Periodic boundary value problems for nonlinear impulsive fractional differential equation

In this paper, we investigate the existence and uniqueness of solution of the periodic boundary value problem for nonlinear impulsive fractional differential equation involving Riemann-Liouville fractional derivative by using Banach contraction principle.

For clarity and brevity, we restrict our attention to BVPs with one impulse, the difference between the theory of one or an arbitrary number of impulses is quite minimal.
In [1], the author investigated the existence and uniqueness of solution to initial value problems for a class of fractional differential equations D α u(t) = f (t, u(t)), t ∈ (0, T ] 0 < α < 1, (1.4) by using the method of upper and lower solutions and its associated monotone iterative.
In [2], the existence and uniqueness of solution of the following fractional differential equation with periodic boundary value condition lim was discussed by using the fixed point theorem of Schaeffer and the Banach contraction principle.
Impulsive differential equations are now recognized as an excellent source of models to simulate process and phenomena observed in control theory, physics, chemistry, population dynamics, biotechnology, industrial robotic, optimal control, etc. [26,27].Periodic boundary value for impulsive differential equation has drawn much attention, see [28][29][30][31].Anti-periodic problems constitute an important class of boundary value problems and have recently received considerable attention.The recent results on anti-periodic BVPs or impulsive anti-periodic BVPs of fractional differential equations can be found in [32][33][34][35].But till now, the theory of boundary value problems for nonlinear fractional differential equations is still in the initial stages.For some recent work on impulsive fractional differential equations, see [36][37][38][39][40][41] and the references therein.
To the best of the authors knowledge, no one has studied the existence of solutions for BVP (1.1)- (1.3).The purpose of this paper is to study the existence and uniqueness of solution of the periodic boundary value problem for nonlinear impulsive fractional differential equation involving Riemann-Liouville fractional derivative by using Banach contraction principle.

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used In order to define the solution of (1.1)-(1.3)we shall consider the space there exist x(t − 1 ) and x(t + 1 ) with x(t − 1 ) = x(t 1 )}.
Proof.By Theorem 3.2 in [2], we have that For each t < t 1 , we have Using the identities Similarly, we can obtain that Thus, we have In consequence, Next we prove that the solution of BVP (2.1) By (2.5) and (2.6)-(2.7),we get that v(t) ≡ 0 for any t ).On the other hand, lim t→t − 1 t 1−α v(t) = 0. Thus, we obtain that v(t 1 ) = 0. Hence, u 1 (t) = u 2 (t) for each t ∈ (0, 1].Moreover, by (2.7), we Therefore, Remark 2.2.For α = 1, Lemma 2.1 reduces to the one for a first order linear impulsive boundary value problem holds, then the operator A : Proof.Define the operator A as follows : Here, we only prove that (t − t 1 ) Similarly, we can prove that and tend to zero as τ 1 → τ 2 by (4.13) and (4.14) in [2].
exist.Thus, A : For convenience, set ) ) Theorem 3.2.Suppose that (H 1 ) and the following condition hold: (H 2 ) there exist positive constant k, and l such that

Then the problem (1.1)-(1.3) has a unique solution in
Proof.By (H 1 ) and Lemma 3.1, we have A : , and each t ∈ [0, t 1 ], we obtain by (H 2 ) that From the expression of G λ,α (t, s), and (4.21) in [2], we have And EJQTDE, 2011 No. 3, p. 9 Moreover, we have On the other hand, for each x, y ∈ P C 1−α [0, 1] and t ∈ (t 1 , 1], we obtain that (3.17)By (4.21) in [2], we have Moreover, we have which implies that A is a contraction (by (3.10)).Therefore it has a unique fixed point.

An example
The following illustrative example will demonstrate the effectiveness of our main result.So that the conditions (H 1 ) and (H 2 ) hold.Moreover, we have E α,α (λ) = E 0.9,0.9 throughout this paper.Let C(a, b] (C[a, b]) be the Banach space of all continuous real functions defined on (a, b] ([a, b]).EJQTDE, 2011 No. 3, p. 2