Fractional Langevin Equations with Nonseparated Integral Boundary Conditions

Fractional derivatives give an excellent description of memory and hereditary properties of different processes. Properties of the fractional derivatives make the fractionalorder 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:


Introduction
Fractional derivatives give an excellent description of memory and hereditary properties of different processes. Properties of the fractional derivatives make the fractionalorder models more useful and practical than the classical integral-order models.
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][11][12][13][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][20][21][22][23][24][25]). In [26], the authors studied a nonlinear Langevin equation involving two frac-tional 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: Þds, where 0 < α < 1, 1 < β ≤ 2, p > 0, λ, μ, σ 1 , σ 2 , σ 3 ∈ ℝ * with μ ≠ −1, and c D β , c D α are the Caputo fractional derivatives and f : ½0, 1 × ℝ × ℝ ⟶ ℝ and g, h, k : ½0, 1 × ℝ ⟶ ℝ 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.

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 α > 0 with the lower limit zero for a function f can be defined as Definition 2 (see [5]). The 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, and t > 0.
Theorem 3 (see [30]). Let M be a bounded, closed, convex, and nonempty subset of a Banach space X. Let A and B be operators such that (I) Ax + By ∈ M whenever x, y ∈ M (II) A is compact and continuous (III) B is a contraction mapping Then, there exists z ∈ M such that z = Az + Bz.

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Proof. By applying Lemma 5, we have where a 0 , a 1 , a 2 ∈ ℝ.

Main Results
Denote by X the Banach space of all continuous functions from ½0, 1 ⟶ ℝ endowed with norm kxk = sup fjxðtÞj: t ∈ ½0, 1g.
By Lemma 6, we transform problem (1) into a fixed point problem as x = Px, where P : X ⟶ X is given by for all x 1 , x 2 , y 1 , y 2 ∈ ℝ, t ∈ ½0, 1. ðH 2 Þ-there exist positive constants q 3 , q 4 , q 5 such that Then there exist a unique solution for boundary value problem (1) provided that r 1 < 1, where and A i = max

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Let B r = fx ∈ X : kxk ≤ rg the ball with radius r, where r ≥ ðr 2 /ð1 − r 1 ÞÞ, with Then, B r is a closed, convex, and nonempty subset of the Banach space X.
Our aim is to prove that the operator P has a unique fixed point on B r .We show that PB r ⊆ B r . For which implies that Now, for x, y ∈ B r and for t ∈ ½0 ; 1, Since r 1 < 1, then operator P is a contraction mapping. Therefore, boundary value problem (1) has a unique solution.