Boundary Value Problems for First-Order Impulsive Functional 𝑞-Integrodifference Equations

Abstract

We discuss the existence and uniqueness of solutions for a first-order boundary value problem for impulsive functional -integrodifference equations. The main results are obtained with the aid of some classical fixed point theorems. Illustrative examples are also presented.

1. Introduction

In this paper, we study the boundary value problem for impulsive functional -integro-difference equation of the following the form: where , ,  , is a continuous function, , for ,  ,   are real constants, and for .

The notions of -derivative and -integral on finite intervals were introduced recently by the authors in [1]. Their basic properties were studied and applications existence and uniqueness results were proved for initial value problems for first- and second-order impulsive -difference equations. In this paper, we continue the study on this new subject by considering the boundary value problem (1).

The book by Kac and Cheung [2] covers many of the fundamental aspects of the quantum calculus. In recent years, the topic of -calculus has attracted the attention of several researchers and a variety of new results can be found in the papers [3–15] and the references cited therein. On the other hand, for some monographs on the impulsive differential equations we refer to [16–18].

The rest of this paper is organized as follows. In Section 2, we recall the notions of -derivative and -integral on finite intervals and present a preliminary result which will be used in this paper. In Section 3, we will consider the existence results for problem (1) while in Section 4, we will give examples to illustrate our main results.

2. Preliminaries

In this section, we recall the notions of -derivative and -integral on finite intervals. For a fixed let be an interval and let be a constant. We define -derivative of a function at a point as follows.

Definition 1. Assume is a continuous function and let . Then the expression is called the -derivative of function at .

We say that is -differentiable on provided that exists for all . Note that if and in (3), then , where is the well-known -derivative of the function defined by

In addition, we should define the higher -derivative of functions.

Definition 2. Let be a continuous function; we call the second-order -derivative provided that is -differentiable on with . Similarly, we define higher order -derivative .

The -integral is defined as follows.

Definition 3. Assume is a continuous function. Then the -integral is defined by for . Moreover, if , then the definite -integral is defined by

Note that if and , then (5) reduces to -integral of a function , defined by

For the basic properties of -derivative and -integral we refer to [1].

Let , , and for . Let be continuous everywhere except for some at which and exist and , .   is a Banach space with the norm .

We now consider the following linear case: where .

Lemma 4. Let . The unique solution of problem (8) is given by with for , where

Proof. For ,  -integrating (8), it follows which leads to For , taking -integral to (8), we have Since , then we have Again -integrating (8) from to , where , then Repeating the above process, for , we obtain In particular, for , we have Further, -integrating (16) from to , it follows Applying the boundary condition of (8), one has Since and for , we have Therefore, Substituting the constant into (16), we obtain (9) as requested.

3. Main Results

In view of Lemma 4, we define an operator by

It should be noticed that problem (1) has solutions if and only if the operator has fixed points.

Our first result is an existence and uniqueness result for the impulsive boundary value problem (1) by using Banach's contraction mapping principle.

Further, for convenience we set

Theorem 5. Assume the following. ) The function is continuous and there exists a constant such that for each and .() The function is continuous and there exist constants such that for each and ,  .()The functions are continuous and there exists a constant such that for each , .
If where is defined by (23), then the boundary value problem (1) has a unique solution on .

Proof. Firstly, we transform the boundary value problem (1) into a fixed point problem, , where the operator is defined by (22). Using the Banach contraction principle, we will show that has a fixed point which is the unique solution of the boundary value problem (1).
Let and be nonnegative constants such that and   =  . By choosing a positive constant as where and defined by (24), we will show that , where a suitable ball is defined by . For , we have which yields .
For any and for each , we have which implies that . Since , is a contraction. Therefore, by Banach's contraction mapping principle, we conclude that has a fixed point which is the unique solution of problem (1).

The second existence result is based on Krasnoselskii's fixed point theorem.

Lemma 6 ((Krasnoselskii's fixed point theorem) [19]). Let be a closed, bounded, convex, and nonempty subset of a Banach space . Let , be the operators such that (a) whenever ; (b) is compact and continuous; (c) is a contraction mapping. Then there exists such that .

We use the following notations:

Theorem 7. Let be a continuous function. Suppose that holds. In addition, we suppose the following: (), , and ,()there exists a constant such that for all ,  for .Then the impulsive functional -integrodifference boundary value problem (1) has at least one solution on provided that

Proof. Let . By choosing a suitable ball , where and , are defined by (32) and (33), respectively; we define the operators and on by For any , we have This implies that .
To show that is a contraction, for , we have From (34), it follows that is a contraction.
Next, the continuity of implies that operator is continuous. Further, is uniformly bounded on by Now we will prove the compactness of . Setting , then for each for some with , we have As , the right hand side above tends to zero independently on . Therefore, the operator is equicontinuous. Since maps bounded subsets into relatively compact subsets, it follows that is relative compact on . Hence, by the Arzelá-Ascoli theorem, is compact on . Thus, all the assumptions of Lemma 6 are satisfied. Hence, by the conclusion of Lemma 6, the impulsive functional -integrodifference boundary value problem (1) has at least one solution on .

Our third existence result is based on Leray-Schauder degree theory. Before proving the result, we set

Theorem 8. Assume that and are continuous functions. In addition we suppose the following. ()There exist constants and such that ()There exist constants and such that ()There exist constants and such that If where is given by (41), then the impulsive functional -integrodifference boundary value problem (1) has at least one solution on .

Proof. We define an operator as in (22) and consider the fixed point problem: We are going to prove that there exists a fixed point satisfying (47). It is sufficient to show that satisfies where . We define It is easy to see that the operator is continuous, uniformly bounded, and equicontinuous. Then, by the Arzelá-Ascoli Theorem, a continuous map defined by is completely continuous. If (48) is true, then the following Leray-Schauder degrees are well defined and by the homotopy invariance of topological degree, it follows that where denotes the identity operator. By the nonzero property of Leray-Schauder degree, for at least one . In order to prove (48), we assume that for some . Then which implies that If , inequality (48) holds. This completes the proof.

4. Examples

In this section, we will give some examples to illustrate our main results.

Example 1. Consider the following boundary value problem for nonlinear first-order impulsive functional -integrodifference equation:

Set ,   for , for , , , , and .

Since then are satisfied with , , , , and . We can show that . Hence, by Theorem 5, the boundary value problem (53) has a unique solution on .

Example 2. Consider the following boundary value problem for nonlinear first-order impulsive functional -integrodifference equation:

Set for ,   for ,   ,  , , and .

Since , and , then are satisfied with , , , , , , and . We can show that . Hence, by Theorem 8, the boundary value problem (56) has at least one solution on .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The research of Jessada Tariboon is supported by King Mongkut’s University of Technology North Bangkok, Thailand. Sotiris K. Ntouyas is a Member of Nonlinear Analysis and Applied Mathematics (NAAM) Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.