Existence of solutions of BVPs for impulsive fractional Langevin equations involving Caputo fractional derivatives

The standard Caputo fractional derivative is generalized for the piecewise continuous functions. A more general boundary value problem for the impulsive Langevin fractional differential equation involving the Caputo fractional derivatives is studied. New existence results for solutions of concerned problems are established.


Introduction
Fractional differential equations have many applications in modeling of physical and chemical processes. In its turn, mathematical aspects of fractional differential equations and methods of their solutions were discussed by many authors, see the text books [9,10,15].
The Langevin equation (first formulated by Langevin in 1908) is found to be an effective tool to describe the evolution of physical phenomena in fluctuating environments [5]. For some new developments on the fractional Langevin equation in physics, see, for example, [1,2,4,6,11,22]. Lizana et al. [11] studied a singleparticle equation of motion starting with a microscopic description of a tracer particle in a one-dimensional many-particle system with a general two-body interaction potential and they have shown that the resulting dynamical equation belongs to the class of fractional Langevin equations using a harmonization technique. In [6], Gambo et al. discussed the Caputo modification of the Hadamard fractional derivative. Ahmad et al. [1,3,4] considered solutions of nonlinear Langevin equation involving two fractional orders. In [7, 14, 17-22, 24, 26, 28, 29], Tariboon et al. studied the existence and uniqueness of solutions of the nonlinear Langevin equation of Hadamard-Caputo-type fractional derivatives with nonlocal fractional integral conditions using a variety of fixed point theorems. Tariboon and Ntouyas [19] discussed the existence and uniqueness of solutions for Langevin impulsive q-difference equations with boundary conditions. In recent years, some authors have studied solvability or existence and uniqueness of solutions of boundary value problems (BVPs for short) for impulsive Langevin fractional differential equations see [25,27].
The first purpose of this paper is to provide a method to convert boundary value problems for impulsive Langevin fractional differential equations involving two fractional derivatives to integral equations. Then we establish existence results for solutions of BVP(1.4) by using Schauder's fixed point theorem [12] under some suitable assumptions. It is noted that the lower point of the fractional differential equations involved is 0 which is different from those ones used in [13].
The remainder of the paper is organized as follows: In Section 2, the related definitions are introduced firstly. Then we seek continuous solutions of a class of linear Langevin fractional differential equations and we also seek piecewise continuous solutions of a class of linear Langevin fractional differential equations). In Section 3, the equivalent integral equations of BVP(1.4) are presented. Finally in Section 4, we establish sufficient conditions for the existence of solutions of BVP(1.4).

Preliminary results
In this section, we firstly present some necessary definitions from the fractional calculus theory which can be found in the literature [8,15]. Then we get exact continuous solutions of a class of fractional Landevin equations.
Thirdly, we get exact piecewise continuous solutions of a class of impulsive fractional Langevin equations.
Let the Gamma and beta functions Γ(α) , B(p, q), and the Mitag-Leffler function E α,δ (x) be defined by

Definition 2.1 (page 69 in [8])
Let −∞ < a < b < +∞ . The Riemann-Liouville fractional integrals I α a + g and I α b − g of order α ∈ C(R(α) > 0) are defined by respectively. These integrals are called the left side and the right side fractional integrals.
Let us define Then P C 0 (0, 1] is a Banach space.

Now, we seek continuous solutions of linear Langevin fractional differential equations (LFDEs for short)
with the Caputo fractional derivatives and the Riemann-Liouville fractional derivatives, respectively.

Lemma 2.9 x is a solution of (2.1) if and only if there exist constants
Proof The proof follows from [16,23] in Section 3 by using the Laplace transform [8] and is omitted. 2 Now, we seek piecewise continuous solutions of linear impulsive Langevin fractional differential equations (ILFDEs for short) with the Caputo fractional derivatives.

Lemma 2.10 x is a piecewise continuous solution of (2.3) if and only if there exist
The proof is very long since the careful computation is needed. We divide it into the following two steps.
Step 1. Note that the starting point of the derivatives is 0 similar to [13]. We prove that x satisfies We note that Hence, (2.4) holds for k = 0 . Now suppose that (2.4) holds for k = 0, 1, · · · , ω , i.e. there exist constants We also note that (2.7) We will prove that (2.4) holds for k = ω + 1 . Then by mathematical induction method, (2.4) holds for all k ∈ IN m 0 . Then this step is completed. In order to get the exact expression of x on (t ω+1 , t ω+2 ] , we suppose that there exists Φ such that (2.8) Using Definition 2.2, (2.6), and (2.8), we get by direct computation that Using Definition 2.2, (2.6), and (2.8), we get by direct computation that It follows that (2.10) On the other hand, we have for t ∈ (t ω+1 , t ω+2 ] that Using (2.10), (2.11), and direct computation, we get that
Note that the starting point of the derivatives is 0 similar to [13]. We prove that x is a piecewise continuous solution of (2.3) if x satisfies (2.4).

Equivalent integral equations of BVP(1.4)
In this section, we present equivalent integral equations of BVP (1.4) by using Lemma 2.2. For ease of expression, denote Then by direct computation, we get ) satisfy the following iterative equations: Suppose that (a)-(c) hold and Θ = 0, Ξ = 0. Then BVP(1.4) is equivalent to the following integral equation Suppose that x is a solution of BVP(1.4). From Lemma 2.2 (choose l = n = 1 , ρ = α, ϱ = β , replacing By Direct computation, for t ∈ (t i , t i+1 ] we can get that

It follows that
By ∆x(t k ) = I(t k , x(t k )) and (3.2), we get and (3.2), using (3.4), we get
On the other hand, if x satisfies (3.1), we can prove that x is a solution of BVP(1.4). The proof is completed. 2

Solvability of BVP(1.4)
In this section, we establish existence results for solutions of BVP(1.4). We list the following assumptions: there exist constants M f , M I ≥ 0 such that Let us denote Suppose that (a)-(c), (H1) hold, Θ = 0, Ξ = 0 . Then BVP(1.4) has at least one solution if there exists ) are defined in Section 3. Define the operator T on ) satisfy the following iterative equations: One sees from the definition of M ν,k that Then the algebraic complement N * i,j of N i,j satisfies
Hence, T Ω 0 ⊂ Ω 0 . Then Schauder's fixed point theorem implies that T has at least one solution in Ω 0 , which is a solution of BVP(1.4). The proof is completed.