overline{U}$ or there is an element $u\in \partial U$ such that $u\in \mu G\left(u\right)$ with $\mu \in (0,1).$

Theorem 1.

Assume that:

$\left({\mathcal{H}}_1\right)$

$\mathcal{F}:[x_1,x_2]\times \mathbb{R}\to \mathcal{P}\left(\mathbb{R}\right)$ has convex, compact values and is $L^1$-Carathéodory, that is, (i) $t⟼\mathcal{F}(t,y)$ is measurable for each $y\in \mathbb{R}$; (ii) $y⟼\mathcal{F}(t,y)$ is upper semicontinuous for almost all $t\in [x_1,x_2];$ (iii) for each $r>0$, there exists ${\varphi }_r\in L^1([x_1,x_2],{\mathbb{R}}^{+})$ such that

$$\parallel \mathcal{F}(t,y)\parallel =sup\left\{\right|v|:v\left(t\right)\in \mathcal{F}(t,y)\}\le {\varphi }_r\left(t\right)$$

for all $y\in \mathbb{R}$ with $\parallel y\parallel \le r$ and for a.e. $t\in [x_1,x_2];$

$\left({\mathcal{H}}_2\right)$

there exists a function $p\in C([x_1,x_2],{\mathbb{R}}^{+})$ and a continuous nondecreasing function $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that

$${\parallel \mathcal{F}(t,y)\parallel }_{\mathcal{P}}:=sup\left\{\right|\chi |:\chi \in \mathcal{F}(t,y)\}\le p\left(t\right)\psi (\parallel y\parallel )foreach(t,y)\in [x_1,x_2]\times \mathbb{R};$$

$\left({\mathcal{H}}_3\right)$

$\left|g(t,y)\right|\le \theta \left(t\right),\forall (t,y)\in [x_1,x_2]\times \mathbb{R},$$\theta \in C([x_1,x_2],{\mathbb{R}}^{+});$

$\left({\mathcal{H}}_4\right)$

there exists a constant $\mathcal{K}>0$ such that

$${\displaystyle \frac{(1-|\alpha \left|{\mathsf{\Phi }}_0\right)\mathcal{K}}{\left|\beta \right|{\mathsf{\Phi }}_1\parallel \theta \parallel +{\mathsf{\Phi }}_2\parallel p\parallel \psi \left(\mathcal{K}\right)}>1,\left|\alpha \right|{\mathsf{\Phi }}_0<1.}$$

Then there exists at least one solution on $[x_1,x_2]$ for the boundary value problem (2)–(3).

Proof. 

We transform the problem (2)–(3) into a fixed point problem by introducing a multi-valued operator ${\mathcal{N}}_{\mathcal{F}}:C([x_1,x_2],\mathbb{R})\to \mathcal{P}\left(C([x_1,x_2],\mathbb{R})\right)$ as

$${\mathcal{N}}_{\mathcal{F}}\left(y\right)=\left\{\begin{array}{cc}& h\in C([x_1,x_2],\mathbb{R}):\hfill \\ & h\left(t\right)=\left\{\begin{array}{c}-\alpha I^{p}y\left(t\right)-\beta I^{\sigma +p}g(t,y\left(t\right))+I^{p+q}v\left(t\right)\hfill \\ +{\omega }_1\left(t\right)\left[\alpha I^{p}y\left(x_2\right)+\beta I^{\sigma +p}g(x_2,y\left(x_2\right))-I^{p+q}v\left(x_2\right)\right]\hfill \\ +{\omega }_2\left(t\right)\left[\alpha I^{p-1}y\left(x_2\right)+\beta I^{\sigma +p-1}g(x_2,y\left(x_2\right))-I^{p+q-1}v\left(x_2\right)\right]\hfill \\ -{\omega }_3\left(t\right)[\alpha \left(I^{p}y\left(\xi \right)+I^{p}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p}g(\xi ,y\left(\xi \right))+I^{\sigma +p}g(\eta ,y\left(\eta \right))\right)\hfill \\ -\left(I^{p+q}v\left(\xi \right)+I^{p+q}v\left(\eta \right)\right)]+{\omega }_4\left(t\right)[\alpha \left(I^{p-1}y\left(\xi \right)+I^{p-1}y\left(\eta \right)\right)\hfill \\ +\beta \left(I^{\sigma +p-1}g(\xi ,y\left(\xi \right))+I^{\sigma +p-1}g(\eta ,y\left(\eta \right)\right)\hfill \\ -\left(I^{p+q-1}v\left(\xi \right)+I^{p+q-1}v\left(\eta \right)\right)],v\in S_{\mathcal{F},y},\hfill \end{array}\right.\hfill \end{array}\right\}$$

where $S_{\mathcal{F},y}=\{f\in L^1([x_1,x_2],\mathbb{R}):f\left(t\right)\in F(t,y)fora.e.t\in [x_1,x_2]\}.$ In order to establish that the problem (2)–(3) has a solution, we need to show that the operator ${\mathcal{N}}_{\mathcal{F}}$ satisfies the hypothesis of Lemma 2. We first show that ${\mathcal{N}}_{\mathcal{F}}$ is convex for each $y\in C([x_1,x_2],\mathbb{R}).$

Let $h_1,h_2\in {\mathcal{N}}_{\mathcal{F}}.$ Then, there exist $v_1,v_2\in S_{\mathcal{F},y}$ such that for each $t\in [x_1,x_2]$, we have

$$\begin{array}{ccc}\hfill h_i\left(t\right)& =& -\alpha I^{p}y\left(t\right)-\beta I^{\sigma +p}g(t,y\left(t\right))+I^{p+q}{v}_i\left(t\right)\hfill \\ & & +{\omega }_1\left(t\right)\left[\alpha I^{p}y\left(x_2\right)+\beta I^{\sigma +p}g(x_2,y\left(x_2\right))-I^{p+q}{v}_i\left(x_2\right)\right]\hfill \\ & & +{\omega }_2\left(t\right)\left[\alpha I^{p-1}y\left(x_2\right)+\beta I^{\sigma +p-1}g(x_2,y\left(x_2\right))-I^{p+q-1}{v}_i\left(x_2\right)\right]\hfill \\ & & -{\omega }_3\left(t\right)[\alpha \left(I^{p}y\left(\xi \right)+I^{p}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p}g(\xi ,y\left(\xi \right))+I^{\sigma +p}g(\eta ,y\left(\eta \right))\right)\hfill \\ & & -\left(I^{p+q}{v}_i\left(\xi \right)+I^{p+q}{v}_i\left(\eta \right)\right)]\hfill \\ & & +{\omega }_4\left(t\right)[\alpha \left(I^{p-1}y\left(\xi \right)+I^{p-1}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p-1}g(\xi ,y\left(\xi \right))+I^{\sigma +p-1}g(\eta ,y\left(\eta \right)\right)\hfill \\ & & -\left(I^{p+q-1}{v}_i\left(\xi \right)+I^{p+q-1}{v}_i\left(\eta \right)\right)].\hfill \end{array}$$

Letting $0\le \kappa \le 1,$ for each $t\in [x_1,x_2]$, we have

$$\begin{array}{ccc}& & [\kappa h_1+(1-\kappa )h_2]\left(t\right)\hfill \\ & =& -\alpha I^{p}y\left(t\right)-\beta I^{\sigma +p}g(t,y\left(t\right))+I^{p+q}[\kappa v_1\left(t\right)+(1-\kappa )v_2\left(t\right)]\hfill \\ & & +{\omega }_1\left(t\right)\left[\alpha I^{p}y\left(x_2\right)+\beta I^{\sigma +p}g(x_2,y\left(x_2\right))-I^{p+q}[\kappa v_1\left(x_2\right)+(1-\kappa )v_2\left(x_2\right)]\right]\hfill \\ & & +{\omega }_2\left(t\right)\left[\alpha I^{p-1}y\left(x_2\right)+\beta I^{\sigma +p-1}g(x_2,y\left(x_2\right))-I^{p+q-1}[\kappa v_1\left(x_2\right)+(1-\kappa )v_2\left(x_{2}s\right)]\left(x_2\right)\right]\hfill \\ & & -{\omega }_3\left(t\right)[\alpha \left(I^{p}y\left(\xi \right)+I^{p}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p}g(\xi ,y\left(\xi \right))+I^{\sigma +p}g(\eta ,y\left(\eta \right))\right)\hfill \\ & & -\left(I^{p+q}[\kappa v_1\left(\xi \right)+(1-\kappa )v_2\left(\xi \right)]+I^{p+q}[\kappa v_1\left(\eta \right)+(1-\kappa )v_2\left(\eta \right)]\right)]\hfill \\ & & +{\omega }_4\left(t\right)[\alpha \left(I^{p-1}y\left(\xi \right)+I^{p-1}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p-1}g(\xi ,y\left(\xi \right))+I^{\sigma +p-1}g(\eta ,y\left(\eta \right)\right)\hfill \\ & & -\left(I^{p+q-1}[\kappa v_1\left(\xi \right)+(1-\kappa )v_2\left(\xi \right)]+I^{p+q-1}[\kappa v_1\left(\eta \right)+(1-\kappa )v_2\left(\eta \right)]\right)].\hfill \end{array}$$

Consequently $\kappa h_1+(1-\kappa )h_2\in {\mathcal{N}}_{\mathcal{F}}\left(y\right),$ since $\mathcal{F}$ has convex values. Next we show that ${\mathcal{N}}_{\mathcal{F}}$ maps in $C([x_1,x_2],\mathbb{R})$ bounded sets into bounded sets. Let $B_r=\{y\in C([x_1,x_2],\mathbb{R}):\parallel y\parallel \le r\}$ be a bounded set in $C([x_1,x_2],\mathbb{R})$ for a positive number $r.$ Then for each $h\in {\mathcal{N}}_{\mathcal{F}}\left(y\right),y\in B_r,$ there exists $v\in S_{\mathcal{F},y}$ such that

$$\begin{array}{ccc}\hfill h\left(t\right)& =& -\alpha I^{p}y\left(t\right)-\beta I^{\sigma +p}g(t,y\left(t\right))+I^{p+q}v\left(t\right)\hfill \\ & & +{\omega }_1\left(t\right)\left[\alpha I^{p}y\left(x_2\right)+\beta I^{\sigma +p}g(x_2,y\left(x_2\right))-I^{p+q}v\left(x_2\right)\right]\hfill \\ & & +{\omega }_2\left(t\right)\left[\alpha I^{p-1}y\left(x_2\right)+\beta I^{\sigma +p-1}g(x_2,y\left(x_2\right))-I^{p+q-1}v\left(x_2\right)\right]\hfill \\ & & -{\omega }_3\left(t\right)[\alpha \left(I^{p}y\left(\xi \right)+I^{p}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p}g(\xi ,y\left(\xi \right))+I^{\sigma +p}g(\eta ,y\left(\eta \right))\right)\hfill \\ & & -\left(I^{p+q}v\left(\xi \right)+I^{p+q}v\left(\eta \right)\right)]\hfill \\ & & +{\omega }_4\left(t\right)[\alpha \left(I^{p-1}y\left(\xi \right)+I^{p-1}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p-1}g(\xi ,y\left(\xi \right))+I^{\sigma +p-1}g(\eta ,y\left(\eta \right)\right)\hfill \\ & & -\left(I^{p+q-1}v\left(\xi \right)+I^{p+q-1}v\left(\eta \right)\right)].\hfill \end{array}$$

Then, for $t\in [x_1,x_2],$ we have

$$\begin{array}{ccc}\hfill \left|h\right(t\left)\right|& \le & \left|\alpha \right|I^p\left|y\left(t\right)\right|+\left|\beta \right|I^{\sigma +p}\left|g(t,y\left(t\right))\right|+I^{p+q}\left|v\left(t\right)\right|\hfill \\ & & +|{\omega }_1\left(t\right)|\left[\left|\alpha \right|I^p|y\left(x_2\right)|+|\beta |I^{\sigma +p}|g(x_2,y\left(x_2\right))|+I^{p+q}\left|v\left(x_2\right)\right|\right]\hfill \\ & & +|{\omega }_2\left(t\right)|\left[\left|\alpha \right|I^{p-1}|y\left(x_2\right)|+|\beta |I^{\sigma +p-1}|g(x_2,y\left(x_2\right))|+I^{p+q-1}\left|v\left(x_2\right)\right|\right]\hfill \\ & & +|{\omega }_3\left(t\right)\left|\right[\left|\alpha \right|\left(I^p\left|y\left(\xi \right)\right|+I^p\left|y\left(\eta \right)\right|\right)+\left|\beta \right|\left(I^{\sigma +p}\left|g(\xi ,y\left(\xi \right))\right|+I^{\sigma +p}\left|g(\eta ,y\left(\eta \right))\right|\right)\hfill \\ & & +\left(I^{p+q}\left|v\left(\xi \right)\right|+I^{p+q}\left|v\left(\eta \right)\right|\right)]\hfill \\ & & +|{\omega }_4\left(t\right)\left|\right[\left|\alpha \right|\left(I^{p-1}\left|y\left(\xi \right)\right|+I^{p-1}\left|y\left(\eta \right)\right|\right)+\left|\beta \right|\left(I^{\sigma +p-1}\left|g(\xi ,y\left(\xi \right))\right|+I^{\sigma +p-1}\left|g\right(\eta ,y\left(\eta \right)|\right)\hfill \\ & & +\left(I^{p+q-1}\left|v\left(\xi \right)\right|+I^{p+q-1}\left|v\left(\eta \right)\right|\right)]\hfill \\ & \le & \left|\alpha \right|\frac{{(x_2-x_1)}^p}{\mathsf{\Gamma }(p+1)}\parallel y\parallel +\left|\beta \right|\frac{{(x_2-x_1)}^{p+q}}{\mathsf{\Gamma }(p+q+1)}\parallel \theta \parallel +\frac{{(x_2-x_1)}^{p+q}}{\mathsf{\Gamma }(p+q+1)}\parallel p\parallel \psi (\parallel y\parallel )\hfill \\ & & +{\tilde{\omega }}_1\left[\left|\alpha \right|\frac{{(x_2-x_1)}^p}{\mathsf{\Gamma }(p+1)}\parallel y\parallel +\left|\beta \right|\frac{{(x_2-x_1)}^{\sigma +p}}{\mathsf{\Gamma }(\sigma +p+1)}\parallel \theta \parallel +\frac{{(x_2-x_1)}^{p+q}}{\mathsf{\Gamma }(p+q+1)}\parallel p\parallel \psi (\parallel y\parallel )\right]\hfill \\ & & +{\tilde{\omega }}_2\left[\left|\alpha \right|\frac{{(x_2-x_1)}^{p-1}}{\mathsf{\Gamma }\left(p\right)}\parallel y\parallel +\left|\beta \right|\frac{{(x_2-x_1)}^{p+q-1}}{\mathsf{\Gamma }(p+q)}\parallel \theta \parallel +\frac{{(x_2-x_1)}^{p+q-1}}{\mathsf{\Gamma }(p+q)}\parallel p\parallel \psi (\parallel y\parallel )\right]\hfill \\ & & +{\tilde{\omega }}_3\left[\right|\alpha |\left(\frac{{(\xi -x_1)}^p}{\mathsf{\Gamma }(p+1)}+\frac{{(\eta -x_1)}^p}{\mathsf{\Gamma }(p+1)}\right)\parallel y\parallel +\left|\beta \right|\left(\frac{{(\xi -x_1)}^{\sigma +p}}{\mathsf{\Gamma }(\sigma +p+1)}+\frac{{(\eta -x_1)}^{\sigma +p}}{\mathsf{\Gamma }(\sigma +p+1)}\right)\parallel \theta \parallel \hfill \\ & & +\left(\frac{{(\xi -x_1)}^{p+q}}{\mathsf{\Gamma }(p+q+1)}+\frac{{(\eta -x_1)}^{p+q}}{\mathsf{\Gamma }(p+q+1)}\right)\parallel p\parallel \psi (\parallel y\parallel )]\hfill \\ & & +{\tilde{\omega }}_4\left[\right|\alpha |\left(\frac{{(\xi -x_1)}^{p-1}}{\mathsf{\Gamma }\left(p\right)}+\frac{{(\eta -x_1)}^{p-1}}{\mathsf{\Gamma }\left(p\right)}\right)\parallel y\parallel +\left|\beta \right|\left(\frac{{(\xi -x_1)}^{\sigma +p-1}}{\mathsf{\Gamma }(\sigma +p)}+\frac{{(\eta -x_1)}^{\sigma +p-1}}{\mathsf{\Gamma }(\sigma +p)}\right)\parallel \theta \parallel \hfill \\ & & +\left(\frac{{(\xi -x_1)}^{p+q-1}}{\mathsf{\Gamma }(p+q)}+\frac{{(\eta -x_1)}^{p+q-1}}{\mathsf{\Gamma }(p+q)}\right)\parallel p\parallel \psi (\parallel y\parallel )]\hfill \\ & \le & \left|\alpha \right|{\mathsf{\Phi }}_0\parallel y\parallel +\left|\beta \right|{\mathsf{\Phi }}_1\parallel \theta \parallel +{\mathsf{\Phi }}_2\parallel p\parallel \psi (\parallel y\parallel ),\hfill \end{array}$$

and consequently

$$\parallel h\parallel \le \left|\alpha \right|{\mathsf{\Phi }}_{0}r+\left|\beta \right|{\mathsf{\Phi }}_1\parallel \theta \parallel +{\mathsf{\Phi }}_2\parallel p\parallel \psi \left(r\right).$$

Now we demonstrate that ${\mathcal{N}}_{\mathcal{F}}$ maps bounded sets into equicontinuous sets of $C([x_1,x_2],\mathbb{R}).$ For $t_1,t_2\in \left[x_1{x}_2\right],$$y\in B_r$ and for each $h\in {\mathcal{N}}_{\mathcal{F}},$ we have

$$\begin{array}{ccc}& & |h\left(t_2\right)-h\left(t_1\right)|\hfill \\ & \le & |-\alpha {\int }_{x_1}^{t_1}\frac{{(t_2-u)}^{p-1}-{(t_1-u)}^{p-1}}{\mathsf{\Gamma }\left(p\right)}y\left(u\right)du-\alpha {\int }_{t_1}^{t_2}\frac{{(t_2-u)}^{p-1}}{\mathsf{\Gamma }\left(p\right)}y\left(u\right)du\hfill \\ & & -\beta {\int }_{x_1}^{t_1}\frac{{(t_2-u)}^{\sigma +p-1}-{(t_1-u)}^{\alpha +p-1}}{\mathsf{\Gamma }(\sigma +p)}g(u,y\left(u\right))du-\beta {\int }_{t_1}^{t_2}\frac{{(t_2-u)}^{\sigma +p-1}}{\mathsf{\Gamma }(\sigma +p)}g(u,y\left(u\right))du\hfill \\ & & +{\int }_{x_1}^{t_1}\frac{{(t_2-u)}^{p+q-1}-{(t_1-u)}^{p+q-1}}{\mathsf{\Gamma }(p+q)}v\left(u\right)du+{\int }_{t_1}^{t_2}\frac{{(t_2-u)}^{p+q-1}}{\mathsf{\Gamma }(p+q)}v\left(u\right)du|\hfill \\ & & +|{\omega }_1\left(t_2\right)-{\omega }_1\left(t_1\right)|\left[\left|\alpha \right|I^p|y\left(x_2\right)|+|\beta |I^{\sigma +p}|g(x_2,y\left(x_2\right))|+I^{p+q}\left|v\left(x_2\right)\right|\right]\hfill \\ & & +|{\omega }_2\left(t_2\right)-{\omega }_2\left(t_1\right)|\left[\left|\alpha \right|I^{p-1}|y\left(x_2\right)|+|\beta |I^{\sigma +p-1}|g(x_2,y\left(x_2\right))|+I^{p+q-1}\left|v\left(x_2\right)\right|\right]\hfill \\ & & +|{\omega }_3\left(t_2\right)-{\omega }_3\left(t_1\right)\left|\right[\left|\alpha \right|\left(I^p\left|y\left(\xi \right)\right|+I^p\left|y\left(\eta \right)\right|\right)+\left|\beta \right|\left(I^{\sigma +p}\left|g(\xi ,y\left(\xi \right))\right|+I^{\sigma +p}\left|g(\eta ,y\left(\eta \right))\right|\right)\hfill \\ & & +\left(I^{p+q}\left|v\left(\xi \right)\right|+I^{p+q}\left|v\left(\eta \right)\right|\right)]\hfill \\ & & +|{\omega }_4\left(t_2\right)-{\omega }_4\left(t_1\right)\left|\right[\left|\alpha \right|\left(I^{p-1}\left|y\left(\xi \right)\right|+I^{p-1}\left|y\left(\eta \right)\right|\right)+\left|\beta \right|\left(I^{\sigma +p-1}\left|g(\xi ,y\left(\xi \right))\right|+I^{\sigma +p-1}\left|g\right(\eta ,y\left(\eta \right)|\right)\hfill \\ & & +\left(I^{p+q-1}\left|v\left(\xi \right)\right|+I^{p+q-1}\left|v\left(\eta \right)\right|\right)]\hfill \\ & \le & \frac{\left|\alpha \right|r}{\mathsf{\Gamma }(p+1)}\left(|{(t_2-x_1)}^p-{(t_1-x_1)}^p|+2{(t_2-t_1)}^p\right)\hfill \\ & & +\frac{\left|\beta \right|\parallel \theta \parallel }{\mathsf{\Gamma }(\sigma +p+1)}\left(|{(t_2-x_1)}^{\sigma +p}-{(t_1-x_1)}^{\sigma +p}|+2{(t_2-t_1)}^{\sigma +p}\right)\hfill \\ & & +\frac{\parallel p\parallel \psi \left(r\right)}{\mathsf{\Gamma }(p+q+1)}\left(|{(t_2-x_1)}^{p+q}-{(t_1-x_1)}^{p+q}|+2{(t_2-t_1)}^{p+q}\right)\hfill \\ & & +|{\omega }_1\left(t_2\right)-{\omega }_1\left(t_1\right)|\left[\left|\alpha \right|\frac{r{(x_2-x_1)}^p}{\mathsf{\Gamma }(p+1)}+\left|\beta \right|\frac{\parallel \theta \parallel {(x_2-x_1)}^{\sigma +p}}{\mathsf{\Gamma }(\sigma +p+1)}+\frac{\parallel p\parallel \psi \left(r\right){(x_2-x_1)}^{p+q}}{\mathsf{\Gamma }(p+q+1)}\right]\hfill \\ & & +|{\omega }_2\left(t_2\right)-{\omega }_2\left(t_1\right)|\left[\left|\alpha \right|\frac{r{(x_2-x_1)}^{p-1}}{\mathsf{\Gamma }\left(p\right)}+\left|\beta \right|\frac{\parallel \theta \parallel {(x_2-x_1)}^{\sigma +p-1}}{\mathsf{\Gamma }(\sigma +p)}+\frac{\parallel p\parallel \psi \left(r\right){(x_2-x_1)}^{p+q-1}}{\mathsf{\Gamma }(p+q)}\right]\hfill \\ & & +|{\omega }_3\left(t_2\right)-{\omega }_3\left(t_1\right)\left|\right[\left|\alpha \right|r\left(\frac{{(\xi -x_1)}^p}{\mathsf{\Gamma }(p+1)}+\frac{{(\eta -x_1)}^p}{\mathsf{\Gamma }(p+1)}\right)+\left|\beta \right|\parallel \theta \parallel \left(\frac{{(\xi -x_1)}^{\sigma +p}}{\mathsf{\Gamma }(\sigma +p+1)}+\frac{{(\eta -x_1)}^{\sigma +p}}{\mathsf{\Gamma }(\sigma +p+1)}\right)\hfill \\ & & +\parallel p\parallel \psi \left(r\right)\left(\frac{{(\xi -x_1)}^{p+q}}{\mathsf{\Gamma }(p+q+1)}+\frac{{(\eta -x_1)}^{p+q}}{\mathsf{\Gamma }(p+q+1)}\right)]\hfill \\ & & +|{\omega }_4\left(t_2\right)-{\omega }_4\left(t_1\right)\left|\right[\left|\alpha \right|r\left(\frac{{(\xi -x_1)}^{p-1}}{\mathsf{\Gamma }\left(p\right)}+\frac{{(\eta -x_1)}^{p-1}}{\mathsf{\Gamma }\left(p\right)}\right)\hfill \\ & & +\left|\beta \right|\parallel \theta \parallel \left(\frac{{(\xi -x_1)}^{\sigma +p-1}}{\mathsf{\Gamma }(\sigma +p)}+\frac{{(\eta -x_1)}^{\sigma +p-1}}{\mathsf{\Gamma }(\sigma +p)}\right)\hfill \\ & & +\parallel p\parallel \psi \left(r\right)\left(\frac{{(\xi -x_1)}^{p+q-1}}{\mathsf{\Gamma }(p+q)}+\frac{{(\eta -x_1)}^{p+q-1}}{\mathsf{\Gamma }(p+q)}\right)].\hfill \end{array}$$

Clearly the right hand of the above inequality tends to zero independently of $y\in B_r$ as $t_1\to t_2$. Hence ${\mathcal{N}}_{\mathcal{F}}:C([x_1,x_2],\mathbb{R})\to \mathcal{P}\left(C([x_1,x_2],\mathbb{R})\right)$ is completely continuous, by Arzelá-Ascoli theorem.

Next we show that the operator ${\mathcal{N}}_{\mathcal{F}}$ is upper semicontinuous. It is enough to establish that ${\mathcal{N}}_{\mathcal{F}}$ has a closed graph, because from (Proposition 1.2 [31]) we know that if an operator is completely continuous and has a closed graph, then it is upper semi-continuous.

Let $y_n\to y_{*}$, $h_n\in {\mathcal{N}}_{\mathcal{F}}\left(y_n\right)$ and $h_n\to h_{*}$. We need to show that $h_{*}\in {\mathcal{N}}_{\mathcal{F}}\left(y_{*}\right)$. Now $h_n\in {\mathcal{N}}_{\mathcal{F}}\left(y_n\right)$ implies that there exists $v_n\in S_{\mathcal{F},y_n}$ such that for each $t\in [x_1,x_2]$,

$$\begin{array}{ccc}\hfill h_n\left(t\right)& =& -\alpha I^{p}y\left(t\right)-\beta I^{\sigma +p}g(t,y\left(t\right))+I^{p+q}{v}_n\left(t\right)\hfill \\ & & +{\omega }_1\left(t\right)\left[\alpha I^{p}y\left(x_2\right)+\beta I^{\sigma +p}g(x_2,y\left(x_2\right))-I^{p+q}{v}_n\left(x_2\right)\right]\hfill \\ & & +{\omega }_2\left(t\right)\left[\alpha I^{p-1}y\left(x_2\right)+\beta I^{\sigma +p-1}g(x_2,y\left(x_2\right))-I^{p+q-1}{v}_n\left(x_2\right)\right]\hfill \\ & & -{\omega }_3\left(t\right)[\alpha \left(I^{p}y\left(\xi \right)+I^{p}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p}g(\xi ,y\left(\xi \right))+I^{\sigma +p}g(\eta ,y\left(\eta \right))\right)\hfill \\ & & -\left(I^{p+q}{v}_n\left(\xi \right)+I^{p+q}{v}_n\left(\eta \right)\right)]\hfill \\ & & +{\omega }_4\left(t\right)[\alpha \left(I^{p-1}y\left(\xi \right)+I^{p-1}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p-1}g(\xi ,y\left(\xi \right))+I^{\sigma +p-1}g(\eta ,y\left(\eta \right)\right)\hfill \\ & & -\left(I^{p+q-1}{v}_n\left(\xi \right)+I^{p+q-1}{v}_n\left(\eta \right)\right)].\hfill \end{array}$$

Hence it is enough to prove that there exists $v_{*}\in S_{\mathcal{F},y_{*}}$ such that for each $t\in [x_1,x_2]$,

$$\begin{array}{ccc}\hfill h_{*}\left(t\right)& =& -\alpha I^{p}y\left(t\right)-\beta I^{\sigma +p}g(t,y\left(t\right))+I^{p+q}{v}_{*}\left(t\right)\hfill \\ & & +{\omega }_1\left(t\right)\left[\alpha I^{p}y\left(x_2\right)+\beta I^{\sigma +p}g(x_2,y\left(x_2\right))-I^{p+q}{v}_{*}\left(x_2\right)\right]\hfill \\ & & +{\omega }_2\left(t\right)\left[\alpha I^{p-1}y\left(x_2\right)+\beta I^{\sigma +p-1}g(x_2,y\left(x_2\right))-I^{p+q-1}{v}_{*}\left(x_2\right)\right]\hfill \\ & & -{\omega }_3\left(t\right)[\alpha \left(I^{p}y\left(\xi \right)+I^{p}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p}g(\xi ,y\left(\xi \right))+I^{\sigma +p}g(\eta ,y\left(\eta \right))\right)\hfill \\ & & -\left(I^{p+q}{v}_{*}\left(\xi \right)+I^{p+q}{v}_{*}\left(\eta \right)\right)]\hfill \\ & & +{\omega }_4\left(t\right)[\alpha \left(I^{p-1}y\left(\xi \right)+I^{p-1}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p-1}g(\xi ,y\left(\xi \right))+I^{\sigma +p-1}g(\eta ,y\left(\eta \right)\right)\hfill \\ & & -\left(I^{p+q-1}{v}_{*}\left(\xi \right)+I^{p+q-1}{v}_{*}\left(\eta \right)\right)].\hfill \end{array}$$

Consider the linear operator $\mathsf{\Theta }:L^1([x_1,x_2],\mathbb{R})\to C([x_1,x_2],\mathbb{R})$ given by

$$\begin{array}{ccc}\hfill v\to \mathsf{\Theta }\left(v\right)\left(t\right)& =& -\alpha I^{p}y\left(t\right)-\beta I^{\sigma +p}g(t,y\left(t\right))+I^{p+q}v\left(t\right)\hfill \\ & & +{\omega }_1\left(t\right)\left[\alpha I^{p}y\left(x_2\right)+\beta I^{\sigma +p}g(x_2,y\left(x_2\right))-I^{p+q}v\left(x_2\right)\right]\hfill \\ & & +{\omega }_2\left(t\right)\left[\alpha I^{p-1}y\left(x_2\right)+\beta I^{\sigma +p-1}g(x_2,y\left(x_2\right))-I^{p+q-1}v\left(x_2\right)\right]\hfill \\ & & -{\omega }_3\left(t\right)[\alpha \left(I^{p}y\left(\xi \right)+I^{p}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p}g(\xi ,y\left(\xi \right))+I^{\sigma +p}g(\eta ,y\left(\eta \right))\right)\hfill \\ & & -\left(I^{p+q}v\left(\xi \right)+I^{p+q}v\left(\eta \right)\right)]\hfill \\ & & +{\omega }_4\left(t\right)[\alpha \left(I^{p-1}y\left(\xi \right)+I^{p-1}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p-1}g(\xi ,y\left(\xi \right))+I^{\sigma +p-1}g(\eta ,y\left(\eta \right)\right)\hfill \\ & & -\left(I^{p+q-1}v\left(\xi \right)+I^{p+q-1}v\left(\eta \right)\right)].\hfill \end{array}$$

Observe that $\parallel h_n-h_{*}\parallel \to 0$ as $n\to \infty ,$ and thus, it follows from a closed graph lemma [32], that $\mathsf{\Theta }\circ S_{\mathcal{F},y}$ is a closed graph operator. Moreover, we have

$$h_n\in \mathsf{\Theta }\left(S_{\mathcal{F},y_n}\right).$$

Since $y_n\to y_{*}$, the closed graph lemma [32] implies that

$$\begin{array}{ccc}\hfill h_{*}\left(t\right)& =& -\alpha I^{p}y\left(t\right)-\beta I^{\sigma +p}g(t,y\left(t\right))+I^{p+q}{v}_{*}\left(t\right)\hfill \\ & & +{\omega }_1\left(t\right)\left[\alpha I^{p}y\left(x_2\right)+\beta I^{\sigma +p}g(x_2,y\left(x_2\right))-I^{p+q}{v}_{*}\left(x_2\right)\right]\hfill \\ & & +{\omega }_2\left(t\right)\left[\alpha I^{p-1}y\left(x_2\right)+\beta I^{\sigma +p-1}g(x_2,y\left(x_2\right))-I^{p+q-1}{v}_{*}\left(x_2\right)\right]\hfill \\ & & -{\omega }_3\left(t\right)[\alpha \left(I^{p}y\left(\xi \right)+I^{p}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p}g(\xi ,y\left(\xi \right))+I^{\sigma +p}g(\eta ,y\left(\eta \right))\right)\hfill \\ & & -\left(I^{p+q}{v}_{*}\left(\xi \right)+I^{p+q}{v}_{*}\left(\eta \right)\right)]\hfill \\ & & +{\omega }_4\left(t\right)[\alpha \left(I^{p-1}y\left(\xi \right)+I^{p-1}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p-1}g(\xi ,y\left(\xi \right))+I^{\sigma +p-1}g(\eta ,y\left(\eta \right)\right)\hfill \\ & & -\left(I^{p+q-1}{v}_{*}\left(\xi \right)+I^{p+q-1}{v}_{*}\left(\eta \right)\right)],\hfill \end{array}$$

for some $v_{*}\in S_{\mathcal{F},y_{*}}.$

In the final step we demonstrate that there exists an open set $\mathcal{U}\subseteq C([x_1,x_2],\mathbb{R})$ with $y\notin {\mathcal{N}}_{\mathcal{F}}\left(y\right)$ for any $\lambda \in (0,1)$ and all $y\in \partial \mathcal{U}.$ Let $y\in \lambda {\mathcal{N}}_{\mathcal{F}}\left(y\right)$ for some $\lambda \in (0,1).$ Let $\lambda \in (0,1)$ and $y\in \lambda {\mathcal{N}}_{\mathcal{F}}\left(y\right).$ Then there exists $v\in L^1([x_1,x_2],\mathbb{R})$ with $v\in S_{\mathcal{F},y}$ such that, for $t\in [x_1,x_2]$, we have

$$\begin{array}{ccc}\hfill y\left(t\right)& =& -\lambda \alpha I^{p}y\left(t\right)-\lambda \beta I^{\sigma +p}g(t,y\left(t\right))+\lambda I^{p+q}v\left(t\right)\hfill \\ & & +\lambda {\omega }_1\left(t\right)\left[\alpha I^{p}y\left(x_2\right)+\beta I^{\sigma +p}g(x_2,y\left(x_2\right))-I^{p+q}v\left(x_2\right)\right]\hfill \\ & & +\lambda {\omega }_2\left(t\right)\left[\alpha I^{p-1}y\left(x_2\right)+\beta I^{\sigma +p-1}g(x_2,y\left(x_2\right))-I^{p+q-1}v\left(x_2\right)\right]\hfill \\ & & -\lambda {\omega }_3\left(t\right)[\alpha \left(I^{p}y\left(\xi \right)+I^{p}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p}g(\xi ,y\left(\xi \right))+I^{\sigma +p}g(\eta ,y\left(\eta \right))\right)\hfill \\ & & -\left(I^{p+q}v\left(\xi \right)+I^{p+q}v\left(\eta \right)\right)]\hfill \\ & & +\lambda {\omega }_4\left(t\right)[\alpha \left(I^{p-1}y\left(\xi \right)+I^{p-1}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p-1}g(\xi ,y\left(\xi \right))+I^{\sigma +p-1}g(\eta ,y\left(\eta \right)\right)\hfill \\ & & -\left(I^{p+q-1}v\left(\xi \right)+I^{p+q-1}v\left(\eta \right)\right)].\hfill \end{array}$$

Following the computation above, when proving that ${\mathcal{N}}_{\mathcal{F}}$ maps bounded sets into bounded sets, we have for each $t\in [x_1,x_2],$

$$\begin{array}{c}\hfill \left|y\left(t\right)\right|\le \left|\alpha \right|{\mathsf{\Phi }}_0\parallel y\parallel +\left|\beta \right|{\mathsf{\Phi }}_1\parallel \theta \parallel +{\mathsf{\Phi }}_2\parallel p\parallel \psi (\parallel y\parallel ),\end{array}$$

or

$$(1-|\alpha |{\mathsf{\Phi }}_0)\parallel y\parallel \le \left|\beta \right|{\mathsf{\Phi }}_1\parallel \theta \parallel +{\mathsf{\Phi }}_2\parallel p\parallel \psi (\parallel y\parallel ).$$

Consequently, we have

$${\displaystyle \frac{(1-|\alpha \left|{\mathsf{\Phi }}_0\right)\parallel y\parallel }{\left|\beta \right|{\mathsf{\Phi }}_1\parallel \theta \parallel +{\mathsf{\Phi }}_2\parallel p\parallel \psi (\parallel y\parallel )}\le 1.}$$

By $\left({\mathcal{H}}_3\right)$, there exists $\mathcal{K}$ such that $\parallel y\parallel \ne \mathcal{K}$. Let us set

$$\mathcal{U}=\{y\in C([x_1,x_2],\mathbb{R}):\parallel y\parallel <\mathcal{K}\}.$$

Notice that the operator ${\mathcal{N}}_{\mathcal{F}}:\overline{\mathcal{U}}\to \mathcal{P}\left(C\right([x_1,x_2],$$\mathbb{R}\left)\right)$ is a compact, upper semicontunuous multi-valued map with convex closed values. There is no $y\in \partial \mathcal{U}$ such that $y\in \lambda {\mathcal{N}}_{\mathcal{F}}\left(y\right)$ for some $\lambda \in (0,1)$, by the choice of $\mathcal{U}.$ Consequently, we deduce that ${\mathcal{N}}_{\mathcal{F}}$ has a fixed point $y\in \overline{\mathcal{U}},$ by the nonlinear alternative of Leray–Schauder type (Lemma 2), which is a solution of the boundary value problem (2)–(3). The proof is completed. □

Theorem 2.

Assume that $\left({\mathcal{H}}_1\right)$ and $\left({\mathcal{H}}_2\right)$ hold. In addition, we suppose that:

${\left({\mathcal{H}}_3\right)}^{*}$

There exists a positive constant L such that

$$|g(t,y)-g(t,\overline{y})|\le L|y-\overline{y}|,foreacht\in [x_1,x_2]andy,\overline{y}\in \mathbb{R};$$

${\left({\mathcal{H}}_4\right)}^{*}$

there exists a constant ${\mathcal{K}}_1>0$ such that

$${\displaystyle \frac{(1-|\alpha |{\mathsf{\Phi }}_0-\left|\beta \right|L{\mathsf{\Phi }}_1){\mathcal{K}}_1}{\left|\beta \right|{\mathsf{\Phi }}_1{L}_0+{\mathsf{\Phi }}_2\parallel p\parallel \psi \left({\mathcal{K}}_1\right)}>1,\left|\alpha \right|{\mathsf{\Phi }}_0+\left|\beta \right|L{\mathsf{\Phi }}_1<1,}$$

where $L_0={sup}_{t\in [x_1,x_2]}\left|g(t,0)\right|.$

Then there exists at least one solution on $[x_1,x_2]$ for the boundary value problem (2)–(3).

Proof. 

Note that ${\left({\mathcal{H}}_3\right)}^{*}$ implies

$$\left|g(t,y)\right|\le |g(t,y)-g(t,0)|+\left|g(t,0)\right|\le L\parallel y\parallel +L_0.$$

The rest of the proof is similar to that of Theorem 1 and is omitted. □

3.2. The Lipschitz Case

An existence result for the boundary value problem (2)–(3) is proved in this subsection, in the case when $\mathcal{F}$ has nonconvex values, by applying a fixed point theorem for multivalued contractive maps due to Covitz and Nadler [33].

Let $(X,d)$ be a metric space induced from the normed space $(X;\parallel ·\parallel )$. Consider $H_d:\mathcal{P}\left(X\right)\times \mathcal{P}\left(X\right)\to \mathbb{R}\cup \{\infty \}$ given by

$$H_d(A,B)=max\{\underset{a\in A}{sup}d(a,B),\underset{b\in B}{sup}d(A,b)\},$$

where $d(A,b)={inf}_{a\in A}d(a;b)$ and $d(a,B)={inf}_{b\in B}d(a;b)$. Then $({\mathcal{P}}_{b,cl}\left(X\right),$$H_d)$ is a metric space (see [34]).

Definition 4.

A multivalued operator $N:X\to {\mathcal{P}}_{cl}\left(X\right)$ is called

$\left(\mathit{a}\right)$

$\theta –$Lipschitz if and only if there exists $\theta >0$ such that

$$H_d(N\left(x\right),N\left(y\right))\le \theta d(x,y)foreachx,y\in X;$$

$\left(\mathit{b}\right)$

a contraction if and only if it is $\theta –$Lipschitz with $\theta <1$.

In the next lemma we denote by $FixN$ the fixed point set of the multivalued operator $N.$

Lemma 3

(Covitz and Nadler [33]). Let $(X,d)$ be a complete metric space. If $N:X\to {\mathcal{P}}_{cl}\left(X\right)$ is a contraction, then $FixN\ne \varnothing$.

Theorem 3.

Assume that ${\left({\mathcal{H}}_3\right)}^{*}$ and the following conditions hold:

$\left({\mathcal{C}}_1\right)$

$\mathcal{F}:[x_1,x_2]\times \mathbb{R}\to {\mathcal{P}}_{cp}\left(\mathbb{R}\right)$ is such that $\mathcal{F}(·,y):[x_1,x_2]\to {\mathcal{P}}_{cp}\left(\mathbb{R}\right)$ is measurable for each $y\in \mathbb{R}$.

$\left({\mathcal{C}}_2\right)$

$H_d(\mathcal{F}(t,y),\mathcal{F}(t,\overline{y}))\le m\left(t\right)|y-\overline{y}|$ for almost all $t\in [x_1,x_2]$ and $y,\overline{y}\in \mathbb{R}$ with $m\in C([x_1,x_2],{\mathbb{R}}^{+})$ and $d(0,\mathcal{F}(t,0\left)\right)\le m\left(t\right)$ for almost all $t\in [x_1,x_2]$.

Then the boundary value problem (2)–(3) has at least one solution on $[x_1,x_2]$ if

$$\left|\alpha \right|{\mathsf{\Phi }}_0+\left|\beta \right|{\mathsf{\Phi }}_{1}L+{\mathsf{\Phi }}_2\parallel m\parallel <1.$$

Proof. 

Consider the operator ${\mathcal{N}}_{\mathcal{F}}:C([x_1,x_2],\mathbb{R})\to \mathcal{P}\left(C([x_1,x_2],\mathbb{R})\right)$ defined in Theorem 1 at the beginning of the proof. We show that the operator ${\mathcal{N}}_{\mathcal{F}}$ fulfills the assumptions of Lemma 3. Note that since the set-valued map $\mathcal{F}(·,y(·\left)\right)$ is measurable by the measurable selection theorem (e.g., (Theorem III.6 [35])) and it admits a measurable selection $v:[x_1,x_2]\to \mathbb{R}$. Moreover, by the assumption $\left({\mathcal{C}}_2\right),$ we have

$$\left|v\right(t\left)\right|\le m\left(t\right)+m\left(t\right)\left|y\right(t\left)\right|,$$

i.e., $v\in L^1([x_1,x_2],\mathbb{R})$ and hence $\mathcal{F}$ is integrably bounded. Therefore, $S_{\mathcal{F},y}\ne \varnothing$. Moreover ${\mathcal{N}}_{\mathcal{F}}\left(y\right)\in {\mathcal{P}}_{cl}\left(C\right([x_1,x_2],$$\mathbb{R}\left)\right)$ for each $y\in C([x_1,x_2],\mathbb{R})$.

Let ${\left\{u_n\right\}}_{n\ge 0}\in {\mathcal{N}}_{\mathcal{F}}\left(y\right)$ be such that $u_n\to u(n\to \infty )$ in $C([x_1,x_2],\mathbb{R}).$ Then $u\in C([x_1,x_2],\mathbb{R})$ and there exists $v_n\in S_{\mathcal{F},y_n}$ such that, for each $t\in [x_1,x_2]$,

$$\begin{array}{ccc}\hfill u_n\left(t\right)& =& -\alpha I^{p}y\left(t\right)-\beta I^{\sigma +p}g(t,y\left(t\right))+I^{p+q}{v}_n\left(t\right)\hfill \\ & & +{\omega }_1\left(t\right)\left[\alpha I^{p}y\left(x_2\right)+\beta I^{\sigma +p}g(x_2,y\left(x_2\right))-I^{p+q}{v}_n\left(x_2\right)\right]\hfill \\ & & +{\omega }_2\left(t\right)\left[\alpha I^{p-1}y\left(x_2\right)+\beta I^{\sigma +p-1}g(x_2,y\left(x_2\right))-I^{p+q-1}{v}_n\left(x_2\right)\right]\hfill \\ & & -{\omega }_3\left(t\right)[\alpha \left(I^{p}y\left(\xi \right)+I^{p}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p}g(\xi ,y\left(\xi \right))+I^{\sigma +p}g(\eta ,y\left(\eta \right))\right)\hfill \\ & & -\left(I^{p+q}{v}_n\left(\xi \right)+I^{p+q}{v}_n\left(\eta \right)\right)]\hfill \\ & & +{\omega }_4\left(t\right)[\alpha \left(I^{p-1}y\left(\xi \right)+I^{p-1}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p-1}g(\xi ,y\left(\xi \right))+I^{\sigma +p-1}g(\eta ,y\left(\eta \right)\right)\hfill \\ & & -\left(I^{p+q-1}{v}_n\left(\xi \right)+I^{p+q-1}{v}_n\left(\eta \right)\right)].\hfill \end{array}$$

As $\mathcal{F}$ has compact values, we pass onto a subsequence (if necessary) to obtain that $v_n$ converges to v in $L^1([x_1,x_2],\mathbb{R}).$ Thus $v\in S_{\mathcal{F},y}$ and for each $t\in [x_1,x_2]$, we have

$$\begin{array}{ccc}\hfill u_n\left(t\right)\to u\left(t\right)& =& -\alpha I^{p}y\left(t\right)-\beta I^{\sigma +p}g(t,y\left(t\right))+I^{p+q}v\left(t\right)\hfill \\ & & +{\omega }_1\left(t\right)\left[\alpha I^{p}y\left(x_2\right)+\beta I^{\sigma +p}g(x_2,y\left(x_2\right))-I^{p+q}v\left(x_2\right)\right]\hfill \\ & & +{\omega }_2\left(t\right)\left[\alpha I^{p-1}y\left(x_2\right)+\beta I^{\sigma +p-1}g(x_2,y\left(x_2\right))-I^{p+q-1}v\left(x_2\right)\right]\hfill \\ & & -{\omega }_3\left(t\right)[\alpha \left(I^{p}y\left(\xi \right)+I^{p}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p}g(\xi ,y\left(\xi \right))+I^{\sigma +p}g(\eta ,y\left(\eta \right))\right)\hfill \\ & & -\left(I^{p+q}v\left(\xi \right)+I^{p+q}v\left(\eta \right)\right)]\hfill \\ & & +{\omega }_4\left(t\right)[\alpha \left(I^{p-1}y\left(\xi \right)+I^{p-1}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p-1}g(\xi ,y\left(\xi \right))+I^{\sigma +p-1}g(\eta ,y\left(\eta \right)\right)\hfill \\ & & -\left(I^{p+q-1}v\left(\xi \right)+I^{p+q-1}v\left(\eta \right)\right)].\hfill \end{array}$$

Hence, $u\in {\mathcal{N}}_{\mathcal{F}}\left(y\right).$

Next we show that there exists $0<\delta <1$ such that

$$H_d({\mathcal{N}}_{\mathcal{F}}\left(y\right),{\mathcal{N}}_{\mathcal{F}}\left(\overline{y}\right))\le \delta \parallel y-\overline{y}\parallel \mathrm{for}\mathrm{each}y,\overline{y}\in AC([x_1,x_2],\mathbb{R}).$$

Let $y,\overline{y}\in AC([x_1,x_2],\mathbb{R})$ and $h_1\in {\mathcal{N}}_{\mathcal{F}}\left(y\right)$. Then there exists $v_1\left(t\right)\in \mathcal{F}(t,y\left(t\right))$ such that, for each $t\in [x_1,x_2]$,

$$\begin{array}{ccc}\hfill h_1\left(t\right)& =& -\alpha I^{p}y\left(t\right)-\beta I^{\sigma +p}g(t,y\left(t\right))+I^{p+q}{v}_1\left(t\right)\hfill \\ & & +{\omega }_1\left(t\right)\left[\alpha I^{p}y\left(x_2\right)+\beta I^{\sigma +p}g(x_2,y\left(x_2\right))-I^{p+q}{v}_1\left(x_2\right)\right]\hfill \\ & & +{\omega }_2\left(t\right)\left[\alpha I^{p-1}y\left(x_2\right)+\beta I^{\sigma +p-1}g(x_2,y\left(x_2\right))-I^{p+q-1}{v}_1\left(x_2\right)\right]\hfill \\ & & -{\omega }_3\left(t\right)[\alpha \left(I^{p}y\left(\xi \right)+I^{p}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p}g(\xi ,y\left(\xi \right))+I^{\sigma +p}g(\eta ,y\left(\eta \right))\right)\hfill \\ & & -\left(I^{p+q}{v}_1\left(\xi \right)+I^{p+q}{v}_1\left(\eta \right)\right)]\hfill \\ & & +{\omega }_4\left(t\right)[\alpha \left(I^{p-1}y\left(\xi \right)+I^{p-1}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p-1}g(\xi ,y\left(\xi \right))+I^{\sigma +p-1}g(\eta ,y\left(\eta \right)\right)\hfill \\ & & -\left(I^{p+q-1}{v}_1\left(\xi \right)+I^{p+q-1}{v}_1\left(\eta \right)\right)].\hfill \end{array}$$

By $\left({\mathcal{C}}_2\right)$, we have

$$H_d(\mathcal{F}(t,y),\mathcal{F}(t,\overline{y}))\le m\left(t\right)|y\left(t\right)-\overline{y}\left(t\right)|.$$

Therefore there exists $w\in \mathcal{F}(t,\overline{y}\left(t\right))$ such that

$$|v_1\left(t\right)-w|\le m\left(t\right)|y\left(t\right)-\overline{y}\left(t\right)|,t\in [x_1,x_2].$$

Define $U:[x_1,x_2]\to \mathcal{P}\left(\mathbb{R}\right)$ by

$$U\left(t\right)=\{w\in \mathbb{R}:|v_1\left(t\right)-w|\le m\left(t\right)|y\left(t\right)-\overline{y}\left(t\right)\left|\right\}.$$

Since $U\left(t\right)\cap \mathcal{F}(t,\overline{y}\left(t\right))$ is measurable (Proposition III.4 [35]), there exists a function $v_2\left(t\right)$ which is a measurable selection for U. Hence $v_2\left(t\right)\in \mathcal{F}(t,\overline{y}\left(t\right))$ and for each $t\in [x_1,x_2]$, we have $|v_1\left(t\right)-v_2\left(t\right)|\le m\left(t\right)|y\left(t\right)-\overline{y}\left(t\right)|$.

For each $t\in [x_1,x_2]$, let us define

$$\begin{array}{ccc}\hfill h_2\left(t\right)& =& -\alpha I^{p}y\left(t\right)-\beta I^{\sigma +p}g(t,y\left(t\right))+I^{p+q}{v}_2\left(t\right)\hfill \\ & & +{\omega }_1\left(t\right)\left[\alpha I^{p}y\left(x_2\right)+\beta I^{\sigma +p}g(x_2,y\left(x_2\right))-I^{p+q}{v}_2\left(x_2\right)\right]\hfill \\ & & +{\omega }_2\left(t\right)\left[\alpha I^{p-1}y\left(x_2\right)+\beta I^{\sigma +p-1}g(x_2,y\left(x_2\right))-I^{p+q-1}{v}_2\left(x_2\right)\right]\hfill \\ & & -{\omega }_3\left(t\right)[\alpha \left(I^{p}y\left(\xi \right)+I^{p}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p}g(\xi ,y\left(\xi \right))+I^{\sigma +p}g(\eta ,y\left(\eta \right))\right)\hfill \\ & & -\left(I^{p+q}{v}_2\left(\xi \right)+I^{p+q}{v}_2\left(\eta \right)\right)]\hfill \\ & & +{\omega }_4\left(t\right)[\alpha \left(I^{p-1}y\left(\xi \right)+I^{p-1}y\left(\eta \right)\right)+\beta \left(I^{\sigma +p-1}g(\xi ,y\left(\xi \right))+I^{\sigma +p-1}g(\eta ,y\left(\eta \right)\right)\hfill \\ & & -\left(I^{p+q-1}{v}_2\left(\xi \right)+I^{p+q-1}{v}_2\left(\eta \right)\right)].\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}& & \left|h_1\left(t\right)-h_2\left(t\right)\right|\hfill \\ & \le & \left|\alpha \right|I^p|y\left(t\right)-\overline{y}\left(t\right)|+|\beta |I^{\sigma +p}|g(t,y\left(t\right))-g(t,\overline{y}\left(t\right))|+I^{p+q}|v_2\left(t\right)-v_1\left(t\right)|\hfill \\ & & +{\tilde{\omega }}_1\left[\left|\alpha \right|I^p|y\left(x_2\right)-\overline{y}\left(x_2\right)|+|\beta |I^{\sigma +p}|g(x_2,y\left(x_2\right))-g(x_2,\overline{y}\left(x_2\right))|+I^{p+q}|v_2\left(x_2\right)-v_1\left(x_2\right)|\right]\hfill \\ & & +{\tilde{\omega }}_2\left[\left|\alpha \right|I^{p-1}|y\left(x_2\right)-\overline{y}\left(x_2\right)|+|\beta |I^{\sigma +p-1}|g(x_2,y\left(x_2\right))-g(x_2,\overline{y}\left(x_2\right))|+I^{p+q-1}|v_2\left(x_2\right)-v_1\left(x_2\right)|\right]\hfill \\ & & +{\tilde{\omega }}_3\left[\right|\alpha |\left(I^p|y\left(\xi \right)-\overline{y}\left(\xi \right)|+I^p|y\left(\eta \right)-\overline{y}\left(\eta \right)|\right)+\left|\beta \right|(I^{\sigma +p}|g(\xi ,y\left(\xi \right))-g(\xi ,\overline{y}\left(\xi \right))|\hfill \\ & & +I^{\sigma +p}|g(\eta ,y\left(\eta \right))-g(\eta ,\overline{y}\left(\eta \right))|)+\left(I^{p+q}|v_2\left(\xi \right)-v_1\left(\xi \right)|+I^{p+q}|v_2\left(\eta \right)-v_1\left(\eta \right)|\right)]\hfill \\ & & +{\tilde{\omega }}_4[\left|\alpha \right|\left(I^{p-1}|y\left(\xi \right)-\overline{y}\left(\xi \right)|+I^{p-1}|y\left(\eta \right)-\overline{y}\left(\eta \right)|\left(\eta \right)\right)\hfill \\ & & +\left|\beta \right|\left(I^{\sigma +p-1}|g(\xi ,y\left(\xi \right))-g(\xi ,\overline{y}\left(\xi \right))|+I^{\sigma +p-1}|g(\eta ,y\left(\eta \right))-g(\eta ,\overline{y}\left(\eta \right))|\right)\hfill \\ & & +\left(I^{p+q-1}|v_2\left(\xi \right)-v_1\left(\xi \right)|+I^{p+q-1}|v_2\left(\eta \right)-v_1\left(\eta \right)|\right)]\hfill \\ & \le & \left(\right|\alpha |{\mathsf{\Phi }}_0+\left|\beta \right|{\mathsf{\Phi }}_{1}L+{\mathsf{\Phi }}_2\parallel m\parallel )\parallel y-\overline{y}\parallel .\hfill \end{array}$$

Hence

$$\parallel h_1-h_2\parallel \le \left(\right|\alpha |{\mathsf{\Phi }}_0+\left|\beta \right|{\mathsf{\Phi }}_{1}L+{\mathsf{\Phi }}_2\parallel m\parallel )\parallel y-\overline{y}\parallel .$$

Interchanging the roles of y and $\overline{y}$, we obtain

$$H_d({\mathcal{N}}_{\mathcal{F}}\left(y\right),{\mathcal{N}}_{\mathcal{F}}\left(\overline{y}\right))\le \left(\right|\alpha |{\mathsf{\Phi }}_0+\left|\beta \right|{\mathsf{\Phi }}_{1}L+{\mathsf{\Phi }}_2\parallel m\parallel )\parallel y-\overline{y}\parallel .$$

Since ${\mathcal{N}}_{\mathcal{F}}$ is a contraction, it follows by Lemma 3 that ${\mathcal{N}}_{\mathcal{F}}$ has a fixed point x which is a solution of (2)–(3). This completes the proof. □

4. Examples

Consider the following boundary value problem:

$$\left\{\begin{array}{c}\hfill {}^c{D}^{\frac{7}{5}}\left[\left({}^c{D}^{\frac{3}{2}}+\frac{9}{166}\right)y\left(t\right)+\frac{8}{25}{I}^{\frac{9}{5}}g(t,y\left(t\right))\right]\in \mathcal{F}(t,y\left(t\right)),t\in [0,1],\\ \\ \hfill y\left(0\right)+y\left(1\right)=0,y^{\prime }\left(0\right)+y^{\prime }\left(1\right)=0,y\left(\xi \right)+y\left(\eta \right)=0,y^{\prime }\left(\xi \right)+y^{\prime }\left(\eta \right)=0,\end{array}\right.$$

(11)

where $g(t,y\left(t\right))=\frac{t^2+2}{800}\left(siny\left(t\right)+\frac{y^2\left(t\right)}{2+y^2\left(t\right)}\right).$ As defined in the problem (2)–(3), we take $\mathcal{F}(t,y\left(t\right))=\left[e^{-t}(\frac{y}{2\pi }{tan}^{-1}y+\frac{1}{4}),\frac{1}{20}{t}^{2}siny\right],$$x_1=0,x_2=1,q=\frac{7}{5},p=\frac{3}{2},\sigma =\frac{9}{5},\alpha =\frac{9}{166},\beta =\frac{8}{25},\xi =\frac{1}{2},\eta =\frac{2}{3}.$ Using the given data, we find that ${\gamma }_1\simeq 0.752253,{\gamma }_2\simeq 0.300902,{\gamma }_3=1,{\gamma }_4\simeq 1.12838,{\gamma }_5\simeq 0.675433,{\gamma }_6\simeq 0.162385,{\gamma }_7\simeq 1.16667,{\gamma }_8\simeq 1.71920,{\delta }_1\simeq 1.22029,{\delta }_2\simeq -0.95628,{\nu }_1\simeq 0.581948,{\nu }_2\simeq -1.47573,{\nu }_3\simeq 1.52423,{\nu }_4\simeq 4.47574,{\nu }_5\simeq 1.66729,{\nu }_6\simeq -1.29431,{\nu }_7\simeq -2.01176,{\nu }_8\simeq 0.705473,{\nu }_9\simeq -0.373137,{\rho }_1\simeq 0.613606,{\rho }_2\simeq -0.048519,{\rho }_3\simeq -9.06169,{\rho }_4\simeq -0.192012,{\tilde{\omega }}_1\simeq 0.613606,{\tilde{\omega }}_2\simeq 0.048523,{\tilde{\omega }}_3\simeq 9.06169,{\tilde{\omega }}_4\simeq 0.192012,{\mathsf{\Phi }}_0\simeq 7.71932,{\mathsf{\Phi }}_1\simeq 0.615362,{\mathsf{\Phi }}_2\simeq 1.16454,\theta \left(t\right)=\frac{t^2+2}{400}$, $p\left(t\right)=\frac{e^{-t}}{4}$ and $\psi (\parallel y\parallel )=(1+\parallel y\parallel )$. Moreover, $\left|\alpha \right|{\mathsf{\Phi }}_0\simeq 0.418517<1$. Using the condition $\left({\mathcal{H}}_4\right)$, we find that $\mathcal{K}>1.007797.$ Clearly the hypothesis of Theorem 1 is satisfied. Therefore, there exists at least one solution for the problem (11) on $[0,1]$.

In order to illustrate Theorem 3, we take

$$g(t,y)=\frac{e^{-t}}{\sqrt{t^2+49}}siny+cost,$$

(12)

and as defined in the problem (2)–(3), we choose

$$\mathcal{F}(t,y\left(t\right))=\left[\frac{1+sin\left|y\right|}{{(17+t)}^2},\left(\frac{t^2+1}{45}\right)\left(\frac{1+3\left|y\right|}{1+2\left|y\right|}\right)\right].$$

(13)

Clearly $\mathcal{F}$ is measurable for all $y\in \mathbb{R}$ and that

$$H_d(\mathcal{F}(t,y),\mathcal{F}(t,\overline{y}))\le \left(\frac{t^2+1}{45}\right)|y-\overline{y}|,y,\overline{y}\in \mathbb{R},t\in [0,1].$$

Letting $m\left(t\right)=(t^2+1)/45$, we have $\parallel m\parallel =2/45$ and a $d(0,\mathcal{F}(t,0\left)\right)\le m\left(t\right)$, $t\in [0,1]$. Further, $L=1/7$ as $|g(t,y)-g(t,\overline{y})|\le \frac{1}{7}|y-\overline{y}|$ and $\left|\alpha \right|{\mathsf{\Phi }}_0+\left|\beta \right|{\mathsf{\Phi }}_{1}L+{\mathsf{\Phi }}_2\parallel m\parallel \approx 0.49845<1.$ Thus all the assumptions of Theorem 3 hold true and consequently its conclusion applies to the problem (11) with the values of g and $\mathcal{F}$ given by (12) and (13).

5. Conclusions

We have discussed the existence of solutions for fractional differential inclusions involving two Caputo fractional derivatives of different orders and a Riemann–Liouville type integral nonlinearity, equipped with a new class of anti-periodic boundary conditions. In case of convex-valued case, we make use of the Leray–Schauder nonlinear alternative for multivalued maps to derive the existence result for the problem at hand, while the case of non-convex multi-valued map relies on Covitz and Nadler fixed point theorem for contractive maps. In the given configuration, our results are new and contribute to the existing literature on the topic. Moreover, by taking $\beta =0$ in the obtained results, we get the ones for the following multi-valued problem:

$$\begin{array}{ccc}& & {}^c{D}^q{(}^c{D}^p+\alpha )y\left(t\right))\in \mathcal{F}(t,y\left(t\right)),1<p,q\le 2,t\in [x_1,x_2],\hfill \\ & & y\left(x_1\right)+y\left(x_2\right)=0,y^{\prime }\left(x_1\right)+y^{\prime }\left(x_2\right)=0,y\left(\xi \right)+y\left(\eta \right)=0,y^{\prime }\left(\xi \right)+y^{\prime }\left(\eta \right)=0,\hfill \end{array}$$

which are indeed new.