On two fractional differential inclusions

Abstract

We investigate in this manuscript the existence of solution for two fractional differential inclusions. At first we discuss the existence of solution of a class of fractional hybrid differential inclusions. To illustrate our results we present an illustrative example. We study the existence and dimension of the solution set for some fractional differential inclusions.

Background

As you know, fractional dynamical systems be used in modeling of some real processes and there are many published works about the existence of solutions for many fractional differential equations (see for example Baleanu et al. 2013a, b, c; Chai 2013 and the references therein) and inclusions (see for example, Benchohra and Hamidi 2010; Agarwal et al. 2013; Ahmad et al. 2013; Nieto et al. 2013; Ouahab 2008; Phung and Truong 2013; Bragdi et al. 2013 and the references therein). For finding more details about elementary notions and definitions of fractional differential equations and inclusions one can study well-known books (see for example Aubin and Ceuina 1984; Deimling 1992; Kilbas et al. 2006; Kisielewicz 1991; Podlubny 1999). Recently, it has been published many useful works about modeling of fractional differential equations via providing different applications in some fields (see for example Atangana 2016; Atangana and Alkahtani 2016; Atangana and Koca 2016a, b). In this article, we first review the existence solution for the fractional hybrid derivative inclusion ${}^c{D}^{\mathit{\alpha }}\left(\frac{y(s)}{g(s,y(s),I^{{\mathit{\alpha }}_1}y(s),\cdots ,I^{{\mathit{\alpha }}_n}y(s))}\right)\in G(s,y(s),I^{{\mathit{\beta }}_1}y(s),\cdots ,I^{{\mathit{\beta }}_k}y(s))$ with boundary conditions $y(0)=y_0$ and $y(1)=y_1$, where $1<\mathit{\alpha }\le 2$, ${\mathit{\alpha }}_1,\cdots ,{\mathit{\alpha }}_n>0$, ${\mathit{\beta }}_1,\cdots ,{\mathit{\beta }}_k>0$, $y_0,y_1\in \mathbb{R}$, ${}^c{D}^{\mathit{\alpha }}$ denotes Caputo fractional derivative of order $\mathit{\alpha }$, $g:J\times {\mathbb{R}}^n\to \mathbb{R}-\{0\}$ is continuous and $G:J\times {\mathbb{R}}^k\to \mathcal{P}(\mathbb{R})$ is a multifunction via some properties. Also, we review existence and dimension of the solution set of fractional derivative inclusion

$$\begin{array}{c}\hfill {}^c{D}^{\mathit{\alpha }}y(s)\in G\left(s,y(s),(\mathit{\varphi }y)(s),(\mathit{\psi }y)(s),{}^c{D}^{{\mathit{\beta }}_1}y(s),\cdots ,{}^c{D}^{{\mathit{\beta }}_k}y(s),I^{{\mathit{\gamma }}_1}y(s),\cdots ,I^{{\mathit{\gamma }}_k}y(s)\right)\end{array}$$

with boundary condition $y(0)+{\sum }_{i=1}^k{}^c{D}^{{\mathit{\beta }}_i}y(1)+{\sum }_{i=1}^k{I}^{{\mathit{\gamma }}_i}y(1)=0$, where $0<{\mathit{\beta }}_i<\mathit{\alpha }\le 1$ and $0<{\mathit{\gamma }}_i<1$ for $i=1,\cdots ,k$, $G:J\times {\mathbb{R}}^{2k+3}\to \mathcal{P}(\mathbb{R})$ is a multifunction via some properties, $\mathit{\gamma },\mathit{\lambda }:J\times J\to [0,\infty )$ are two mappings with the properties ${sup}_{s\in J}|{\int }_0^s\mathit{\lambda }(s,t)ds|<\infty$ and ${sup}_{s\in J}|{\int }_0^s\mathit{\gamma }(s,t)ds|<\infty$ and the functions $\mathit{\varphi }$ and $\mathit{\psi }$ are defined by $(\mathit{\varphi }y)(s)={\int }_0^s\mathit{\gamma }(s,t)y(t)dt$ and $(\mathit{\psi }y)(s)={\int }_0^s\mathit{\lambda }(s,t)y(t)dt$.

Preliminaries

Suppose that $(\mathcal{X},d)$ be a metric space. Denote by $\mathcal{P}(\mathcal{X})$ and $2^{\mathcal{X}}$ the class of all subsets and the class of all nonempty subsets of $\mathcal{X}$ respectively. Here, ${\mathcal{P}}_{cl}(\mathcal{X})$, ${\mathcal{P}}_{bd}(\mathcal{X})$, ${\mathcal{P}}_{cv}(\mathcal{X})$ and ${\mathcal{P}}_{cp}(\mathcal{X})$ denote the class of all closed, bounded, convex and compact subsets of $\mathcal{X}$ respectively. A mapping $\mathcal{Q}:\mathcal{X}\to 2^{\mathcal{X}}$ is called a multifunction on $\mathcal{X}$ and $x\in \mathcal{X}$ is called a fixed point of $\mathcal{Q}$ whenever $x\in \mathcal{Q}x$ (Deimling 1992). A multifunction $\mathcal{Q}:\mathcal{X}\to \mathcal{P}(\mathcal{X})$ is called lower semi-continuous whenever the set ${\mathcal{Q}}^{-1}(A):=\{x\in \mathcal{X}:\mathcal{Q}x\cap A\ne \mathrm{\varnothing }\}$ is open for each open subset A of $\mathcal{X}$ (Kisielewicz 1991). If the set $\{x\in \mathcal{X}:\mathcal{Q}x\subset A\}$ is open for each open set A of $\mathcal{X}$, then we say that $\mathcal{Q}$ is upper semi-continuous (Kisielewicz 1991). A multifunction $\mathcal{Q}:\mathcal{X}\to \mathcal{P}(\mathcal{X})$ is called compact whenever $\overline{\mathcal{Q}(S)}$ is a compact for each bounded subsets S of $\mathcal{X}$ (Aubin and Ceuina 1984). A multifunction $\mathcal{Q}:J\to {\mathcal{P}}_{cl}(\mathbb{R})$ is said to be measurable whenever the function $s\mapsto d(y,\mathcal{Q}(s))=inf\{|y-z|:z\in \mathcal{Q}(s)\}$ is measurable for all $y\in \mathbb{R}$ and $s\in J=[0,1]$ (Deimling 1992). The Pompeiu–Hausdorff metric $H=H_d$ on $2^{\mathcal{X}}\times 2^{\mathcal{X}}$ into $[0,\infty )$ is defined by $H(\mathcal{A},\mathcal{B})=max\{{sup}_{a\in \mathcal{A}}d(a,\mathcal{B}),{sup}_{b\in \mathcal{B}}d(\mathcal{A},b)\}$, where $d(\mathcal{A},b)={inf}_{a\in \mathcal{A}}d(a;b)$ (Berinde and Pacurar 2013). Then $({\mathcal{P}}_{bd,cl}(\mathcal{X}),H)$ is a metric space and $({\mathcal{P}}_{cl}(\mathcal{X}),H)$ is a generalized metric space (Berinde and Pacurar 2013). A multifunction $\mathcal{Q}:\mathcal{X}\to {\mathcal{P}}_{cl}(\mathcal{X})$ is called a contraction whenever there exists $\mathit{\gamma }\in (0,1)$ such that $H_d(\mathcal{Q}(x),\mathcal{Q}(y))\le \mathit{\gamma }d(x,y)$ for all $x,y\in \mathcal{X}$ (Covitz and Nadler 1970). Covitz and Nadler (1970) proved that each closed valued contractive multifunction on a complete metric space has a fixed point. We say that $\mathcal{Q}:J\times {\mathbb{R}}^k\to 2^{\mathbb{R}}$ is a Caratheodory multifunction whenever $s\mapsto \mathcal{Q}(s,x_1,x_2,\dots ,x_k)$ is measurable for all $x_1,x_2,\dots ,x_k\in \mathbb{R}$ and $(x_1,x_2,\dots ,x_k)\mapsto \mathcal{Q}(s,x_1,x_2,\dots ,x_k)$ is an upper semi-continuous map for almost all $s\in J$ (see Aubin and Ceuina 1984; Deimling 1992; Kisielewicz 1991). Also, a Caratheodory multifunction $\mathcal{Q}:J\times {\mathbb{R}}^k\to 2^{\mathbb{R}}$ is called $L^1$-Caratheodory whenever for each $\mathit{\rho }>0$ there exists ${\mathit{\varphi }}_{\mathit{\rho }}\in L^1(J,{\mathbb{R}}^{+})$ such that

$$\begin{array}{c}\hfill \Vert \mathcal{Q}(s,x_1,x_2,\cdots ,x_k)\Vert =sup\{|v|:v\in \mathcal{Q}(s,x_1,x_2,\cdots ,x_k)\}\le {\mathit{\varphi }}_{\mathit{\rho }}(s)\end{array}$$

for all $|x_1|,|x_2|,\cdots ,|x_k|\le \mathit{\rho }$ and for almost all $s\in J$ (see Aubin and Ceuina 1984; Deimling 1992; Kisielewicz 1991).

Lemma 1

(Deimling 1992) If$G:\mathcal{X}\to {\mathcal{P}}_{cl}(\mathcal{Y})$is upper semi-continuous, thenGr(G) is a closed subset of$\mathcal{X}\times \mathcal{Y}$. IfGis completely continuous and has a closed graph, then it is upper semi-continuous.

Lemma 2

(Lasota and Opial 1965) Suppose that$\mathcal{X}$is a Banach space,$G:J\times \mathcal{X}\to {\mathcal{P}}_{cp,cv}(\mathcal{X})$an$L^1$-Caratheodory multivalued and$\mathrm{\Theta }$a linear continuous mapping from$L^1(J,\mathcal{X})$to$C(J,\mathcal{X})$. Then the mapping$\mathrm{\Theta }o{S}_G:C(J,\mathcal{X})\to {\mathcal{P}}_{cp,cv}C(J,\mathcal{X})$defined by$(\mathrm{\Theta }o{S}_G)(x)=\mathrm{\Theta }(S_{G,x})$is a closed graph mapping in$C(J,\mathcal{X})\times C(J,\mathcal{X})$.

Theorem 3

(Dhage 2006) Suppose that$\mathcal{X}$is a Banach algebra space,$S\in {\mathcal{P}}_{bd,cl,cv}(\mathcal{X})$and$\mathcal{A}:S\to {\mathcal{P}}_{cl,cv,bd}(\mathcal{X})$and$\mathcal{B}:S\to {\mathcal{P}}_{cp,cv}(\mathcal{X})$two multifunctions satisfying the following conditions

$1.\mathcal{A}$is Lipschitz with a Lipschitz constantk,

$2.\mathcal{B}$is upper semi-continuous and compact,

$3.\mathcal{A}x\mathcal{B}x$is a convex subsetSfor all$x\in S$,

$4.Mk<1$, where$M=\Vert \mathcal{B}(S)\Vert =sup\{\Vert \mathcal{B}x\Vert :x\in S\}.$

Then, there exists$y\in S$such that$y\in \mathcal{A}y\mathcal{B}y$.

Lemma 4

(Agarwal et al. 2013) Suppose that$G:[0,1]\to {\mathcal{P}}_{cp,cv}(\mathbb{R})$is a measurable map such that the Lebesgue measure$\mathit{\mu }$of the set$\{s:dimG(s)<1\}$is zero. Then there are arbitrarily many linearly independent measurable selections$x_1(.),\cdots ,x_m(.)$ofG.

Theorem 5

(Agarwal et al. 2013) Suppose thatCis a nonempty closed convex subset of Banach space$\mathcal{X}$. Let$G:C\to {\mathcal{P}}_{cp,cv}(C)$b a$\mathit{\gamma }$-contraction. If$dimG(x)\ge m$for all$x\in C$, then$dimFix(G)\ge m$.

Main results

First, we review the fractional hybrid differential inclusion

$$\begin{array}{c}\hfill {}^c{D}^{\mathit{\alpha }}\left(\frac{y(s)}{g(s,y(s),I^{{\mathit{\alpha }}_1}y(s),\cdots ,I^{{\mathit{\alpha }}_n}y(s))}\right)\in G\left(s,y(s),I^{{\mathit{\beta }}_1}y(s),\cdots ,I^{{\mathit{\beta }}_k}y(s)\right)\end{array}$$ 1

with the boundary conditions $y(0)=y_0$ and $y(1)=y_1$, where $1<\mathit{\alpha }\le 2$, ${\mathit{\alpha }}_1,\cdots ,{\mathit{\alpha }}_n>0$, ${\mathit{\beta }}_1,\cdots ,{\mathit{\beta }}_k>0$, $y_0,y_1\in \mathbb{R}$, ${}^c{D}^{\mathit{\alpha }}$ denotes Caputo fractional derivative of order $\mathit{\alpha }$, $g:J\times {\mathbb{R}}^n\to \mathbb{R}-\{0\}$ is continuous and $G:J\times {\mathbb{R}}^k\to \mathcal{P}(\mathbb{R})$ is a multifunction via some properties.

Lemma 6

Suppose that$x\in C(J,\mathbb{R})$, $\mathit{\alpha }\in (1,2]$and${\mathit{\alpha }}_1,\cdots ,{\mathit{\alpha }}_n>0$. The unique solution of the fractional differential problem${}^c{D}^{\mathit{\alpha }}\left(\frac{y(s)}{g(s,y(s),I^{{\mathit{\alpha }}_1}y(s),\cdots ,I^{{\mathit{\alpha }}_n}y(s))}\right)=x(s)$with the boundary value conditions$y(0)=y_0$and$y(1)=y_1$is given by

$$\begin{array}{cc}\hfill y(s)& =g(s,y(s),I^{{\mathit{\alpha }}_1}y(s),\cdots ,I^{{\mathit{\alpha }}_n}y(s))\hfill \\ \hfill & \times \left[I^{\mathit{\alpha }}x(s)+\frac{(1-s)y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}+\frac{s{y}_1}{f(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}-s{I}^{\mathit{\alpha }}x(1)\right].\hfill \end{array}$$

Proof

The general solution of the equation${}^c{D}^{\mathit{\alpha }}\left(\frac{y(s)}{g(s,y(s),I^{{\mathit{\alpha }}_1}y(s),\cdots ,I^{{\mathit{\alpha }}_n}y(s))}\right)=x(s)$ is $y(s)=g(s,y(s),I^{{\mathit{\alpha }}_1}y(s),\dots ,I^{{\mathit{\alpha }}_n}y(s))\left[I^{\mathit{\alpha }}x(s)+c_0+c_{1}s\right],$ where $c_0,c_1\in \mathbb{R}$ are arbitrary constants (see Kilbas et al. 2006; Podlubny 1999). By using the boundary conditions, we get $y(0)=g(0,y(0),{\underbrace{0,\dots ,0}}_n)c_0=y_0$ and $y(1)=g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))(I^{\mathit{\alpha }}x(1)+c_0+c_1)=y_1$. Hence, $c_0=\frac{y_0}{g(0,y(0),{\underbrace{0,\dots ,0}}_n)}$ and $c_1=\frac{y_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}-I^{\mathit{\alpha }}x(1)-\frac{y_0}{g(0,y(0),{\underbrace{0,\dots ,0}}_n)}.$ This completes the proof. $\square$

$y\in \mathcal{Y}=C(J,\mathbb{R})$ is solution for the problem (1) whenever it satisfies the boundary conditions and there exists a function $v\in S_{G,y}$ such that

$$\begin{array}{cc}\hfill y(s)& =g(s,y(s),I^{{\mathit{\alpha }}_1}y(s),\cdots ,I^{{\mathit{\alpha }}_n}y(s))\hfill \\ \hfill & \times \left[I^{\mathit{\alpha }}v(s)+\frac{(1-s)y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}+\frac{s{y}_1}{f(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}-s{I}^{\mathit{\alpha }}v(1)\right].\hfill \end{array}$$

where $S_{G,y}=\{v\in L^1[0,1]:v(s)\in G(s,y(s),I^{{\mathit{\beta }}_1}y(s),\cdots ,I^{{\mathit{\beta }}_k}y(s))\text{for}\text{almost}\text{all}s\in J\}$.

Theorem 7

Let$G:J\times {\mathbb{R}}^{k+1}\to {\mathcal{P}}_{cp,cv}(\mathbb{R})$be a Caratheodory multifunction, $g:J\times {\mathbb{R}}^{n+1}\to \mathbb{R}-\{0\}$is a continuous and bounded function with boundKand there exist continuous functions$p,m:J\to (0,\infty )$such that$\Vert G(s,y_1,y_2,\dots ,y_k)\Vert \le m(s)$and$|g(s,y_1,y_2,\dots ,y_{n+1})-g(s,x_1,x_2,\dots ,x_{n+1})|\le p(s){\sum }_{i=1}^{n+1}|y_i-x_i|$for all$s\in J$. If

$$\begin{array}{cc}& {\Vert p\Vert }_{\infty }\left(1+\frac{1}{\mathrm{\Gamma }({\mathit{\alpha }}_1+1)}+\frac{1}{\mathrm{\Gamma }({\mathit{\alpha }}_2+1)}+\cdots +\frac{1}{\mathrm{\Gamma }({\mathit{\alpha }}_n+1)}\right)\left(\frac{{2\Vert m\Vert }_{\infty }}{\mathrm{\Gamma }(\mathit{\alpha }+1)}+\left|\frac{y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}\right|\right.\hfill \\ \hfill & \left.+\left|\frac{y_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}\right|\right)<1,\hfill \end{array}$$

then the problem (1) has a solution.

Proof

Define $S=\{y\in \mathcal{Y}:\Vert y\Vert \le L\}$, where

$$\begin{array}{c}\hfill L=K\left(\frac{{2\Vert m\Vert }_{\infty }}{\mathrm{\Gamma }(\mathit{\alpha }+1)}+\left|\frac{y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}\right|+\left|\frac{y_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}\right|\right).\end{array}$$

clearly S is a closed, bounded and convex subset of the Banach algebra space $\mathcal{Y}$. Now, consider the multivalued operators $\mathcal{A},\mathcal{B}:S\to \mathcal{P}(\mathcal{Y})$ by

$$\begin{array}{c}\hfill \mathcal{A}y(s)=\left\{g(s,y(s),I^{{\mathit{\alpha }}_1}y(s),\dots ,I^{{\mathit{\alpha }}_n}y(s))\right\}\end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{B}y(s)& =\left\{u\in \mathcal{Y}:\text{there}\text{exists}v\in S_{G,y}\text{such}\text{that}u(s)=I^{\mathit{\alpha }}v(s)+\frac{(1-s)y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}\right.\hfill \\ \hfill & \left.+\frac{s{y}_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}-s{I}^{\mathit{\alpha }}v(1)\text{for}\text{all}s\in J\right\}.\hfill \end{array}$$

Thus, the problem (1) is tantamount to the problem $y\in \mathcal{A}(y)\mathcal{B}(y)$. We prove that the multifunctions $\mathcal{A}$ and $\mathcal{B}$ well-defined the conditions of Theorem 3. Note that, the operator $\mathcal{B}=\mathit{\theta }\circ S_G$, where $\mathit{\theta }$ is the continuous linear operator on $L^1(J,\mathbb{R})$ into $\mathcal{Y}$ defined by

$$\begin{array}{c}\hfill \mathit{\theta }v(s)=I^{\mathit{\alpha }}v(s)+\frac{(1-s)y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}+\frac{s{y}_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}-s{I}^{\mathit{\alpha }}v(1).\end{array}$$

Let $y\in S$ be arbitrary and $\{v_n\}$ a sequence in $S_{G,y}$. Then, $v_n(s)\in G(s,y(s),I^{{\mathit{\beta }}_1}y(s),\dots ,I^{{\mathit{\beta }}_k}y(s))$ for almost $s\in J$. Because $G(s,y(s),I^{{\mathit{\beta }}_1}y(s),\dots ,I^{{\mathit{\beta }}_k}y(s))$ is compact for all $s\in J$, there exists a convergent subsequence of $\{v_n(s)\}$ (we show it again $\{v_n(s)\}$) to some $v\in S_{G,y}$. Since $\mathit{\theta }$ is continuous, $\mathit{\theta }{v}_n(s)\to \mathit{\theta }v(s)$ pointwise on J. Because we will show that the convergence is uniform, we have to prove that$\{\mathit{\theta }{v}_n\}$ is an equi-continuous sequence. Suppose that $\mathit{\tau }<s\in J$. So, we have

$$\begin{array}{cc}\hfill |\mathit{\theta }{v}_n(s)-\mathit{\theta }{v}_n(\mathit{\tau })|& \le \frac{{\Vert m\Vert }_{\infty }(s^{\mathit{\alpha }}-{\mathit{\tau }}^{\mathit{\alpha }})}{\mathrm{\Gamma }(\mathit{\alpha }+1)})+(s-\mathit{\tau })|\frac{y_0}{g(0,y(0),\underset{n}{\underbrace{0,\cdots ,0}})}\hfill \\ \hfill & +\frac{y_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\cdots ,I^{{\mathit{\alpha }}_n}y(1))}-I^{\mathit{\alpha }}{v}_n(1)|.\hfill \end{array}$$

Hence the right hand of above inequalities tends to 0 as $s\to \mathit{\tau }$ and so the sequence $\{\mathit{\theta }{v}_n\}$ is equi-continuous. By using the Arzela–Ascoli theorem, it has a uniformly convergent subsequence. Thus, there is a subsequence of $\{v_n\}$ (we show it again by $\{v_n\}$) such that $\mathit{\theta }{v}_n\to \mathit{\theta }v$. Hence, $\mathit{\theta }v\in \mathit{\theta }(S_{G,y})$. Thus, $\mathcal{B}=\mathit{\theta }(S_{G,y})$ is compact for all $y\in S$. Now, we show that $\mathcal{B}y$ is convex for all $y\in S$. Let $y\in S$ and $u,u^{\prime }\in \mathcal{B}y$. Choose $v,v^{\prime }\in S_{G,y}$ such that

$$\begin{array}{c}\hfill u(s)=I^{\mathit{\alpha }}v(s)+\frac{(1-s)y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}+\frac{s{y}_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}-s{I}^{\mathit{\alpha }}v(1),\\ \hfill u^{\prime }(s)=I^{\mathit{\alpha }}{v}^{\prime }(s)+\frac{(1-s)y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}+\frac{s{y}_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}-s{I}^{\mathit{\alpha }}{v}^{\prime }(1)\end{array}$$

for almost all $s\in J$. Let $0\le \mathit{\lambda }\le 1$. Then, we have

$$\begin{array}{cc}\hfill \mathit{\lambda }u(s)+(1-\mathit{\lambda })u^{\prime }(s)& =\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}[\mathit{\lambda }v(t)+(1-\mathit{\lambda })v^{\prime }(t)]ds+\frac{(1-s)y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}\hfill \\ \hfill & +\frac{s{y}_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}-\frac{s}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^1{(1-t)}^{\mathit{\alpha }-1}[\mathit{\lambda }v(t)+(1-\mathit{\lambda })v^{\prime }(t)]dt.\hfill \end{array}$$

Since G is convex valued, $\mathit{\lambda }u+(1-\mathit{\lambda })u^{\prime }\in \mathcal{B}y$. Cleary, $\mathcal{A}$ is bounded, closed and convex valued. We prove that $\mathcal{A}y\mathcal{B}y$ is a convex subset of S for all $y\in S$. Suppose that $y\in S$ and $u,u^{\prime }\in \mathcal{A}y\mathcal{B}y$. Choose $v,v^{\prime }\in S_{G,y}$ such that

$$\begin{array}{cc}\hfill u(s)& =g\left(s,y(s),I^{{\mathit{\alpha }}_1}y(s),\dots ,I^{{\mathit{\alpha }}_n}y(s)\right)\left[I^{\mathit{\alpha }}v(s)+\frac{(1-s)y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}+\frac{s{y}_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}-s{I}^{\mathit{\alpha }}v(1)\right],\hfill \\ \hfill u^{\prime }(s)& =g\left(s,y(s),I^{{\mathit{\alpha }}_1}y(s),\dots ,I^{{\mathit{\alpha }}_n}y(s)\right)\left[I^{\mathit{\alpha }}{v}^{\prime }(s)+\frac{(1-s)y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}+\frac{s{y}_1}{g(1,s(1),I^{{\mathit{\alpha }}_1}s(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}-s{I}^{\mathit{\alpha }}{v}^{\prime }(1)\right]\hfill \end{array}$$

for almost all $s\in J$. Hence, we get

$$\begin{array}{cc}\hfill \mathit{\lambda }u(s)+(1-\mathit{\lambda })u^{\prime }(s)& =g\left(s,y(s),I^{{\mathit{\alpha }}_1}y(s),\dots ,I^{{\mathit{\alpha }}_n}y(s)\right)\left[\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}[\mathit{\lambda }v(t)+(1-\mathit{\lambda })v^{\prime }(t)\right]dt\hfill \\ \hfill & +\frac{(1-s)y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}+\frac{s{y}_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}-\frac{s}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^1{(1-t)}^{\mathit{\alpha }-1}[\mathit{\lambda }v(t)+(1-\mathit{\lambda })v^{\prime }(t)]dt.\hfill \end{array}$$

Since G is convex valued, $\mathit{\lambda }u+(1-\mathit{\lambda })u^{\prime }\in \mathcal{A}y\mathcal{B}y$. So, $\mathcal{A}y\mathcal{B}y$ is convex subset of $\mathcal{Y}$ for all $y\in \mathcal{Y}$. But, we have

$$\begin{array}{cc}\hfill |u(s)|& =\left|g\left(s,y(s),I^{{\mathit{\alpha }}_1}y(s),\dots ,I^{{\mathit{\alpha }}_n}y(s)\right)\left[I^{\mathit{\alpha }}v(s)+\frac{(1-s)y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}+\frac{s{y}_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}\right.\right.\hfill \\ \hfill & \left.\left.-s{I}^{\mathit{\alpha }}v(1)\right]\right|\le K\left(\frac{{2\Vert m\Vert }_{\infty }}{\mathrm{\Gamma }(\mathit{\alpha }+1)}+\left|\frac{y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}\right|+\left|\frac{y_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}\right|\right)=L\hfill \end{array}$$

for all $s\in J$. So, $u\in S$ and $\mathcal{A}y\mathcal{B}y$ is a convex subset of S for all $y\in S$. Here, We show that operator $\mathcal{B}$ is compact. For showing this, it is enough to prove that $\mathcal{B}(S)$ is uniformly bounded and equi-continuous. Let $u\in \mathcal{B}(S)$. Choose $v\in S_{G,y}$ such that $u(s)=g(s,y(s),I^{{\mathit{\alpha }}_1}y(s),\dots ,I^{{\mathit{\alpha }}_n}y(s))\left[I^{\mathit{\alpha }}v(s)+\frac{(1-s)y_0}{g(0,y(0),{\underbrace{0,\dots ,0}}_n)}+\frac{s{y}_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}-s{I}^{\mathit{\alpha }}v(1)\right]$ for some $y\in S$. Hence,

$$\begin{array}{c}\hfill |u(s)|\le \left(\frac{{2\Vert m\Vert }_{\infty }}{\mathrm{\Gamma }(\mathit{\alpha }+1)}+\left|\frac{y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}\right|+\left|\frac{y_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}\right|\right)\end{array}$$

and so ${\Vert u\Vert }_{\infty }={max}_{s\in J}|u(s)|\le \left(\frac{{2\Vert m\Vert }_{\infty }}{\mathrm{\Gamma }(\mathit{\alpha }+1)}+|\frac{y_0}{g(0,y(0),{\underbrace{0,\dots ,0}}_n)}|+|\frac{y_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}|\right).$ In this part, prove that $\mathcal{B}$ maps S to equi-continuous subsets of $\mathcal{Y}$. Suppose that $s,\mathit{\tau }\in J$ with $\mathit{\tau }<s$, $y\in S$ and $u\in \mathcal{B}y$. Choose $v\in S_{G,y}$ such that $u(s)=I^{\mathit{\alpha }}v(s)+\frac{(1-s)y_0}{g(0,y(0),{\underbrace{0,\dots ,0}}_n)}+\frac{s{y}_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}-s{I}^{\mathit{\alpha }}v(1)$. Then, we have

$$\begin{array}{cc}\hfill |u(s)-u(\mathit{\tau })|& \le \left(\frac{{\Vert m\Vert }_{\infty }(s^{\mathit{\alpha }}-{\mathit{\tau }}^{\mathit{\alpha }})}{\mathrm{\Gamma }(\mathit{\alpha }+1)}\right)+(s-\mathit{\tau })\left|\frac{y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}\right.\hfill \\ \hfill & \left.+\frac{y_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}-I^{\mathit{\alpha }}v(1)\right|.\hfill \end{array}$$

So the right side of inequality towards to 0 as $s\to \mathit{\tau }$. Hence by using the Arzela–Ascoli theorem, $\mathcal{B}$ is compact. Here, we show that $\mathcal{B}$ has a closed graph. Suppose that $y_n\in S$ and $u_n\in \mathcal{B}{y}_n$ for all n such that $y_n\to y^{\prime }$ and $u_n\to u^{\prime }$. We show that $u^{\prime }\in \mathcal{B}{y}^{\prime }$. For each natural number n, choose $v_n\in S_{G,y_n}$ such that

$$\begin{array}{c}\hfill u_n(s)=I^{\mathit{\alpha }}{v}_n(s)+\frac{(1-s)y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}+\frac{s{y}_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}-s{I}^{\mathit{\alpha }}{v}_n(1)\end{array}$$

for all $s\in J$. Again, consider the continuous linear operator $\mathit{\theta }:L^1(J,\mathbb{R})\to \mathcal{Y}$ such that

$$\begin{array}{c}\hfill \mathit{\theta }(v)(s)=u(s)=I^{\mathit{\alpha }}v(s)+\frac{(1-s)y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}+\frac{s{y}_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}-s{I}^{\mathit{\alpha }}v(1).\end{array}$$

By using Lemma 2, $\mathit{\theta }o{S}_G$ is a closed graph operator. Since $y_n\to y^{\prime }$ and $u_n\in \mathit{\theta }(S_{G,y_n})$ for all n, there is $v^{\prime }\in S_{G,y^{\prime }}$ such that

$$\begin{array}{c}\hfill u^{\prime }(s)=I^{\mathit{\alpha }}{v}^{\prime }(s)+\frac{(1-s)y_0}{g(0,y(0),\underset{n}{\underbrace{0,\dots ,0}})}+\frac{s{y}_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1),\dots ,I^{{\mathit{\alpha }}_n}y(1))}-s{I}^{\mathit{\alpha }}{v}^{\prime }(1).\end{array}$$

Hence, $u^{\prime }\in \mathcal{B}{y}^{\prime }$. This implies that, $\mathcal{B}$ has a closed graph and thus the operator $\mathcal{B}$ is upper semi-continuous. Now, we show that $\mathcal{A}$ is a contractive multifunction. Note that,

$$\begin{array}{cc}\hfill H(\mathcal{A}y,\mathcal{A}x)& =\Vert \mathcal{A}y-\mathcal{A}x\Vert =\underset{s\in J}{max}\left|g(s,y(s),I^{{\mathit{\alpha }}_1}y(s),\dots ,I^{{\mathit{\alpha }}_n}y(s))-g(s,x(s),I^{{\mathit{\alpha }}_1}x(s),\dots ,I^{{\mathit{\alpha }}_n}x(s))\right|\hfill \\ \hfill & \le \underset{s\in J}{max}(|p(s)||y(s)-x(s)|)\left(1+\frac{1}{\mathrm{\Gamma }({\mathit{\alpha }}_1+1)}+\frac{1}{\mathrm{\Gamma }({\mathit{\alpha }}_2+1)}+\cdots +\frac{1}{\mathrm{\Gamma }({\mathit{\alpha }}_n+1)}\right)\hfill \\ \hfill & ={\Vert p\Vert }_{\infty }\left(1+\frac{1}{\mathrm{\Gamma }({\mathit{\alpha }}_1+1)}+\frac{1}{\mathrm{\Gamma }({\mathit{\alpha }}_2+1)}+\cdots +\frac{1}{\mathrm{\Gamma }({\mathit{\alpha }}_n+1)}\right){\Vert y-x\Vert }_{\infty }\hfill \end{array}$$

for all $x,y\in \mathcal{Y}$. So, $\mathcal{A}$ and $\mathcal{B}$ satisfy the conditions of Theorem 3 and thus the operator inclusions $y\in \mathcal{A}y\mathcal{A}y$ has a solution in S. Therefore, the problem (1) has a solution. $\square$

To illustrate our main results, we present the following example:

Example 1

Here, we investigation the problem

$$\begin{array}{c}\hfill {}^c{D}^{\frac{1}{2}}\left(\frac{y(s)}{\frac{{(s+1)}^2}{60}siny(s)+\frac{|I^{\sqrt{2}}y(s)|}{1+|I^{\sqrt{2}}y(s)|}+3}\right)\in \left[-1,s^{2}siny(s)+cos(I^{\frac{1}{4}}y(s))+1\right](\ast )\end{array}$$

with the boundary conditions $y(0)=\frac{\mathit{\pi }}{2}$ and $y(1)=0$. Put $\mathit{\alpha }=\frac{1}{2}$, ${\mathit{\alpha }}_1=\sqrt{2}$, ${\mathit{\beta }}_1=\frac{1}{4}$, $n=k=1$, $y_0=\frac{\mathit{\pi }}{2}$, $y_1=0$, $g(s,y,x)=\frac{{(s+1)}^2}{60}siny+\frac{|x|}{1+|x|}+3$, $G(s,y,x)=[-1,s^{2}siny+cosx+1]$, $m(s)=s^2+2$ and $p(s)=\frac{{(s+1)}^2}{60}$ for $s\in [0,1]$. Note that, $\Vert G(s,y,x)\Vert \le s^2+2$,

$$\begin{array}{c}\hfill \left|g(s,y,x)-g(s,y^{\prime },x^{\prime })\right|\le \frac{{(s+1)}^2}{60}\left(\left|y-y^{\prime }|+|x-x^{\prime }\right|\right)\end{array}$$

and ${\Vert p\Vert }_{\infty }\left(1+\frac{1}{\mathrm{\Gamma }({\mathit{\alpha }}_1+1)}\right)\left(\frac{{2\Vert m\Vert }_{\infty }}{\mathrm{\Gamma }(\mathit{\alpha }+1)}+|\frac{y_0}{g(0,y(0),0)}|+|\frac{y_1}{g(1,y(1),I^{{\mathit{\alpha }}_1}y(1))}|\right)=0.8784698182<1$. By using the Theorem 7, the problem $(\ast )$ has a solution.

Now, we review existence and dimension of the solution set of the fractional differential inclusion problem

$$\begin{array}{c}\hfill {}^c{D}^{\mathit{\alpha }}y(s)\in G\left(s,y(s),(\mathit{\varphi }y)(s),(\mathit{\psi }y)(s),{}^c{D}^{{\mathit{\beta }}_1}y(s),\cdots ,{}^c{D}^{{\mathit{\beta }}_k}y(s),I^{{\mathit{\gamma }}_1}y(s),\cdots ,I^{{\mathit{\gamma }}_k}y(s)\right)\end{array}$$ 2

with boundary condition $y(0)+{\sum }_{i=1}^k{}^c{D}^{{\mathit{\beta }}_i}y(1)+{\sum }_{i=1}^k{I}^{{\mathit{\gamma }}_i}y(1)=0$, where $0<{\mathit{\beta }}_i<\mathit{\alpha }\le 1$ and $0<{\mathit{\gamma }}_i<1$ for $i=1,\cdots ,k$, $G:J\times {\mathbb{R}}^{2k+3}\to \mathcal{P}(\mathbb{R})$ is a multifunction via some properties, $\mathit{\gamma },\mathit{\lambda }:J\times J\to [0,\infty )$ are two mappings with the properties ${sup}_{s\in J}|{\int }_0^s\mathit{\lambda }(s,t)ds|<\infty$ and ${sup}_{s\in J}|{\int }_0^s\mathit{\gamma }(s,t)ds|<\infty$ and the functions $\mathit{\varphi }$ and $\mathit{\psi }$ are defined by $(\mathit{\varphi }y)(s)={\int }_0^s\mathit{\gamma }(s,t)y(t)dt$ and $(\mathit{\psi }y)(s)={\int }_0^s\mathit{\lambda }(s,t)y(t)dt$.

Lemma 8

Suppose that$v\in C(J,\mathbb{R})$, $\mathit{\alpha }\in (0,1]$and${\mathit{\beta }}_i,{\mathit{\gamma }}_i\in (0,1)$with$\mathit{\alpha }-{\mathit{\beta }}_i>0$for$1\le i\le k$. Then solution of the problem${}^c{D}^{\mathit{\alpha }}y(s)=v(s)$with the boundary condition$y(0)+{\sum }_{i=1}^k{}^c{D}^{{\mathit{\beta }}_i}y(1)+{\sum }_{i=1}^k{I}^{{\mathit{\gamma }}_i}y(1)=0$ is

$$\begin{array}{cc}\hfill y(s)& =\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}v(t)dt+\frac{-1}{1+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}}\sum _{i=1}^k\left(\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}v(t)dt\right.\hfill \\ \hfill & \left.+\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }+{\mathit{\gamma }}_i-1}v(t)dt\right).\hfill \end{array}$$

Proof

The general solution of the problem ${}^c{D}^{\mathit{\alpha }}y(s)=v(s)$ is formed by

$$\begin{array}{c}\hfill y(s)=I^{\mathit{\alpha }}v(s)+c_0=\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}v(t)dt+c_0,\end{array}$$

where $c_0$ is arbitrary constant and $t\in J$ (Podlubny 1999). Thus, we obtain

$$\begin{array}{c}\hfill {}^c{D}^{{\mathit{\beta }}_i}y(s)=I^{\mathit{\alpha }-{\mathit{\beta }}_i}v(s)=\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_{0}s{(s-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}v(t)dt\end{array}$$

and

$$\begin{array}{c}\hfill I^{{\mathit{\gamma }}_i}y(s)=I^{\mathit{\alpha }+{\mathit{\gamma }}_i}v(s)+\frac{c_0{s}^{{\mathit{\gamma }}_i}}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}=\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}{\int }_0^s{(s-t)}^{\mathit{\alpha }+{\mathit{\gamma }}_i-1}v(t)dt+\frac{c_0{s}^{{\mathit{\gamma }}_i}}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}\end{array}$$

for all $1\le i\le k$. By using the boundary condition, we get

$$\begin{array}{cc}\hfill c_0& =\frac{-1}{1+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}}\sum _{i=1}^k\left(\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}v(t)dt\right.\hfill \\ \hfill & \left.+\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }+{\mathit{\gamma }}_i-1}v(t)dt\right).\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill y(s)& =\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}v(t)dt+\frac{-1}{1+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}}\sum _{i=1}^k\left(\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}v(t)dt\right.\hfill \\ \hfill & \left.+\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }+{\mathit{\gamma }}_i-1}v(t)dt\right).\hfill \end{array}$$

This completes the proof. $\square$

An element $y\in C(J,\mathbb{R})$ is a solution for the problem (2) whenever it satisfies the boundary condition and there is a function $v\in L^1(J)$ such that

$$\begin{array}{c}\hfill v(s)\in G(s,y(s),(\mathit{\varphi }y)(s),(\mathit{\psi }y)(s),{}^c{D}^{{\mathit{\beta }}_1}y(s),\cdots ,{}^c{D}^{{\mathit{\beta }}_k}y(s),I^{{\mathit{\gamma }}_1}y(s),\cdots ,I^{{\mathit{\gamma }}_k}y(s))\end{array}$$

for almost all $s\in J$ and

$$\begin{array}{cc}\hfill y(s)& =\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}v(t)dt+\frac{-1}{1+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}}\sum _{i=1}^k\left(\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}v(t)dt\right.\hfill \\ \hfill & +\left.\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }+{\mathit{\gamma }}_i-1}v(t)dt\right).\hfill \end{array}$$

Put $\mathcal{Y}=\{y:y,{}^c{D}^{{\mathit{\beta }}_i}y\in C(J,\mathbb{R})foreachi\in \{1,\cdots ,k\}\}$ with the norm

$$\begin{array}{c}\hfill \Vert y\Vert =\underset{s\in J}{sup}|y(s)|+\sum _{i=1}^k\underset{s\in J}{sup}{|}^c{D}^{{\mathit{\beta }}_i}y(s)|.\end{array}$$

$(\mathcal{Y},\Vert .\Vert )$ is a Banach space (Su 2009). Define selection set of G at $y\in \mathcal{Y}$ by

$$\begin{array}{c}\hfill S_{G,y}:=\left\{v\in L^1(J,\mathbb{R}):v(s)\in G(s,y(s),(\mathit{\varphi }y)(s),(\mathit{\psi }y)(s),{}^c{D}^{{\mathit{\beta }}_1}y(s),\cdots ,{}^c{D}^{{\mathit{\beta }}_k}y(s),\right.\\ \hfill \left.I^{{\mathit{\gamma }}_1}y(s),\cdots ,I^{{\mathit{\gamma }}_k}y(s))\text{for}\text{almost all}s\in J\right\}.\end{array}$$

Theorem 9

Suppose that$m\in L^1(J,{\mathbb{R}}^{+})$, $l=({\mathrm{\Lambda }}_1+{\sum }_{i=1}^k{\mathrm{\Lambda }}_2^i)<1$, where

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_1={\Vert m\Vert }_1\left(1+{\mathit{\lambda }}_0+{\mathit{\gamma }}_0+\sum _{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}\right)\left(\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}+\frac{1}{1+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}}\sum _{i=1}^k\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}+\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}\right),\end{array}$$

${\mathrm{\Lambda }}_2^i={\Vert m\Vert }_1\left(1+{\mathit{\lambda }}_0+{\mathit{\gamma }}_0+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}\right)\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}$for$i=1,\cdots ,k$, $F:J\times {\mathbb{R}}^{2k+3}\to {\mathcal{P}}_{cv,cp}(\mathbb{R})$is a multifunction such that the map$s⊢G(s,y_1,y_2,\dots {,}_{2k+3})$is measurable,

$$\begin{array}{c}\hfill \Vert G(s,y_1,y_2,\dots {,}_{2k+3})\Vert =sup\{|v|:v\in G(s,y_1,y_2,\dots {,}_{2k+3})\}\le m(s)\end{array}$$

and$H(G(s,y_1,y_2,\dots ,y_{2k+3})),G(s,x_1,x_2,\dots ,x_{2k+3}))\le m(s){\sum }_{i=1}^{2k+3}(|y_i-x_i|)$for almost all$s\in J$and$\in x_1,x_2,\dots ,x_{2k+3},y_1,y_2,\dots ,y_{2k+3}\in \mathbb{R}$. Then the inclusion problem (2) has a solution.

Proof

Note that, the multivalued map

$$\begin{array}{c}\hfill s⊢G\left(s,y(s),(\mathit{\varphi }y)(s),(\mathit{\psi }y)(s),{}^c{D}^{{\mathit{\beta }}_1}y(s),\cdots ,{}^c{D}^{{\mathit{\beta }}_k}y(s),I^{{\mathit{\gamma }}_1}y(s),\cdots ,I^{{\mathit{\gamma }}_k}y(s)\right)\end{array}$$

is measurable and closed valued for all $y\in \mathcal{Y}$. Hence, it has a measurable selection and so the set $S_{G,y}$ is nonempty. Now, consider the operator $\mathrm{{\rm Y}}:\mathcal{Y}\to 2^{\mathcal{Y}}$ defined by

$$\begin{array}{c}\hfill \mathrm{{\rm Y}}(y)=\left\{\mathit{\xi }\in X:\text{there is}v\in S_{G,y}:\mathit{\xi }(s)=\mathit{\nu }(s)\text{for all}s\in J\right\},\end{array}$$

where

$$\begin{array}{cc}\hfill \mathit{\nu }(s)& =\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}v(t)dt+\frac{-1}{1+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}}\sum _{i=1}^k\left(\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}v(t)dt\right.\hfill \\ \hfill & \left.+\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }+{\mathit{\gamma }}_i-1}v(t)dt\right)\hfill \end{array}$$

for all $s\in J$. Here, we prove that $\mathrm{{\rm Y}}(y)$ is a closed subset of $\mathcal{Y}$ for each $y\in \mathcal{Y}$. Let $y\in \mathcal{Y}$ and ${\{u_n\}}_{n\ge 1}$ be a sequence in $\mathrm{{\rm Y}}(y)$ with $u_n\to u$. For each n, choose $v_n\in S_{G,y}$ such that

$$\begin{array}{cc}\hfill u_n(s)& =\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}{v}_n(t)dt+\frac{-1}{1+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}}\sum _{i=1}^k\left(\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}{v}_n(t)dt\right.\hfill \\ \hfill & +\left.\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }+{\mathit{\gamma }}_i-1}{v}_n(t)dt\right)\hfill \end{array}$$

for almost all $s\in J$. because G has compact values, ${\{v_n\}}_{n\ge 1}$ has a subsequence which converges to a $v\in L^1(J,\mathbb{R})$. We write it again by ${\{v_n\}}_{n\ge 1}$. Clearly $v\in S_{G,y}$ and

$$\begin{array}{cc}\hfill u_n(s)\to u(s)& =\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}v(t)dt+\frac{-1}{1+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}}\sum _{i=1}^k\left(\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}v(t)dt\right.\hfill \\ \hfill & +\left.\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }+{\mathit{\gamma }}_i-1}v(t)dt\right)\hfill \end{array}$$

for all $s\in J$. This implies that $u\in \mathrm{{\rm Y}}(x)$. Thus, the multifunction $\mathrm{{\rm Y}}$ has closed values. Now, we show that $\mathrm{{\rm Y}}$ is a contractive multifunction with constant $l=({\mathrm{\Lambda }}_1+{\sum }_{i=1}^k{\mathrm{\Lambda }}_2^i)<1$. Suppose that $x,y\in \mathcal{Y}$ and $h_1\in \mathrm{{\rm Y}}(x)$. Consider $v_1\in S_{G,x}$ such that

$$\begin{array}{cc}\hfill h_1(s)& =\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}{v}_1(t)dt+\frac{-1}{1+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}}\sum _{i=1}^k\left(\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}{v}_1(t)dt\right.\hfill \\ \hfill & \left.+\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }+{\mathit{\gamma }}_i-1}{v}_1(t)dt\right)\hfill \end{array}$$

for almost all $s\in J$. Since

$$\begin{array}{cc}& H\left(G\left(s,y(s),(\mathit{\varphi }y)(s),(\mathit{\psi }y)(s),{}^c{D}^{{\mathit{\beta }}_1}y(s),\cdots ,{}^c{D}^{{\mathit{\beta }}_k}y(s),I^{{\mathit{\gamma }}_1}y(s),\cdots ,I^{{\mathit{\gamma }}_k}y(s)\right),\right.\hfill \\ \hfill & \left.G\left(s,x(s),(\mathit{\varphi }x)(s),(\mathit{\psi }x)(s),{}^c{D}^{{\mathit{\beta }}_1}x(s),\cdots ,{}^c{D}^{{\mathit{\beta }}_k}x(s),I^{{\mathit{\gamma }}_1}x(s),\cdots ,I^{{\mathit{\gamma }}_k}x(s)\right)\right)\hfill \\ \hfill & \le m(t)\left(|y(s)-x(s)|+|(\mathit{\varphi }y)(s)-(\mathit{\varphi }x)(s)|+|(\mathit{\psi }y)(s)-(\mathit{\psi }x)(s)|\right.\hfill \\ \hfill & \left.+\sum _{i=1}^k\left|{}^c{D}^{{\mathit{\beta }}_i}y(s)-{}^c{D}^{{\mathit{\beta }}_i}x(s)\right|+\sum _{i=1}^k\left|I^{{\mathit{\gamma }}_i}y(s)-I^{{\mathit{\gamma }}_i}x(s)\right|\right)\hfill \end{array}$$

for almost all $s\in J$, there is

$$\begin{array}{c}\hfill \mathit{\nu }\in \left(G\left(s,y(s),(\mathit{\varphi }y)(s),(\mathit{\psi }y)(s),{}^c{D}^{{\mathit{\beta }}_1}y(s),\cdots ,{}^c{D}^{{\mathit{\beta }}_k}y(s),I^{{\mathit{\gamma }}_1}y(s),\cdots ,I^{{\mathit{\gamma }}_k}y(s)\right)\right)\end{array}$$

such that

$$\begin{array}{cc}\hfill |v_1(s)-\mathit{\nu }|& \le m(s)\left(|y(s)-x(s)|+|(\mathit{\varphi }y)(s)-(\mathit{\varphi }x)(s)|+|(\mathit{\psi }y)(s)-(\mathit{\psi }x)(s)|\right.\hfill \\ \hfill & \left.+\sum _{i=1}^k\left|{}^c{D}^{{\mathit{\beta }}_i}y(s)-{}^c{D}^{{\mathit{\beta }}_i}x(s)\right|+\sum _{i=1}^k\left|I^{{\mathit{\gamma }}_i}y(s)-I^{{\mathit{\gamma }}_i}x(s)\right|\right)\hfill \end{array}$$

for almost all $s\in J$. Define the multifunction $\mathrm{\Delta }:J\to 2^{\mathbb{R}}$ by

$$\begin{array}{cc}\hfill \mathrm{\Delta }(s)& =\left\{\mathit{\nu }\in \mathbb{R}:|v_1(s)-\mathit{\nu }|\le m(s)\left(|y(s)-x(s)|+|(\mathit{\varphi }y)(s)-(\mathit{\varphi }x)(s)|+|(\mathit{\psi }y)(s)-(\mathit{\psi }x)(s)|\right.\right.\hfill \\ \hfill & \left.\left.+\sum _{i=1}^k\left|{}^c{D}^{{\mathit{\beta }}_i}y(s)-{}^c{D}^{{\mathit{\beta }}_i}x(s)\right|+\sum _{i=1}^k\left|I^{{\mathit{\gamma }}_i}y(s)-I^{{\mathit{\gamma }}_i}x(s)\right|\right)\text{for almost all}s\in J\right\}.\hfill \end{array}$$

The multifunction

$$\begin{array}{c}\hfill \mathrm{\Delta }(.)\bigcap \left(G\left(.,y(.),(\mathit{\varphi }y)(.),(\mathit{\psi }y)(.),{}^c{D}^{{\mathit{\beta }}_1}y(.),{}^c{D}^{{\mathit{\beta }}_2}y(.),\cdots ,{}^c{D}^{{\mathit{\beta }}_k}y(.),I^{{\mathit{\gamma }}_1}y(.),I^{{\mathit{\gamma }}_2}y(.),\dots ,I^{{\mathit{\gamma }}_k}y(.)\right)\right)\end{array}$$

is measurable. Thus, we can choose $v_2\in S_{G,y}$ such that

$$\begin{array}{cc}\hfill |v_1(s)-v_2(s)|& \le m(s)\left(|y(s)-x(s)|+|(\mathit{\varphi }y)(s)-(\mathit{\varphi }x)(s)|+|(\mathit{\psi }y)(s)-(\mathit{\psi }x)(s)|\right.\hfill \\ \hfill & \left.+\sum _{i=1}^k\left|{}^c{D}^{{\mathit{\beta }}_i}y(s)-{}^c{D}^{{\mathit{\beta }}_i}x(s)\right|+\sum _{i=1}^k\left|I^{{\mathit{\gamma }}_i}y(s)-I^{{\mathit{\gamma }}_i}x(s)\right|\right)\hfill \end{array}$$

for almost all $s\in J$. Now, define $h_2\in \mathrm{{\rm Y}}(y)$ by

$$\begin{array}{cc}\hfill h_2(s)& =\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}{v}_2(t)dt+\frac{-1}{1+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}}\sum _{i=1}^k\left(\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}{v}_2(t)dt\right.\hfill \\ \hfill & \left.+\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }+{\mathit{\gamma }}_i-1}{v}_2(t)dt\right).\hfill \end{array}$$

So

$$\begin{array}{cc}\hfill |h_1(s)-h_2(s)|& \le \frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}|v_1(t)-v_2(t)|dt\hfill \\ \hfill & +\frac{1}{1+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}}\sum _{i=1}^k\left(\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}|v_1(t)-v_2(t)|dt\right.\hfill \\ \hfill & \left.+\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }+{\mathit{\gamma }}_i-1}|v_1(t)-v_2(t)|dt\right)\le {\mathrm{\Lambda }}_1\Vert y-x\Vert \hfill \end{array}$$

and $|{}^c{D}^{{\mathit{\beta }}_i}{h}_1(s)-{}^c{D}^{{\mathit{\beta }}_i}{h}_2(s)|\le \frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_0^s{(s-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}|v_1(t)-v_2(t)|\le {\mathrm{\Lambda }}_2^i\Vert x-y\Vert$ and so we get $\Vert h_1-h_2\Vert \le ({\mathrm{\Lambda }}_1+{\sum }_{i=1}^k{\mathrm{\Lambda }}_2^i)\Vert x-y\Vert =l\Vert x-y\Vert$. This implies that the multifunction N is a contraction via closed values. By using the well-known theorem of Covitz and Nadler, N has a fixed point which is a solution for the inclusion problem (2). $\square$

Lemma 10

Suppose that$m\in L^1(J,{\mathbb{R}}^{+})$, $G:J\times {\mathbb{R}}^{2k+3}\to {\mathcal{P}}_{cv,cp}(\mathbb{R})$is a multifunction such that the map$s⊢G(s,y_1,y_2,\dots ,y_{2k+3})$is measurable,

$$\begin{array}{c}\hfill \Vert G(s,y_1,y_2,\dots ,y_{2k+3})\Vert =sup\{|v|:v\in G(s,y_1,y_2,\dots ,y_{2k+3})\}\le m(s)\end{array}$$

for almost all$s\in J$and$\in y_1,y_2,\dots ,y_{2k+3}\in \mathbb{R}$and$\mathrm{{\rm Y}}:\mathcal{Y}\to \mathcal{P}(\mathcal{Y})$is defined by

$$\begin{array}{c}\hfill \mathrm{{\rm Y}}(y)=\left\{\mathit{\xi }\in \mathcal{Y}:\text{there is}v\in S_{G,y}:\mathit{\xi }(s)=\mathit{\nu }(s)\text{for all}s\in J\right\},\end{array}$$

where

$$\begin{array}{cc}\hfill \mathit{\nu }(s)& =\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}v(t)dt+\frac{-1}{1+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}}\sum _{i=1}^k\left(\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}v(t)dt\right.\hfill \\ \hfill & +\left.\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }+{\mathit{\gamma }}_i-1}v(t)dt\right).\hfill \end{array}$$

Then, $\mathrm{{\rm Y}}(y)\in {\mathcal{P}}_{cp.cv}(\mathcal{Y})$for all$y\in \mathcal{Y}$.

Proof

Note that the operator $\mathrm{{\rm Y}}=\mathit{\theta }\circ S_G$, where $\mathit{\theta }$ is the continuous linear operator on $L^1(J,\mathbb{R})$ into $\mathcal{Y}$ which is defined by

$$\begin{array}{cc}\hfill \mathit{\theta }v(s)& =\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}v(t)dt+\frac{-1}{1+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}}\sum _{i=1}^k\left(\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}v(t)dt\right.\hfill \\ \hfill & +\left.\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }+{\mathit{\gamma }}_i-1}v(t)dt\right).\hfill \end{array}$$

Suppose that $y\in \mathcal{Y}$ and $\{v_n\}$ is a sequence in $S_{G,y}$. so,

$$\begin{array}{c}\hfill v_n(s)\in G\left(s,y(s),(\mathit{\varphi }y)(s),(\mathit{\psi }y)(s),{}^c{D}^{{\mathit{\beta }}_1}y(s),\cdots ,{}^c{D}^{{\mathit{\beta }}_k}y(s),I^{{\mathit{\gamma }}_1}y(s),\cdots ,I^{{\mathit{\gamma }}_k}y(s)\right)\end{array}$$

for almost $s\in J$. Since

$$\begin{array}{c}\hfill G\left(s,y(s),(\mathit{\varphi }y)(s),(\mathit{\psi }y)(s),{}^c{D}^{{\mathit{\beta }}_1}y(s),\cdots ,{}^c{D}^{{\mathit{\beta }}_k}y(s),I^{{\mathit{\gamma }}_1}y(s),\cdots ,I^{{\mathit{\gamma }}_k}y(s)\right)\end{array}$$

is compact for all $s\in J$, there exists a convergent subsequence of $\{v_n(s)\}$ (we show it again by $\{v_n(s)\}$) which converges to some $v\in S_{G,y}$. Since $\mathit{\theta }$ is continuous, $\mathit{\theta }{v}_n(s)\to \mathit{\theta }v(s)$ pointwise on J. Because we show that the convergence is uniform, we must prove that $\{\mathit{\theta }{v}_n\}$ is an equi-continuous sequence. Let $\mathit{\tau }<s\in J$. Then

$$\begin{array}{c}\hfill |\mathit{\theta }{v}_n(s)-\mathit{\theta }{v}_n(\mathit{\tau })|=\left|\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}{v}_n(t)dt-\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^{\mathit{\tau }}{(\mathit{\tau }-t)}^{\mathit{\alpha }-1}{v}_n(t)dt\right|\\ \hfill \le \left|\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^{\mathit{\tau }}({(s-t)}^{\mathit{\alpha }-1}-{(\mathit{\tau }-t)}^{\mathit{\alpha }-1})v_n()dt|+|\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_{\mathit{\tau }}^s{(s-t)}^{\mathit{\alpha }-1}{v}_n(t)dt\right|.\end{array}$$

Note that, the right side of the inequality towards to zero when $\mathit{\tau }\to s$. So, the sequence $\{\mathit{\theta }{v}_n\}$ is equi-continuous and so by using the Arzela–Ascoli theorem there is a uniformly convergent subsequence. Thus, there exists a subsequence of $\{v_n\}$ (we show it again by $\{v_n\}$) such that $\mathit{\theta }{v}_n\to \mathit{\theta }v$. This implies that $\mathit{\theta }v\in \mathit{\theta }(S_{G,y})$. Hence, $\mathrm{{\rm Y}}y=\mathit{\theta }(S_{G,y})$ is compact for all $y\in \mathcal{Y}$. Now, we prove that $\mathrm{{\rm Y}}y$ is convex for each $y\in \mathcal{Y}$. Let $h,h^{\prime }\in \mathrm{{\rm Y}}y$. Choose $v,v^{\prime }\in S_{G,y}$ such that

$$\begin{array}{cc}\hfill h(s)& =\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}v(t)dt+\frac{-1}{1+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}}\sum _{i=1}^k\left(\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}v(t)dt\right.\hfill \\ \hfill & \left.+\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }+{\mathit{\gamma }}_i-1}v(t)dt\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill h^{\prime }(s)& =\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}{v}^{\prime }(t)dt+\frac{-1}{1+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}}\sum _{i=1}^k\left(\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}{v}^{\prime }(t)dt\right.\hfill \\ \hfill & \left.+\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }+{\mathit{\gamma }}_i-1}{v}^{\prime }(t)dt\right)\hfill \end{array}$$

for almost all $s\in J$. Let $0\le \mathit{\lambda }\le 1$. Then, we have

$$\begin{array}{cc}\hfill \mathit{\lambda }h(s)+(1-\mathit{\lambda })h^{\prime }(s)& =\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}(\mathit{\lambda }v(t)+(1-\mathit{\lambda })v^{\prime }(t))ds\hfill \\ \hfill & +\frac{-1}{1+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}}\sum _{i=1}^k\left(\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}(\mathit{\lambda }v(t)+(1-\mathit{\lambda })v^{\prime }(t))dt\right.\hfill \\ \hfill & \left.+\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }+{\mathit{\gamma }}_i-1}(\mathit{\lambda }v(t)+(1-\mathit{\lambda })v^{\prime }(t))dt\right).\hfill \end{array}$$

Since $S_{G,y}$ is convex, $\mathit{\lambda }h+(1-\mathit{\lambda })h^{\prime }\in \mathrm{{\rm Y}}y$. This completes the proof. $\square$

One can check that the fixed point set of $\mathrm{{\rm Y}}$ is equal to the set of all solutions of the problem (2).

Theorem 11

Suppose that$m\in L^1(J,{\mathbb{R}}^{+})$, $G:J\times {\mathbb{R}}^{2k+3}\to {\mathcal{P}}_{cv,cp}(\mathbb{R})$is a multifunction such that the map$s⊢G(s,y_1,y_2,\dots ,y_{2k+3})$is measurable,

$$\begin{array}{c}\hfill H\left(G(s,y_1,y_2,\dots ,y_{2k+3}),G(s,x_1,x_2,\dots ,x_{2k+3})\right)\le m(s)\sum _{i=1}^{2k+3}|y_i-x_i|\end{array}$$

and$\Vert G(s,y_1,y_2,\dots ,y_{2k+3})\Vert =sup\{|v|:v\in G(s,y_1,y_2,\dots ,y_{2k+3})\}\le m(s)$for almost all$s\in J$and$x_1,x_2,\dots ,x_{2k+3},y_1,y_2,y_{2k+3}\in \mathbb{R}$. If Lebesgue measure of the set

$$\begin{array}{c}\hfill \left\{s:dimG(s,y_1,y_2,\dots ,y_{2k+3})<1\text{for some}{y}_1,y_2,\dots ,y_{2k+3}\in \mathbb{R}\right\}\end{array}$$

is zero and $l<1$, then the set of all solutions of the problem (2) is infinite dimensional, wherelis defined in Theorem9.

Proof

Define the operator $\mathrm{{\rm Y}}$ by

$$\begin{array}{c}\hfill \mathrm{{\rm Y}}(y)=\left\{\mathit{\xi }\in \mathcal{Y}:\text{there exists}v\in S_{G,y}\text{such that}\mathit{\xi }(s)=\mathit{\nu }(s)\text{for all}s\in J\right\},\end{array}$$

where

$$\begin{array}{cc}\hfill \mathit{\nu }(s)& =\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}v(t)dt+\frac{-1}{1+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}}\sum _{i=1}^k\left(\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}v(t)dt\right.\hfill \\ \hfill & \left.+\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }+{\mathit{\gamma }}_i-1}v(t)dt\right).\hfill \end{array}$$

By using Lemma 10, $\mathrm{{\rm Y}}x\in {\mathcal{P}}_{cp,cv}(\mathcal{Y})$ for each $y\in \mathcal{Y}$. Like to the Theorem 9, $\mathrm{{\rm Y}}$ is contraction. we show that $dim\mathrm{{\rm Y}}y>m$ for each $y\in \mathcal{Y}$ and $m\ge 1$. Let $y\in \mathcal{Y}$ and

$$\begin{array}{c}\hfill F(s)=G(s,y(s),(\mathit{\varphi }y)(s),(\mathit{\psi }y)(s),{}^c{D}^{{\mathit{\beta }}_1}y(s),{}^c{D}^{{\mathit{\beta }}_2}y(s),\cdots ,{}^c{D}^{{\mathit{\beta }}_k}y(s),I^{{\mathit{\gamma }}_1}y(s),I^{{\mathit{\gamma }}_2}y(s),\dots ,I^{{\mathit{\gamma }}_k}y(s))\end{array}$$

for all $s\in J$. By using Lemma 4, there exist linearly independent measurable selections $v_1,\cdots ,v_m$ for F. Put

$$\begin{array}{cc}\hfill h_i(s)& =\frac{1}{\mathrm{\Gamma }(\mathit{\alpha })}{\int }_0^s{(s-t)}^{\mathit{\alpha }-1}{v}_i(t)dt+\frac{-1}{1+{\sum }_{i=1}^k\frac{1}{\mathrm{\Gamma }({\mathit{\gamma }}_i+1)}}\sum _{i=1}^k\left(\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }-{\mathit{\beta }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }-{\mathit{\beta }}_i-1}{v}_i(t)dt\right.\hfill \\ \hfill & \left.+\frac{1}{\mathrm{\Gamma }(\mathit{\alpha }+{\mathit{\gamma }}_i)}{\int }_0^1{(1-t)}^{\mathit{\alpha }+{\mathit{\gamma }}_i-1}v{(t)}_{i}dt\right)\hfill \end{array}$$

for $i=1,\cdots ,m$. Assume that ${\sum }_{i=1}^m{a}_i{h}_i(s)=0$ for almost $s\in J$. By using the Caputo derivatives, we get ${\sum }_{i=1}^m{a}_i{v}_i(s)=0$ for almost $s\in J$. Hence, $a_1=\cdots ,a_n=0$. This implies that $h_1,\cdots ,h_m$ are linearly independent. Thus, $dim\mathrm{{\rm Y}}y\ge m$. Now by using Theorem 5, the set of fixed points of $\mathrm{{\rm Y}}$ is infinite dimensional. $\square$

Conclusions

The existence of solution for fractional differential inclusions is an important task which can be used successfully in solving real world problems from many fields of science and engineering. Thus, in our paper we analyze firstly the existence of solution of a given class of fractional hybrid differential inclusions. An example was give in order to show the reported results Secondly we concentrate our attention on proving the existence and dimension of the solution set for some fractional differential inclusions. These results are useful for the numerical studies involving the investigated equations.