Existence of Solutions for Nonlinear Impulsive Fractional Differential Equations with 𝑝-Laplacian Operator

Abstract

This paper studies the existence of solutions for a nonlinear boundary value problem of impulsive fractional differential equations with -Laplacian operator. Our results are based on some standard fixed point theorems. Examples are given to show the applicability of our results.

1. Introduction

Fractional calculus (differentiation and integration of arbitrary order) has proved to be an important tool in the modeling of dynamical systems associated with phenomena such as fractals and chaos. In fact, this branch of calculus has found its applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and image processing, polymer rheology, regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electrodynamics of complex medium, viscoelasticity and damping, control theory, wave propagation, percolation, identification, and fitting of experimental data. Fractional differential equations serve as an excellent tool for the description of hereditary properties of various materials and processes. The interest in the study of fractional-order differential equations lies in the fact that fractional-order models are found to be more accurate than the classical integer-order models; that is, there are more degrees of freedom in the fractional-order models. In consequence, the subject of fractional differential equations is gaining more and more attention. For some recent development on the topic, see [1–11] and the references therein.

Since the -Laplacian operator and fractional calculus arise from many applied fields such as turbulent filtration in porous media, blood flow problems, rheology, modeling of viscoplasticity, and material science, it is worth studying the fractional -Laplacian differential equations. The research of boundary value problems for -Laplacian equations of fractional order has just begun in recent years; see [12–16].

The theory of impulsive differential equations is adequate mathematical models for description of evolution processes characterized by the combination of a continuous and jumps change of their state. Impulsive differential equations have become an important area of research in recent years of the needs of modern technology, engineering, economic and physics. Moreover, impulsive differential equations are richer in applications compared to the corresponding theory of ordinary differential equations. Many of the mathematical problems encountered in the study of impulsive differential equations cannot be treated with the usual techniques within the standard framework of ordinary differential equations. For the introduction of the basic theory of impulsive equations, see [17–22] and the references therein. It is worthwhile mentioning that impulsive differential equations of fractional order have not been much studied and many aspects of these equations are yet to be explored. The recent results on impulsive fractional differential equations can be found in [23–30].

The investigation on fractional -Laplacian impulsive differential equations has not been appreciated well enough. As far as we know, there is only one paper that has considered nonlinear fractional impulsive differential equation with -Laplacian operator; see [31].

Motivated by the above mentioned work on -Laplacian and impulsive boundary value problems of fractional order, in this study, we consider the following problem: where is the Caputo fractional derivative, is a -Laplacian operator, , , is invertible, and where , , , with , , , , , where and denote the right and the left limits of at , respectively. have a similar meaning for .

This paper is organized as follows. In Section 2, we provide some necessary definitions and preliminary lemmas which are key tools for our main results. We give and prove our main results in Section 3. Finally, in Section 4, we give some examples to demonstrate our main results.

2. Preliminaries

In this section, we present auxiliary lemmas which will be used later.

Let , , , and we introduce the following spaces.

with the norm , and with the norm . Obviously, and are Banach spaces.

Definition 1. For a continuous function , the Caputo derivative of fractional order is defined as where denotes the integer part of real number .

Definition 2. The Riemann-Liouville fractional integral of order is defined as provided the integral exists.

Lemma 3 (see [7]). Let ; then the differential equation has solution , , , .

Lemma 4 (see [7]). Let , ; then

Definition 5. A function with its Caputo derivative of order existing on is a solution of (1) if it satisfies (1).
We need the following known results to prove the existence of solutions for (1).

Theorem 6 (see [32]). Let be a Banach space. Assume that is an open bounded subset of with and let be a completely continuous operator such that Then has a fixed point in .

Theorem 7 (see [32]). Let be a Banach space. Assume that is a completely continuous operator and the set is bounded. Then has a fixed point in .

Lemma 8. For a given , a function is a solution of the impulsive boundary value problem of fractional order if and only if is a solution of the impulsive fractional integral equation where

Proof. Let be a solution of (7). Then, for , there exist constants such that
For , there exist constants such that Then we have In view of and , we have
Consequently,
By a similar process, we can get
By conditions and , we have Substituting the value of in (10) and (16), we can get (8). Conversely, assume that is a solution of the impulsive fractional integral equation (8); then by a direct computation, it follows that the solution given by (8) satisfies (7). This completes the proof.

3. Main Results

For the sake of convenience, we set Define the operator as Then the problem (1) has a solution if and only if the operator has a fixed point.

Lemma 9. The operator defined by (19) is completely continuous.

Proof. It is obvious that is continuous in view of the continuity of , , and . Let be bounded. Then, there exist positive constants such that , and , . Thus, , and we have Since , therefore there exists a positive constant , such that , which implies that the operator is uniformly bounded.
On the other hand, for any , , we have Hence, for , , , we have which implies that is equicontinuous on all , . Thus, by the Arzela-Ascoli theorem, the operator is completely continuous.

Theorem 10. Let , , and , and then the problem (1) has at least one solution.

Proof. Since , , and , therefore there exists a constant such that , , and for , where satisfy the inequality
Let us set and take such that ; that is, . Then, by the process used to obtain (20), we have Thus, it follows that , . Therefore, by Theorem 6, the operator has at least one fixed point, which in turn implies that the problem (1) has at least one solution .

Theorem 11. Assume that there exist positive constants such that Then the problem (1) has at least one solution.

Proof. Let us show that the set is bounded. Let , and then , . For any , we have Combining (25) and (26) and employing the procedure used to obtain (20), we obtain which implies that for any . So, the set is bounded. Thus, by Theorem 7, the operator has at least one fixed point. Hence the problem (1) has at least one solution.

4. Examples

Example 1. For , consider the following fractional order impulsive boundary value problem: Here , , , , , , , and . Clearly Furthermore, in this case, given by (23) satisfy the inequality Thus all the assumptions of Theorem 10 hold. Hence, the conclusion of Theorem 10 applies and the impulsive fractional boundary value problem (28) has at least one solution.

Example 2. Consider the following fractional order impulsive boundary value problem: Here , , , , , , , , and .
In this case, , , , and the conditions of Theorem 11 can readily be verified. Thus, by the conclusion of Theorem 11, the problem (28) has at least one solution.

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

All authors contributed equally in this paper. They read and approved the paper.

Acknowledgment

The authors would like to thank the reviewers for their valuable comments and suggestions on the paper.