Existence results for fractional integral inclusions via nonlinear alternative for contractive maps

The theory of fractional differential equations and inclusions has developed into an important field of investigation due to its extensive applications in numerous branches of physics, economics, and engineering sciences [1–4]. The nonlocal behavior exhibited by a fractional-order differential operator makes it distinct from the integer-order differential operator. It means that the future state of a dynamical system or process involving fractional derivatives depends on its current state as well its past states. In fact, differential equations of arbitrary order are capable of describing memory and hereditary properties of several materials and processes. This characteristic of fractional calculus has contributed to its popularity and has convinced many researchers of the need to shift their focus from classical integer-order models to fractional-order models. There has been a great surge in developing new theoretical aspects such as periodicity, asymptotic behavior, and numerical methods for fractional equations. For some recent work on the topic, see [5–18] and the references cited therein.

where $0<\delta \le 1$, $2<\alpha -\delta <3$, $\beta >0$, $D^{(\cdot )}$ denotes the Riemann-Liouville fractional derivative of order $(\cdot )$, $F,G:[0,1]\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$ are multivalued maps, $\mathcal{P}(\mathbb{R})$ is the family of all nonempty subsets of ℝ and A, B are real constants.

We establish two new existence results for the problem (1.1). The first result relies on a nonlinear alternative for contractive maps, while in the second result, we shall combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semi-continuous multivalued maps with nonempty closed and decomposable values.

The paper is organized as follows. In Section 2, we recall some preliminary facts that we need in the sequel and Section 3 deals with the main results.

Let us recall some basic definitions of the fractional calculus [1–3].

where $[q]$ denotes the integer part of the real number q.

provided the integral exists.

To define the solutions of the problem (1.1), we need the following lemma. Though the proof of this lemma involves standard arguments, we trace its proof for the convenience of the reader.

Notice that $\alpha \ne \delta +{\eta }^{\alpha -\delta }$ in view of the given values of the parameters involved in the expression. This completes the proof. □

Let $\mathcal{C}=C([0,1],\mathbb{R})$ denote the Banach space of all continuous functions from $[0,1]\to \mathbb{R}$ endowed with the norm defined by $\parallel x\parallel =sup\{|x(t)|,t\in [0,1]\}$.

To establish the main results of this paper, we use the following form of the nonlinear alternative for contractive maps [[19], Corollary 3.8].

for all $u\in \mathbb{R}$ with $\parallel u\parallel \le \rho$.

Lemma 2.6 (Lasota and Opial [20])

is a closed graph operator in $C([0,1],\mathbb{R})\times C([0,1],\mathbb{R})$.

Before presenting the main results, we define the solutions of the boundary value problem (1.1).

Theorem 3.2 Assume that

(H₁) $F:[0,1]\times \mathbb{R}\to {\mathcal{P}}_{cp,c}(\mathbb{R})$ is an $L^1$ Carathéodory multivalued map;

for all $x,y\in C([0,1],\mathbb{R})$ and $Z_1\parallel k\parallel <1$, where $Z_1$ is given by (3.2);

(H₃) $G:[0,1]\times \mathbb{R}\to {\mathcal{P}}_{cp,c}(\mathbb{R})$ is an $L^1$ Carathéodory multivalued map;

for all $x\in \mathbb{R}$;

where $Z_1$, $Z_2$ are given by (3.2) and (3.3), respectively, and $F_0={\int }_0^1\parallel F(t,0)\parallel dt$.

Then the problem (1.1) has a solution on $[0,1]$.

for $f\in S_{F,x}$, $g\in S_{G,x}$, where Q is given by (3.1).

for $g\in S_{G,x}$.

Observe that $\mathcal{N}={\mathcal{N}}_1+{\mathcal{N}}_2$. We shall show that the operators ${\mathcal{N}}_1$ and ${\mathcal{N}}_2$ satisfy all the conditions of Theorem 2.4 on $[0,1]$. For the sake of clarity, we split the proof into a sequence of steps and claims.

Step 1. We show that ${\mathcal{N}}_1$ is a multivalued contraction on $C([0,1],\mathbb{R})$.

has nonempty values and is measurable. Let $v_2$ be a measurable selection for U (which exists by Kuratowski-Ryll-Nardzewski’s selection theorem [21, 22]). Then $v_2\in F(t,y)$ and $|v_1(t)-v_2(t)|\le k(t)\parallel x-y\parallel$ a.e. on $[0,1]$.

Step 2. We shall show that the operator ${\mathcal{N}}_2$ is u.s.c. and compact. It is well known [[23], Proposition 1.2] that if an operator is completely continuous and has a closed graph, then it is u.s.c. Therefore we will prove that ${\mathcal{N}}_2$ is completely continuous and has a closed graph. This step involves several claims.

Claim I ${\mathcal{N}}_2$ maps bounded sets into bounded sets in $C([0,1],\mathbb{R})$.

Let $B_r=\{x\in C([0,1],\mathbb{R}):\parallel x\parallel \le r\}$ be a bounded set in $C([0,1],\mathbb{R})$.

Hence ${\mathcal{N}}_2$ is bounded.

Claim II ${\mathcal{N}}_2$ maps bounded sets into equicontinuous sets.

Obviously the right hand side of the above inequality tends to zero independently of $x\in B_r$ as $t_1-t_2\to 0$. Therefore it follows by the Arzelá-Ascoli theorem that ${\mathcal{N}}_2:C([0,1],\mathbb{R})\to \mathcal{P}(C([0,1],\mathbb{R}))$ is completely continuous.

Claim III Next we prove that ${\mathcal{N}}_2$ has a closed graph.

for some $v_{\ast }\in S_{G,x_{\ast }}$.

Hence ${\mathcal{N}}_2$ has a closed graph (and therefore it has closed values). In consequence, ${\mathcal{N}}_2$ is compact valued.