SOLVABILITY FOR IMPULSIVE FRACTIONAL LANGEVIN EQUATION∗

We investigate impulsive fractional Langevin equation involving two fractional Caputo derivatives with boundary value conditions. By Banach contraction mapping principle and Krasnoselskii’s fixed point theorem, some results on the existence and uniqueness of solution are obtained.


Introduction
The states of many processes are often subject to instantaneous perturbations and experience abrupt changes at certain moments of time. The model of impulsive differential equations is better than pure continuous-time or discrete-time model for describing those processes [3,16,17]. Although fractional differential equation is developing rapidly owing to its wide applications of science and engineering in recent decades [1,8,9,13,15,18,[20][21][22][23]25], the study of fractional impulsive differential equations has been started quite recently ( [17,25]).
There are some ways to consider the concept of a solution to fractional differential equations with impulses. In 2012, Fečkan et al. [5] gave a new concept which is to keep the lower limit t 0 of the fractional derivative for all t ≥ t 0 but consider different initial conditions on each interval (t k , t k+1 ). Fractional derivative provides an excellent instrument for the description of memory and hereditary properties of processes. This is the main advantage of fractional derivatives in comparison with classical integer derivatives [15]. This concept can reflect that fractional derivatives have global property and the memory accumulated by the long time effects in the whole process including impulsive moments. This approach is used in some papers (for example, [2, 5-7, 14, 19, 24]).
In this paper, also in this way, we consider the boundary value problem of twoterm Caputo fractional impulsive Langevin equation where D δ and D δ−1 are the standard Caputo fractional derivatives with the lower limit zero and 1 < δ ≤ 2, J = [0, 1], f : J × R → R is a given continuous function, . The Langevin equation was introduced by Langevin in 1908 to give an elaborate description of Brownian motion. It has been widely used to describe the evolution of physical phenomena in fluctuating environments [4]. The nonlinear fractional Langevin equation involving two fractional derivatives as a kind of generalization of Langevin equation has been studied by many researchers [10][11][12]. In addition, A. Kilbas et al. [8] considered the fractional differential equation with two fractional derivatives of the type: where λ ∈ R, D α and D β denote the Caputo fractional derivatives with the lower limit zero. However, there are less results about multi-term fractional impulsive differential equations and no paper considered the solution for two-term Caputo fractional impulsive Langevin equation with boundary conditions (1.1)-(1.3). What's more, the equation we studied can reduce to single-term fractional differential equations by letting parameter λ = 0 and reduce to classical Langevin equation by letting order δ = 2. In this article, we will study the existence and uniqueness of solution for BVP (1.1)-(1.3), using Banach contraction mapping principle and Krasnoselskii's fixed point theorem.
The paper is organized as follows. In Section 2, we recall some necessary concepts and results and present preliminary results. In Section 3, some results on the existence and uniqueness of solution are obtained. Two examples are given in Section 4.

Preliminaries
In this section, we give some definitions and lemmas which are required for building our theorems.
where n is the smallest integer greater than or equal to α, provided that the right side is pointwise defined on [0, +∞).

Definition 2.3 ( [15]
). The Caputo fractional derivative of order α > 0 of a func-tion f : [0, +∞) → R is given by where n is the smallest integer greater than or equal to α, provided that the right side is pointwise defined on [0, +∞).
(ii) A is compact and continuous; (iii) B is a contraction mapping.
Then there exists z ∈ M such that z = Az + Bz.

Lemma 2.3. Let h : J → R be continuous. A function u is a solution of the boundary value problem
if and only if u ∈ P C(J, R) is a solution of the integral equation

G(t, s)h(s)ds.
Conversely, assume that u(t) is a solution of (2.3), we can easily show that u(t) is the solution of (2.1)-(2.2). The proof is complete.
For convenience, we denote H = sup t∈[0,1] For the forthcoming analysis, we need the following hypotheses (H1) There exists a constant L > 0 such that 1], and all u, v ∈ R; (H2) There exists a integrable function µ : [0, 1] → R + such that

Existence and uniqueness of solution
In this section, we will show the existence and uniqueness of solution for boundary value problems (1.1)-(1.3) by Banach contraction mapping principle and Krasnoselskii's fixed point theorem. Proof. Define operator T : P C(J, R) → P C(J, R) by Then T is well-defined and u ∈ P C(J, R) is a solution to the BVP (1 .1)-(1.3), if and only if u is a fixed point of T . It is easy to verify that T u ∈ P C(J, R) by Lebesgue's dominated convergence theorem.
For all u, v ∈ P C(J, R), t ∈ [0, 1], by (H1), we have Hence, T is a contraction mapping and there exists a unique fixed point according to Banach contraction mapping principle. Therefore, (1.1)-(1.3) has a unique solution. Proof. Define operators E and F from P C(J, R) into itself by for u ∈ P C(J, R). It is easy to verify that E and F are continuous on P C(J, R) by Lebesgue's dominated convergence theorem. Since |λ|H < 1, we can take r > 0 Then B r is a nonempty bounded closed convex subset in P C(J, R). For any u, v ∈ B r , k = 1, · · · , m, we have On the other hand, E is a contraction mapping since for all t ∈ [0, 1], Next we prove F is compact. Take any bounded subset B η = {u ∈ P C(J, R) : ||u|| ≤ η}. According to (3.3), F (B η ) is bounded. Taking τ 1 , τ 2 ∈ [0, t 1 ] with τ 1 < τ 2 , for any u ∈ B η , we have As τ 1 → τ 2 , the right side of the above inequality tends to zero. In general, using the same way, for τ 1 , τ 2 ∈ (t k , t k+1 ], we can get |F u(τ 2 ) − F u(τ 1 )| → 0 as τ 1

Examples
Example 4.1. Consider the following BVP for two-term fractional impulsive Langevin equation