# On a fractional differential inclusion via a new integral boundary condition

## Abstract

In this paper, we investigate the existence of solution for two systems of fractional differential inclusions via some integral boundary value conditions. For this purpose, we use an endpoint result for multifunctions which has been proved in 2010 by Amini-Harandi (Nonlinear Anal. 72:132-134, 2010). Finally, we give an example for illustrating one of our results.

## 1 Introduction

As we know, diverse classes of fractional differential equations have been studied by researchers (see for example,  and the references therein). Much attention has been devoted to the fractional differential inclusions (see for example,  and the references therein). Also, there have been provided many applications of this field (see for example, [33, 34] and ).

It is the aim of this paper to investigate the existence of solutions for two systems of fractional differential inclusions, subject to some integral boundary value conditions. In this respect, we use an endpoint result for multifunctions due to Amini-Harandi, . We provide an example for illustrating one of our results.

## 2 Preliminaries

As is well known, the Riemann-Liouville fractional integral of order $\alpha >0$ of a function $f:\left(0,\mathrm{\infty }\right)\to \mathbb{R}$ is given by ${I}^{\alpha }f\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}f\left(s\right)\phantom{\rule{0.2em}{0ex}}ds$, provided the right side is pointwise defined on $\left(0,\mathrm{\infty }\right)$ (see [10, 13] and ). The Caputo fractional derivative of order α for a continuous function f is defined by ${}^{c}D^{\alpha }f\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(n-\alpha \right)}{\int }_{0}^{t}\frac{{f}^{\left(n\right)}\left(s\right)}{{\left(t-s\right)}^{\alpha -n+1}}\phantom{\rule{0.2em}{0ex}}ds$, where $n=\left[\alpha \right]+1$ (see [10, 13] and ).

Recall that a multifunction $G:J\to {P}_{cl}\left(\mathbb{R}\right)$ is said to be measurable whenever the function $t↦d\left(y,G\left(t\right)\right)$ is measurable for all $y\in \mathbb{R}$, where $J=\left[0,1\right]$ .

Let $\left(X,d\right)$ be a metric space. We have the well-known Pompeiu-Hausdorff metric (see )

${H}_{d}:{2}^{X}×{2}^{X}\to \left[0,\mathrm{\infty }\right),\phantom{\rule{1em}{0ex}}{H}_{d}\left(A,B\right)=max\left\{\underset{a\in A}{sup}d\left(a,B\right),\underset{b\in B}{sup}d\left(A,b\right)\right\},$

where $d\left(A,b\right)={inf}_{a\in A}d\left(a,b\right)$. Then $\left(CB\left(X\right),{H}_{d}\right)$ is a metric space and $\left(C\left(X\right),{H}_{d}\right)$ is a generalized metric space, where $CB\left(X\right)$ is the set of closed and bounded subsets of X and $C\left(X\right)$ is the set of closed subsets of X (see ).

Let $T:X\to {2}^{X}$ be a multifunction. An element $x\in X$ is called an endpoint of T whenever $Tx=\left\{x\right\}$ . Also, we say that T has the approximate endpoint property whenever ${inf}_{x\in X}{sup}_{y\in Tx}d\left(x,y\right)=0$ . A function $g:\mathbb{R}\to \mathbb{R}$ is called upper semi-continuous whenever ${lim sup}_{n\to \mathrm{\infty }}g\left({\lambda }_{n}\right)\le g\left(\lambda \right)$ for all sequences ${\left\{{\lambda }_{n}\right\}}_{n\ge 1}$ with ${\lambda }_{n}\to \lambda$ .

In 2010, Amini-Harandi proved the next result .

Lemma 2.1 Let $\psi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ be an upper semi-continuous function such that $\psi \left(t\right) and ${lim inf}_{t\to \mathrm{\infty }}\left(t-\psi \left(t\right)\right)>0$, for all $t>0$, $\left(X,d\right)$ a complete metric space and $T:X\to CB\left(X\right)$ a multifunction such that ${H}_{d}\left(Tx,Ty\right)\le \psi \left(d\left(x,y\right)\right)$ for all $x,y\in X$. Then T has a unique endpoint if and only if T has approximate end point property.

In 2011, Ahmad et al. investigated the fractional inclusion problem ${}^{c}D^{\alpha }x\left(t\right)\in F\left(t,x\left(t\right)\right)$, via the integral boundary conditions ${x}^{j}\left(0\right)-{\lambda }_{j}{x}^{j}\left(T\right)={\mu }_{j}{\int }_{0}^{1}{g}_{j}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds$ for $j=0,1,2$, where F is a multifunction (see for more details ).

In this paper, we are going to extend the problem in a sense. In this respect, we first investigate the existence of solution for the fractional differential inclusion problem

${}^{c}D^{\alpha }x\left(t\right)\in F\left(t,x\left(t\right),{x}^{\prime }\left(t\right),{x}^{″}\left(t\right)\right),$
(1)

via integral boundary value conditions

$\left\{\begin{array}{l}x\left(0\right)+x\left(\eta \right)+x\left(1\right)={\int }_{0}^{1}{g}_{0}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds,\\ {}^{c}D^{\beta }x\left(0\right)+{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{\beta }x\left(\eta \right)+{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{\beta }x\left(1\right)={\int }_{0}^{1}{g}_{1}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds,\\ {}^{c}D^{\gamma }x\left(0\right)+{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{\gamma }x\left(\eta \right)+{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{\gamma }x\left(1\right)={\int }_{0}^{1}{g}_{2}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds,\end{array}$
(2)

where $t\in J$, $2<\alpha \le 3$, $0<\eta ,\beta <1$, $1<\gamma <2$, and $F:J×\mathbb{R}×\mathbb{R}×\mathbb{R}\to {P}_{cp}\left(\mathbb{R}\right)$ is a multifunction, ${g}_{1},{g}_{2},{g}_{3}:J×\mathbb{R}\to \mathbb{R}$ are continuous functions and ${}^{c}D^{q}$ is the standard Caputo differentiation. Here, ${P}_{cp}\left(\mathbb{R}\right)$ is the set of all compact subsets of .

Also, we investigate the existence of solution for the fractional differential inclusion problem

${}^{c}D^{\alpha }x\left(t\right)\in F\left(t,x\left(t\right),{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{1}}x\left(t\right),\dots ,{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{n}}x\left(t\right)\right),$
(3)

via integral boundary value conditions

$\left\{\begin{array}{l}{x}^{\prime }\left(0\right)+b{x}^{\prime }\left(1\right)={\sum }_{i=1}^{n}{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{i}}x\left(\eta \right),\\ x\left(0\right)+ax\left(1\right)={\sum }_{i=1}^{n}{I}^{{\gamma }_{i}}x\left(\eta \right),\end{array}$
(4)

where $t\in J=\left[0,1\right]$, $1<\alpha \le 2$, $0<\eta ,{\gamma }_{i}<1$, $\alpha -{\gamma }_{i}\ge 1$ for all $1\le i\le n$, $a>{\sum }_{i=1}^{n}\frac{{\eta }^{{\gamma }_{i}+1}}{\mathrm{\Gamma }\left({\gamma }_{i}+2\right)}$, $b>{\sum }_{i=1}^{n}\frac{{\eta }^{1-{\gamma }_{i}}}{\mathrm{\Gamma }\left(2-{\gamma }_{i}\right)}$, $n\ge 1$, and $F:J×{\mathbb{R}}^{n+1}\to P\left(\mathbb{R}\right)$ is a multifunction.

## 3 Main results

Now, we are ready to state and prove our main results. First, we give the following one.

Lemma 3.1 Let $v\in C\left(J,\mathbb{R}\right)$, $\alpha \in \left(2,3\right]$, $\beta \in \left(0,1\right)$, $\gamma \in \left(1,2\right)$ and ${g}_{0},{g}_{1},{g}_{2}:J×\mathbb{R}\to \mathbb{R}$ be continuous functions. The unique solution of the fractional differential problem

${}^{c}D^{\alpha }x\left(t\right)=v\left(t\right)$
(5)

via the boundary value conditions (2) is given by

$\begin{array}{rl}x\left(t\right)=& \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\left(\alpha -1\right)}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+\frac{1}{3}{\int }_{0}^{1}{g}_{0}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{1}{3\mathrm{\Gamma }\left(\alpha \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]\\ +\frac{3\mathrm{\Gamma }\left(2-\beta \right)t-\left(\eta +1\right)\mathrm{\Gamma }\left(2-\beta \right)}{3\left({\eta }^{1-\beta }+1\right)}{\int }_{0}^{1}{g}_{1}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\left(\eta +1\right)\mathrm{\Gamma }\left(2-\beta \right)-3\mathrm{\Gamma }\left(2-\beta \right)t}{3\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(\alpha -\beta \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\beta -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\beta -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]\\ +\left(\frac{2\left(\eta +1\right)\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)-\left({\eta }^{2}+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}\\ +\frac{-6\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)t+3\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(3-\beta \right){t}^{2}}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}\right)\\ ×{\int }_{0}^{1}{g}_{2}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\left(\frac{\left({\eta }^{2}+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\beta \right)-2\left(\eta +1\right)\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\\ +\frac{6\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)t-3\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(3-\beta \right)\left({\eta }^{1-\beta }+1\right){t}^{2}}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\right)\\ ×\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\gamma -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\gamma -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right].\end{array}$

Proof It is known that the general solution of (5) is

$x\left(t\right)={I}^{\alpha }v\left(t\right)+{c}_{0}+{c}_{1}t+{c}_{2}{t}^{2},$

that is

$x\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{c}_{0}+{c}_{1}t+{c}_{2}{t}^{2},$
(6)

where ${c}_{0}$, ${c}_{1}$, ${c}_{2}$ are real arbitrary constants (see [10, 13] and ). Thus,

${}^{c}D^{\beta }x\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha -\beta \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -\beta -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+\frac{{c}_{1}{t}^{1-\beta }}{\mathrm{\Gamma }\left(2-\beta \right)}+\frac{2{c}_{2}{t}^{2-\beta }}{\mathrm{\Gamma }\left(3-\beta \right)}$

and ${}^{c}D^{\gamma }x\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha -\gamma \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -\gamma -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+\frac{2{c}_{2}{t}^{2-\gamma }}{\mathrm{\Gamma }\left(3-\gamma \right)}$. Hence,

$\begin{array}{c}\begin{array}{rl}x\left(0\right)+x\left(\eta \right)+x\left(1\right)=& 3{c}_{0}+\left(1+\eta \right){c}_{1}+\left(1+{\eta }^{2}\right){c}_{2}\\ +\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right],\end{array}\hfill \\ \begin{array}{r}{}^{c}D^{\beta }x\left(0\right)+{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{\beta }x\left(\eta \right)+{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{\beta }x\left(1\right)\\ \phantom{\rule{1em}{0ex}}={c}_{1}\frac{{\eta }^{1-\beta }+1}{\mathrm{\Gamma }\left(2-\beta \right)}+{c}_{2}\frac{2\left({\eta }^{2-\beta }+1\right)}{\mathrm{\Gamma }\left(3-\beta \right)}\\ \phantom{\rule{2em}{0ex}}+\frac{1}{\mathrm{\Gamma }\left(\alpha -\beta \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\beta -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\beta -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]\end{array}\hfill \end{array}$

and

$\begin{array}{c}{}^{c}D^{\gamma }x\left(0\right)+{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{\gamma }x\left(\eta \right)+{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{\gamma }x\left(1\right)\hfill \\ \phantom{\rule{1em}{0ex}}={c}_{2}\frac{2\left({\eta }^{2-\gamma }+1\right)}{\mathrm{\Gamma }\left(3-\gamma \right)}+\frac{1}{\mathrm{\Gamma }\left(\alpha -\gamma \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\gamma -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\gamma -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right].\hfill \end{array}$

By using the boundary conditions, we obtain

$\begin{array}{c}\begin{array}{r}3{c}_{0}+\left(1+\eta \right){c}_{1}+\left(1+{\eta }^{2}\right){c}_{2}\\ \phantom{\rule{1em}{0ex}}={\int }_{0}^{1}{g}_{0}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right],\end{array}\hfill \\ \begin{array}{r}{c}_{1}\frac{{\eta }^{1-\beta }+1}{\mathrm{\Gamma }\left(2-\beta \right)}+{c}_{2}\frac{2\left({\eta }^{2-\beta }+1\right)}{\mathrm{\Gamma }\left(3-\beta \right)}\\ \phantom{\rule{1em}{0ex}}={\int }_{0}^{1}{g}_{1}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{1}{\mathrm{\Gamma }\left(\alpha -\beta \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\beta -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\beta -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]\end{array}\hfill \end{array}$

and

$\begin{array}{c}{c}_{2}\frac{2\left({\eta }^{2-\gamma }+1\right)}{\mathrm{\Gamma }\left(3-\gamma \right)}\hfill \\ \phantom{\rule{1em}{0ex}}={\int }_{0}^{1}{g}_{2}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{1}{\mathrm{\Gamma }\left(\alpha -\gamma \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\gamma -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\beta -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right].\hfill \end{array}$

This is a linear system of equations of triangular form, having ${c}_{0}$, ${c}_{1}$, and ${c}_{2}$ as unknowns. We solve by back substitution and find

$\begin{array}{rl}{c}_{0}=& \frac{1}{3}{\int }_{0}^{1}{g}_{0}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{1}{3\mathrm{\Gamma }\left(\alpha \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]\\ -\frac{\mathrm{\Gamma }\left(2-\beta \right)\left(\eta +1\right)}{3\left({\eta }^{1-\beta }+1\right)}\\ ×{\int }_{0}^{1}{g}_{1}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds+\frac{\left(\eta +1\right)\mathrm{\Gamma }\left(2-\beta \right)}{3\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(\alpha -\beta \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\beta -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ +{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\beta -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]\\ +\frac{2\left(\eta +1\right)\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)-\left({\eta }^{2}+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}\\ ×{\int }_{0}^{1}{g}_{2}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\left(\frac{\left({\eta }^{2}+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\beta \right)-2\left(\eta +1\right)\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\right)\\ ×\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\gamma -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\gamma -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right],\\ {c}_{1}=& \frac{\mathrm{\Gamma }\left(2-\beta \right)}{\left({\eta }^{1-\beta }+1\right)}{\int }_{0}^{1}{g}_{1}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{\mathrm{\Gamma }\left(2-\beta \right)}{\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(\alpha -\beta \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\beta -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ +{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\beta -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]-\frac{\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)}{\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}{\int }_{0}^{1}{g}_{2}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)}{\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\\ ×\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\gamma -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\gamma -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right],\end{array}$

and

$\begin{array}{rl}{c}_{2}=& \frac{\mathrm{\Gamma }\left(3-\gamma \right)}{2\left({\eta }^{2-\gamma }+1\right)}{\int }_{0}^{1}{g}_{2}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{\mathrm{\Gamma }\left(3-\gamma \right)}{2\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\\ ×\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\gamma -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\gamma -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right].\end{array}$

Now, we replace ${c}_{0}$, ${c}_{1}$, and ${c}_{2}$ in (6) and find the solution $x\left(t\right)$ as we stated. This completes the proof. □

Let $X={C}^{2}\left(\left[0,1\right]\right)$ endowed with the norm $\parallel x\parallel ={sup}_{t\in J}|x\left(t\right)|+{sup}_{t\in J}|{x}^{\prime }\left(t\right)|+{sup}_{t\in J}|{x}^{″}\left(t\right)|$. Then $\left(X,\parallel \cdot \parallel \right)$ is a Banach space. For $x\in X$, define

For the study of problem (1) and (2), we shall consider the following conditions.

(H1) $F:J×\mathbb{R}×\mathbb{R}×\mathbb{R}\to {P}_{cp}\left(\mathbb{R}\right)$ is an integrable bounded multifunction such that $F\left(\cdot ,x,y,z\right):\left[0,1\right]\to {P}_{cp}\left(\mathbb{R}\right)$ is measurable for all $x,y,z\in \mathbb{R}$;

(H2) ${g}_{0},{g}_{1},{g}_{2}:J×\mathbb{R}\to \mathbb{R}$ be continuous functions, $\psi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ a nondecreasing upper semi-continuous map such that ${lim inf}_{t\to \mathrm{\infty }}\left(t-\psi \left(t\right)\right)>0$ and $\psi \left(t\right) for all $t>0$;

(H3) There exist $m,{m}_{0},{m}_{1},{m}_{2}\in C\left(J,\left[0,\mathrm{\infty }\right)\right)$ such that

${H}_{d}\left(F\left(t,{x}_{1},{x}_{2},{x}_{3}\right),F\left(t,{y}_{1},{y}_{2},{y}_{3}\right)\right)\le \frac{1}{{\mathrm{\Lambda }}_{1}+{\mathrm{\Lambda }}_{2}+{\mathrm{\Lambda }}_{3}}m\left(t\right)\psi \left(|{x}_{1}-{y}_{1}|+|{x}_{2}-{y}_{2}|+|{x}_{3}-{y}_{3}|\right)$

and $|{g}_{j}\left(t,x\right)-{g}_{j}\left(t,y\right)|\le \frac{1}{{\mathrm{\Lambda }}_{1}+{\mathrm{\Lambda }}_{2}+{\mathrm{\Lambda }}_{3}}{m}_{j}\left(t\right)\psi \left(|x-y|\right)$ for all $t\in J$, $x,y,{x}_{1},{x}_{2},{x}_{3},{y}_{1},{y}_{2},{y}_{3}\in \mathbb{R}$ and $j=0,1,2$, where

$\begin{array}{rl}{\mathrm{\Lambda }}_{1}=& \left[\frac{{\parallel m\parallel }_{\mathrm{\infty }}}{\mathrm{\Gamma }\left(\alpha +1\right)}+\frac{{\parallel {m}_{0}\parallel }_{\mathrm{\infty }}}{3}+\frac{2{\parallel m\parallel }_{\mathrm{\infty }}}{3\mathrm{\Gamma }\left(\alpha +1\right)}+\frac{5\mathrm{\Gamma }\left(2-\beta \right){\parallel {m}_{1}\parallel }_{\mathrm{\infty }}}{3}+\frac{10\mathrm{\Gamma }\left(2-\beta \right){\parallel m\parallel }_{\mathrm{\infty }}}{3\mathrm{\Gamma }\left(\alpha -\beta +1\right)}\\ +\frac{10\left(2\mathrm{\Gamma }\left(2-\beta \right)+\mathrm{\Gamma }\left(3-\beta \right)\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\parallel {m}_{2}\parallel }_{\mathrm{\infty }}\mathrm{\Gamma }\left(\alpha -\gamma +1\right)+2{\parallel m\parallel }_{\mathrm{\infty }}\right)}{3\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right],\\ {\mathrm{\Lambda }}_{2}=& \left[\frac{{\parallel m\parallel }_{\mathrm{\infty }}}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{2\mathrm{\Gamma }\left(2-\beta \right){\parallel m\parallel }_{\mathrm{\infty }}}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}\\ +\frac{\left(2\mathrm{\Gamma }\left(2-\beta \right)+\mathrm{\Gamma }\left(3-\beta \right)\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\parallel {m}_{2}\parallel }_{\mathrm{\infty }}\mathrm{\Gamma }\left(\alpha -\gamma +1\right)+2{\parallel m\parallel }_{\mathrm{\infty }}\right)}{\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right],\end{array}$

and ${\mathrm{\Lambda }}_{3}=\left[\frac{{\parallel m\parallel }_{\mathrm{\infty }}}{\mathrm{\Gamma }\left(\alpha -1\right)}+\frac{\mathrm{\Gamma }\left(3-\gamma \right)\left({\parallel {m}_{2}\parallel }_{\mathrm{\infty }}\mathrm{\Gamma }\left(\alpha -\gamma +1\right)+2{\parallel m\parallel }_{\mathrm{\infty }}\right)}{\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right]$, and finally

(H4) $N:X\to {2}^{X}$ is given by

where

$\begin{array}{rl}w\left(t\right)=& \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\left(\alpha -1\right)}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+\frac{1}{3}{\int }_{0}^{1}{g}_{0}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{1}{3\mathrm{\Gamma }\left(\alpha \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ +{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]+\frac{3\mathrm{\Gamma }\left(2-\beta \right)t-\left(\eta +1\right)\mathrm{\Gamma }\left(2-\beta \right)}{3\left({\eta }^{1-\beta }+1\right)}{\int }_{0}^{1}{g}_{1}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\left(\eta +1\right)\mathrm{\Gamma }\left(2-\beta \right)-3\mathrm{\Gamma }\left(2-\beta \right)t}{3\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(\alpha -\beta \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\beta -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\beta -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]\\ +\left(\frac{2\left(\eta +1\right)\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)-\left({\eta }^{2}+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}\\ +\frac{-6\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)t+3\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(3-\beta \right){t}^{2}}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}\right)\\ ×{\int }_{0}^{1}{g}_{2}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\left(\frac{\left({\eta }^{2}+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\beta \right)-2\left(\eta +1\right)\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\\ +\frac{6\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)t-3\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(3-\beta \right)\left({\eta }^{1-\beta }+1\right){t}^{2}}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\right)\\ ×\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\gamma -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\gamma -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right].\end{array}$

Theorem 3.1 Assume that (H1)-(H4) are satisfied. If the multifunction N has the approximate endpoint property, then the boundary value inclusion problem (1) and (2) has a solution.

Proof We show that the multifunction $N:X\to P\left(X\right)$ has a endpoint which is a solution of the problem (1) and (2).

Note that the multivalued map $t⊢F\left(t,x\left(t\right),{x}^{\prime }\left(t\right),{x}^{″}\left(t\right)\right)$ is measurable and has closed values for all $x\in X$. Hence, it has measurable selection and so ${S}_{F,x}$ is nonempty for all $x\in X$. First, we show that $N\left(x\right)$ is closed subset of X for all $x\in X$.

Let $x\in X$ and ${\left\{{u}_{n}\right\}}_{n\ge 1}$ be a sequence in $N\left(x\right)$ with ${u}_{n}\to u$. For each $n\in \mathbb{N}$, choose ${v}_{n}\in {S}_{F,x}$ such that

$\begin{array}{rl}{u}_{n}\left(t\right)=& \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\left(\alpha -1\right)}{v}_{n}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+\frac{1}{3}{\int }_{0}^{1}{g}_{0}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{1}{3\mathrm{\Gamma }\left(\alpha \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -1}{v}_{n}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ +{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -1}{v}_{n}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]+\frac{3\mathrm{\Gamma }\left(2-\beta \right)t-\left(\eta +1\right)\mathrm{\Gamma }\left(2-\beta \right)}{3\left({\eta }^{1-\beta }+1\right)}{\int }_{0}^{1}{g}_{1}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\left(\eta +1\right)\mathrm{\Gamma }\left(2-\beta \right)-3\mathrm{\Gamma }\left(2-\beta \right)t}{3\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(\alpha -\beta \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\beta -1}{v}_{n}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ +{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\beta -1}{v}_{n}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]\\ +\left(\frac{2\left(\eta +1\right)\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)-\left({\eta }^{2}+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}\\ +\frac{-6\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)t+3\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(3-\beta \right){t}^{2}}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}\right)\\ ×{\int }_{0}^{1}{g}_{2}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\left(\frac{\left({\eta }^{2}+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\beta \right)-2\left(\eta +1\right)\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\\ +\frac{6\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)t-3\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(3-\beta \right)\left({\eta }^{1-\beta }+1\right){t}^{2}}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\right)\\ ×\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\gamma -1}{v}_{n}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\gamma -1}{v}_{n}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]\end{array}$

for all $t\in J$.

Since F has compact values, ${\left\{{v}_{n}\right\}}_{n\ge 1}$ has a subsequence which converges to some $v\in {L}^{1}\left[0,1\right]$. We denote this subsequence again by ${\left\{{v}_{n}\right\}}_{n\ge 1}$.

It is easy to check that $v\in {S}_{F,x}$ and

$\begin{array}{r}{u}_{n}\left(t\right)\to u\left(t\right)\\ \phantom{\rule{1em}{0ex}}=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\left(\alpha -1\right)}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+\frac{1}{3}{\int }_{0}^{1}{g}_{0}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}-\frac{1}{3\mathrm{\Gamma }\left(\alpha \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]\\ \phantom{\rule{2em}{0ex}}+\frac{3\mathrm{\Gamma }\left(2-\beta \right)t-\left(\eta +1\right)\mathrm{\Gamma }\left(2-\beta \right)}{3\left({\eta }^{1-\beta }+1\right)}{\int }_{0}^{1}{g}_{1}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+\frac{\left(\eta +1\right)\mathrm{\Gamma }\left(2-\beta \right)-3\mathrm{\Gamma }\left(2-\beta \right)t}{3\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(\alpha -\beta \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\beta -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\beta -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]\\ \phantom{\rule{2em}{0ex}}+\left(\frac{2\left(\eta +1\right)\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)-\left({\eta }^{2}+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}\\ \phantom{\rule{2em}{0ex}}+\frac{-6\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)t+3\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(3-\beta \right){t}^{2}}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}\right)\\ \phantom{\rule{2em}{0ex}}×{\int }_{0}^{1}{g}_{2}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+\left(\frac{\left({\eta }^{2}+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\beta \right)-2\left(\eta +1\right)\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\\ \phantom{\rule{2em}{0ex}}+\frac{6\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)t-3\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(3-\beta \right)\left({\eta }^{1-\beta }+1\right){t}^{2}}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\right)\\ \phantom{\rule{2em}{0ex}}×\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\gamma -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\gamma -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]\end{array}$

for all $t\in J$. This implies that $u\in N\left(x\right)$ and so N has closed values.

Since F is a compact multivalued map, it is easy to check that $N\left(x\right)$ is a bounded set for all $x\in X$.

Now, we show that ${H}_{d}\left(N\left(x\right),N\left(y\right)\right)\le \psi \left(\parallel x-y\parallel \right)$.

Let $x,y\in X$ and ${h}_{1}\in N\left(y\right)$. Choose ${v}_{1}\in {S}_{F,y}$ such that

$\begin{array}{rl}{h}_{1}\left(t\right)=& \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\left(\alpha -1\right)}{v}_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+\frac{1}{3}{\int }_{0}^{1}{g}_{0}\left(s,y\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{1}{3\mathrm{\Gamma }\left(\alpha \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -1}{v}_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ +{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -1}{v}_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]+\frac{3\mathrm{\Gamma }\left(2-\beta \right)t-\left(\eta +1\right)\mathrm{\Gamma }\left(2-\beta \right)}{3\left({\eta }^{1-\beta }+1\right)}{\int }_{0}^{1}{g}_{1}\left(s,y\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\left(\eta +1\right)\mathrm{\Gamma }\left(2-\beta \right)-3\mathrm{\Gamma }\left(2-\beta \right)t}{3\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(\alpha -\beta \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\beta -1}{v}_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ +{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\beta -1}{v}_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]\\ +\left(\frac{2\left(\eta +1\right)\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)-\left({\eta }^{2}+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}\\ +\frac{-6\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)t+3\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(3-\beta \right){t}^{2}}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}\right)\\ ×{\int }_{0}^{1}{g}_{2}\left(s,y\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\left(\frac{\left({\eta }^{2}+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\beta \right)-2\left(\eta +1\right)\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\\ +\frac{6\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)t-3\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(3-\beta \right)\left({\eta }^{1-\beta }+1\right){t}^{2}}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\right)\\ ×\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\gamma -1}{v}_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\gamma -1}{v}_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]\end{array}$

for almost all $t\in J$.

Since

$\begin{array}{r}{H}_{d}\left(F\left(t,x\left(t\right),{x}^{\prime }\left(t\right),{x}^{″}\left(t\right)\right),F\left(t,y\left(t\right),{y}^{\prime }\left(t\right),{y}^{″}\left(t\right)\right)\right)\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{{\mathrm{\Lambda }}_{1}+{\mathrm{\Lambda }}_{2}+{\mathrm{\Lambda }}_{3}}m\left(t\right)\psi \left(|x\left(t\right)-y\left(t\right)|+|{x}^{\prime }\left(t\right)-{y}^{\prime }\left(t\right)|+|{x}^{″}\left(t\right)-{y}^{″}\left(t\right)|\right)\end{array}$

for all $t\in J$, there exists $w\in F\left(t,x\left(t\right),{x}^{\prime }\left(t\right),{x}^{″}\left(t\right)\right)$ such that

$|{v}_{1}\left(t\right)-w|\le \frac{1}{{\mathrm{\Lambda }}_{1}+{\mathrm{\Lambda }}_{2}+{\mathrm{\Lambda }}_{3}}m\left(t\right)\psi \left(|x\left(t\right)-y\left(t\right)|+|{x}^{\prime }\left(t\right)-{y}^{\prime }\left(t\right)|+|{x}^{″}\left(t\right)-{y}^{″}\left(t\right)|\right)$

for all $t\in J$.

Consider the multivalued map $U:J\to P\left(\mathbb{R}\right)$ defined by

$\begin{array}{rl}U\left(t\right)=& \left\{w\in \mathbb{R}:|{v}_{1}\left(t\right)-w|\\ \le \frac{1}{{\mathrm{\Lambda }}_{1}+{\mathrm{\Lambda }}_{2}+{\mathrm{\Lambda }}_{3}}m\left(t\right)\psi \left(|x\left(t\right)-y\left(t\right)|+|{x}^{\prime }\left(t\right)-{y}^{\prime }\left(t\right)|+|{x}^{″}\left(t\right)-{y}^{″}\left(t\right)|\right)\right\}.\end{array}$

Since ${v}_{1}$ and $\phi =m\psi \left(|x-y|+|{x}^{\prime }-{y}^{\prime }|+|{x}^{″}-{y}^{″}|\right)\left(\frac{1}{{\mathrm{\Lambda }}_{1}+{\mathrm{\Lambda }}_{2}+{\mathrm{\Lambda }}_{3}}\right)$ are measurable, the multifunction $U\left(\cdot \right)\cap F\left(\cdot ,x\left(\cdot \right),{x}^{\prime }\left(\cdot \right),{x}^{″}\left(\cdot \right)\right)$ is measurable.

Choose ${v}_{2}\left(t\right)\in F\left(t,x\left(t\right),{x}^{\prime }\left(t\right),{x}^{″}\left(t\right)\right)$ such that

$\begin{array}{r}|{v}_{1}\left(t\right)-{v}_{2}\left(t\right)|\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{{\mathrm{\Lambda }}_{1}+{\mathrm{\Lambda }}_{2}+{\mathrm{\Lambda }}_{3}}m\left(t\right)\psi \left(|x\left(t\right)-y\left(t\right)|+|{x}^{\prime }\left(t\right)-{y}^{\prime }\left(t\right)|+|{x}^{″}\left(t\right)-{y}^{″}\left(t\right)|\right)\end{array}$

for all $t\in J$.

Now, consider the element ${h}_{2}\in N\left(x\right)$, which is defined by

$\begin{array}{rl}{h}_{2}\left(t\right)=& \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\left(\alpha -1\right)}{v}_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+\frac{1}{3}{\int }_{0}^{1}{g}_{0}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{1}{3\mathrm{\Gamma }\left(\alpha \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -1}{v}_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ +{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -1}{v}_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]+\frac{3\mathrm{\Gamma }\left(2-\beta \right)t-\left(\eta +1\right)\mathrm{\Gamma }\left(2-\beta \right)}{3\left({\eta }^{1-\beta }+1\right)}{\int }_{0}^{1}{g}_{1}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\frac{\left(\eta +1\right)\mathrm{\Gamma }\left(2-\beta \right)-3\mathrm{\Gamma }\left(2-\beta \right)t}{3\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(\alpha -\beta \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\beta -1}{v}_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ +{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\beta -1}{v}_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]\\ +\left(\frac{2\left(\eta +1\right)\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)-\left({\eta }^{2}+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}\\ +\frac{-6\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)t+3\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(3-\beta \right){t}^{2}}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}\right)\\ ×{\int }_{0}^{1}{g}_{2}\left(s,x\left(s\right)\right)\phantom{\rule{0.2em}{0ex}}ds\\ +\left(\frac{\left({\eta }^{2}+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\beta \right)-2\left(\eta +1\right)\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\\ +\frac{6\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)t-3\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(3-\beta \right)\left({\eta }^{1-\beta }+1\right){t}^{2}}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\right)\\ ×\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\gamma -1}{v}_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\gamma -1}{v}_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\right]\end{array}$

for all $t\in J$. Thus,

$\begin{array}{r}|{h}_{1}\left(t\right)-{h}_{2}\left(t\right)|\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\left(\alpha -1\right)}|{v}_{1}\left(s\right)-{v}_{2}\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+\frac{1}{3}{\int }_{0}^{1}|{g}_{0}\left(s,y\left(s\right)\right)-{g}_{0}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds+\frac{1}{3\mathrm{\Gamma }\left(\alpha \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -1}|{v}_{1}\left(s\right)-{v}_{2}\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -1}|{v}_{1}\left(s\right)-{v}_{2}\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds\right]+|\frac{3\mathrm{\Gamma }\left(2-\beta \right)t-\left(\eta +1\right)\mathrm{\Gamma }\left(2-\beta \right)}{3\left({\eta }^{1-\beta }+1\right)}|\\ \phantom{\rule{2em}{0ex}}×{\int }_{0}^{1}|{g}_{1}\left(s,y\left(s\right)\right)-{g}_{1}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds+|\frac{\left(\eta +1\right)\mathrm{\Gamma }\left(2-\beta \right)-3\mathrm{\Gamma }\left(2-\beta \right)t}{3\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(\alpha -\beta \right)}|\\ \phantom{\rule{2em}{0ex}}×\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\beta -1}|{v}_{1}\left(s\right)-{v}_{2}\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\beta -1}|{v}_{1}\left(s\right)-{v}_{2}\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds\right]\\ \phantom{\rule{2em}{0ex}}+|\left(\frac{2\left(\eta +1\right)\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)-\left({\eta }^{2}+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}\\ \phantom{\rule{2em}{0ex}}+\frac{-6\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)t+3\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(3-\beta \right){t}^{2}}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}\right)|\\ \phantom{\rule{2em}{0ex}}×{\int }_{0}^{1}|{g}_{2}\left(s,y\left(s\right)\right)-{g}_{2}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+|\left(\frac{\left({\eta }^{2}+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\beta \right)-2\left(\eta +1\right)\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\\ \phantom{\rule{2em}{0ex}}+\frac{6\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)t-3\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(3-\beta \right)\left({\eta }^{1-\beta }+1\right){t}^{2}}{6\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\right)|\\ \phantom{\rule{2em}{0ex}}×\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\gamma -1}|{v}_{1}\left(s\right)-{v}_{2}\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\gamma -1}|{v}_{1}\left(s\right)-{v}_{2}\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds\right]\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{{\mathrm{\Lambda }}_{1}+{\mathrm{\Lambda }}_{2}+{\mathrm{\Lambda }}_{3}}\psi \left(\parallel x-y\parallel \right)\left[\frac{{\parallel m\parallel }_{\mathrm{\infty }}}{\mathrm{\Gamma }\left(\alpha +1\right)}+\frac{{\parallel {m}_{0}\parallel }_{\mathrm{\infty }}}{3}+\frac{2{\parallel m\parallel }_{\mathrm{\infty }}}{3\mathrm{\Gamma }\left(\alpha +1\right)}+\frac{5\mathrm{\Gamma }\left(2-\beta \right){\parallel {m}_{1}\parallel }_{\mathrm{\infty }}}{3}\\ \phantom{\rule{2em}{0ex}}+\frac{10\mathrm{\Gamma }\left(2-\beta \right){\parallel m\parallel }_{\mathrm{\infty }}}{3\mathrm{\Gamma }\left(\alpha -\beta +1\right)}\\ \phantom{\rule{2em}{0ex}}+\frac{10\left(2\mathrm{\Gamma }\left(2-\beta \right)+\mathrm{\Gamma }\left(3-\beta \right)\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\parallel {m}_{2}\parallel }_{\mathrm{\infty }}\mathrm{\Gamma }\left(\alpha -\gamma +1\right)+2{\parallel m\parallel }_{\mathrm{\infty }}\right)}{3\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right]\\ \phantom{\rule{1em}{0ex}}=\frac{{\mathrm{\Lambda }}_{1}}{{\mathrm{\Lambda }}_{1}+{\mathrm{\Lambda }}_{2}+{\mathrm{\Lambda }}_{3}}\psi \left(\parallel x-y\parallel \right),\\ |{h}_{1}^{\prime }\left(t\right)-{h}_{2}^{\prime }\left(t\right)|\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\mathrm{\Gamma }\left(\alpha -1\right)}{\int }_{0}^{t}{\left(t-s\right)}^{\left(\alpha -2\right)}|{v}_{1}\left(s\right)-{v}_{2}\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+\frac{\mathrm{\Gamma }\left(2-\beta \right)}{\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(\alpha -\beta \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\beta -1}|{v}_{1}\left(s\right)-{v}_{2}\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\beta -1}|{v}_{1}\left(s\right)-{v}_{2}\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds\right]\\ \phantom{\rule{2em}{0ex}}+|\left(\frac{\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)+\mathrm{\Gamma }\left(3-\gamma \right)\left({\eta }^{1-\beta }+1\right)\mathrm{\Gamma }\left(3-\beta \right)t}{\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)}|\\ \phantom{\rule{2em}{0ex}}×{\int }_{0}^{1}|{g}_{2}\left(s,y\left(s\right)\right)-{g}_{2}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+|\frac{\left({\eta }^{2-\beta }+1\right)\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(2-\beta \right)-\mathrm{\Gamma }\left(3-\gamma \right)\mathrm{\Gamma }\left(3-\beta \right)\left({\eta }^{1-\beta }+1\right)t}{\left({\eta }^{1-\beta }+1\right)\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\right)|\\ \phantom{\rule{2em}{0ex}}×\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\gamma -1}|{v}_{1}\left(s\right)-{v}_{2}\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\gamma -1}|{v}_{1}\left(s\right)-{v}_{2}\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds\right]\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{{\mathrm{\Lambda }}_{1}+{\mathrm{\Lambda }}_{2}+{\mathrm{\Lambda }}_{3}}\psi \left(\parallel x-y\parallel \right)\left[\frac{{\parallel m\parallel }_{\mathrm{\infty }}}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{2\mathrm{\Gamma }\left(2-\beta \right){\parallel m\parallel }_{\mathrm{\infty }}}{\mathrm{\Gamma }\left(\alpha -\beta +1\right)}\\ \phantom{\rule{2em}{0ex}}+\frac{\left(2\mathrm{\Gamma }\left(2-\beta \right)+\mathrm{\Gamma }\left(3-\beta \right)\right)\mathrm{\Gamma }\left(3-\gamma \right)\left({\parallel {m}_{2}\parallel }_{\mathrm{\infty }}\mathrm{\Gamma }\left(\alpha -\gamma +1\right)+2{\parallel m\parallel }_{\mathrm{\infty }}\right)}{\mathrm{\Gamma }\left(3-\beta \right)\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right]\\ \phantom{\rule{1em}{0ex}}=\frac{{\mathrm{\Lambda }}_{2}}{{\mathrm{\Lambda }}_{1}+{\mathrm{\Lambda }}_{2}+{\mathrm{\Lambda }}_{3}}\psi \left(\parallel x-y\parallel \right),\end{array}$

and

$\begin{array}{r}|{h}_{1}^{\prime \prime }\left(t\right)-{h}_{2}^{\prime \prime }\left(t\right)|\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\mathrm{\Gamma }\left(\alpha -2\right)}{\int }_{0}^{t}{\left(t-s\right)}^{\left(\alpha -3\right)}|{v}_{1}\left(s\right)-{v}_{2}\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+\frac{\mathrm{\Gamma }\left(3-\gamma \right)}{\left({\eta }^{2-\gamma }+1\right)}{\int }_{0}^{1}|{g}_{2}\left(s,y\left(s\right)\right)-{g}_{2}\left(s,x\left(s\right)\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+\frac{\mathrm{\Gamma }\left(3-\gamma \right)}{\left({\eta }^{2-\gamma }+1\right)\mathrm{\Gamma }\left(\alpha -\gamma \right)}\left[{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -\gamma -1}|{v}_{1}\left(s\right)-{v}_{2}\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+{\int }_{0}^{\eta }{\left(\eta -s\right)}^{\alpha -\gamma -1}|{v}_{1}\left(s\right)-{v}_{2}\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds\right]\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{{\mathrm{\Lambda }}_{1}+{\mathrm{\Lambda }}_{2}+{\mathrm{\Lambda }}_{3}}\psi \left(\parallel x-y\parallel \right)\left[\frac{{\parallel m\parallel }_{\mathrm{\infty }}}{\mathrm{\Gamma }\left(\alpha -1\right)}+\frac{\mathrm{\Gamma }\left(3-\gamma \right)\left({\parallel {m}_{2}\parallel }_{\mathrm{\infty }}\mathrm{\Gamma }\left(\alpha -\gamma +1\right)+2{\parallel m\parallel }_{\mathrm{\infty }}\right)}{\mathrm{\Gamma }\left(\alpha -\gamma +1\right)}\right]\\ \phantom{\rule{1em}{0ex}}=\frac{{\mathrm{\Lambda }}_{3}}{{\mathrm{\Lambda }}_{1}+{\mathrm{\Lambda }}_{2}+{\mathrm{\Lambda }}_{3}}\psi \left(\parallel x-y\parallel \right).\end{array}$

Hence,

$\begin{array}{rl}\parallel {h}_{1}-{h}_{2}\parallel & =\underset{t\in J}{sup}|{h}_{1}\left(t\right)-{h}_{2}\left(t\right)|+\underset{t\in J}{sup}|{h}_{1}^{\prime }\left(t\right)-{h}_{2}^{\prime }\left(t\right)|+\underset{t\in J}{sup}|{h}_{1}^{\prime \prime }\left(t\right)-{h}_{2}^{\prime \prime }\left(t\right)|\\ \le \frac{1}{{\mathrm{\Lambda }}_{1}+{\mathrm{\Lambda }}_{2}+{\mathrm{\Lambda }}_{3}}\psi \left(\parallel x-y\parallel \right)\left({\mathrm{\Lambda }}_{1}+{\mathrm{\Lambda }}_{2}+{\mathrm{\Lambda }}_{3}\right)=\psi \left(\parallel x-y\parallel \right).\end{array}$

Thus, it is easy to get ${H}_{d}\left(N\left(x\right),N\left(y\right)\right)\le \psi \left(\parallel x-y\parallel \right)$ for all $x,y\in X$.

Since the multifunction N has approximate endpoint property, by using Lemma 2.1 there exists ${x}^{\ast }\in X$ such that $N\left({x}^{\ast }\right)=\left\{{x}^{\ast }\right\}$. Hence by using Lemma 3.1, ${x}^{\ast }$ is a solution of the problem (1) and (2). □

Now, we investigate the existence of solution for the fractional differential inclusion problem

${}^{c}D^{\alpha }x\left(t\right)\in F\left(t,x\left(t\right),{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{1}}x\left(t\right),\dots ,{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{n}}x\left(t\right)\right),$

via integral boundary value conditions

${x}^{\prime }\left(0\right)+b{x}^{\prime }\left(1\right)=\sum _{i=1}^{n}{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{i}}x\left(\eta \right),\phantom{\rule{2em}{0ex}}x\left(0\right)+ax\left(1\right)=\sum _{i=1}^{n}{I}^{{\gamma }_{i}}x\left(\eta \right),$

where $t\in J=\left[0,1\right]$, $1<\alpha \le 2$, $n\ge 2$, $0<\eta ,{\gamma }_{i}<1$, $\alpha -{\gamma }_{i}\ge 1$ for all $1\le i\le n$, $a>{\sum }_{i=1}^{n}\frac{{\eta }^{{\gamma }_{i}+1}}{\mathrm{\Gamma }\left({\gamma }_{i}+2\right)}$, $b>{\sum }_{i=1}^{n}\frac{{\eta }^{1-{\gamma }_{i}}}{\mathrm{\Gamma }\left(2-{\gamma }_{i}\right)}$ and $F:J×{\mathbb{R}}^{n+1}\to P\left(\mathbb{R}\right)$ is a multifunction.

Lemma 3.2 Let $v\in C\left(J,\mathbb{R}\right)$, $\alpha \in \left(1,2\right]$, $\eta \in \left(0,1\right)$, $n\ge 2$ and ${\beta }_{i}\in \left(0,1\right)$ for $i=1,\dots ,n$. The unique solution of the fractional differential problem ${}^{c}D^{\alpha }x\left(t\right)=v\left(t\right)$ via the boundary value conditions

$x\left(0\right)+ax\left(1\right)=\sum _{i=1}^{n}{I}^{{\beta }_{i}}x\left(\eta \right),\phantom{\rule{2em}{0ex}}{x}^{\prime }\left(0\right)+b{x}^{\prime }\left(1\right)=\sum _{i=1}^{n}{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\beta }_{i}}x\left(\eta \right),$

with $a>{\sum }_{i=1}^{n}\frac{{\eta }^{{\beta }_{i}+1}}{\mathrm{\Gamma }\left({\beta }_{i}+2\right)}$, $b>{\sum }_{i=1}^{n}\frac{{\eta }^{1-{\beta }_{i}}}{\mathrm{\Gamma }\left(2-{\beta }_{i}\right)}$ is

$x\left(t\right)={\int }_{0}^{1}G\left(t,s\right)v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds,$

where $G\left(t,s\right)$ is the Green function given by

$\begin{array}{rl}G\left(t,s\right)=& \frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{1}{A}\sum _{i=1}^{n}\frac{{\left(\eta -s\right)}^{\alpha +{\beta }_{i}-1}}{\mathrm{\Gamma }\left(\alpha +{\beta }_{i}\right)}-\frac{a}{A\mathrm{\Gamma }\left(\alpha \right)}{\left(1-s\right)}^{\alpha -1}-\frac{1}{AB}\left(a-\sum _{i=1}^{n}\frac{{\eta }^{{\beta }_{i}+2}}{\mathrm{\Gamma }\left({\beta }_{i}+3\right)}\right)\\ ×\sum _{i=1}^{n}\frac{{\left(\eta -s\right)}^{\alpha -{\beta }_{i}-1}}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}\right)}-\frac{b}{AB}\left(a-\sum _{i=1}^{n}\frac{{\eta }^{{\beta }_{i}+2}}{\mathrm{\Gamma }\left({\beta }_{i}+3\right)}\right)\frac{{\left(1-s\right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}+\frac{t}{B}\sum _{i=1}^{n}\frac{{\left(\eta -s\right)}^{\alpha -{\beta }_{i}-1}}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}\right)}\\ -\frac{bt}{B\mathrm{\Gamma }\left(\alpha -1\right)}{\left(1-s\right)}^{\alpha -2}\end{array}$

whenever $0\le s\le \eta \le t\le 1$,

$\begin{array}{rl}G\left(t,s\right)=& \frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}-\frac{a}{A\mathrm{\Gamma }\left(\alpha \right)}{\left(1-s\right)}^{\alpha -1}\\ -\frac{b}{AB}\left(a-\sum _{i=1}^{n}\frac{{\eta }^{{\beta }_{i}+2}}{\mathrm{\Gamma }\left({\beta }_{i}+3\right)}\right)\frac{{\left(1-s\right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}-\frac{bt}{B\mathrm{\Gamma }\left(\alpha -1\right)}{\left(1-s\right)}^{\alpha -2}\end{array}$

whenever $0\le \eta \le s\le t\le 1$,

$G\left(t,s\right)=-\frac{a}{A\mathrm{\Gamma }\left(\alpha \right)}{\left(1-s\right)}^{\alpha -1}-\frac{b}{AB}\left(a-\sum _{i=1}^{n}\frac{{\eta }^{{\beta }_{i}+2}}{\mathrm{\Gamma }\left({\beta }_{i}+3\right)}\right)\frac{{\left(1-s\right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}-\frac{bt}{B\mathrm{\Gamma }\left(\alpha -1\right)}{\left(1-s\right)}^{\alpha -2}$

whenever $0\le \eta \le s\le t\le 1$,

$\begin{array}{rl}G\left(t,s\right)=& \frac{{\left(t-s\right)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{1}{A}\sum _{i=1}^{n}\frac{{\left(\eta -s\right)}^{\alpha +{\beta }_{i}-1}}{\mathrm{\Gamma }\left(\alpha +{\beta }_{i}\right)}-\frac{a}{A\mathrm{\Gamma }\left(\alpha \right)}{\left(1-s\right)}^{\alpha -1}-\frac{1}{AB}\left(a-\sum _{i=1}^{n}\frac{{\eta }^{{\beta }_{i}+2}}{\mathrm{\Gamma }\left({\beta }_{i}+3\right)}\right)\\ ×\sum _{i=1}^{n}\frac{{\left(\eta -s\right)}^{\alpha -{\beta }_{i}-1}}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}\right)}-\frac{b}{AB}\left(a-\sum _{i=1}^{n}\frac{{\eta }^{{\beta }_{i}+2}}{\mathrm{\Gamma }\left({\beta }_{i}+3\right)}\right)\frac{{\left(1-s\right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}\\ +\frac{t}{B}\sum _{i=1}^{n}\frac{{\left(\eta -s\right)}^{\alpha -{\beta }_{i}-1}}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}\right)}-\frac{bt}{B\mathrm{\Gamma }\left(\alpha -1\right)}{\left(1-s\right)}^{\alpha -2}\end{array}$

whenever $0\le s\le t\le \eta \le 1$,

$\begin{array}{rl}G\left(t,s\right)=& \frac{1}{A}\sum _{i=1}^{n}\frac{{\left(\eta -s\right)}^{\alpha +{\beta }_{i}-1}}{\mathrm{\Gamma }\left(\alpha +{\beta }_{i}\right)}-\frac{a}{A\mathrm{\Gamma }\left(\alpha \right)}{\left(1-s\right)}^{\alpha -1}-\frac{1}{AB}\left(a-\sum _{i=1}^{n}\frac{{\eta }^{{\beta }_{i}+2}}{\mathrm{\Gamma }\left({\beta }_{i}+3\right)}\right)\\ ×\sum _{i=1}^{n}\frac{{\left(\eta -s\right)}^{\alpha -{\beta }_{i}-1}}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}\right)}-\frac{b}{AB}\left(a-\sum _{i=1}^{n}\frac{{\eta }^{{\beta }_{i}+2}}{\mathrm{\Gamma }\left({\beta }_{i}+3\right)}\right)\frac{{\left(1-s\right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}+\frac{t}{B}\sum _{i=1}^{n}\frac{{\left(\eta -s\right)}^{\alpha -{\beta }_{i}-1}}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}\right)}\\ -\frac{bt}{B\mathrm{\Gamma }\left(\alpha -1\right)}{\left(1-s\right)}^{\alpha -2}\end{array}$

whenever $0\le t\le s\le \eta \le 1$ and

$G\left(t,s\right)=-\frac{a}{A\mathrm{\Gamma }\left(\alpha \right)}{\left(1-s\right)}^{\alpha -1}-\frac{b}{AB}\left(a-\sum _{i=1}^{n}\frac{{\eta }^{{\beta }_{i}+2}}{\mathrm{\Gamma }\left({\beta }_{i}+3\right)}\right)\frac{{\left(1-s\right)}^{\alpha -2}}{\mathrm{\Gamma }\left(\alpha -1\right)}-\frac{bt}{B\mathrm{\Gamma }\left(\alpha -1\right)}{\left(1-s\right)}^{\alpha -2}$

whenever $0\le t\le \eta \le s\le 1$, where $A=1+a-{\sum }_{i=1}^{n}\frac{{\eta }^{{\beta }_{i}+1}}{\mathrm{\Gamma }\left({\beta }_{i}+2\right)}$ and $B=1+b-{\sum }_{i=1}^{n}\frac{{\eta }^{1-{\beta }_{i}}}{\mathrm{\Gamma }\left(2-{\beta }_{i}\right)}$.

Proof It is known that the general solution of the equation ${}^{c}D^{\alpha }x\left(t\right)=v\left(t\right)$ is

$x\left(t\right)={I}^{\alpha }v\left(t\right)+{c}_{0}+{c}_{1}t=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\left(\alpha -1\right)}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{c}_{0}+{c}_{1}t,$

where ${c}_{0},{c}_{1}\in \mathbb{R}$ are arbitrary constants (see [10, 13] and ). Thus,

$\begin{array}{c}{}^{c}D^{{\beta }_{i}}x\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}\right)}{\int }_{0}^{t}{\left(t-s\right)}^{\left(\alpha -{\beta }_{i}-1\right)}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+\frac{{c}_{1}{t}^{1-{\beta }_{i}}}{\mathrm{\Gamma }\left(2-{\beta }_{i}\right)},\hfill \\ {I}^{{\beta }_{i}}x\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha +{\beta }_{i}\right)}{\int }_{0}^{t}{\left(t-s\right)}^{\left(\alpha +{\beta }_{i}-1\right)}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+\frac{{c}_{0}{t}^{1+{\beta }_{i}}}{\mathrm{\Gamma }\left(2+{\beta }_{i}\right)}+\frac{{c}_{1}{t}^{2+{\beta }_{i}}}{\mathrm{\Gamma }\left(3+{\beta }_{i}\right)},\hfill \end{array}$

and ${x}^{\prime }\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha -1\right)}{\int }_{0}^{t}{\left(t-s\right)}^{\left(\alpha -2\right)}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+{c}_{1}$. Hence,

$x\left(0\right)+ax\left(1\right)=\left(a+1\right){c}_{0}+a{c}_{1}+\frac{a}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{1}{\left(1-s\right)}^{\left(\alpha -1\right)}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds$

and

${x}^{\prime }\left(0\right)+b{x}^{\prime }\left(1\right)=\left(1+b\right){c}_{1}+\frac{b}{\mathrm{\Gamma }\left(\alpha -1\right)}{\int }_{0}^{1}{\left(1-s\right)}^{\left(\alpha -2\right)}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds.$

By using the boundary conditions, we obtain

$\begin{array}{r}{c}_{0}\left(1+a-\sum _{i=1}^{n}\frac{{\eta }^{{\beta }_{i}+1}}{\mathrm{\Gamma }\left({\beta }_{i}+2\right)}\right)+{c}_{1}\left(a-\sum _{i=1}^{n}\frac{{\eta }^{{\beta }_{i}+2}}{\mathrm{\Gamma }\left({\beta }_{i}+3\right)}\right)\\ \phantom{\rule{1em}{0ex}}=\sum _{i=1}^{n}{\int }_{0}^{\eta }\frac{{\left(\eta -s\right)}^{\alpha +{\beta }_{i}-1}}{\mathrm{\Gamma }\left({\beta }_{i}+\alpha \right)}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{a}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\end{array}$

and

${c}_{1}\left(1+b-\sum _{i=1}^{n}\frac{{\eta }^{1-{\beta }_{i}}}{\mathrm{\Gamma }\left(2-{\beta }_{i}\right)}\right)=\sum _{i=1}^{n}{\int }_{0}^{\eta }\frac{{\left(\eta -s\right)}^{\alpha -{\beta }_{i}-1}}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}\right)}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{b}{\mathrm{\Gamma }\left(\alpha -1\right)}{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -2}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds.$

Thus,

$\begin{array}{rl}{c}_{0}=& \frac{1}{A}\sum _{i=1}^{n}{\int }_{0}^{\eta }\frac{{\left(\eta -s\right)}^{\alpha +{\beta }_{i}-1}}{\mathrm{\Gamma }\left(\alpha +{\beta }_{i}\right)}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{a}{A\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{1}{AB}\left(a-\sum _{i=1}^{n}\frac{{\eta }^{{\beta }_{i}+2}}{\mathrm{\Gamma }\left({\beta }_{i}+3\right)}\right)\\ ×\sum _{i=1}^{n}{\int }_{0}^{\eta }\frac{{\left(\eta -s\right)}^{\alpha -{\beta }_{i}-1}}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}\right)}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{b}{AB\mathrm{\Gamma }\left(\alpha -1\right)}\left(a-\sum _{i=1}^{n}\frac{{\eta }^{{\beta }_{i}+2}}{\mathrm{\Gamma }\left({\beta }_{i}+3\right)}\right){\int }_{0}^{1}{\left(1-s\right)}^{\alpha -2}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds,\\ {c}_{1}=& \frac{1}{B}\sum _{i=1}^{n}{\int }_{0}^{\eta }\frac{{\left(\eta -s\right)}^{\alpha -{\beta }_{i}-1}}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}\right)}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{b}{B\mathrm{\Gamma }\left(\alpha -1\right)}{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -2}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds.\end{array}$

Hence,

$\begin{array}{rl}x\left(t\right)=& \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+\frac{1}{A}\sum _{i=1}^{n}{\int }_{0}^{\eta }\frac{{\left(\eta -s\right)}^{\alpha +{\beta }_{i}-1}}{\mathrm{\Gamma }\left(\alpha +{\beta }_{i}\right)}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{a}{A\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -1}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{1}{AB}\left(a-\sum _{i=1}^{n}\frac{{\eta }^{{\beta }_{i}+2}}{\mathrm{\Gamma }\left({\beta }_{i}+3\right)}\right)\sum _{i=1}^{n}{\int }_{0}^{\eta }\frac{{\left(\eta -s\right)}^{\alpha -{\beta }_{i}-1}}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}\right)}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{b}{AB\mathrm{\Gamma }\left(\alpha -1\right)}\left(a-\sum _{i=1}^{n}\frac{{\eta }^{{\beta }_{i}+2}}{\mathrm{\Gamma }\left({\beta }_{i}+3\right)}\right)\\ ×{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -2}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+\frac{t}{B}\sum _{i=1}^{n}{\int }_{0}^{\eta }\frac{{\left(\eta -s\right)}^{\alpha -{\beta }_{i}-1}}{\mathrm{\Gamma }\left(\alpha -{\beta }_{i}\right)}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ -\frac{tb}{B\mathrm{\Gamma }\left(\alpha -1\right)}{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -2}v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ =& {\int }_{0}^{1}G\left(t,s\right)v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds.\end{array}$

This completes the proof. □

Suppose that endowed with the norm $\parallel x\parallel ={sup}_{t\in J}|x\left(t\right)|+{\sum }_{i=1}^{n}{sup}_{t\in J}|{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{i}}x\left(t\right)|$. Then $\left(X,\parallel \cdot \parallel \right)$ is a Banach space . For $x\in X$, define

Now, put

$\begin{array}{rl}{L}_{1}=& \frac{1}{\mathrm{\Gamma }\left(\alpha +1\right)}+\frac{1}{A}\sum _{i=1}^{n}\frac{{\eta }^{\alpha +{\gamma }_{i}}}{\mathrm{\Gamma }\left(\alpha +{\gamma }_{i}+1\right)}+\frac{a}{A\mathrm{\Gamma }\left(\alpha +1\right)}+\frac{1}{AB}\left(|a-\sum _{i=1}^{n}\frac{{\eta }^{{\gamma }_{i}+2}}{\mathrm{\Gamma }\left({\gamma }_{i}+3\right)}|\right)\\ ×\sum _{i=1}^{n}\frac{{\eta }^{\alpha -{\gamma }_{i}}}{\mathrm{\Gamma }\left(\alpha -{\gamma }_{i}+1\right)}+\frac{b}{AB\mathrm{\Gamma }\left(\alpha \right)}\left(|a-\sum _{i=1}^{n}\frac{{\eta }^{{\gamma }_{i}+2}}{\mathrm{\Gamma }\left({\gamma }_{i}+3\right)}|\right)\\ +\frac{1}{B}\sum _{i=1}^{n}\frac{{\eta }^{\alpha -{\gamma }_{i}}}{\mathrm{\Gamma }\left(\alpha -{\gamma }_{i}+1\right)}+\frac{b}{B\mathrm{\Gamma }\left(\alpha \right)}\end{array}$

and ${L}_{2}^{j}=\frac{1}{\mathrm{\Gamma }\left(\alpha -{\gamma }_{j}+1\right)}+\frac{1}{B\mathrm{\Gamma }\left(2-{\gamma }_{j}\right)}{\sum }_{i=1}^{n}\frac{{\eta }^{\alpha -{\gamma }_{i}}}{\mathrm{\Gamma }\left(\alpha -{\gamma }_{i}+1\right)}+\frac{b}{B\mathrm{\Gamma }\left(2-{\gamma }_{j}\right)\mathrm{\Gamma }\left(\alpha \right)}$ for all $1\le j\le n$.

Theorem 3.2 Let $\psi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ a nondecreasing upper semi-continuous map such that ${lim inf}_{t\to \mathrm{\infty }}\left(t-\psi \left(t\right)\right)>0$ and $\psi \left(t\right) for all $t>0$, $F:J×{\mathbb{R}}^{n+1}\to {P}_{cp}\left(\mathbb{R}\right)$ a multifunction such that $F\left(\cdot ,{x}_{1},{x}_{2},\dots ,{x}_{n+1}\right):\left[0,1\right]\to {P}_{cp}\left(\mathbb{R}\right)$ is measurable and integrable bounded for all ${x}_{1},{x}_{2},\dots ,{x}_{n+1}\in \mathbb{R}$. Assume that there exists $m\in C\left(J,\left[0,\mathrm{\infty }\right)\right)$ such that

$\begin{array}{c}{H}_{d}\left(F\left(t,{x}_{1},{x}_{2},\dots ,{x}_{n+1}\right)-F\left(t,{y}_{1},{y}_{2},\dots ,{y}_{n+1}\right)\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le m\left(t\right)\psi \left(|{x}_{1}-{y}_{1}|+|{x}_{2}-{y}_{2}|+\cdots +|{x}_{n+1}-{y}_{n+1}|\right)\left(\frac{1}{{\parallel m\parallel }_{\mathrm{\infty }}\left({L}_{1}+{\sum }_{j=1}^{n}{L}_{2}^{j}\right)}\right).\hfill \end{array}$

Define $\mathrm{\Omega }:X\to {2}^{X}$ by

If the multifunction Ω has the approximate endpoint property, then the boundary value inclusion problem (3) and (4) has a solution.

Proof We show that the multifunction $\mathrm{\Omega }:X\to P\left(X\right)$ has a endpoint which is a solution of the problem (3) and (4).

First, we show that $\mathrm{\Omega }\left(x\right)$ is closed subset of X for all $x\in X$.

Let $x\in X$ and ${\left\{{u}_{n}\right\}}_{n\ge 1}$ be a sequence in $\mathrm{\Omega }\left(x\right)$ with ${u}_{n}\to u$. For each $n\in \mathbb{N}$, choose ${v}_{n}\in {S}_{F,x}$ such that ${u}_{n}\left(t\right)={\int }_{0}^{1}G\left(t,s\right){v}_{n}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds$ for all $t\in J$. Since F has compact values, ${\left\{{v}_{n}\right\}}_{n\ge 1}$ has a subsequence which converges to some $v\in {L}^{1}\left[0,1\right]$. We denote this subsequence again by ${\left\{{v}_{n}\right\}}_{n\ge 1}$.

It is easy to check that $v\in {S}_{F,x}$ and ${u}_{n}\left(t\right)\to u\left(t\right)={\int }_{0}^{1}G\left(t,s\right)v\left(s\right)\phantom{\rule{0.2em}{0ex}}ds$ for all $t\in J$. This implies that $u\in \mathrm{\Omega }\left(x\right)$ and so Ω has closed values. Since F is a compact multivalued map, it is easy to check that $\mathrm{\Omega }\left(x\right)$ is a bounded set for all $x\in X$.

Now, we show that for all $x,y\in X$, ${H}_{d}\left(\mathrm{\Omega }\left(x\right),\mathrm{\Omega }\left(y\right)\right)\le \psi \left(\parallel x-y\parallel \right)$.

Let $x,y\in X$ and ${h}_{1}\in \mathrm{\Omega }\left(y\right)$. Choose ${v}_{1}\in {S}_{F,y}$ such that ${h}_{1}\left(t\right)={\int }_{0}^{1}G\left(t,s\right){v}_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds$ for almost all $t\in J$. Since

$\begin{array}{c}{H}_{d}\left(F\left(t,x\left(t\right),{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{1}}x\left(t\right),\dots ,{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{n}}x\left(t\right)\right),F\left(t,y\left(t\right),{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{1}}y\left(t\right),\dots ,{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{n}}y\left(t\right)\right)\right)\hfill \\ \phantom{\rule{1em}{0ex}}\le m\left(t\right)\psi \left(|x\left(t\right)-y\left(t\right)|+|{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{1}}x\left(t\right)-{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{1}}y\left(t\right)|+\cdots +|{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{n}}x\left(t\right)-{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{n}}y\left(t\right)|\right)\hfill \\ \phantom{\rule{2em}{0ex}}×\left(\frac{1}{{\parallel m\parallel }_{\mathrm{\infty }}\left({L}_{1}+{\sum }_{j=1}^{n}{L}_{2}^{j}\right)}\right)\hfill \end{array}$

for all $t\in J$, there exists $w\in F\left(t,x\left(t\right),{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{1}}x\left(t\right),\dots ,{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{n}}x\left(t\right)\right)$ such that

$\begin{array}{rl}|{v}_{1}\left(t\right)-w|\le & m\left(t\right)\psi \left(|x\left(t\right)-y\left(t\right)|+|{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{1}}x\left(t\right)-{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{1}}y\left(t\right)|+\cdots \\ +|{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{n}}x\left(t\right)-{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{n}}y\left(t\right)|\right)\left(\frac{1}{{\parallel m\parallel }_{\mathrm{\infty }}\left({L}_{1}+{\sum }_{j=1}^{n}{L}_{2}^{j}\right)}\right)\end{array}$

for all $t\in J$. Consider the multivalued map $U:J\to P\left(\mathbb{R}\right)$ defined by the rule

$\begin{array}{rl}U\left(t\right)=& \left\{w\in \mathbb{R}:|{v}_{1}\left(t\right)-w|\le m\left(t\right)\psi \left(|x\left(t\right)-y\left(t\right)|+|{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{1}}x\left(t\right)-{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{1}}y\left(t\right)|+\cdots \\ +|{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{n}}x\left(t\right)-{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{n}}y\left(t\right)|\right)\left(\frac{1}{{\parallel m\parallel }_{\mathrm{\infty }}\left({L}_{1}+{\sum }_{j=1}^{n}{L}_{2}^{j}\right)}\right)\right\}.\end{array}$

Since ${v}_{1}$ and

$\phi =m\psi \left(|x-y|+|{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{1}}x-{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{1}}y|+\cdots +|{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{n}}x-{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{n}}y|\right)\left(\frac{1}{{\parallel m\parallel }_{\mathrm{\infty }}\left({L}_{1}+{\sum }_{j=1}^{n}{L}_{2}^{j}\right)}\right)$

are measurable, the multifunction

$U\left(\cdot \right)\cap F\left(t,x\left(\cdot \right),{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{1}}x\left(\cdot \right),\dots ,{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{n}}x\left(\cdot \right)\right)$

is measurable.

Choose ${v}_{2}\left(t\right)\in F\left(t,x\left(t\right),{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{1}}x\left(t\right),\dots ,{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{n}}x\left(t\right)\right)$ such that

$\begin{array}{rl}|{v}_{1}\left(t\right)-{v}_{2}\left(t\right)|\le & m\left(t\right)\psi \left(|x\left(t\right)-y\left(t\right)|+|{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{1}}x\left(t\right)-{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{1}}y\left(t\right)|+\cdots \\ +|{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{n}}x\left(t\right)-{\phantom{\rule{0.1em}{0ex}}}^{c}{D}^{{\gamma }_{n}}y\left(t\right)|\right)\left(\frac{1}{{\parallel m\parallel }_{\mathrm{\infty }}\left({L}_{1}+{\sum }_{j=1}^{n}{L}_{2}^{j}\right)}\right)\end{array}$

for all $t\in J$. Now, consider the element ${h}_{2}\in \mathrm{\Omega }\left(x\right)$ which is defined by ${h}_{2}\left(t\right)={\int }_{0}^{1}G\left(t,s\right){v}_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds$ for all $t\in J$.

Thus,

$\begin{array}{r}|{h}_{1}\left(t\right)-{h}_{2}\left(t\right)|\\ \phantom{\rule{1em}{0ex}}=|{\int }_{0}^{1}G\left(t,s\right){v}_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds-{\int }_{0}^{1}G\left(t,s\right){v}_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds|\\ \phantom{\rule{1em}{0ex}}=|\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{v}_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+\frac{1}{A}\sum _{i=1}^{n}{\int }_{0}^{\eta }\frac{{\left(\eta -s\right)}^{\alpha +{\gamma }_{i}-1}}{\mathrm{\Gamma }\left(\alpha +{\gamma }_{i}\right)}{v}_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{a}{A\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -1}{v}_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}-\frac{1}{AB}\left(a-\sum _{i=1}^{n}\frac{{\eta }^{{\gamma }_{i}+2}}{\mathrm{\Gamma }\left({\gamma }_{i}+3\right)}\right)\\ \phantom{\rule{2em}{0ex}}×\sum _{i=1}^{n}{\int }_{0}^{\eta }\frac{{\left(\eta -s\right)}^{\alpha -{\gamma }_{i}-1}}{\mathrm{\Gamma }\left(\alpha -{\gamma }_{i}\right)}{v}_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{b}{AB\mathrm{\Gamma }\left(\alpha -1\right)}\left(a-\sum _{i=1}^{n}\frac{{\eta }^{{\gamma }_{i}+2}}{\mathrm{\Gamma }\left({\gamma }_{i}+3\right)}\right)\\ \phantom{\rule{2em}{0ex}}×{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -2}{v}_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+\frac{t}{B}\sum _{i=1}^{n}{\int }_{0}^{\eta }\frac{{\left(\eta -s\right)}^{\alpha -{\gamma }_{i}-1}}{\mathrm{\Gamma }\left(\alpha -{\gamma }_{i}\right)}{v}_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds-\frac{tb}{B\mathrm{\Gamma }\left(\alpha -1\right)}{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -2}{v}_{1}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}-\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{v}_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}-\frac{1}{A}\sum _{i=1}^{n}{\int }_{0}^{\eta }\frac{{\left(\eta -s\right)}^{\alpha +{\gamma }_{i}-1}}{\mathrm{\Gamma }\left(\alpha +{\gamma }_{i}\right)}{v}_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+\frac{a}{A\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -1}{v}_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+\frac{1}{AB}\left(a-\sum _{i=1}^{n}\frac{{\eta }^{{\gamma }_{i}+2}}{\mathrm{\Gamma }\left({\gamma }_{i}+3\right)}\right)\\ \phantom{\rule{2em}{0ex}}×\sum _{i=1}^{n}{\int }_{0}^{\eta }\frac{{\left(\eta -s\right)}^{\alpha -{\gamma }_{i}-1}}{\mathrm{\Gamma }\left(\alpha -{\gamma }_{i}\right)}{v}_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+\frac{b}{AB\mathrm{\Gamma }\left(\alpha -1\right)}\left(a-\sum _{i=1}^{n}\frac{{\eta }^{{\gamma }_{i}+2}}{\mathrm{\Gamma }\left({\gamma }_{i}+3\right)}\right)\\ \phantom{\rule{2em}{0ex}}×{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -2}{v}_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}-\frac{t}{B}\sum _{i=1}^{n}{\int }_{0}^{\eta }\frac{{\left(\eta -s\right)}^{\alpha -{\gamma }_{i}-1}}{\mathrm{\Gamma }\left(\alpha -{\gamma }_{i}\right)}{v}_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds+\frac{tb}{B\mathrm{\Gamma }\left(\alpha -1\right)}{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -2}{v}_{2}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds|\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}|{v}_{1}\left(s\right)-{v}_{2}\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+\frac{1}{A}\sum _{i=1}^{n}{\int }_{0}^{\eta }\frac{{\left(\eta -s\right)}^{\alpha +{\gamma }_{i}-1}}{\mathrm{\Gamma }\left(\alpha +{\gamma }_{i}\right)}|{v}_{1}\left(s\right)-{v}_{2}\left(s\right)|\phantom{\rule{0.2em}{0ex}}ds\\ \phantom{\rule{2em}{0ex}}+\frac{a}{A\mathrm{\Gamma }\left(\alpha \right)}{\int }_{0}^{1}{\left(1-}^{}\end{array}$