Coupled System of Nonlinear Fractional Langevin Equations with Multipoint and Nonlocal Integral Boundary Conditions

&is research paper is about the existence and uniqueness of the coupled system of nonlinear fractional Langevin equations with multipoint and nonlocal integral boundary conditions. &e Caputo fractional derivative is used to formulate the fractional differential equations, and the fractional integrals mentioned in the boundary conditions are due to Atangana–Baleanu and Katugampola. &e existence of solution has been proven by two main fixed-point theorems: O’Regan’s fixed-point theorem and Krasnoselskii’s fixed-point theorem. By applying Banach’s fixed-point theorem, we proved the uniqueness result for the concerned problem. &is research paper highlights the examples related with theorems that have already been proven.


Introduction
Recently, many mathematical fields have been developed rapidly via fractional calculus. Different applications can be described by fractional equations involving fractional derivatives. Fractional calculus was an essential element in many recently published articles, such as a fractional biological population model, a fractional SISR-SI malaria disease model, a fractional Biswas-Milovic model, fractional wave equations, fractional reaction-diffusion equations, and nonlinear fractional shock wave equations. More recent published articles related with fractional calculus can be clearly found in [1][2][3][4][5][6][7][8][9]. Fractional differential equations have obtained a remarkable reputation among the mathematicians due to rapid development which is applicable in many fields such as mathematics, chemistry, and electronics. For more details, we refer to [10][11][12][13][14][15][16]. e coupled systems of fractional differential equations are mainly significant because such systems occur frequently in various scientific applications (see [17][18][19]).
e Langevin equations (first formulated by Langevin in 1908) have been done with accuracy in order to have a full description of evolution of physical phenomena in fluctuating environment [20]. ere is a clear progress on fractional Langevin equations in physics (see [21,22]). New results on Langevin equations under the variety of boundary value conditions have been published [23][24][25][26].
Different forms of fractional integral have been identified and employed in many different applications. ree of the fractional integrals will be used: Riemann-Liouville [10], Atangana and Baleanu [27,28], and Ntouyas et al. [29,30].
Recent paper [31] has discussed the existence and uniqueness of solutions obtained from boundary value conditions for nonlinear fractional differential equations for Riemann-Liouville type under the generalized nonlocal integral boundary condition. In addition, the authors in [32] have studied existence and uniqueness of the solution for a certain class of ordinary differential equations of Atangana-Baleanu fractional derivative.
In this paper, we modify the boundary value conditions of coupled systems of Langevin fractional differential equations of Caputo type into new boundary value conditions. So, we deal with the following coupled systems of nonlinear fractional Langevin equations of α and β fractional orders: supplemented by the following: where c D is the Caputo fractional derivative of order 0 < α i ≤ 1 and 1 < β i ≤ 2 for i � 1, 2. AB I and ρ I c are Atangana-Baleanu, and Katugampola fractional integrals, respectively. ρ i > 0 and λ i , From the definitions of fractional integrals mentioned in the next section, it is worth pointing out that the fractional integral of Katugampola is a generalization for Riemann-Liouville fractional integral (ρ ⟶ 1) and Hadamard fractional integral (ρ ⟶ 0). Also, the fractional integral of Atangana-Baleanu contains the Riemann-Liouville fractional integral and when c � 0, we recover the initial function and if c � 1, we obtain the ordinary integral. ese motivate us to choose these fractional integrals in our boundary conditions. Furthermore, to the extent of our knowledge, this is the first paper that discusses the existence and uniqueness of the solutions to coupled systems of fractional Langevin equations involving the nonlocal integro-multipoint of Atangana-Baleanu type and the nonlocal integral of Katugampola type as boundary value conditions. e research article has been organized as follows. In the second section, we introduce some main concepts and essential lemma. In the third one, the main results show the existence and uniqueness of solutions to (1) and (2)

Basic Concepts and Relevant Lemmas
We will deduce the main outcomes by the following preliminary concepts in fractional calculus.
Definition 1 (see [33]). For n ∈ N, let f ∈ C n [0, ∞), then the Caputo fractional derivative of order β for a continuous function f is defined by provided the right-hand side exists.
Definition 2 (see [33]). e Riemann-Liouville fractional integral of order ω for a continuous function f: (0, ∞) ⟶ R is given by provided the integral exists.
Definition 4 (see [35]). e AB "Atangana-Baleanu" fractional integral AB I c is defined by has a solution given by where and for i � 1, 2 Proof. Clearly, by direct computation, both (11) and (12) are solutions to (1). Conversely, by using Lemma 1, the general solution of (1) can be given as

Mathematical Problems in Engineering
For i � 1, both first and second boundary conditions, we obtain c 21 � 0 and c 01 � Γ(α 1 + 1) . By substituting c 11 and c 01 in (15), we get (11). Similarly, in case of i � 2, the proof is done.
Proof. For each i � 1, 2, we have ese mean that S i (t) is increasing on (0, t 0 ) and decreasing on (t 0 , 1) and so e proof is done.

Main Results
For convenience, we simplify the following expressions: Define the operator T: where

Mathematical Problems in Engineering
By splitting both (26) and (27), we have Forthcoming theorems proof need more convenient dialog for the operator T. So, it is required to rewrite it as follows: where

Existence via O'Regan's eorem.
e first main result counts on O'Regan's theorem which proves the existence of the solutions for (1) and (2).
en, either (i) G has a fixed point y ∈ V or (ii) There exists a point y ∈zV and θ ∈ (0, 1) with y � θG (y).
where Λ i for each i � 1, 2, 3, 4, are defined in (21)- (24), respectively. From (29) which are defined in both (30) and (31) as two separate operators, first of all, we will show that T is uniformly bounded on V r . Indeed, By taking the norm over both sides, we obtain Similarly, erefore, T 1 is uniformly bounded: T 2 is bounded, i.e., ∀(x 1 , x 2 ) ∈ V r , and we have By taking the norm over both sides, it yields Similarly, we can deduce that which implies that us, Now, we shall show that T 1 is completely continuous, ∀t 1 , t 2 ∈ [0, 1], then we have 6 Mathematical Problems in Engineering Similarly, which shows the independence of the pair (x 1 , x 2 ) and (t 2 − t 1 ) ⟶ 0. We conclude that T 1 is equicontinuous. By Arzelá-Ascoli theorem, T 1 (V r ) is relatively compact. us, T 1 is completely continuous. We shall show that T 2 is a contraction mapping. Indeed, ∀t ∈ [0, 1] and for any ( So, we can write which clarifies that T 2 is a contraction mapping via (3.11). e last step is to show the first case of Lemma 5. By the way of contradiction, so we suppose ∃ θ ∈ (0, 1) and (x 1 , x 2 ) ∈ zV r such that (x 1 , x 2 ) � θT(x 1 , x 2 ). en, we have ‖(x 1 , x 2 )‖ � r, where consequently, equivalently, which clearly contradicts (E 2 ). Hence, the operator T has at least one fixed point (x 1 , x 2 ) ∈ V r . is ensures that (1) x, y), 0 < t < 1,

Mathematical Problems in Engineering
Obviously, From the shown data above, we have Clearly, by applying eorem 1 we have Hence, our example possess at least one solution on [0, 1].

Uniqueness via Banach Fixed-Point eorem.
e last result in this paper is about uniqueness criteria for the solution of (1) and (2), which can be achieved by Banach's fixed-point theorem.
Proof. Set F 1 � sup t∈[0,1] f 1 (t, 0, 0) and F 2 � sup t∈[0,1] f 2 (t, 0, 0) and r 0 > 0 such that First of all, we will show that T(B r 0 ) ⊆ B r 0 where For all (x 1 , x 2 ) ∈ B r 0 , we have Likewise, erefore, we obtain (68) Next, we shall show that the operator T is a contraction operator on [0, 1]. Indeed, for any distinct two pairs (x 1 , x 2 ), (x 1 ′ , x 2 ′ ) ∈ X 1 × X 2 , we see that Other results can be considered if we take μ 1 � μ 2 � 0, then the boundary conditions will be x 1 (0) � 0, c D α 1 x 1 (0) � Γ α 1 + 1 ρ 1 I c 1 x 1 η 1 , m 1 j�1 a j 1 x 1 ξ j 1 � 0, In the future, in the case of obtaining theorems related with the existence and uniqueness of solutions under certain boundary conditions, both how can they be used to prove existence and uniqueness of solutions to the given problem and what are the conditions have to be considered.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.