Impulsive Problems for Fractional Differential Equations with Nonlocal Boundary Value Conditions

and Applied Analysis 3 Lemma 4 (see [2, 5]). Let α > 0. Then one has I α 0 + c D α u (t) = u (t) + c 0 + c 1 t + c 2 t 2 + ⋅ ⋅ ⋅ + c n−1 t n−1 , (8) where c i ∈ R, i = 0, 1, . . . , n − 1, and n = [α] + 1. Lemma 5 (see [29, Lemma 2.9]). Let y ∈ C(J, R) satisfy the following inequality: 󵄨󵄨󵄨󵄨y (t) 󵄨󵄨󵄨󵄨 ≤ p1 + p2 ∫ t 0 (t − s) q−1󵄨󵄨󵄨󵄨y(s) 󵄨󵄨󵄨󵄨 λ ds


Introduction
Fractional differential equations have recently proved to be strong tools in the modeling of many physical phenomena.It draws a great application in nonlinear oscillations of earthquakes, many physical phenomena such as seepage flow in porous media, and fluid dynamic traffic model.For more details on fractional calculus theory, one can see the monographs of Diethelm [1], Kilbas et al. [2], Lakshmikantham et al. [3], Miller and Ross [4], Podlubny [5], and Tarasov [6].Fractional differential equations involving the Riemann-Liouville fractional derivative or the Caputo fractional derivative have been paid more and more attentions (see, e.g., [7][8][9][10][11][12][13]).
The impulsive differential equations arise from the real world problems to describe the dynamics of processes in which sudden, discontinuous jumps occur.Such processes are naturally seen in biology, physics, engineering, and so forth.Due to their significance, many authors have established the solvability of impulsive differential equations.For the general theory and applications of such equations we refer the interested readers to see the papers [14][15][16][17] and references therein.
In [27], by a fixed point theorem due to O'Regan, the authors established sufficient conditions for the existence of at least one solution for the problem (1).
The rest of this paper is organized as follows.In Section 2, we will give some lemmas which are essential to prove our main results.In Section 3, we give the main results.The first result is based on the Banach contraction principle, the second result is based on Schaefer's fixed point theorem via a generalized hybrid singular Gronwall inequality, and the third result is based on a nonlinear alternative of Leray-Schauder type.In Section 4, some examples are offered to demonstrate the application of our main results.

Preliminaries
At first, we present the necessary definitions for the fractional calculus theory.
Taking derivative of (13), we can get Conversely, taking ( 13) and ( 14) into (12), we can easily get the equation and all the impulse conditions and boundary value conditions are satisfied.So we complete the proof of Lemma 9.
Consider the operator  :  1 (,) →  1 (, ) defined by where Then we have Clearly,  is well defined.

Main Results
This section deals with the existence of solutions for problem (4).Before stating and proving the main results, we make the following hypotheses.
In order to get the second main result, we replace ( 2 )  with ( 2 ).
Proof.According to Lemma 6, if we want to get the solution of problem (4), we only need to consider the fixed point of operator , which is defined by (28).We divide the proof into four steps.
Step 4.There exists a priori bound.
Next we show that the set () = { ∈  1 (, ) :  = , for some  ∈ (0, 1]}, is bounded. Consider ∀ ∈ ,  ∈ (); we have Then by Lemma As a consequence of Schaefer's fixed point theorem (Lemma 6), we deduce that  has at least one fixed point which means that the problem (4) has at least one solution.
Next we apply the nonlinear alternative of Leray-Schauder type to get Theorem 12.We give the following hypothesis ( 3 )  .