Journal of Function Spaces
Volume 2018 (2018), Article ID 2503915, 11 pages
https://doi.org/10.1155/2018/2503915

Research Article

Impulsive Fractional Differential Equations with -Laplacian Operator in Banach Spaces

Jingjing Tan,¹ Kemei Zhang,² and Meixia Li1,3

¹School of Mathematics and Information Science, Weifang University, Shandong 261061, China
²School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
³College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China

Correspondence should be addressed to Jingjing Tan; tanjingjing1108@163.com

Received 1 May 2018; Accepted 18 October 2018; Published 15 November 2018

Academic Editor: Yong H. Wu

Copyright © 2018 Jingjing Tan et al. 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 a class of boundary value problem (BVP) with multiple point boundary conditions of impulsive p-Laplacian operator fractional differential equations. We establish the sufficient conditions for the existence of solutions in Banach spaces. Our analysis relies on the Kuratowski noncompactness measure and the Sadovskii fixed point theorem. An example is given to demonstrate the main results.

1. Introduction

With the development of the theory of fractional order calculus, fractional differential equations are getting more and more extensively used (see[1–15]). For instance, the impulsive fractional differential equations are widely used in various scientific fields, such as the problem of dynamics of populations subject to abrupt changes, harvesting, diseases, and so on. Lakshmikantham et al. [16], Bainov and Simeonov [17], and Benchohra et al. [18] have done in-depth studies on this issue. Moreover, the p-Laplacian operator differential equation was first proposed by Leibenson [19] in order to study the problem of turbulent flow in a porous medium. He converted this problem into the existence of solution of the following differential equation:

is the p-Laplacian operator, is the inverse function of with , and satisfy . In recent years, many results about the solutions of the p-Laplacian operator fractional differential equation BVP have been obtained (see [20–26]). The research of the solutions of the BVP with p-Laplacian operator and with impulsive has been attracting increasing interest.

Zhao and Gong [27] study the solution of the following impulsive fractional differential equations with generalized periodic boundary conditions:

By means of the Schauder and Guo- Krasnosel’skii fixed point theorem, they get the existence of single and multiple positive solutions of the above BVP. By the technique of the Guo-Krasnosel’skii fixed point theorem and the Leggett-Williams theorem, Wang et al. [28] obtained the results of the following BVP.

The results about the BVP with multiple point boundary conditions of impulsive p-Laplacian operator fractional differential equations are few, especially in Banach space.

In this paper, we study the following BVP with multiple point boundary conditions of impulsive p-Laplacian operator fractional differential equations in Banach space :

where is the zero element of , and are the fractional derivatives of order , , , with . , , and are continuous. The impulsive point set satisfies ; we denote , . We denote by and the right and left limits of at the point , i.e., and . We establish the existence of solution to BVP (4). by the technique of the Kuratowski noncompactness measure and the Sadovskii fixed point theorem. The main innovations of this paper are as follows. Firstly, we study the impulsive fractional differential equations with the p-Laplacian Operator. Secondly, we study the BVP in Banach space. Thirdly, the nonlinear term contains the derivatives .

The paper is organized as follows. In Section 2, we recall some definitions and lemmas. In Section 3, the main results of this paper are discussed. Finally, one example is given in Section 4.

2. Preliminaries

First, we recall some definitions and preliminary.

Definition 1 (see [29]). Let , .
The fractional integral of order of is given by

the fractional derivative of order of is given by

where , denotes the integer part of and the right side integral is pointwise defined on .

Lemma 2 (see[30, 31]). Let be integrable, , . Then
(1)

where , , .
(2)

Definition 3 (see [32]). Let be a bounded set in a real Banach space ; the Kuratowski noncompactness measure of is given by

where denote the diameters of .

Remark 4. From the definition, it is obvious that .

Lemma 5 (see [33]). If is bounded and equicontinuous, then is continuous on and , , where for each .

Definition 6 (see [32]). Let and be real Banach spaces, , and be a continuous and bounded operator; is called a k-set contraction operator if there exists a constant , for any bounded set in , such that .

Remark 7 (see [34]). is called a strict set contraction operator if . It is clear that a strict set contraction operator is a condensing operator.
In the following, we define the basic space of this paper. Denote

and it is easy to see that is Banach space with the norm

The basic space used in this paper is . The Kuratowski noncompactness measures in , , and DC(I) are denoted by , , and , respectively.

The following Sadovskii fixed point theorem is needed for the proof of our main results.

Lemma 8 ((Sadovskii)(see [32])). Let be a bounded, closed, and convex subset of the Banach space . If the operator is condensing, then has a fixed point in .

3. Existence of Solutions

Before proceeding further, let us give some denotations as follows:

For simplicity of presentation, we list some conditions.

are nonnegative functions and satisfy

For any , , is uniformly continuous on .

For any ,, , and , .

For all and all bounded subsets , there exist such that

with .

Lemma 9. Given , the following BVP,

has a unique solution satisfying the following.

Proof. From (15) and Lemma 2, we know

Because of , we can obtain that . According to the definition of the p-Laplacian operator it follows that

Therefore

If , then

For and , we have

When , we can also obtain

and taking the boundary condition of (15) into consideration, some tedious manipulation yields the following.

Substituting (24) and (25) into (19), we can get (16), which implies that the solution of BVP (15) is given by (16).

From Lemma 9, we can establish the following conclusion.

Lemma 10. If is satisfied, then BVP (4) has a unique solution satisfying

Proof. The proof is almost identical to Lemma 9, with the major change being the substitution of for .

Remark 11. From Lemma 10, we can get the conclusion that the solutions to the BVP (4) are equivalent to the fixed point of the following operator:

Taking derivative to both sides of (27), we have

Lemma 12. If , , and are satisfied, then operator is continuous and bounded.

Proof. ​​
Step 1. For any , we prove that is well defined and . From condition , we have

Together with the definition of operator , we have

As above, from (28), a tedious calculation gives

From (30) and (31), we have that is well defined and for any .
Step 2. We prove that is bounded. For any , from (29)−(31), we get

So is bounded operator.
Step 3. It is time to prove that is continuous. Let , and for any , , there exists , when , satisfying that . So is a bounded subset of ; let such that ; taking limit, we obtain . From , we know that, for any and , there exists ; when , we have

Taking , for any , , and , according to (27), (28), we have

Thus So is continuous.

Lemma 13. If are satisfied, is a bounded subset of . Then and are equicontinuous on .

Proof. In fact, from the boundedness of , that is, for any , there exists such that . Suppose that with ; by the mean value theorem, we have the following.

Let

From (37),

For the case of , using the same methods we can also get (38). So is equicontinuous on . An argument similar to the one used in shows that

So is equicontinuous on .

Lemma 14 (see [35]). Let condition be satisfied and be a bounded subset of . Then

Theorem 15. Let conditions be satisfied. Then BVP (4) has at least one solution belonging to .

Proof. By Remark 11, we need only to show that the operator has a fixed point in . Let

where

Step 1. we prove that . In fact, for any , by (27), we have

From (43) and (44), we know . Thus, from Lemma 12, follows.
Step 2. It is time to prove that is a strict set contraction operator. Let . It is easy to see that is nonempty, bounded, convex, and closed subset of . According to Lemma 13, and are equicontinuous on , so and are equicontinuous on . From the definition of , we get and , and by Lemma 12, is bounded and continuous. From , it follows that are equicontinuous on . For any and , by and Lemma 2, we obtain

Since is arbitrary, we obtain

Using similar methods, we can also show that

Form (46) and (47), using Lemma 14 we get

Obviously, , From Remark 7, is a condensing operator too. It follows from the Sadovskii fixed point theorem that has at least one fixed point in ; therefore, BVP(4) has at least one solution in .

Remark 16. If , we can get the following result.

Corollary 17. Let and be satisfied. Then BVP (4) has at least one solution in .

Proof. Letting in Theorem 15, we can prove the desired result.

Corollary 18. Suppose the following assumptions holds:
are nonnegative functions and satisfy

for any , , is uniformly continuous on ;
for any ,, , and ;
for all and all bounded subsets , there exists such that

with . Then the following BVP,

has at least one solution belonging to .

Proof. The proofs are similar to Theorem 15, except the need for substitution of for nonlinear term .

4. Illustrative Example

Let and equipped with the norm . The following BVP

can be regarded as a problem with the form of BVP (4), where

Clearly , , , and

Let , , and . According to and , we have

where and . So, all the conditions of Theorem 15 are satisfied. Therefore, BVP (52) has at least one solution.

5. Conclusion

In this paper, we present some sufficient conditions which ensure the existence of solution to BVP (4) in Banach spaces. Through construction space , using the technique of the Kuratowski noncompactness measure and the Sadovskii fixed point theorem, we obtain some new existence criteria for BVP (4). As far as we know, only a few papers have dealt with the boundary value problem for impulsive p-Laplacian fractional differential equations, especially in Banach space.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors wrote, read, and approved the final manuscript.

Acknowledgments

This project is supported by Shandong Provincial Natural Science Foundation (Grant No. ZR2017LA002), Weifang Science and Technology Development Projects (Grant No. 2017GX025), and Doctoral Research Foundation of Weifang University (Grant No. 2017BS02).

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