Fractional Langevin Equations with Nonseparated Integral Boundary Conditions

Abstract

In this paper, we discuss the existence of solutions for nonlinear fractional Langevin equations with nonseparated type integral boundary conditions. The Banach fixed point theorem and Krasnoselskii fixed point theorem are applied to establish the results. Some examples are provided for the illustration of the main work.

1. Introduction

Fractional derivatives give an excellent description of memory and hereditary properties of different processes. Properties of the fractional derivatives make the fractional-order models more useful and practical than the classical integral-order models.

Several researchers in the recent years have employed the fractional calculus as a way of describing natural phenomena in different fields such as physics, biology, finance, economics, and bioengineering (for more details see [1–9] and many other references).

With the recent outstanding development in fractional differential equations, the Langevin equation has been considered a part of fractional calculus, and thus, important results have been elaborated (see [10–14]).

The Langevin equation was first introduced by Langevin in 1908; it is a fundamental theory of the Brownian motion to describe the evolution of physical phenomena in fluctuating environments [15, 16]. The fractional model of the Langevin equation as a generalization of the classical one gives a fractional Gaussian process parametrized by two indices, and this fractional model is more flexible for modeling fractal processes [17, 18].

The fractional Langevin equation is extensively studied in the literature from both the theoretical and numerical point of views (for more details see [19–25]). In [26], the authors studied a nonlinear Langevin equation involving two fractional orders in different intervals. In [27], the authors discussed the existence theory for a nonlinear Langevin equations with nonlocal multipoint and multistrip boundary conditions. In [28], fractional Langevin equations with nonlocal integral boundary conditions have been investigated by Salem et al. In [14], an antiperiodic boundary value problem for the Langevin equation involving two fractional orders has been studied.

Recently, in [29], the authors discussed the nonlinear fractional differential equations with nonseparated type integral boundary conditions; however, the fractional Langevin equations involving nonseparated integral boundary conditions have not been investigated yet; that is why, in this work and motivated by all the works cited above, we study the existence and uniqueness of the fractional Langevin equations with nonseparated integral boundary conditions as follows: where , , , with , and are the Caputo fractional derivatives and and are given continuous functions.

This paper is divided into four sections, in which the second provides some notations and basic known results, in the third section, we study the existence and uniqueness of solutions to problem (1), and in the fourth section, we give two examples to illustrate our results.

2. Preliminaries and Notations

In this section, we give some notation, definitions, and lemma which are needed throughout this paper.

Definition 1 (see [5]). The fractional integral of order with the lower limit zero for a function can be defined as

Definition 2 (see [5]). The Caputo derivative of order with the lower limit zero for a function can be defined as where , , and .

Theorem 3 (see [30]). Let be a bounded, closed, convex, and nonempty subset of a Banach space . Let and be operators such that (I) whenever (II) is compact and continuous(III) is a contraction mappingThen, there exists such that .

Lemma 4 (see [5]). Let ; then, the following relations hold:

Lemma 5 (see [5]). Let and . If is a continuous function, then, we have

Lemma 6. Let . Then, a unique solution of the bondary value problem is given by where

Proof. By applying Lemma 5, we have where .
Using the condition , we obtain By conditions and , we have

Substituting the value of , , and , we obtain the desired results. And by direct computation, one can obtain the converse of the lemma.

3. Main Results

Denote by the Banach space of all continuous functions from endowed with norm .

By Lemma 6, we transform problem (1) into a fixed point problem as , where is given by

Theorem 2. Suppose that and are continuous functions satisfying
—there exist positive constants such that for all , .
—there exist positive constants such that Then there exist a unique solution for boundary value problem (1) provided that , where and for .

Proof. We set , , , .
Let the ball with radius , where , with

Then, is a closed, convex, and nonempty subset of the Banach space .

Our aim is to prove that the operator has a unique fixed point on .We show that .

For , , we have which implies that

Now, for and for ,

Thus,

Since , then operator is a contraction mapping. Therefore, boundary value problem (1) has a unique solution.

Theorem 7. Assume that and hold, is a continuous function. Further, we suppose
— with .Then, boundary value problem (1) has at least one solution on if and , where

Proof. Consider the closed ball with the radius , where We introduce the decomposition , where For , we have which implies that

Thus, .

For , we have which implies that since , then is a contraction.

Next, we show that is compact and continuous. Continuity of implies that the operator is continuous.

Since, , therefore, is uniformly bounded on . Suppose that . We have

As , the above expression tends to be zero independently from . This implies that is relatively compact on . Then, by the Arzela-Ascoli theorem, operator is compact on .

Therefore, according to the Krasnoselskii fixed point theorem, problem (1) has at least one solution on .

4. Example

Example 8. Consider the following boundary value problem: We choose , , , , , , , , and the continuous functions Clearly, , , and ,
Then, we have .
Thus, all the assumptions of Theorem 2 hold. Then, problem (29) has a unique solution.

Example 9. Consider the following boundary value problem: Here, , , , , , , , , and the continuous functions Clearly, , , and ; thus, we have and .
Then, problem (31) has a least one solution.

5. Conclusion

In this paper, we studied the existence and uniqueness of fractional Langevin equations with nonseparated integral boundary conditions. First of all, we transformed the problem into an equivalent fixed point problem; second of all, we utilized the Banach contraction principle and the Krasnoselskii fixed point theorem to prove the existence and uniqueness of solutions.

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 contributed equally to the writing of this paper. All authors read and approved the final manuscript.