On a fractional differential inclusion via a new integral boundary condition

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.

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].

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]).

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.

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.

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

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. □

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 $t>0$;

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

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 $x\in X$.

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 ${\{v_n\}}_{n\ge 1}$.

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 $H_d(N(x),N(y))\le \psi (\parallel x-y\parallel )$.