Solvability for Second Order Nonlinear Impulsive Boundary Value Problems

In this paper, we are concerned with the solvability for a class of second order nonlinear impulsive boundary value problem. New criteria are established based on Schaefer's fixed-point theorem. An example is presented to illustrate our main result. Our results essentially extend and complement some previous known results. anti-periodic boundary conditions.


Introduction
Impulsive differential equations play a very important role in understanding mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology, economics and so on, see [1,2,8,10,17].About wide applications of the theory of impulsive differential equations to different areas, we refer the readers to monographs [5,7,18,19] and the references therein.Some resent works on periodic and anti-periodic nonlinear impulsive boundary value problems can be found in [6,12,20,21].
Recently, J. Chen, C. Tisdell, and R. Yuan in [4] studied the following first order impulsive nonlinear periodic boundary value problem where T > 0 and f : [0, T ] × R n → R n is continuous on (t, u) ∈ [0, T ] \ {t 1 } × R n .The authors studied the existence of solutions to the problem (1.1) in view of differential inequalities and Schaefer's fixed-point theorem.Their results extend those of [9,14] in the sense that they allow superlinear growth in nonlinearity f (t, p) in p .
About further investigation, in 2007, Bai and Yang in [3] presented the existence results for the following second-order impulsive periodic boundary value problems Inspired by [3,4], in this paper, we investigate the following second order impulsive nonlinear boundary value problems a i , b i are constants for i = 1, 2, ..., k, β, γ are constants satisfying |β| ≥ 1, |γ| ≥ 1.Notice that our results not only extend some known results from the nonimpulsive case [16] to the impulsive case, or from single impulse [3] to multiple impulses, but also extend those of [11] in the sense that we allow superlinear growth of f (t, u, v) in u and v .Furthermore, the impulsive boundary-value problem reduces to a periodic boundary value problem [15,22] for β = γ = 1, p = q ≡ 0, and anti-periodic boundary value problem [21] for β = γ = −1, p = q ≡ 0. Hence, the problem (1.3) can be considered as a generalization of periodic and anti-periodic boundary value problems.
We shall establish the existence of solutions for impulsive BVP (1.3) by means of well-known Schaefer's fixed-point theorem.The rest of paper is organized as follows.In section 2, we present some definitions and lemmas, and the fixed point theorem which is key to our proof.In section 3, the new existence theorem of (1.3) is stated.An example is given in the last section to demonstrate the application of our main result.

Preliminaries
First, we introduce and denote the Banach space P C([0, T ], R n ) by u is left continuous at t = t i , the right − hand limit u(t + i ) exists} with the norm where • is the usual Euclidean norm.
We denote the Banach space P C 1 ([0, T ]; R n ) by with the norm The following fixed-point theorem due to Schaefer, is essential in the proof of our main result.
Lemma 2.1.Let E be a normed linear space and Φ : E → E be a compact operator.Suppose that the set and Proof.The proof is similar to that of Lemma 2.1 in [13], so we omit it here.
Remark 2.3.It follows from Lemma 2.2 that In order to prove our main results, we present a useful lemma in this section.Consider the following impulsive boundary value problem where φ 1 , φ 2 satisfies (2.1), (2.2) respectively, and EJQTDE, 2009 No. 41, p. 3 Proof.Since φ 1 , φ 2 are two linearly independent solutions of the equation we know the solutions of (2.9) can be presented as where c 1 , c 2 are any constants. Let Employing the method of variation of parameter, by some calculation, we get So the solution of (2.10) can be given as Next, we consider (2.11) It is easy to know the solution of (2.11) is as the following form (2.12) Finally, we consider the solution of (2.4).Substituting (2.12) into u(0) = βu(T ), u (0) = γu (T ), we have EJQTDE, 2009 No. 41, p. 4 (2.13) By some calculations, we get Hence, the problem (2.4) has the unique solution In view of Lemma 2.4, we easily know that u is a fixed point of operator A iff u is a solution to the impulsive periodic boundary problem (1.3).
Proof.This is similar to that of Lemma 3.2 in [4].
For convenience, let Now we are in the position to present our main results.

Main results
Theorem 3.1.Suppose that f : Then BVP (1.3) has at least one solution.
Proof.Let u ∈ P C([0, T ], R n ) be such that u = λAu for some λ ∈ (0, 1).That is, A similar calculation yields an estimate on u : differentiating both sides of the integration and taking norms yields, for each t ∈ [0, T ], we have Thus, we conclude that As a result, set S is bounded.Applying Scheafer's fixed-point theorem, the problem (3.2) has at least one fixed point, which means that (1.3) has at least one solution.We complete the proof.
A similar discuss as Theorem 3.1 leads to the following result.
Remark 3.2.If the condition (3.1) is replaced by and all the other assumptions are satisfied in Theorem 3.1, then the problem (1.3) has at least one solution.

An example
In this section, an example is given to highlight our main result.Thus, condition (3.3) holds.By Remark 3.2, we conclude that the solvability of (4.1) follows.