Discrete Dynamics in Nature and Society
Volume 2019 (2019), Article ID 1323176, 9 pages
https://doi.org/10.1155/2019/1323176

Research Article

Existence Results for Impulsive Fractional Differential Inclusions with Two Different Caputo Fractional Derivatives

Dongdong Gao and Jianli Li

Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP), Ministry of Education of China, School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China

Correspondence should be addressed to Jianli Li; ljianli18@163.com

Received 22 October 2018; Accepted 10 January 2019; Published 18 February 2019

Academic Editor: Marek Galewski

Copyright © 2019 Dongdong Gao and Jianli Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we study the impulsive fractional differential inclusions with two different Caputo fractional derivatives and nonlinear integral boundary value conditions. Under certain assumptions, new criteria to guarantee the impulsive fractional impulsive fractional differential inclusion has at least one solution are established by using Bohnenblust-Karlin’s fixed point theorem. Also, some previous results will be significantly improved.

1. Introduction

In this paper, we consider the following fractional differential inclusions with impulsive effects:

where , , and represent the different Caputo fractional derivatives of orders and , respectively. is a multivalued map, is the family of all nonempty subsets of , and is a given continuous function. , are real constants and is a given positive parameter. and represent the right and left limits of at

As an extension of integer-order differential equations, fractional-order differential equations have been of great interest since the equations involving fractional derivatives always have better effects in applications than the traditional differential equations of integer order. Due to these significant applications in various sciences, such as physics, engineering, chemistry, and biology, fractional differential equations have received much attention by researchers during the past two decades. Up to now, fractional boundary value problems are still heated research topics. That is why, more and more considerations by many people have been paid to study the existence of solutions for fractional boundary value problems; we refer readers to [1–12].

However, the articles of fractional boundary value problems with two different Caputo fractional derivatives are not many. More precisely, in [10], the authors have studied the following impulsive fractional Langevin equations with two different Caputo fractional derivatives:

where is a given function, and , , , , and represent the right and left limits of at

Then, in [11], the authors considered the following nonlinear Langevin inclusions with two different Caputo fractional derivatives:

where , is a real number, is the Riemann-Liouville fractional integral of order , and are constants.

In [12], the author investigates the following impulsive fractional differential equations with two different Caputo fractional derivatives with coefficients:

where , , are real constants.

To the best of our knowledge, integral boundary conditions appear in population dynamics and cellular systems; it has constituted a very interesting and important class of problems. However, fractional boundary value problems with integral boundary conditions have not received so much attention as periodic boundary conditions, so the main aim in this paper is intended as an attempt to establish some criteria of existence of solutions for (1). It is worth pointing out that there was no paper considering the impulsive fractional differential inclusions with two different Caputo fractional derivatives and nonlinear integral conditions by using Bohnenblust-Karlin’s fixed point theorem up to now, so our results are new. Also, we improve some previous results.

The arrangement of the rest paper is as follows. In Section 2, some preliminaries and results which are applied in the later paper are presented. In Section 3, the main proof of theorems will be vividly shown. In Section 4, a corresponding example is given to illustrate the obtained results in Section 3.

2. Preliminaries

In this section, we recall some basic knowledge of definitions and lemmas that we shall use in the rest of the paper.

Let denote a Banach space of continuous functions from into with the norm

for Also, we denote the function space by

with the norm Clearly, is Banach spaces.

Let be a Banach space of measurable functions which are Lebesgue integrable and normed by

Let be a Banach space. We give following notations for convenience: let

and denote the set of all nonempty bounded, closed, and convex subset of

A multivalued map

(i) is convex (closed) valued if is convex (closed) for all ;

(ii) is bounded on bounded sets if is bounded in for any bounded set of ;

(iii) is called upper semicontinuous (u.s.c.) on if, for each , the set is nonempty closed subset of , and if, for each open set of containing , there exists an open neighborhood of such that ;

(iv) is said to be completely continuous if is relatively compact for every bounded subset of ;

(v) is completely continuous with nonempty compact values; then is if and only if has a closed graph; , imply

(vi) has a fixed point if there is such that

Definition 1. A multivalued map is Carathéodory if

(i) is measurable for each ,

(ii) is upper semicontinuous for almost all

Moreover, a Carathéodory function is called Carathéodory if

(iii) for each , there exists such that

for all for

For each , define that the set of selections for by

is nonempty.

Lemma 2 (see [13]). Let be a Banach space. Let be an Carathéodory multivalued map, and let be a linear continuous mapping from to . Then the operator

and

is a closed graph operator in

For more details, please refer to [13–15].

Definition 3. A function is called a solution of (1) if there exists a function with such that , , and

Next, we present the following necessary basic knowledge of fractional calculus theory which is used in the later paper.

Definition 4 (see [4]). The Riemann-Liouville fractional integral of order of a function is given by

provided that the right-hand side is pointwise defined on , where is the gamma function.

Definition 5 (see [4]). The Riemann-Liouville fractional derivative of order of a function is given by

where , provided that the right-hand side is pointwise defined on

Definition 6 (see [4]). The Caputo fractional derivative of order of a function is given by

where , provided that the right-hand side is pointwise defined on

Definition 7 (see [10]). Functions and are called classical and generalized Mittag-Leffler functions, respectively, given by

Lemma 8 (see [10]). Let , and then functions , and are nonnegative and have the following properties.
(i) For any and ,

Moreover,

(ii) For any and , when , we have

(iii) For any and and , we have

Lemma 9 (see [16]). Let For a given with , then the boundary value problem (1) has a unique solution which is defined by the following form:

Finally, we give the following lemma which is greatly important in the proof of our main results.

Lemma 10 (see [17, Bohnenblust-Karlin]). Let be a Banach space, a nonempty subset of , which is bounded, closed, and convex. Suppose is . with closed, convex values, and such that and are compact. Then has a fixed point.

3. Main Results

In order to begin our main results, we also need the following conditions:

(H1)There exists , and a real function such that

 for each (H2) and there exists such that

 for and , where satisfies in which is defined in (1).

For convenience, we denote

Theorem 11. Suppose that (H1) and (H2) hold; then system (1) has at least one solution on .

Proof. We transform problem (1) into a fixed point problem. Consider the operator defined by

for
Next we shall show that satisfies all the assumptions of Lemma 10; that is to say, has a fixed point which is a solution of problem (1). For the sake of convenience, we subdivide the proof into several steps.
Step 1 ( is convex for each ). In fact, assume , then there exist such that, for each , we have

Let Then, for each , we have

Since is convex ( has convex values), so it follows that
Step 2. Let , where

Then is a bounded closed convex set in Thus we need to verify In fact, from Lemma 8, (H1), and (H2), for each , we have

From (28), we have
Step 3 ( is equicontinuous). Let For convenience, we also let ,

and

and then we have

and from Lemma 8, we clearly see the right hand side of the above inequality tends to zero as This implies that is equicontinuous on . As a consequence of Steps 1–3 together with the Ascoli-Arzela theorem, we can conclude that is a compact valued map.
Step 4 ( has a closed graph). Let and Then we need to verify . implies that there exists such that for each we have

and thus we must verify that there exists such that for each we have

Consider the continuous linear operator

then,

By Lemma 2, we know is a closed graph operator.
Also from the definition of we have

Since , Lemma 2 implies that

for some
Therefore, is a compact multivalued map, with convex closed values. By Lemma 10, we have that has a fixed point which is a solution of problem (1).

Corollary 12 .. Assume that (H2) and (H3) hold.
(H3) There exist continuous and bounded functions such that

then problem (1) has at least a solution on

Proof. The proof is the same as Theorem 11 which we can take as

Remark 13. If we let and be a constant function, then the above Corollary 12 improves Theorem 3.1 in [12].

Remark 14. Note that if and , we have and , respectively. Thus, in this paper, let ; our system (1) reduces to [18], so our problem (1) gives generalization of [18].

Remark 15 .. If , the boundary value condition becomes , and our system (1) reduces to [16, 19]. If , the boundary value condition becomes , and our system (1) reduces to [20]. Thus, our problem (1) gives generalizations of [16, 19, 20].

4. An Example

In this part, we will give corresponding example to illustrate the main results in our paper.

Example 1. Consider the following system:

where , and let and Then we let and , and we have

then (H1) and (H2) of Theorem 11 all hold. Hence, system (41) has at least one solution on .

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the NSFC (11571088, 11471109), the Zhejiang Provincial Natural Science Foundation of China (LY14A010024), and Scientific Research Fund of Hunan Provincial Education Department (14A098).

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