Existence of Solution for a Fractional Langevin System with Nonseparated Integral Boundary Conditions

In the recent few decades, fractional differential equations have been studied by many researchers, and this is due to the importance of this field and its applications in many problems of physics, chemistry, biology, and economy (for more details, we refer the readers to [1–6] and many other references therein). In particular, fractional Langevin differential equations have been one of the important subjects in the field of fractional differential equations for their rich history (for more information, see [7–9]). Fractional Langevin equations are applied widely in many domains like engineering, physics, and biology (for more details, we give the following references [10–13]). On the other hand, coupled systems of fractional differential equations are very important to study because they appear naturally in many problems (see [14–18]). Recently, in [19], the existence and uniqueness of solutions for a coupled system of Riemann–Liouville and Hadamard fractional derivatives of Langevin equation with fractional integral conditions were proved. +e existence and uniqueness of the coupled system of nonlinear fractional Langevin equations with multipoint and nonlocal integral boundary conditions have been studied in [20]. So, in this current article, we study the existence and uniqueness of solutions for a coupled system of fractional Langevin equation as follows:


Introduction
In the recent few decades, fractional differential equations have been studied by many researchers, and this is due to the importance of this field and its applications in many problems of physics, chemistry, biology, and economy (for more details, we refer the readers to [1][2][3][4][5][6] and many other references therein).
In particular, fractional Langevin differential equations have been one of the important subjects in the field of fractional differential equations for their rich history (for more information, see [7][8][9]). Fractional Langevin equations are applied widely in many domains like engineering, physics, and biology (for more details, we give the following references [10][11][12][13]).
On the other hand, coupled systems of fractional differential equations are very important to study because they appear naturally in many problems (see [14][15][16][17][18]).
Recently, in [19], the existence and uniqueness of solutions for a coupled system of Riemann-Liouville and Hadamard fractional derivatives of Langevin equation with fractional integral conditions were proved. e existence and uniqueness of the coupled system of nonlinear fractional Langevin equations with multipoint and nonlocal integral boundary conditions have been studied in [20].
So, in this current article, we study the existence and uniqueness of solutions for a coupled system of fractional Langevin equation as follows: subject to the fractional nonseparated integral boundary conditions: x 1 (0) + μ 1 x 1 (1) � σ 11 To our knowledge, coupled fractional Langevin equations involving nonseparated type integral boundary conditions have not been extensively investigated yet. e main results shown in this paper can be viewed as the extension of the results in [21]. is paper is organized as follows. In Section 2, we recall some notations and several known results. In Section 3, we show the existence and uniqueness of solutions to problems (1) and (2). In Section 4, we give two examples to demonstrate the application of our main results.

Preliminaries and Notations
In this section, we introduce some notations, definitions, and lemmas that we need in our proofs later.
Definition 1 (see [3]). e fractional integral of order α > 0 with the lower limit zero for a function f can be defined as Definition 2 (see [3]). e Caputo derivative of order α > 0 with the lower limit zero for a function f can be defined as where n ∈ N, 0 ≤ n − 1 < α < n, t > 0.
Theorem 1 (see [22]). Let M be a bounded, closed, convex, and nonempty subset of a Banach space X. Let A and B be operators such that (ii) A is compact and continuous. (iii) B is a contraction mapping.
en, there exists z ∈ M such that z � Az + Bz.
According to the condition

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Using the facts that Substituting the value of a 01 , a 11 , and a 21 , we obtain Analogously, we can deduce that 4 Journal of Mathematics By direct computation, the converse of the lemma can be easily verified.

Main Results
Let X be a Banach space of all continuous functions from In view of Lemma 3, we define the operator U: Here, For computational convenience, we set Journal of Mathematics A ij (t) , for i � 1, 2, . . . , 5 and j � 1, 2.

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Analogously, we can also have which leads to As r 11 + r 12 < 1, the operator U is a contraction mapping. en, we deduce that systems (1) and (2) (1) and (2) have at least one solution on Proof. We define a bounded closed and convex ball B r′ � ( Let us introduce the decomposition with (28)

Conclusion
In this paper, we have investigated the existence and uniqueness results for a coupled system of nonlinear fractional Langevin equations supplemented with nonseparated integral boundary conditions by using the Banach contraction principle and Krasnoselskii's fixed point theorem. Finally, we gave two examples to prove the validity of our results.

Data Availability
No data were used to support this study.