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, [1–15] and the references therein). Much attention has been devoted to the fractional differential inclusions (see for example, [16–32] and the references therein). Also, there have been provided many applications of this field (see for example, [33, 34] and [35]).

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, [36]. 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 $f:(0,\mathrm{\infty })\to \mathbb{R}$ is given by $I^{\alpha }f(t)=\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t{(t-s)}^{\alpha -1}f(s)ds$, provided the right side is pointwise defined on $(0,\mathrm{\infty })$ (see [10, 13] and [14]). The Caputo fractional derivative of order α for a continuous function f is defined by ${}^{c}D^{\alpha }f(t)=\frac{1}{\mathrm{\Gamma }(n-\alpha )}{\int }_0^t\frac{f^{(n)}(s)}{{(t-s)}^{\alpha -n+1}}ds$, where $n=[\alpha ]+1$ (see [10, 13] and [14]).

Recall that a multifunction $G:J\to P_{cl}(\mathbb{R})$ is said to be measurable whenever the function $t\mapsto d(y,G(t))$ is measurable for all $y\in \mathbb{R}$, where $J=[0,1]$ [37].

Let $(X,d)$ be a metric space. We have the well-known Pompeiu-Hausdorff metric (see [38])

$$H_d:2^X\times 2^X\to [0,\mathrm{\infty }),H_d(A,B)=max\{\underset{a\in A}{sup}d(a,B),\underset{b\in B}{sup}d(A,b)\},$$

where $d(A,b)={inf}_{a\in A}d(a,b)$. Then $(CB(X),H_d)$ is a metric space and $(C(X),H_d)$ is a generalized metric space, where $CB(X)$ is the set of closed and bounded subsets of X and $C(X)$ is the set of closed subsets of X (see [27]).

Let $T:X\to 2^X$ be a multifunction. An element $x\in X$ is called an endpoint of T whenever $Tx=\{x\}$ [36]. Also, we say that T has the approximate endpoint property whenever ${inf}_{x\in X}{sup}_{y\in Tx}d(x,y)=0$ [36]. A function $g:\mathbb{R}\to \mathbb{R}$ is called upper semi-continuous whenever ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}g({\lambda }_n)\le g(\lambda )$ for all sequences ${\{{\lambda }_n\}}_{n\ge 1}$ with ${\lambda }_n\to \lambda$ [36].

In 2010, Amini-Harandi proved the next result [36].

Lemma 2.1 Let $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be an upper semi-continuous function such that $\psi (t)<t$ and ${lim\hspace{0.17em}inf}_{t\to \mathrm{\infty }}(t-\psi (t))>0$, for all $t>0$, $(X,d)$ a complete metric space and $T:X\to CB(X)$ a multifunction such that $H_d(Tx,Ty)\le \psi (d(x,y))$ 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 ${}^{c}D^{\alpha }x(t)\in F(t,x(t))$, via the integral boundary conditions $x^j(0)-{\lambda }_j{x}^j(T)={\mu }_j{\int }_0^1{g}_j(s,x(s))ds$ for $j=0,1,2$, where F is a multifunction (see for more details [20]).

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

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

(1)

via integral boundary value conditions

$$\{\begin{array}{l}x(0)+x(\eta )+x(1)={\int }_0^1{g}_0(s,x(s))ds,\\ {}^{c}D^{\beta }x(0)+{}^c{D}^{\beta }x(\eta )+{}^c{D}^{\beta }x(1)={\int }_0^1{g}_1(s,x(s))ds,\\ {}^{c}D^{\gamma }x(0)+{}^c{D}^{\gamma }x(\eta )+{}^c{D}^{\gamma }x(1)={\int }_0^1{g}_2(s,x(s))ds,\end{array}$$

(2)

where $t\in J$, $2<\alpha \le 3$, $0<\eta ,\beta <1$, $1<\gamma <2$, and $F:J\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to P_{cp}(\mathbb{R})$ is a multifunction, $g_1,g_2,g_3:J\times \mathbb{R}\to \mathbb{R}$ are continuous functions and ${}^{c}D^q$ is the standard Caputo differentiation. Here, $P_{cp}(\mathbb{R})$ is the set of all compact subsets of ℝ.

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

$${}^{c}D^{\alpha }x(t)\in F(t,x(t),{}^c{D}^{{\gamma }_1}x(t),\dots ,{}^c{D}^{{\gamma }_n}x(t)),$$

(3)

via integral boundary value conditions

$$\{\begin{array}{l}{x}^{\prime }(0)+b{x}^{\prime }(1)={\sum }_{i=1}^n{}^c{D}^{{\gamma }_i}x(\eta ),\\ x(0)+ax(1)={\sum }_{i=1}^n{I}^{{\gamma }_i}x(\eta ),\end{array}$$

(4)

where $t\in J=[0,1]$, $1<\alpha \le 2$, $0<\eta ,{\gamma }_i<1$, $\alpha -{\gamma }_i\ge 1$ for all $1\le i\le n$, $a>{\sum }_{i=1}^n\frac{{\eta }^{{\gamma }_i+1}}{\mathrm{\Gamma }({\gamma }_i+2)}$, $b>{\sum }_{i=1}^n\frac{{\eta }^{1-{\gamma }_i}}{\mathrm{\Gamma }(2-{\gamma }_i)}$, $n\ge 1$, and $F:J\times {\mathbb{R}}^{n+1}\to P(\mathbb{R})$ 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 $v\in C(J,\mathbb{R})$, $\alpha \in (2,3]$, $\beta \in (0,1)$, $\gamma \in (1,2)$ and $g_0,g_1,g_2:J\times \mathbb{R}\to \mathbb{R}$ be continuous functions. The unique solution of the fractional differential problem

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

(5)

via the boundary value conditions (2) is given by

$$\begin{array}{rl}x(t)=& \frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t{(t-s)}^{(\alpha -1)}v(s)ds+\frac{1}{3}{\int }_0^1{g}_0(s,x(s))ds\\ -\frac{1}{3\mathrm{\Gamma }(\alpha )}[{\int }_0^1{(1-s)}^{\alpha -1}v(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -1}v(s)ds]\\ +\frac{3\mathrm{\Gamma }(2-\beta )t-(\eta +1)\mathrm{\Gamma }(2-\beta )}{3({\eta }^{1-\beta }+1)}{\int }_0^1{g}_1(s,x(s))ds\\ +\frac{(\eta +1)\mathrm{\Gamma }(2-\beta )-3\mathrm{\Gamma }(2-\beta )t}{3({\eta }^{1-\beta }+1)\mathrm{\Gamma }(\alpha -\beta )}[{\int }_0^1{(1-s)}^{\alpha -\beta -1}v(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -\beta -1}v(s)ds]\\ +(\frac{2(\eta +1)({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )-({\eta }^2+1)\mathrm{\Gamma }(3-\gamma )({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\beta )}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )}\\ +\frac{-6({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )t+3({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(3-\beta )t^2}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )})\\ \times {\int }_0^1{g}_2(s,x(s))ds\\ +(\frac{({\eta }^2+1)\mathrm{\Gamma }(3-\gamma )({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\beta )-2(\eta +1)({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma )}\\ +\frac{6({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )t-3\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(3-\beta )({\eta }^{1-\beta }+1)t^2}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma )})\\ \times [{\int }_0^1{(1-s)}^{\alpha -\gamma -1}v(s)ds{\int }_0^{\eta }{(\eta -s)}^{\alpha -\gamma -1}v(s)ds].\end{array}$$

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

$$x(t)=I^{\alpha }v(t)+c_0+c_{1}t+c_2{t}^2,$$

that is

$$x(t)=\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t{(t-s)}^{\alpha -1}v(s)ds+c_0+c_{1}t+c_2{t}^2,$$

(6)

where $c_0$, $c_1$, $c_2$ are real arbitrary constants (see [10, 13] and [14]). Thus,

$${}^{c}D^{\beta }x(t)=\frac{1}{\mathrm{\Gamma }(\alpha -\beta )}{\int }_0^t{(t-s)}^{\alpha -\beta -1}v(s)ds+\frac{c_1{t}^{1-\beta }}{\mathrm{\Gamma }(2-\beta )}+\frac{2c_2{t}^{2-\beta }}{\mathrm{\Gamma }(3-\beta )}$$

and ${}^{c}D^{\gamma }x(t)=\frac{1}{\mathrm{\Gamma }(\alpha -\gamma )}{\int }_0^t{(t-s)}^{\alpha -\gamma -1}v(s)ds+\frac{2c_2{t}^{2-\gamma }}{\mathrm{\Gamma }(3-\gamma )}$. Hence,

$$\begin{array}{c}\begin{array}{rl}x(0)+x(\eta )+x(1)=& 3c_0+(1+\eta )c_1+(1+{\eta }^2)c_2\\ +\frac{1}{\mathrm{\Gamma }(\alpha )}[{\int }_0^1{(1-s)}^{\alpha -1}v(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -1}v(s)ds],\end{array}\hfill \\ \begin{array}{r}{}^{c}D^{\beta }x(0)+{}^c{D}^{\beta }x(\eta )+{}^c{D}^{\beta }x(1)\\ =c_1\frac{{\eta }^{1-\beta }+1}{\mathrm{\Gamma }(2-\beta )}+c_2\frac{2({\eta }^{2-\beta }+1)}{\mathrm{\Gamma }(3-\beta )}\\ +\frac{1}{\mathrm{\Gamma }(\alpha -\beta )}[{\int }_0^1{(1-s)}^{\alpha -\beta -1}v(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -\beta -1}v(s)ds]\end{array}\hfill \end{array}$$

and

$$\begin{array}{c}{}^{c}D^{\gamma }x(0)+{}^c{D}^{\gamma }x(\eta )+{}^c{D}^{\gamma }x(1)\hfill \\ =c_2\frac{2({\eta }^{2-\gamma }+1)}{\mathrm{\Gamma }(3-\gamma )}+\frac{1}{\mathrm{\Gamma }(\alpha -\gamma )}[{\int }_0^1{(1-s)}^{\alpha -\gamma -1}v(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -\gamma -1}v(s)ds].\hfill \end{array}$$

By using the boundary conditions, we obtain

$$\begin{array}{c}\begin{array}{r}3c_0+(1+\eta )c_1+(1+{\eta }^2)c_2\\ ={\int }_0^1{g}_0(s,x(s))ds-\frac{1}{\mathrm{\Gamma }(\alpha )}[{\int }_0^1{(1-s)}^{\alpha -1}v(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -1}v(s)ds],\end{array}\hfill \\ \begin{array}{r}{c}_1\frac{{\eta }^{1-\beta }+1}{\mathrm{\Gamma }(2-\beta )}+c_2\frac{2({\eta }^{2-\beta }+1)}{\mathrm{\Gamma }(3-\beta )}\\ ={\int }_0^1{g}_1(s,x(s))ds-\frac{1}{\mathrm{\Gamma }(\alpha -\beta )}[{\int }_0^1{(1-s)}^{\alpha -\beta -1}v(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -\beta -1}v(s)ds]\end{array}\hfill \end{array}$$

and

$$\begin{array}{c}{c}_2\frac{2({\eta }^{2-\gamma }+1)}{\mathrm{\Gamma }(3-\gamma )}\hfill \\ ={\int }_0^1{g}_2(s,x(s))ds-\frac{1}{\mathrm{\Gamma }(\alpha -\gamma )}[{\int }_0^1{(1-s)}^{\alpha -\gamma -1}v(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -\beta -1}v(s)ds].\hfill \end{array}$$

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

$$\begin{array}{rl}{c}_0=& \frac{1}{3}{\int }_0^1{g}_0(s,x(s))ds-\frac{1}{3\mathrm{\Gamma }(\alpha )}[{\int }_0^1{(1-s)}^{\alpha -1}v(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -1}v(s)ds]\\ -\frac{\mathrm{\Gamma }(2-\beta )(\eta +1)}{3({\eta }^{1-\beta }+1)}\\ \times {\int }_0^1{g}_1(s,x(s))ds+\frac{(\eta +1)\mathrm{\Gamma }(2-\beta )}{3({\eta }^{1-\beta }+1)\mathrm{\Gamma }(\alpha -\beta )}[{\int }_0^1{(1-s)}^{\alpha -\beta -1}v(s)ds\\ +{\int }_0^{\eta }{(\eta -s)}^{\alpha -\beta -1}v(s)ds]\\ +\frac{2(\eta +1)({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )-({\eta }^2+1)\mathrm{\Gamma }(3-\gamma )({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\beta )}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )}\\ \times {\int }_0^1{g}_2(s,x(s))ds\\ +\left(\frac{({\eta }^2+1)\mathrm{\Gamma }(3-\gamma )({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\beta )-2(\eta +1)({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma )}\right)\\ \times [{\int }_0^1{(1-s)}^{\alpha -\gamma -1}v(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -\gamma -1}v(s)ds],\\ c_1=& \frac{\mathrm{\Gamma }(2-\beta )}{({\eta }^{1-\beta }+1)}{\int }_0^1{g}_1(s,x(s))ds-\frac{\mathrm{\Gamma }(2-\beta )}{({\eta }^{1-\beta }+1)\mathrm{\Gamma }(\alpha -\beta )}[{\int }_0^1{(1-s)}^{\alpha -\beta -1}v(s)ds\\ +{\int }_0^{\eta }{(\eta -s)}^{\alpha -\beta -1}v(s)ds]-\frac{({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )}{({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )}{\int }_0^1{g}_2(s,x(s))ds\\ +\frac{({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )}{({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma )}\\ \times [{\int }_0^1{(1-s)}^{\alpha -\gamma -1}v(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -\gamma -1}v(s)ds],\end{array}$$

and

$$\begin{array}{rl}{c}_2=& \frac{\mathrm{\Gamma }(3-\gamma )}{2({\eta }^{2-\gamma }+1)}{\int }_0^1{g}_2(s,x(s))ds-\frac{\mathrm{\Gamma }(3-\gamma )}{2({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(\alpha -\gamma )}\\ \times [{\int }_0^1{(1-s)}^{\alpha -\gamma -1}v(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -\gamma -1}v(s)ds].\end{array}$$

Now, we replace $c_0$, $c_1$, and $c_2$ in (6) and find the solution $x(t)$ as we stated. This completes the proof. □

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

$$S_{F,x}=\{v\in L^1[0,1]:v(t)\in F(t,x(t),x^{\prime }(t),x^{″}(t))\text{ for almost all }t\in [0,1]\}.$$

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

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

(H2) $g_0,g_1,g_2:J\times \mathbb{R}\to \mathbb{R}$ be continuous functions, $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ a nondecreasing upper semi-continuous map such that ${lim\hspace{0.17em}inf}_{t\to \mathrm{\infty }}(t-\psi (t))>0$ and $\psi (t)<t$ for all $t>0$;

(H3) There exist $m,m_0,m_1,m_2\in C(J,[0,\mathrm{\infty }))$ such that

$$H_d(F(t,x_1,x_2,x_3),F(t,y_1,y_2,y_3))\le \frac{1}{{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3}m(t)\psi (|x_1-y_1|+|x_2-y_2|+|x_3-y_3|)$$

and $|g_j(t,x)-g_j(t,y)|\le \frac{1}{{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3}{m}_j(t)\psi (|x-y|)$ 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

$$\begin{array}{rl}{\mathrm{\Lambda }}_1=& [\frac{{\parallel m\parallel }_{\mathrm{\infty }}}{\mathrm{\Gamma }(\alpha +1)}+\frac{{\parallel m_0\parallel }_{\mathrm{\infty }}}{3}+\frac{2{\parallel m\parallel }_{\mathrm{\infty }}}{3\mathrm{\Gamma }(\alpha +1)}+\frac{5\mathrm{\Gamma }(2-\beta ){\parallel m_1\parallel }_{\mathrm{\infty }}}{3}+\frac{10\mathrm{\Gamma }(2-\beta ){\parallel m\parallel }_{\mathrm{\infty }}}{3\mathrm{\Gamma }(\alpha -\beta +1)}\\ +\frac{10(2\mathrm{\Gamma }(2-\beta )+\mathrm{\Gamma }(3-\beta ))\mathrm{\Gamma }(3-\gamma )({\parallel m_2\parallel }_{\mathrm{\infty }}\mathrm{\Gamma }(\alpha -\gamma +1)+2{\parallel m\parallel }_{\mathrm{\infty }})}{3\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma +1)}],\\ {\mathrm{\Lambda }}_2=& [\frac{{\parallel m\parallel }_{\mathrm{\infty }}}{\mathrm{\Gamma }(\alpha )}+\frac{2\mathrm{\Gamma }(2-\beta ){\parallel m\parallel }_{\mathrm{\infty }}}{\mathrm{\Gamma }(\alpha -\beta +1)}\\ +\frac{(2\mathrm{\Gamma }(2-\beta )+\mathrm{\Gamma }(3-\beta ))\mathrm{\Gamma }(3-\gamma )({\parallel m_2\parallel }_{\mathrm{\infty }}\mathrm{\Gamma }(\alpha -\gamma +1)+2{\parallel m\parallel }_{\mathrm{\infty }})}{\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma +1)}],\end{array}$$

and ${\mathrm{\Lambda }}_3=[\frac{{\parallel m\parallel }_{\mathrm{\infty }}}{\mathrm{\Gamma }(\alpha -1)}+\frac{\mathrm{\Gamma }(3-\gamma )({\parallel m_2\parallel }_{\mathrm{\infty }}\mathrm{\Gamma }(\alpha -\gamma +1)+2{\parallel m\parallel }_{\mathrm{\infty }})}{\mathrm{\Gamma }(\alpha -\gamma +1)}]$, and finally

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

$$N(x)=\{h\in X:\text{ there exist }v\in S_{F,x}\text{ such that }h(t)=w(t)\text{ for all }t\in J\},$$

where

$$\begin{array}{rl}w(t)=& \frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t{(t-s)}^{(\alpha -1)}v(s)ds+\frac{1}{3}{\int }_0^1{g}_0(s,x(s))ds-\frac{1}{3\mathrm{\Gamma }(\alpha )}[{\int }_0^1{(1-s)}^{\alpha -1}v(s)ds\\ +{\int }_0^{\eta }{(\eta -s)}^{\alpha -1}v(s)ds]+\frac{3\mathrm{\Gamma }(2-\beta )t-(\eta +1)\mathrm{\Gamma }(2-\beta )}{3({\eta }^{1-\beta }+1)}{\int }_0^1{g}_1(s,x(s))ds\\ +\frac{(\eta +1)\mathrm{\Gamma }(2-\beta )-3\mathrm{\Gamma }(2-\beta )t}{3({\eta }^{1-\beta }+1)\mathrm{\Gamma }(\alpha -\beta )}[{\int }_0^1{(1-s)}^{\alpha -\beta -1}v(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -\beta -1}v(s)ds]\\ +(\frac{2(\eta +1)({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )-({\eta }^2+1)\mathrm{\Gamma }(3-\gamma )({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\beta )}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )}\\ +\frac{-6({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )t+3({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(3-\beta )t^2}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )})\\ \times {\int }_0^1{g}_2(s,x(s))ds\\ +(\frac{({\eta }^2+1)\mathrm{\Gamma }(3-\gamma )({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\beta )-2(\eta +1)({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma )}\\ +\frac{6({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )t-3\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(3-\beta )({\eta }^{1-\beta }+1)t^2}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma )})\\ \times [{\int }_0^1{(1-s)}^{\alpha -\gamma -1}v(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -\gamma -1}v(s)ds].\end{array}$$

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 $N:X\to P(X)$ has a endpoint which is a solution of the problem (1) and (2).

Note that the multivalued map $t⊢F(t,x(t),x^{\prime }(t),x^{″}(t))$ 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 $N(x)$ is closed subset of X for all $x\in X$.

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

$$\begin{array}{rl}{u}_n(t)=& \frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t{(t-s)}^{(\alpha -1)}{v}_n(s)ds+\frac{1}{3}{\int }_0^1{g}_0(s,x(s))ds-\frac{1}{3\mathrm{\Gamma }(\alpha )}[{\int }_0^1{(1-s)}^{\alpha -1}{v}_n(s)ds\\ +{\int }_0^{\eta }{(\eta -s)}^{\alpha -1}{v}_n(s)ds]+\frac{3\mathrm{\Gamma }(2-\beta )t-(\eta +1)\mathrm{\Gamma }(2-\beta )}{3({\eta }^{1-\beta }+1)}{\int }_0^1{g}_1(s,x(s))ds\\ +\frac{(\eta +1)\mathrm{\Gamma }(2-\beta )-3\mathrm{\Gamma }(2-\beta )t}{3({\eta }^{1-\beta }+1)\mathrm{\Gamma }(\alpha -\beta )}[{\int }_0^1{(1-s)}^{\alpha -\beta -1}{v}_n(s)ds\\ +{\int }_0^{\eta }{(\eta -s)}^{\alpha -\beta -1}{v}_n(s)ds]\\ +(\frac{2(\eta +1)({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )-({\eta }^2+1)\mathrm{\Gamma }(3-\gamma )({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\beta )}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )}\\ +\frac{-6({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )t+3({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(3-\beta )t^2}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )})\\ \times {\int }_0^1{g}_2(s,x(s))ds\\ +(\frac{({\eta }^2+1)\mathrm{\Gamma }(3-\gamma )({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\beta )-2(\eta +1)({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma )}\\ +\frac{6({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )t-3\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(3-\beta )({\eta }^{1-\beta }+1)t^2}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma )})\\ \times [{\int }_0^1{(1-s)}^{\alpha -\gamma -1}{v}_n(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -\gamma -1}{v}_n(s)ds]\end{array}$$

for all $t\in J$.

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

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

$$\begin{array}{r}{u}_n(t)\to u(t)\\ =\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t{(t-s)}^{(\alpha -1)}v(s)ds+\frac{1}{3}{\int }_0^1{g}_0(s,x(s))ds\\ -\frac{1}{3\mathrm{\Gamma }(\alpha )}[{\int }_0^1{(1-s)}^{\alpha -1}v(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -1}v(s)ds]\\ +\frac{3\mathrm{\Gamma }(2-\beta )t-(\eta +1)\mathrm{\Gamma }(2-\beta )}{3({\eta }^{1-\beta }+1)}{\int }_0^1{g}_1(s,x(s))ds\\ +\frac{(\eta +1)\mathrm{\Gamma }(2-\beta )-3\mathrm{\Gamma }(2-\beta )t}{3({\eta }^{1-\beta }+1)\mathrm{\Gamma }(\alpha -\beta )}[{\int }_0^1{(1-s)}^{\alpha -\beta -1}v(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -\beta -1}v(s)ds]\\ +(\frac{2(\eta +1)({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )-({\eta }^2+1)\mathrm{\Gamma }(3-\gamma )({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\beta )}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )}\\ +\frac{-6({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )t+3({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(3-\beta )t^2}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )})\\ \times {\int }_0^1{g}_2(s,x(s))ds\\ +(\frac{({\eta }^2+1)\mathrm{\Gamma }(3-\gamma )({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\beta )-2(\eta +1)({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma )}\\ +\frac{6({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )t-3\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(3-\beta )({\eta }^{1-\beta }+1)t^2}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma )})\\ \times [{\int }_0^1{(1-s)}^{\alpha -\gamma -1}v(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -\gamma -1}v(s)ds]\end{array}$$

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

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

Now, we show that $H_d(N(x),N(y))\le \psi (\parallel x-y\parallel )$.

Let $x,y\in X$ and $h_1\in N(y)$. Choose $v_1\in S_{F,y}$ such that

$$\begin{array}{rl}{h}_1(t)=& \frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t{(t-s)}^{(\alpha -1)}{v}_1(s)ds+\frac{1}{3}{\int }_0^1{g}_0(s,y(s))ds-\frac{1}{3\mathrm{\Gamma }(\alpha )}[{\int }_0^1{(1-s)}^{\alpha -1}{v}_1(s)ds\\ +{\int }_0^{\eta }{(\eta -s)}^{\alpha -1}{v}_1(s)ds]+\frac{3\mathrm{\Gamma }(2-\beta )t-(\eta +1)\mathrm{\Gamma }(2-\beta )}{3({\eta }^{1-\beta }+1)}{\int }_0^1{g}_1(s,y(s))ds\\ +\frac{(\eta +1)\mathrm{\Gamma }(2-\beta )-3\mathrm{\Gamma }(2-\beta )t}{3({\eta }^{1-\beta }+1)\mathrm{\Gamma }(\alpha -\beta )}[{\int }_0^1{(1-s)}^{\alpha -\beta -1}{v}_1(s)ds\\ +{\int }_0^{\eta }{(\eta -s)}^{\alpha -\beta -1}{v}_1(s)ds]\\ +(\frac{2(\eta +1)({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )-({\eta }^2+1)\mathrm{\Gamma }(3-\gamma )({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\beta )}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )}\\ +\frac{-6({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )t+3({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(3-\beta )t^2}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )})\\ \times {\int }_0^1{g}_2(s,y(s))ds\\ +(\frac{({\eta }^2+1)\mathrm{\Gamma }(3-\gamma )({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\beta )-2(\eta +1)({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma )}\\ +\frac{6({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )t-3\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(3-\beta )({\eta }^{1-\beta }+1)t^2}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma )})\\ \times [{\int }_0^1{(1-s)}^{\alpha -\gamma -1}{v}_1(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -\gamma -1}{v}_1(s)ds]\end{array}$$

for almost all $t\in J$.

Since

$$\begin{array}{r}{H}_d(F(t,x(t),x^{\prime }(t),x^{″}(t)),F(t,y(t),y^{\prime }(t),y^{″}(t)))\\ \le \frac{1}{{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3}m(t)\psi (|x(t)-y(t)|+|x^{\prime }(t)-y^{\prime }(t)|+|x^{″}(t)-y^{″}(t)|)\end{array}$$

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

$$|v_1(t)-w|\le \frac{1}{{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3}m(t)\psi (|x(t)-y(t)|+|x^{\prime }(t)-y^{\prime }(t)|+|x^{″}(t)-y^{″}(t)|)$$

for all $t\in J$.

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

$$\begin{array}{rl}U(t)=& \{w\in \mathbb{R}:|v_1(t)-w|\\ \le \frac{1}{{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3}m(t)\psi (|x(t)-y(t)|+|x^{\prime }(t)-y^{\prime }(t)|+|x^{″}(t)-y^{″}(t)|)\}.\end{array}$$

Since $v_1$ and $\phi =m\psi (|x-y|+|x^{\prime }-y^{\prime }|+|x^{″}-y^{″}|)(\frac{1}{{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3})$ are measurable, the multifunction $U(\cdot )\cap F(\cdot ,x(\cdot ),x^{\prime }(\cdot ),x^{″}(\cdot ))$ is measurable.

Choose $v_2(t)\in F(t,x(t),x^{\prime }(t),x^{″}(t))$ such that

$$\begin{array}{r}|v_1(t)-v_2(t)|\\ \le \frac{1}{{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3}m(t)\psi (|x(t)-y(t)|+|x^{\prime }(t)-y^{\prime }(t)|+|x^{″}(t)-y^{″}(t)|)\end{array}$$

for all $t\in J$.

Now, consider the element $h_2\in N(x)$, which is defined by

$$\begin{array}{rl}{h}_2(t)=& \frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t{(t-s)}^{(\alpha -1)}{v}_2(s)ds+\frac{1}{3}{\int }_0^1{g}_0(s,x(s))ds-\frac{1}{3\mathrm{\Gamma }(\alpha )}[{\int }_0^1{(1-s)}^{\alpha -1}{v}_2(s)ds\\ +{\int }_0^{\eta }{(\eta -s)}^{\alpha -1}{v}_2(s)ds]+\frac{3\mathrm{\Gamma }(2-\beta )t-(\eta +1)\mathrm{\Gamma }(2-\beta )}{3({\eta }^{1-\beta }+1)}{\int }_0^1{g}_1(s,x(s))ds\\ +\frac{(\eta +1)\mathrm{\Gamma }(2-\beta )-3\mathrm{\Gamma }(2-\beta )t}{3({\eta }^{1-\beta }+1)\mathrm{\Gamma }(\alpha -\beta )}[{\int }_0^1{(1-s)}^{\alpha -\beta -1}{v}_2(s)ds\\ +{\int }_0^{\eta }{(\eta -s)}^{\alpha -\beta -1}{v}_2(s)ds]\\ +(\frac{2(\eta +1)({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )-({\eta }^2+1)\mathrm{\Gamma }(3-\gamma )({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\beta )}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )}\\ +\frac{-6({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )t+3({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(3-\beta )t^2}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )})\\ \times {\int }_0^1{g}_2(s,x(s))ds\\ +(\frac{({\eta }^2+1)\mathrm{\Gamma }(3-\gamma )({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\beta )-2(\eta +1)({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma )}\\ +\frac{6({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )t-3\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(3-\beta )({\eta }^{1-\beta }+1)t^2}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma )})\\ \times [{\int }_0^1{(1-s)}^{\alpha -\gamma -1}{v}_2(s)ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -\gamma -1}{v}_2(s)ds]\end{array}$$

for all $t\in J$. Thus,

$$\begin{array}{r}|h_1(t)-h_2(t)|\\ \le \frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t{(t-s)}^{(\alpha -1)}|v_1(s)-v_2(s)|ds\\ +\frac{1}{3}{\int }_0^1|g_0(s,y(s))-g_0(s,x(s))|ds+\frac{1}{3\mathrm{\Gamma }(\alpha )}[{\int }_0^1{(1-s)}^{\alpha -1}|v_1(s)-v_2(s)|ds\\ +{\int }_0^{\eta }{(\eta -s)}^{\alpha -1}|v_1(s)-v_2(s)|ds]+\left|\frac{3\mathrm{\Gamma }(2-\beta )t-(\eta +1)\mathrm{\Gamma }(2-\beta )}{3({\eta }^{1-\beta }+1)}\right|\\ \times {\int }_0^1|g_1(s,y(s))-g_1(s,x(s))|ds+\left|\frac{(\eta +1)\mathrm{\Gamma }(2-\beta )-3\mathrm{\Gamma }(2-\beta )t}{3({\eta }^{1-\beta }+1)\mathrm{\Gamma }(\alpha -\beta )}\right|\\ \times [{\int }_0^1{(1-s)}^{\alpha -\beta -1}|v_1(s)-v_2(s)|ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -\beta -1}|v_1(s)-v_2(s)|ds]\\ +\left|\right(\frac{2(\eta +1)({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )-({\eta }^2+1)\mathrm{\Gamma }(3-\gamma )({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\beta )}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )}\\ +\frac{-6({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )t+3({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(3-\beta )t^2}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )}\left)\right|\\ \times {\int }_0^1|g_2(s,y(s))-g_2(s,x(s))|ds\\ +\left|\right(\frac{({\eta }^2+1)\mathrm{\Gamma }(3-\gamma )({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\beta )-2(\eta +1)({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma )}\\ +\frac{6({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )t-3\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(3-\beta )({\eta }^{1-\beta }+1)t^2}{6({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma )}\left)\right|\\ \times [{\int }_0^1{(1-s)}^{\alpha -\gamma -1}|v_1(s)-v_2(s)|ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -\gamma -1}|v_1(s)-v_2(s)|ds]\\ \le \frac{1}{{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3}\psi (\parallel x-y\parallel )[\frac{{\parallel m\parallel }_{\mathrm{\infty }}}{\mathrm{\Gamma }(\alpha +1)}+\frac{{\parallel m_0\parallel }_{\mathrm{\infty }}}{3}+\frac{2{\parallel m\parallel }_{\mathrm{\infty }}}{3\mathrm{\Gamma }(\alpha +1)}+\frac{5\mathrm{\Gamma }(2-\beta ){\parallel m_1\parallel }_{\mathrm{\infty }}}{3}\\ +\frac{10\mathrm{\Gamma }(2-\beta ){\parallel m\parallel }_{\mathrm{\infty }}}{3\mathrm{\Gamma }(\alpha -\beta +1)}\\ +\frac{10(2\mathrm{\Gamma }(2-\beta )+\mathrm{\Gamma }(3-\beta ))\mathrm{\Gamma }(3-\gamma )({\parallel m_2\parallel }_{\mathrm{\infty }}\mathrm{\Gamma }(\alpha -\gamma +1)+2{\parallel m\parallel }_{\mathrm{\infty }})}{3\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma +1)}]\\ =\frac{{\mathrm{\Lambda }}_1}{{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3}\psi (\parallel x-y\parallel ),\\ |h_1^{\prime }(t)-h_2^{\prime }(t)|\\ \le \frac{1}{\mathrm{\Gamma }(\alpha -1)}{\int }_0^t{(t-s)}^{(\alpha -2)}|v_1(s)-v_2(s)|ds\\ +\frac{\mathrm{\Gamma }(2-\beta )}{({\eta }^{1-\beta }+1)\mathrm{\Gamma }(\alpha -\beta )}[{\int }_0^1{(1-s)}^{\alpha -\beta -1}|v_1(s)-v_2(s)|ds\\ +{\int }_0^{\eta }{(\eta -s)}^{\alpha -\beta -1}|v_1(s)-v_2(s)|ds]\\ +\left|\right(\frac{({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )+\mathrm{\Gamma }(3-\gamma )({\eta }^{1-\beta }+1)\mathrm{\Gamma }(3-\beta )t}{({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )}|\\ \times {\int }_0^1|g_2(s,y(s))-g_2(s,x(s))|ds\\ +\left|\frac{({\eta }^{2-\beta }+1)\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(2-\beta )-\mathrm{\Gamma }(3-\gamma )\mathrm{\Gamma }(3-\beta )({\eta }^{1-\beta }+1)t}{({\eta }^{1-\beta }+1)({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma )}\right)|\\ \times [{\int }_0^1{(1-s)}^{\alpha -\gamma -1}|v_1(s)-v_2(s)|ds+{\int }_0^{\eta }{(\eta -s)}^{\alpha -\gamma -1}|v_1(s)-v_2(s)|ds]\\ \le \frac{1}{{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3}\psi (\parallel x-y\parallel )[\frac{{\parallel m\parallel }_{\mathrm{\infty }}}{\mathrm{\Gamma }(\alpha )}+\frac{2\mathrm{\Gamma }(2-\beta ){\parallel m\parallel }_{\mathrm{\infty }}}{\mathrm{\Gamma }(\alpha -\beta +1)}\\ +\frac{(2\mathrm{\Gamma }(2-\beta )+\mathrm{\Gamma }(3-\beta ))\mathrm{\Gamma }(3-\gamma )({\parallel m_2\parallel }_{\mathrm{\infty }}\mathrm{\Gamma }(\alpha -\gamma +1)+2{\parallel m\parallel }_{\mathrm{\infty }})}{\mathrm{\Gamma }(3-\beta )\mathrm{\Gamma }(\alpha -\gamma +1)}]\\ =\frac{{\mathrm{\Lambda }}_2}{{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3}\psi (\parallel x-y\parallel ),\end{array}$$

and

$$\begin{array}{r}|h_1^{\prime \prime }(t)-h_2^{\prime \prime }(t)|\\ \le \frac{1}{\mathrm{\Gamma }(\alpha -2)}{\int }_0^t{(t-s)}^{(\alpha -3)}|v_1(s)-v_2(s)|ds\\ +\frac{\mathrm{\Gamma }(3-\gamma )}{({\eta }^{2-\gamma }+1)}{\int }_0^1|g_2(s,y(s))-g_2(s,x(s))|ds\\ +\frac{\mathrm{\Gamma }(3-\gamma )}{({\eta }^{2-\gamma }+1)\mathrm{\Gamma }(\alpha -\gamma )}[{\int }_0^1{(1-s)}^{\alpha -\gamma -1}|v_1(s)-v_2(s)|ds\\ +{\int }_0^{\eta }{(\eta -s)}^{\alpha -\gamma -1}|v_1(s)-v_2(s)|ds]\\ \le \frac{1}{{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3}\psi (\parallel x-y\parallel )[\frac{{\parallel m\parallel }_{\mathrm{\infty }}}{\mathrm{\Gamma }(\alpha -1)}+\frac{\mathrm{\Gamma }(3-\gamma )({\parallel m_2\parallel }_{\mathrm{\infty }}\mathrm{\Gamma }(\alpha -\gamma +1)+2{\parallel m\parallel }_{\mathrm{\infty }})}{\mathrm{\Gamma }(\alpha -\gamma +1)}]\\ =\frac{{\mathrm{\Lambda }}_3}{{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3}\psi (\parallel x-y\parallel ).\end{array}$$

Hence,

$$\begin{array}{rl}\parallel h_1-h_2\parallel & =\underset{t\in J}{sup}|h_1(t)-h_2(t)|+\underset{t\in J}{sup}|h_1^{\prime }(t)-h_2^{\prime }(t)|+\underset{t\in J}{sup}|h_1^{\prime \prime }(t)-h_2^{\prime \prime }(t)|\\ \le \frac{1}{{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3}\psi (\parallel x-y\parallel )({\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3)=\psi (\parallel x-y\parallel ).\end{array}$$

Thus, it is easy to get $H_d(N(x),N(y))\le \psi (\parallel x-y\parallel )$ 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 $N(x^{\ast })=\{x^{\ast }\}$. 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

$${}^{c}D^{\alpha }x(t)\in F(t,x(t),{}^c{D}^{{\gamma }_1}x(t),\dots ,{}^c{D}^{{\gamma }_n}x(t)),$$

via integral boundary value conditions

$$x^{\prime }(0)+b{x}^{\prime }(1)=\sum _{i=1}^n{}^c{D}^{{\gamma }_i}x(\eta ),x(0)+ax(1)=\sum _{i=1}^n{I}^{{\gamma }_i}x(\eta ),$$

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

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

$$x(0)+ax(1)=\sum _{i=1}^n{I}^{{\beta }_i}x(\eta ),x^{\prime }(0)+b{x}^{\prime }(1)=\sum _{i=1}^n{}^c{D}^{{\beta }_i}x(\eta ),$$

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

$$x(t)={\int }_0^{1}G(t,s)v(s)ds,$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$x(t)=I^{\alpha }v(t)+c_0+c_{1}t=\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t{(t-s)}^{(\alpha -1)}v(s)ds+c_0+c_{1}t,$$

where $c_0,c_1\in \mathbb{R}$ are arbitrary constants (see [10, 13] and [14]). Thus,

$$\begin{array}{c}{}^{c}D^{{\beta }_i}x(t)=\frac{1}{\mathrm{\Gamma }(\alpha -{\beta }_i)}{\int }_0^t{(t-s)}^{(\alpha -{\beta }_i-1)}v(s)ds+\frac{c_1{t}^{1-{\beta }_i}}{\mathrm{\Gamma }(2-{\beta }_i)},\hfill \\ I^{{\beta }_i}x(t)=\frac{1}{\mathrm{\Gamma }(\alpha +{\beta }_i)}{\int }_0^t{(t-s)}^{(\alpha +{\beta }_i-1)}v(s)ds+\frac{c_0{t}^{1+{\beta }_i}}{\mathrm{\Gamma }(2+{\beta }_i)}+\frac{c_1{t}^{2+{\beta }_i}}{\mathrm{\Gamma }(3+{\beta }_i)},\hfill \end{array}$$

and $x^{\prime }(t)=\frac{1}{\mathrm{\Gamma }(\alpha -1)}{\int }_0^t{(t-s)}^{(\alpha -2)}v(s)ds+c_1$. Hence,

$$x(0)+ax(1)=(a+1)c_0+a{c}_1+\frac{a}{\mathrm{\Gamma }(\alpha )}{\int }_0^1{(1-s)}^{(\alpha -1)}v(s)ds$$

and

$$x^{\prime }(0)+b{x}^{\prime }(1)=(1+b)c_1+\frac{b}{\mathrm{\Gamma }(\alpha -1)}{\int }_0^1{(1-s)}^{(\alpha -2)}v(s)ds.$$

By using the boundary conditions, we obtain

$$\begin{array}{r}{c}_0(1+a-\sum _{i=1}^n\frac{{\eta }^{{\beta }_i+1}}{\mathrm{\Gamma }({\beta }_i+2)})+c_1(a-\sum _{i=1}^n\frac{{\eta }^{{\beta }_i+2}}{\mathrm{\Gamma }({\beta }_i+3)})\\ =\sum _{i=1}^n{\int }_0^{\eta }\frac{{(\eta -s)}^{\alpha +{\beta }_i-1}}{\mathrm{\Gamma }({\beta }_i+\alpha )}v(s)ds-\frac{a}{\mathrm{\Gamma }(\alpha )}{\int }_0^1{(1-s)}^{\alpha -1}v(s)ds\end{array}$$

and

$$c_1(1+b-\sum _{i=1}^n\frac{{\eta }^{1-{\beta }_i}}{\mathrm{\Gamma }(2-{\beta }_i)})=\sum _{i=1}^n{\int }_0^{\eta }\frac{{(\eta -s)}^{\alpha -{\beta }_i-1}}{\mathrm{\Gamma }(\alpha -{\beta }_i)}v(s)ds-\frac{b}{\mathrm{\Gamma }(\alpha -1)}{\int }_0^1{(1-s)}^{\alpha -2}v(s)ds.$$

Thus,

$$\begin{array}{rl}{c}_0=& \frac{1}{A}\sum _{i=1}^n{\int }_0^{\eta }\frac{{(\eta -s)}^{\alpha +{\beta }_i-1}}{\mathrm{\Gamma }(\alpha +{\beta }_i)}v(s)ds-\frac{a}{A\mathrm{\Gamma }(\alpha )}{\int }_0^1{(1-s)}^{\alpha -1}v(s)ds\\ -\frac{1}{AB}(a-\sum _{i=1}^n\frac{{\eta }^{{\beta }_i+2}}{\mathrm{\Gamma }({\beta }_i+3)})\\ \times \sum _{i=1}^n{\int }_0^{\eta }\frac{{(\eta -s)}^{\alpha -{\beta }_i-1}}{\mathrm{\Gamma }(\alpha -{\beta }_i)}v(s)ds\\ -\frac{b}{AB\mathrm{\Gamma }(\alpha -1)}(a-\sum _{i=1}^n\frac{{\eta }^{{\beta }_i+2}}{\mathrm{\Gamma }({\beta }_i+3)}){\int }_0^1{(1-s)}^{\alpha -2}v(s)ds,\\ c_1=& \frac{1}{B}\sum _{i=1}^n{\int }_0^{\eta }\frac{{(\eta -s)}^{\alpha -{\beta }_i-1}}{\mathrm{\Gamma }(\alpha -{\beta }_i)}v(s)ds-\frac{b}{B\mathrm{\Gamma }(\alpha -1)}{\int }_0^1{(1-s)}^{\alpha -2}v(s)ds.\end{array}$$

Hence,

$$\begin{array}{rl}x(t)=& \frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t{(t-s)}^{\alpha -1}v(s)ds+\frac{1}{A}\sum _{i=1}^n{\int }_0^{\eta }\frac{{(\eta -s)}^{\alpha +{\beta }_i-1}}{\mathrm{\Gamma }(\alpha +{\beta }_i)}v(s)ds\\ -\frac{a}{A\mathrm{\Gamma }(\alpha )}{\int }_0^1{(1-s)}^{\alpha -1}v(s)ds\\ -\frac{1}{AB}(a-\sum _{i=1}^n\frac{{\eta }^{{\beta }_i+2}}{\mathrm{\Gamma }({\beta }_i+3)})\sum _{i=1}^n{\int }_0^{\eta }\frac{{(\eta -s)}^{\alpha -{\beta }_i-1}}{\mathrm{\Gamma }(\alpha -{\beta }_i)}v(s)ds\\ -\frac{b}{AB\mathrm{\Gamma }(\alpha -1)}(a-\sum _{i=1}^n\frac{{\eta }^{{\beta }_i+2}}{\mathrm{\Gamma }({\beta }_i+3)})\\ \times {\int }_0^1{(1-s)}^{\alpha -2}v(s)ds+\frac{t}{B}\sum _{i=1}^n{\int }_0^{\eta }\frac{{(\eta -s)}^{\alpha -{\beta }_i-1}}{\mathrm{\Gamma }(\alpha -{\beta }_i)}v(s)ds\\ -\frac{tb}{B\mathrm{\Gamma }(\alpha -1)}{\int }_0^1{(1-s)}^{\alpha -2}v(s)ds\\ =& {\int }_0^{1}G(t,s)v(s)ds.\end{array}$$

This completes the proof. □

Suppose that $X=\{x:x,{}^c{D}^{{\gamma }_i}x\in C(J,R)\text{ for all }i=1,\dots ,n\}$ endowed with the norm $\parallel x\parallel ={sup}_{t\in J}|x(t)|+{\sum }_{i=1}^n{sup}_{t\in J}|{}^c{D}^{{\gamma }_i}x(t)|$. Then $(X,\parallel \cdot \parallel )$ is a Banach space [15]. For $x\in X$, define

$$S_{F,x}=\{v\in L^1[0,1]:v(t)\in F(t,x(t),{}^c{D}^{{\gamma }_1}x(t),\dots ,{}^c{D}^{{\gamma }_n}x(t))\text{ for almost all }t\in [0,1]\}.$$

Now, put

$$\begin{array}{rl}{L}_1=& \frac{1}{\mathrm{\Gamma }(\alpha +1)}+\frac{1}{A}\sum _{i=1}^n\frac{{\eta }^{\alpha +{\gamma }_i}}{\mathrm{\Gamma }(\alpha +{\gamma }_i+1)}+\frac{a}{A\mathrm{\Gamma }(\alpha +1)}+\frac{1}{AB}\left(|a-\sum _{i=1}^n\frac{{\eta }^{{\gamma }_i+2}}{\mathrm{\Gamma }({\gamma }_i+3)}|\right)\\ \times \sum _{i=1}^n\frac{{\eta }^{\alpha -{\gamma }_i}}{\mathrm{\Gamma }(\alpha -{\gamma }_i+1)}+\frac{b}{AB\mathrm{\Gamma }(\alpha )}\left(|a-\sum _{i=1}^n\frac{{\eta }^{{\gamma }_i+2}}{\mathrm{\Gamma }({\gamma }_i+3)}|\right)\\ +\frac{1}{B}\sum _{i=1}^n\frac{{\eta }^{\alpha -{\gamma }_i}}{\mathrm{\Gamma }(\alpha -{\gamma }_i+1)}+\frac{b}{B\mathrm{\Gamma }(\alpha )}\end{array}$$

and $L_2^j=\frac{1}{\mathrm{\Gamma }(\alpha -{\gamma }_j+1)}+\frac{1}{B\mathrm{\Gamma }(2-{\gamma }_j)}{\sum }_{i=1}^n\frac{{\eta }^{\alpha -{\gamma }_i}}{\mathrm{\Gamma }(\alpha -{\gamma }_i+1)}+\frac{b}{B\mathrm{\Gamma }(2-{\gamma }_j)\mathrm{\Gamma }(\alpha )}$ for all $1\le j\le n$.

Theorem 3.2 Let $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ a nondecreasing upper semi-continuous map such that ${lim\hspace{0.17em}inf}_{t\to \mathrm{\infty }}(t-\psi (t))>0$ and $\psi (t)<t$ for all $t>0$, $F:J\times {\mathbb{R}}^{n+1}\to P_{cp}(\mathbb{R})$ a multifunction such that $F(\cdot ,x_1,x_2,\dots ,x_{n+1}):[0,1]\to P_{cp}(\mathbb{R})$ is measurable and integrable bounded for all $x_1,x_2,\dots ,x_{n+1}\in \mathbb{R}$. Assume that there exists $m\in C(J,[0,\mathrm{\infty }))$ such that

$$\begin{array}{c}{H}_d(F(t,x_1,x_2,\dots ,x_{n+1})-F(t,y_1,y_2,\dots ,y_{n+1}))\hfill \\ \le m(t)\psi (|x_1-y_1|+|x_2-y_2|+\cdots +|x_{n+1}-y_{n+1}|)\left(\frac{1}{{\parallel m\parallel }_{\mathrm{\infty }}(L_1+{\sum }_{j=1}^n{L}_2^j)}\right).\hfill \end{array}$$

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

$$\mathrm{\Omega }(x)=\{h\in X:\mathit{\text{ there exist }}v\in S_{F,x}\mathit{\text{ such that }}h(t)={\int }_0^{1}G(t,s)v(s)ds\mathit{\text{ for all }}t\in J\}.$$

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 $\mathrm{\Omega }:X\to P(X)$ has a endpoint which is a solution of the problem (3) and (4).

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

Let $x\in X$ and ${\{u_n\}}_{n\ge 1}$ be a sequence in $\mathrm{\Omega }(x)$ with $u_n\to u$. For each $n\in \mathbb{N}$, choose $v_n\in S_{F,x}$ such that $u_n(t)={\int }_0^{1}G(t,s)v_n(s)ds$ for all $t\in J$. Since F has compact values, ${\{v_n\}}_{n\ge 1}$ has a subsequence which converges to some $v\in L^1[0,1]$. We denote this subsequence again by ${\{v_n\}}_{n\ge 1}$.

It is easy to check that $v\in S_{F,x}$ and $u_n(t)\to u(t)={\int }_0^{1}G(t,s)v(s)ds$ for all $t\in J$. This implies that $u\in \mathrm{\Omega }(x)$ and so Ω has closed values. Since F is a compact multivalued map, it is easy to check that $\mathrm{\Omega }(x)$ is a bounded set for all $x\in X$.

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

Let $x,y\in X$ and $h_1\in \mathrm{\Omega }(y)$. Choose $v_1\in S_{F,y}$ such that $h_1(t)={\int }_0^{1}G(t,s)v_1(s)ds$ for almost all $t\in J$. Since

$$\begin{array}{c}{H}_d(F(t,x(t),{}^c{D}^{{\gamma }_1}x(t),\dots ,{}^c{D}^{{\gamma }_n}x(t)),F(t,y(t),{}^c{D}^{{\gamma }_1}y(t),\dots ,{}^c{D}^{{\gamma }_n}y(t)))\hfill \\ \le m(t)\psi (|x(t)-y(t)|+|{}^c{D}^{{\gamma }_1}x(t)-{}^c{D}^{{\gamma }_1}y(t)|+\cdots +|{}^c{D}^{{\gamma }_n}x(t)-{}^c{D}^{{\gamma }_n}y(t)|)\hfill \\ \times \left(\frac{1}{{\parallel m\parallel }_{\mathrm{\infty }}(L_1+{\sum }_{j=1}^n{L}_2^j)}\right)\hfill \end{array}$$

for all $t\in J$, there exists $w\in F(t,x(t),{}^c{D}^{{\gamma }_1}x(t),\dots ,{}^c{D}^{{\gamma }_n}x(t))$ such that

$$\begin{array}{rl}|v_1(t)-w|\le & m(t)\psi (|x(t)-y(t)|+|{}^c{D}^{{\gamma }_1}x(t)-{}^c{D}^{{\gamma }_1}y(t)|+\cdots \\ +|{}^c{D}^{{\gamma }_n}x(t)-{}^c{D}^{{\gamma }_n}y(t)|)\left(\frac{1}{{\parallel m\parallel }_{\mathrm{\infty }}(L_1+{\sum }_{j=1}^n{L}_2^j)}\right)\end{array}$$

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

$$\begin{array}{rl}U(t)=& \{w\in \mathbb{R}:|v_1(t)-w|\le m(t)\psi (|x(t)-y(t)|+|{}^c{D}^{{\gamma }_1}x(t)-{}^c{D}^{{\gamma }_1}y(t)|+\cdots \\ +|{}^c{D}^{{\gamma }_n}x(t)-{}^c{D}^{{\gamma }_n}y(t)|\left)\left(\frac{1}{{\parallel m\parallel }_{\mathrm{\infty }}(L_1+{\sum }_{j=1}^n{L}_2^j)}\right)\right\}.\end{array}$$

Since $v_1$ and

$$\phi =m\psi (|x-y|+|{}^c{D}^{{\gamma }_1}x-{}^c{D}^{{\gamma }_1}y|+\cdots +|{}^c{D}^{{\gamma }_n}x-{}^c{D}^{{\gamma }_n}y|)\left(\frac{1}{{\parallel m\parallel }_{\mathrm{\infty }}(L_1+{\sum }_{j=1}^n{L}_2^j)}\right)$$

are measurable, the multifunction

$$U(\cdot )\cap F(t,x(\cdot ),{}^c{D}^{{\gamma }_1}x(\cdot ),\dots ,{}^c{D}^{{\gamma }_n}x(\cdot ))$$

is measurable.

Choose $v_2(t)\in F(t,x(t),{}^c{D}^{{\gamma }_1}x(t),\dots ,{}^c{D}^{{\gamma }_n}x(t))$ such that

$$\begin{array}{rl}|v_1(t)-v_2(t)|\le & m(t)\psi (|x(t)-y(t)|+|{}^c{D}^{{\gamma }_1}x(t)-{}^c{D}^{{\gamma }_1}y(t)|+\cdots \\ +|{}^c{D}^{{\gamma }_n}x(t)-{}^c{D}^{{\gamma }_n}y(t)|)\left(\frac{1}{{\parallel m\parallel }_{\mathrm{\infty }}(L_1+{\sum }_{j=1}^n{L}_2^j)}\right)\end{array}$$

for all $t\in J$. Now, consider the element $h_2\in \mathrm{\Omega }(x)$ which is defined by $h_2(t)={\int }_0^{1}G(t,s)v_2(s)ds$ for all $t\in J$.

Thus,

$$\begin{array}{r}|h_1(t)-h_2(t)|\\ =|{\int }_0^{1}G(t,s)v_1(s)ds-{\int }_0^{1}G(t,s)v_2(s)ds|\\ =|\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t{(t-s)}^{\alpha -1}{v}_1(s)ds\\ +\frac{1}{A}\sum _{i=1}^n{\int }_0^{\eta }\frac{{(\eta -s)}^{\alpha +{\gamma }_i-1}}{\mathrm{\Gamma }(\alpha +{\gamma }_i)}{v}_1(s)ds-\frac{a}{A\mathrm{\Gamma }(\alpha )}{\int }_0^1{(1-s)}^{\alpha -1}{v}_1(s)ds\\ -\frac{1}{AB}(a-\sum _{i=1}^n\frac{{\eta }^{{\gamma }_i+2}}{\mathrm{\Gamma }({\gamma }_i+3)})\\ \times \sum _{i=1}^n{\int }_0^{\eta }\frac{{(\eta -s)}^{\alpha -{\gamma }_i-1}}{\mathrm{\Gamma }(\alpha -{\gamma }_i)}{v}_1(s)ds-\frac{b}{AB\mathrm{\Gamma }(\alpha -1)}(a-\sum _{i=1}^n\frac{{\eta }^{{\gamma }_i+2}}{\mathrm{\Gamma }({\gamma }_i+3)})\\ \times {\int }_0^1{(1-s)}^{\alpha -2}{v}_1(s)ds\\ +\frac{t}{B}\sum _{i=1}^n{\int }_0^{\eta }\frac{{(\eta -s)}^{\alpha -{\gamma }_i-1}}{\mathrm{\Gamma }(\alpha -{\gamma }_i)}{v}_1(s)ds-\frac{tb}{B\mathrm{\Gamma }(\alpha -1)}{\int }_0^1{(1-s)}^{\alpha -2}{v}_1(s)ds\\ -\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t{(t-s)}^{\alpha -1}{v}_2(s)ds\\ -\frac{1}{A}\sum _{i=1}^n{\int }_0^{\eta }\frac{{(\eta -s)}^{\alpha +{\gamma }_i-1}}{\mathrm{\Gamma }(\alpha +{\gamma }_i)}{v}_2(s)ds+\frac{a}{A\mathrm{\Gamma }(\alpha )}{\int }_0^1{(1-s)}^{\alpha -1}{v}_2(s)ds\\ +\frac{1}{AB}(a-\sum _{i=1}^n\frac{{\eta }^{{\gamma }_i+2}}{\mathrm{\Gamma }({\gamma }_i+3)})\\ \times \sum _{i=1}^n{\int }_0^{\eta }\frac{{(\eta -s)}^{\alpha -{\gamma }_i-1}}{\mathrm{\Gamma }(\alpha -{\gamma }_i)}{v}_2(s)ds+\frac{b}{AB\mathrm{\Gamma }(\alpha -1)}(a-\sum _{i=1}^n\frac{{\eta }^{{\gamma }_i+2}}{\mathrm{\Gamma }({\gamma }_i+3)})\\ \times {\int }_0^1{(1-s)}^{\alpha -2}{v}_2(s)ds\\ -\frac{t}{B}\sum _{i=1}^n{\int }_0^{\eta }\frac{{(\eta -s)}^{\alpha -{\gamma }_i-1}}{\mathrm{\Gamma }(\alpha -{\gamma }_i)}{v}_2(s)ds+\frac{tb}{B\mathrm{\Gamma }(\alpha -1)}{\int }_0^1{(1-s)}^{\alpha -2}{v}_2(s)ds|\\ \le \frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^t{(t-s)}^{\alpha -1}|v_1(s)-v_2(s)|ds\\ +\frac{1}{A}\sum _{i=1}^n{\int }_0^{\eta }\frac{{(\eta -s)}^{\alpha +{\gamma }_i-1}}{\mathrm{\Gamma }(\alpha +{\gamma }_i)}|v_1(s)-v_2(s)|ds\\ +\frac{a}{A\mathrm{\Gamma }(\alpha )}{\int }_0^1{(1-s)}^{\alpha -1}|v_1(s)-v_2(s)|ds\\ +\frac{1}{AB}\left(|a-\sum _{i=1}^n\frac{{\eta }^{{\gamma }_i+2}}{\mathrm{\Gamma }({\gamma }_i+3)}|\right)\sum _{i=1}^n{\int }_0^{\eta }\frac{{(\eta -s)}^{\alpha -{\gamma }_i-1}}{\mathrm{\Gamma }(\alpha -{\gamma }_i)}|v_1(s)-v_2(s)|ds\\ +\frac{b}{AB\mathrm{\Gamma }(\alpha -1)}\left(|a-\sum _{i=1}^n\frac{{\eta }^{{\gamma }_i+2}}{\mathrm{\Gamma }({\gamma }_i+3)}|\right)\\ \times {\int }_0^1{(1-s)}^{\alpha -2}|v_1(s)-v_2(s)|ds+\frac{t}{B}\sum _{i=1}^n{\int }_0^{\eta }\frac{{(\eta -s)}^{\alpha -{\gamma }_i-1}}{\mathrm{\Gamma }(\alpha -{\gamma }_i)}|v_1(s)-v_2(s)|ds\\ +\frac{tb}{B\mathrm{\Gamma }(\alpha -1)}{\int }_0^1{(1-s)}^{\alpha -2}|v_1(s)-v_2(s)|ds\\ \le \left(\frac{L_1}{L_1+{\sum }_{j=1}^n{L}_2^j}\right)\psi (\parallel x-y\parallel ),\end{array}$$

and

$$\begin{array}{r}|{}^c{D}^{{\gamma }_j}{h}_1(t)-{}^c{D}^{{\gamma }_j}{h}_2(t)|\\ \le \frac{1}{\mathrm{\Gamma }(\alpha -{\gamma }_j)}{\int }_0^t{(t-s)}^{\alpha -{\gamma }_j-1}|v_1(s)-v_2(s)|ds+\frac{t^{1-{\gamma }_j}}{B\mathrm{\Gamma }(2-{\gamma }_j)}\\ \times \sum _{i=1}^n{\int }_0^{\eta }\frac{{(\eta -s)}^{\alpha -{\gamma }_i-1}}{\mathrm{\Gamma }(\alpha -{\gamma }_i)}|v_1(s)-v_2(s)|ds\\ +\frac{b{t}^{1-{\gamma }_j}}{B\mathrm{\Gamma }(2-{\gamma }_j)\mathrm{\Gamma }(\alpha -1)}{\int }_0^1{(1-s)}^{\alpha -2}|v_1(s)-v_2(s)|ds\\ \le \left(\frac{L_2^j}{L_1+{\sum }_{j=1}^n{\mathrm{\Lambda }}_2^j}\right)\psi (\parallel x-y\parallel )\end{array}$$

for all $1\le j\le n$. Hence,

$$\begin{array}{rl}\parallel h_1-h_2\parallel & =\underset{t\in J}{sup}|h_1(t)-h_2(t)|+\underset{t\in J}{sup}\sum _{i=1}^n|{}^c{D}^{{\gamma }_i}{h}_1(t)-{}^c{D}^{{\gamma }_i}{h}_2(t)|\\ \le \psi (\parallel x-y\parallel )(\frac{L_1}{L_1+{\sum }_{j=1}^n{L}_2^j}+\sum _{i=1}^n\frac{L_2^i}{L_1+{\sum }_{j=1}^n{L}_2^j})=\psi (\parallel x-y\parallel ).\end{array}$$

Analogously, interchanging the roles of x, y, we obtain $H_d(\mathrm{\Omega }(x),\mathrm{\Omega }(y))\le \psi (\parallel x-y\parallel )$. Since the multifunction Ω has the approximate endpoint property, by using Lemma 3.2 there exists $x^{\ast }\in X$ such that $\mathrm{\Omega }(x^{\ast })=\{x^{\ast }\}$. □

4 Example

Here, we give an example to illustrate our first main result.

Example 4.1 Consider the fractional differential inclusion problem via the integral boundary conditions

$$\{\begin{array}{l}{}^{c}D^{\frac{5}{2}}x(t)\in [0,\frac{t}{1,000}sinx(t)+\frac{1}{1,000}cos{x}^{\prime }(t)+\frac{1}{1,000}\frac{|x^{″}(t)|}{1+|x^{″}(t)|}],\\ x(0)+x(\frac{1}{2})+x(1)={\int }_0^1\frac{s}{300}sinx(s)ds,\\ {}^{c}D^{\frac{1}{2}}x(0)+{}^c{D}^{\frac{1}{2}}x(\frac{1}{2})+{}^c{D}^{\frac{1}{2}}x(1)={\int }_0^1\frac{e^{s-1}}{300}sinx(s)ds,\\ {}^{c}D^{\frac{3}{2}}x(0)+{}^c{D}^{\frac{3}{2}}x(\frac{3}{2})+{}^c{D}^{\frac{3}{2}}x(1)={\int }_0^1\frac{2s+1}{286\pi }sinx(s)ds,\end{array}$$

where $t\in [0,1]$. Define the maps

$$\begin{array}{c}F:[0,1]\times {\mathbb{R}}^3\to P(\mathbb{R}),F(t,x,y,z)=[0,\frac{t}{1,000}sinx+\frac{1}{1,000}cosy+\frac{1}{1,000}\frac{|z|}{1+|z|}],\hfill \\ g_0:[0,1]\times \mathbb{R}\to \mathbb{R},g_0(t,x)=\frac{t}{300}sinx,\hfill \\ g_1:[0,1]\times \mathbb{R}\to \mathbb{R},g_1(t,x)=\frac{e^{t-1}}{300}sinx,\hfill \\ g_2:[0,1]\times \mathbb{R}\to \mathbb{R},g_2(t,x)=\frac{2t+1}{286\pi }sinx,\hfill \end{array}$$

and $N:C^2([0,1])\to 2^{C^2([0,1])}$ by the rule

$$N(x)=\{h\in C^2([0,1]):\text{ there exist }v\in S_{F,x}\text{ such that }h(t)=w(t)\text{ for all }t\in [0,1]\},$$

where

$$\begin{array}{rl}w(t)=& \frac{1}{\mathrm{\Gamma }(\frac{5}{2})}{\int }_0^t{(t-s)}^{\frac{3}{2}}v(s)ds+\frac{1}{3}{\int }_0^1\frac{s}{300}sinx(s)ds\\ -\frac{1}{3\mathrm{\Gamma }(\frac{5}{2})}[{\int }_0^1{(1-s)}^{\frac{3}{2}}v(s)ds+{\int }_0^{\frac{1}{2}}{(\frac{1}{2}-s)}^{\frac{3}{2}}v(s)ds]\\ +\frac{3\mathrm{\Gamma }(\frac{3}{2})t-\frac{3}{2}\mathrm{\Gamma }(\frac{3}{2})}{3({\frac{1}{2}}^{\frac{1}{2}}+1)}{\int }_0^1\frac{e^{s-1}}{300}sinx(s)ds+\frac{\frac{3}{2}\mathrm{\Gamma }(\frac{3}{2})-3\mathrm{\Gamma }(\frac{3}{2})t}{3({\frac{1}{2}}^{\frac{1}{2}}+1)}\\ \times [{\int }_0^1(1-s)v(s)ds+{\int }_0^{\frac{1}{2}}(\frac{1}{2}-s)v(s)ds]\\ +(\frac{3({\frac{1}{2}}^{\frac{5}{2}}+1)\mathrm{\Gamma }{(\frac{3}{2})}^2-\frac{5}{4}\mathrm{\Gamma }(\frac{3}{2})({\frac{1}{2}}^{\frac{1}{2}}+1)\mathrm{\Gamma }(\frac{5}{2})}{6{({\frac{1}{2}}^{\frac{1}{2}}+1)}^2\mathrm{\Gamma }(\frac{5}{2})}\\ +\frac{-6({\frac{1}{2}}^{\frac{3}{2}}+1)\mathrm{\Gamma }{(\frac{3}{2})}^{2}t+3({\frac{1}{2}}^{\frac{1}{2}}+1)\mathrm{\Gamma }(\frac{3}{2})\mathrm{\Gamma }(\frac{5}{2})t^2}{6{({\frac{1}{2}}^{\frac{1}{2}}+1)}^2\mathrm{\Gamma }(\frac{5}{2})}){\int }_0^1\frac{2s+1}{286\pi }sinx(s)ds\\ +(\frac{(\frac{5}{4})\mathrm{\Gamma }(\frac{3}{2})({\frac{1}{2}}^{\frac{1}{2}}+1)\mathrm{\Gamma }(\frac{3}{2})-3({\frac{1}{2}}^{\frac{3}{2}}+1)\mathrm{\Gamma }(\frac{3}{2})\mathrm{\Gamma }(\frac{3}{2})}{6{({\frac{1}{2}}^{\frac{1}{2}}+1)}^2\mathrm{\Gamma }(\frac{5}{2})}\\ +\frac{6({\frac{1}{2}}^{\frac{3}{2}}+1)\mathrm{\Gamma }(\frac{3}{2})\mathrm{\Gamma }(\frac{3}{2})t-3\mathrm{\Gamma }(\frac{3}{2})\mathrm{\Gamma }(\frac{5}{2})({\frac{1}{2}}^{\frac{1}{2}}+1)t^2}{6{({\frac{1}{2}}^{\frac{1}{2}}+1)}^2\mathrm{\Gamma }(\frac{5}{2})})\\ \times [{\int }_0^{1}v(s)ds+{\int }_0^{\frac{1}{2}}v(s)ds].\end{array}$$

Put $m(t)=\frac{3t}{1,000}$, $m_0(t)=\frac{t}{100}$, $m_1(t)=\frac{e^{t-1}}{100}$, $m_2(t)=\frac{2t+1}{100\pi }$, $\psi (t)=\frac{t}{3}$, $\alpha =\frac{5}{2}$, $\beta =\frac{1}{2}$, $\gamma =\frac{3}{2}$, $\eta =\frac{1}{2}$. Then we have ${\mathrm{\Lambda }}_1\cong 0.135$, ${\mathrm{\Lambda }}_2\cong 0.037$, and ${\mathrm{\Lambda }}_3\cong 0.017$.

It is easy to check that

$$H_d(F(t,x_1,x_2,x_3),F(t,y_1,y_2,y_3))\le \frac{1}{{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3}m(t)\psi (|x_1-y_1|+|x_2-y_2|+|x_3-y_3|)$$

and $|g_j(t,x)-g_j(t,y)|\le \frac{1}{{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_3}{m}_j(t)\psi (|x-y|)$ for all $t\in [0,1]$, $x,y,x_1,x_2,x_3,y_1,y_2,y_3\in \mathbb{R}$ and $j=0,1,2$. Note that ${sup}_{x\in N(0)}\parallel x\parallel =0$ and so ${inf}_{x\in C^2([0,1])}{sup}_{y\in N(x)}\parallel x-y\parallel =0$. Hence, N has the approximate endpoint property. Now by using Theorem 3.1, the system of fractional differential inclusions has at least one solution.

5 Concluding remarks

This work contains our dedicated study to develop and improve methods for studying two fractional differential inclusions via integral boundary value conditions. We introduced our result by using an endpoint result for multifunctions, due to Amini-Harandi [36]. This study is motivated by relevant applications for solving many real-world problems which give rise to mathematical models in the sphere of boundary value problems.