Quantum Difference Langevin System with Nonlocal -Derivative Conditions

Abstract

We introduce a new class of boundary value problems for Langevin quantum difference systems. Some new existence and uniqueness results for coupled systems are obtained by using fixed point theorems. The existence and uniqueness of solutions are established by Banach’s contraction mapping principle, while the existence of solutions is derived by using Leray-Schauder’s alternative. The obtained results are well illustrated with the aid of examples.

1. Introduction

Quantum calculus (-calculus) has a rich history and the details of its basic notions, results, and methods can be found in the text [1]. Apart from the traditional treatment of quantum calculus, many interesting questions and problems, especially from theoretical point of view, either remained open or were partially answered. In recent years, the topic has attracted the attention of several researchers and a variety of new results can be found in the papers [2–12]. However, there are many aspects of boundary value problems of quantum difference equations that need attention. For instance, quantum difference Langevin systems with nonlocal -derivative conditions are yet to be addressed.

In this paper, we investigate the sufficient conditions for existence of solutions for quantum difference Langevin system of the formwhere , , are quantum numbers, are constants, are continuous functions, and are fixed points.

The Langevin equation (first formulated by Langevin in 1908) is found to be an effective tool to describe the evolution of physical phenomena in fluctuating environments [13]. For some new developments on the fractional Langevin equation in physics, see, for example, [14–22].

In this paper, we prove existence and uniqueness of solutions by using Banach’s contraction principle and existence of solutions via Leray-Schauder’s alternative.

The paper is organized as follows. In Section 2, we recall some preliminary results from quantum calculus needed in the sequel. Also two basic lemmas are proved. The main existence and uniqueness results are contained in Section 3. Finally, in Section 4, examples illustrating the obtained results are presented.

2. Preliminaries

Let us recall some basic concepts of -calculus [1, 23].

Definition 1. For , we define the -derivative of a real valued function as

The higher order -derivatives are given by

For , we set and define the definite -integral of a function by provided that the series converges.

For , we set

Note that, for , we have , , for some , thus the definite integral is just a finite sum, so no question about convergence is raised.

We note that while, if is continuous at , then In -calculus, the product rule and integration by parts formula are Further, the reversing order of integration is given by In the limit the above results correspond to their counterparts in standard calculus.

Lemma 2. Let be a continuous function and , . Then we have the following: (i) (ii) 

Proof. To prove (i), using the definition of -derivative, we have For , we obtain Next, we will show that (ii) holds. From the reversing order of integration, the double -integral can be reduced to a single integral as Taking the -derivative to the both sides of the above equation, it follows that Since it is easy to see that This completes the proof.

Letwith

Lemma 3. Let and the functions . Then are solutions of the problemif and only if

Proof. Simplifying the first two equations of problem (20) and applying the double quantum integral, we obtainwhere , . From the conditions , , Lemma 2, and , , we have Using the coupled nonlocal boundary conditions and Lemma 2, we get the following system:Solving system (24) for constants and , we haveSubstituting all values of constants , , in (22), we obtain the solutions of system (20) as in (21). The converse follows by direct computation. This completes the proof.

3. Main Results

In this section, we are going to prove the existence and uniqueness of solutions for the Langevin quantum difference system (1) with coupled boundary -derivative conditions by using fixed point theorems. Let be the Banach space of all continuous functions from to with the norm defined by . Obviously is a Banach space. In addition the product space is a Banach space with norm .

In view of Lemma 3, we define an operator by whereIn addition, we set constantswhere

Theorem 4. Assume that are continuous functions and there exist positive constants such that, for all and , In addition, assume thatwhere Then problem (1) has a unique solution on .

Proof. Define and and choose a real number such that Note that from (31) the constant .
Firstly, we will show that , where . For , we haveand in a similar wayThis shows that .
Next, we will prove that the operator is contractive. For any , and , we have Thus,Similarly,It follows from (37) and (38) that From (31), therefore, is a contraction mapping. So, by Banach’s fixed point theorem, the operator has a unique fixed point, which is the unique solution of problem (1). This completes the proof.

In the next result, we prove the existence of solutions for problem (1) by Leray-Schauder alternative.

Lemma 5 ((Leray-Schauder alternative) (see [24])). Let be a nonmed linear space and let be a completely continuous operator (i.e., a map that is restricted to any bounded set in G is compact). Let Then either the set is unbounded or has at least one fixed point.

For convenience, we set constants

Theorem 6. Assume that are continuous functions and there exist real constants and , , such that ; we have If , , , and , then there exists at least one solution for problem (1) on .

Proof. Now we show that the operator is completely continuous. Let where . Then there exist positive constants and such that and a positive real number such that For any , we have In the same way, we deduce that Therefore, is uniformly bounded.
Next, we show that is equicontinuous. Let with . Setting and and for any , we get Similarly, we obtain Then is equicontinuous. So is relatively compact on , and by the Arzelá-Ascoli theorem is completely continuous on .
Finally, it will be verified that the set is bounded. Let ; then . For any , we have Therefore, we obtain So, we have Consequently, for any , where is defined by (42), so that is bounded. Thus, by Lemma 5, the operator has at least one fixed point. Hence, problem (1) has at least one solution on . The proof is completed.

4. Examples

In this section, we present examples to illustrate our result.

Example 1. Consider the following system of Langevin quantum difference equations subject to the coupled nonlocal -derivatives boundary conditions:

Here , , , , , , , , , , , , , , and .

We have Then, the assumption of Theorem 4 is satisfied with , , , , , , , , , , , , and Therefore, we get that Hence, by Theorem 4, problem (54) has a unique solution on .

Example 2. Consider the following system of Langevin quantum difference equations subject to the coupled nonlocal -derivatives boundary conditions:

Here , , , , , , , , , , , , , , and .

So that Then, the assumptions of Theorem 6 are satisfied with , , , , , , , , , , , , , , and Consequently all conditions in Theorem 6 are satisfied. Therefore, problem (1) has at least one solution on .

Competing Interests

The authors declare that they have no competing interests.