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This paper is concerned with the existence of solutions for Sturm-Liouville boundary value problem of a class of second-order impulsive differential equations, under different assumptions on the nonlinearity and impulsive functions, existence criteria of single and multiple solutions are established. The main tools are variational method and critical point theorems. Some examples are also given to illustrate the main results.

Impulsive differential equation is one of the main tools to study the dynamics of processes in which sudden changes occur. The theory of impulsive differential equation has recently received considerable attention, see [

Especially, in [

In [

Tian and Ge in [

In the above-cited articles, analogous results are also given when the impulses are absent, so they cannot reflect the impact of the impulses on the existence of the solutions. In [

Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time. Many systems involve the impulsive condition which depends not only on the derivative of a function but also the function itself. A few papers discussed the solutions of impulsive differential equations involving impulses both on the function and on its derivative by critical point theory; see for example [

Inspired by the above results, in this paper we consider the existence and multiplicity of solution to the following Sturm-Liouville boundary value problem of a class of second-order impulsive differential equations:

We begin by establishing the corresponding variational framework of problem (

Note that when

The rest of this paper is organized as follows. In Section

We define the space:

If

Now we state some lemmas, which are needed in the proof of the main results.

Clearly,

Suppose

Observe that

For

Let

Let

A function

For convenience, denote

If the function

Suppose that

By the definition of

In view of (

If

Therefore,

Let

there are constants

there is an

Let

there are constants

for each finite dimensional subspace

If there exist

Let

From the reflexivity of

Suppose that (

In order to show that problem (

First, it follows from (

By (

Under the same assumptions as Theorem

The proof follows the analogous ideas as that we have developed for Theorem

If we let

When

Assume that

If

By Lemma

The next theorem gives some sufficient conditions that problem (

Suppose

Firstly, we show that

Equation (

From (

For some constant

Finally, by Remark

Suppose (

Firstly, we show that

Next, we verify that

From (

Therefore by the least action principle [

Finally, we show that

In (

Let

If we choose

Consider (

Consider the following problem:

Let

The research of H. R. Sun was supported by NSF of China (10801065), FRFCU (lzujbky-2011-43, lzujbky-2012-k25) and SRF for ROCS, and SEM. The research of J. J. Nieto was partially supported by Ministerio de Ciencia e lnnovación and FEDER, Project MTM2010-15314.