Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited

: We consider the nonlinear fractional Langevin equation involving two fractional orders with initial conditions. Using some basic properties of Prabhakar integral operator, we ﬁnd an equivalent Volterra integral equation with two parameter Mittag–Lefﬂer function in the kernel to the mentioned equation. We used the contraction mapping theorem and Weissinger’s ﬁxed point theorem to obtain existence and uniqueness of global solution in the spaces of Lebesgue integrable functions. The new representation formula of the general solution helps us to ﬁnd the ﬁxed point problem associated with the fractional Langevin equation which its contractivity constant is independent of the friction coefﬁcient. Two examples are discussed to illustrate the feasibility of the main theorems.


Introduction
Dynamical behavior of physical processes are usually represented by differential equations. If the model of physical system in some ways possesses a memory and hereditary properties, for instance, viscoelastic deformation [1], anomalous diffusion [2], stock market [3], bacterial chemotaxis [4] and complex networks [5], relaxation in filled polymer networks [6], relaxation and reaction kinetics of polymers [7], description of mechanical systems subject to damping [8], Behavior of Biomedical Materials [9]; the corresponding models can be described by the fractional differential equations.
Langevin equation is a fundamental theory of the Brownian motion to describe the evolution of physical phenomena in fluctuating environments [10,11]. Fractional Langevin equation as a generalization of classical one gives a fractional Gaussian process parametrized by two indices, which is more flexible for modeling fractal processes [12][13][14][15][16].
The virtually simultaneous development of fractional derivatives, various generalizations of the Langevin equation were proposed and studied by various researchers during recent years. Despite the widespread use of many of the applications [17][18][19][20][21][22], the fractional Langevin equation is extensively studied in literature both from theoretical and numerical points of view. Authors in [23] studied nonlinear fractional Langevin equation involving two fractional orders in different intervals as a generalized form of three point third order nonlocal boundary value problem of nonlinear ordinary differential equations. In [24], the authors have studied fractional Langevin equations with nonlocal integral boundary conditions. Recently, anti-periodic boundary value problem for Langevin equation involving two fractional orders has been studied in [25]. Existence and uniqueness results for coupled and uncoupled systems of fractional Langevin equations of Riemann-Liouville and Hadamard types has been discussed in [26]. Guo et al. [27] gave an efficient numerical method for solving the fractional Langevin equation with or without an external force. Some more recent work on Langevin equation can be found in [28][29][30][31][32][33][34][35][36][37].
In the current paper, we mainly focus on the existence and uniqueness result for the fractional Langevin equation involving two fractional orders: where m − 1 < α ≤ m, n − 1 < β ≤ n, l = max{m, n}, m, n ∈ N, D α is the Caputo fractional derivative, x(t) is the particle displacement, x (i+α) (0) equals D i D α x(0), in the sequential sense, λ ∈ R is the friction coefficient and f : [0, 1] × R → R is a given function which represents a noise term. Based on the criteria specified in [38], the problem (1) is a general form of anomalous systems governed by a generalized Langevin equation with long-range memory. In contrast to the classical Langevin equation, we use D β x(t) and D β D α x(t) instead of the ordinary definition of the velocity and acceleration as the first and second derivatives of the displacement to derive a generalized Langevin equation involving friction memory kernel. For example, if 0 < β ≤ 1, α = 1, then according to the standard definition of the Caputo fractional derivatve operator, we have a special case of generalized Langevin equation involving friction memory kernel equal to λ Γ(1 − β) t β−1 . Based on the calculations in ( [39], Section B), in this case, the resulting motion is in fact subdiffusive. Furthermore, it is worth noting that, if α + β > 2, then we do not have any physical meaning for the main problem. For this case, it is only a valuable problem in the thory of fractional differential equations as a sequential fractional differential equation with initial conditions. As we have seen in the papers cited above about analysis of fractional Langevin equation, using various classical fixed point theorems is a common and useful technique for obtaining the existence and uniqueness results for fractional Langevin equation involving different initial or boundary conditions. In the mentioned papers, the contractivity constant of the fixed point problem associated with the fractional Langevin equation depended on the friction coefficient λ. For example, in the obtained existence and unique results in [33,34], the contractivity constants R 1 , R 2 satisfy the following conditions and where k = 1 1−q(1−α) 1 q and p −1 + q −1 = 1, respectively. As stated in relations (2) and (3), the contractivity constants R 1 , R 2 depend on the friction constant λ. Therefore, from (2), we can not discuss the problems involving the friction constant |λ| ≥ Γ(α + 1). Similarly, from (3), we can not study the problems involving the friction constant |λ| ≥ Γ(α). Note that 0 < 1 − q(1 − α) < 1. Therefore, we cannot discuss the existence and uniqueness of solutions for the problems involving large friction coefficient λ. In this paper, we strive to overcome this major limitation. First we propose a new construction of the general solution for the Equation (1) using two parameter Mittag-Leffler functions and some of the basic properties of Prabhakar operator. This is done in Section 2. Then we obtain a new existence and uniqueness results under some weak conditions by using contractive mapping theorem and Weissinger's fixed point theorem. This is content of Section 3. Two examples are given in Section 4 to illustrate our results.

Preliminaries and Auxiliary Results
In the following section, we apply some technical calculations related to fractional calculus to build a new general solution corresponding to initial value problem (1) which provides an extremely powerful tool for the proof of the main result. Furthermore, we present some preliminaries and notations regarding fractional calculus for the reader's convenience. For details, see [40][41][42][43][44][45][46].
where n − 1 < α ≤ n and n ∈ N, provided that the right-hand-side integral exists and is finite. , is an entire function of order 1 α .

Lemma 1 ([46]
). Let α, β, γ ≥ 0 and x ∈ L 1 [0, 1]. Then The general solution of (1) is given by Proof. Let x(t) be a solution of the problem (1), we have By using the initial conditions for the initial problem (1), we find that a i = ν i +λµ i Γ(i+1) , i = 0, 1, · · · , n − 1. Therefore, we have Now, using the approach of Kilbas et al. ( [40], Section 3.1), the solution of the Equation (5) is given by the following expression . On the other hand, an integration by parts reveals for each i ∈ N. Applying Lemma 1 to the second term in the right-hand side of (7), we conclude which is the desired result. Now, we state Weissinger's fixed point theorem ( [41], Theorem D.7) as a generalization of the so-called contraction mapping theorem which is needed to prove Theorem 3. Theorem 1. Let X to be a Banach space and let θ n ≥ 0 for every n ∈ N ∪ {0} such that ∑ ∞ n=0 θ n converges. Furthermore, assume T : X → X is a nonlinear mapping which satisfies the inequality T n x − T n y ≤ θ n x − y for every n ∈ N and every x, y ∈ X. Then, T has a unique fixed point x * . Moreover, the sequence {T n x 0 } ∞ n=0 converges to this fixed point x * , for any x 0 ∈ X.

Existence and Uniqueness
Our aim in the following section is to deeply investigate the existence and uniqueness results for the main problem (1) in the Lebesgue space. Theorem 2. Let max{1, 1 α+β } ≤ p ≤ ∞, p −1 + q −1 = 1 and the following hypotheses 1-3 hold:

Hypothesis 2. There exists nonnegative a ∈
Then the integral Equation (4) has a unique solution in L q [0, 1].

Proof.
We define the operator T as follows: where 1] |φ(t)|. Note that the generalized Mittag-Leffler functions are entire functions [43,44]. For each x ∈ L q [0, 1], we have Therefore, we have which implies that Note that 1 − q + q(α + β) ≥ 0 because of p ≥ 1 α+β . Therefore, T is a contraction since R < 1. By the Banach contraction principle, T has a unique fixed point, which is the unique solution of the initial problem (1).

Hypothesis 5. There exists L
Then the integral Equation (4) has a unique solution in L q [0, 1].

Illustrative Examples
In this section, some examples are provided to show the applicability of the analytical achievements of the paper.

Conclusions
In this article, we have considered initial value problem of nonlinear fractional Langevin equation involving two fractional orders. As a first step, by applying the tools of fractional calculus and using some basic properties of Prabhakar integral operator, we build a general structure of solutions associated with our proposed model. Once the fixed point operator equation is available, the existence results are established by means of contraction mapping theorem and Weissinger's fixed point theorem. Finally, two examples were presented to support the result.