Open Access

Variational Method to the Impulsive Equation with Neumann Boundary Conditions

Boundary Value Problems20092009:316812

DOI: 10.1155/2009/316812

Received: 28 August 2009

Accepted: 28 September 2009

Published: 11 October 2009

Abstract

We study the existence and multiplicity of classical solutions for second-order impulsive Sturm-Liouville equation with Neumann boundary conditions. By using the variational method and critical point theory, we give some new criteria to guarantee that the impulsive problem has at least one solution, two solutions, and infinitely many solutions under some different conditions, respectively. Some examples are also given in this paper to illustrate the main results.

1. Introduction

In this paper, we consider the boundary value problem of second-order Sturm-Liouville equation with impulsive effects

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq1_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq2_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq3_HTML.gif positive functions, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq4_HTML.gif is a continuous function, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq5_HTML.gif are continuous, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq7_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq8_HTML.gif denote the right and the left limits, respectively, of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq9_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq10_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq11_HTML.gif is the right limit of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq12_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq13_HTML.gif is the left limit of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq14_HTML.gif .

In the recent years, a great deal of work has been done in the study of the existence of solutions for impulsive boundary value problems (IBVPs), by which a number of chemotherapy, population dynamics, optimal control, ecology, industrial robotics, and physics phenomena are described. For the general aspects of impulsive differential equations, we refer the reader to the classical monograph [1]. For some general and recent works on the theory of impulsive differential equations, we refer the reader to [29]. Some classical tools or techniques have been used to study such problems in the literature. These classical techniques include the coincidence degree theory of Mawhin [10], the method of upper and lower solutions with monotone iterative technique [11], and some fixed point theorems in cones [1214].

On the other hand, in the last two years, some researchers have used variational methods to study the existence of solutions for impulsive boundary value problems. Variational method has become a new powerful tool to study impulsive differential equations, we refer the reader to [1520]. More precisely, in [15], the authors studied the following equation with impulsive effects:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq15_HTML.gif is continuous, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq16_HTML.gif , are continuous, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq17_HTML.gif . They essentially proved that IBVP (1.2) has at least two positive solutions via variational method. Recently, in [16], using variational method and critical point theory, Nieto and O'Regan studied the existence of solutions of the following equation:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ3_HTML.gif
(1.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq18_HTML.gif is continuous, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq19_HTML.gif are continuous. They obtained that IBVP (1.3) has at least one solution. Shortly, in [17], authors extended the results of IBVP (1.3).

In [19],Zhou and Li studied the existence of solutions of the following equation:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ4_HTML.gif
(1.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq20_HTML.gif is continuous, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq21_HTML.gif , are continuous. They proved that IBVP (1.4) has at least one solution and infinitely many solutions by using variational method and critical point theorem.

Motivated by the above facts, in this paper, our aim is to study the variational structure of IBVP (1.1) in an appropriate space of functions and obtain the existence and multiplicity of solutions for IBVP (1.1) by using variational method. To the best of our knowledge, there is no paper concerned impulsive differential equation with Neumann boundary conditions via variational method. In addition, this paper is a generalization of [21], in which impulse effects are not involved.

In this paper, we will need the following conditions.

(H1)There is constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq22_HTML.gif such that for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq23_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq24_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq25_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ5_HTML.gif
(1.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq26_HTML.gif .

(H2) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq27_HTML.gif uniformly for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq28_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq29_HTML.gif .

(H3)There exist numbers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq30_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq31_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ6_HTML.gif
(1.6)
(H4)There exist numbers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq32_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq33_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ7_HTML.gif
(1.7)
(H5)There exist numbers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq34_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq35_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ8_HTML.gif
(1.8)
(H6)There exist numbers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq36_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq37_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ9_HTML.gif
(1.9)

This paper is organized as follows. In Section 2, we present some preliminaries. In Section 3, we discuss the existence and multiplicity of classical solutions to IBVP (1.1). Some examples are presented in this section to illustrate our main results in the last section.

2. Preliminaries

Take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq38_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq39_HTML.gif . We transform IBVP (1.1) into the following equivalent form:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ10_HTML.gif
(2.1)

Obviously, the solutions of IBVP (2.1) are solutions of IBVP (1.1). So it suffices to consider IBVP (2.1).

In this section, the following theorem will be needed in our argument. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq40_HTML.gif is a Banach space (in particular a Hilbert space) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq41_HTML.gif . We say that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq42_HTML.gif satisfies the Palais-Smale condition if any sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq43_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq44_HTML.gif is bounded and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq45_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq46_HTML.gif possesses a convergent subsequence in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq47_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq48_HTML.gif be the open ball in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq49_HTML.gif with the radius https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq50_HTML.gif and centered at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq51_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq52_HTML.gif denote its boundary.

Theorem 2.1 ([22, Theorem 38.A]).

For the functional https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq53_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq54_HTML.gif has a solution for which the following hold:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq55_HTML.gif is a real reflexive Banach space;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq56_HTML.gif is bounded and weakly sequentially closed;

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq57_HTML.gif is weakly sequentially lower semicontinuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq58_HTML.gif ; that is, by definition, for each sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq59_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq60_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq61_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq62_HTML.gif , one has https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq63_HTML.gif holds.

Theorem 2.2 ([16, Theorem 2.2]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq64_HTML.gif be a real Banach space and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq65_HTML.gif satisfy the Palais-Smale condition. Assume there exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq66_HTML.gif and a bounded open neighborhood https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq67_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq68_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq69_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ11_HTML.gif
(2.2)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ12_HTML.gif
(2.3)

Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq70_HTML.gif is a critical value of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq71_HTML.gif ; that is, there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq72_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq73_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq74_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq75_HTML.gif

Theorem 2.3 ([23]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq76_HTML.gif be a real Banach space, and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq77_HTML.gif be even satisfying the Palais-Smale condition and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq78_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq79_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq80_HTML.gif is finite dimensional, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq81_HTML.gif satisfies that

(A1)there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq82_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq83_HTML.gif ,

(A2)for each finite dimensional subspace https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq84_HTML.gif , there is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq85_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq86_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq87_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq88_HTML.gif .

Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq89_HTML.gif possesses an unbounded sequence of critical values.

Let us recall some basic knowledge. Denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq90_HTML.gif the Sobolev space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq91_HTML.gif , and consider the inner product

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ13_HTML.gif
(2.4)

which induces the usual norm

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ14_HTML.gif
(2.5)

We also consider the inner product

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ15_HTML.gif
(2.6)

and the norm

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ16_HTML.gif
(2.7)

then the norm https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq92_HTML.gif is equivalent to the usual norm https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq93_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq94_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq95_HTML.gif is reflexive. We define the norm in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq96_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq97_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq98_HTML.gif , respectively.

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq99_HTML.gif , we have that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq100_HTML.gif are absolutely continuous, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq101_HTML.gif , hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq102_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq103_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq104_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq105_HTML.gif is absolutely continuous and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq106_HTML.gif . In this case, the one-side derivatives https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq107_HTML.gif may not exist. As a consequence, we need to introduce a different concept of solution. We say that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq108_HTML.gif is a classical solution of IBVP (2.1) if it satisfies the equation in IBVP (2.1) a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq109_HTML.gif , the limits https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq110_HTML.gif exist and impulsive conditions in IBVP (2.1) hold, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq111_HTML.gif exist and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq112_HTML.gif . Moreover, for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq113_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq114_HTML.gif .

For each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq115_HTML.gif , consider the functional https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq116_HTML.gif defined on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq117_HTML.gif by

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ17_HTML.gif
(2.8)

It is clear that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq118_HTML.gif is differentiable at any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq119_HTML.gif and

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ18_HTML.gif
(2.9)

for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq120_HTML.gif . Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq121_HTML.gif is continuous.

Lemma 2.4.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq122_HTML.gif is a critical point of the functional https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq123_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq124_HTML.gif is a classical solution of IBVP (2.1).

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq125_HTML.gif be a critical point of the functional https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq126_HTML.gif . It shows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ19_HTML.gif
(2.10)
holds for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq127_HTML.gif . Choose any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq128_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq129_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq130_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq131_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq132_HTML.gif . Equation (2.10) implies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ20_HTML.gif
(2.11)
This means, for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq133_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ21_HTML.gif
(2.12)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq134_HTML.gif . Thus https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq135_HTML.gif is a weak solution of the following equation:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ22_HTML.gif
(2.13)
and therefore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq136_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq137_HTML.gif , then (2.13) becomes the following form:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ23_HTML.gif
(2.14)
Then the solution of (2.14) can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ24_HTML.gif
(2.15)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq138_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq139_HTML.gif are two constants. Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq140_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq141_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq142_HTML.gif is a classical solution of (2.13) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq143_HTML.gif satisfies the equation in IBVP (2.1) a.e. on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq144_HTML.gif . By the previous equation, we can easily get that the limits https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq145_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq146_HTML.gif exist. By integrating (2.10), one has
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ25_HTML.gif
(2.16)
and combining with (2.13) we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ26_HTML.gif
(2.17)
Next we will show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq147_HTML.gif satisfies the impulsive conditions in IBVP (2.1). If not, without loss of generality, we assume that there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq148_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ27_HTML.gif
(2.18)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ28_HTML.gif
(2.19)
Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq149_HTML.gif . Substituting them into (2.17), we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ29_HTML.gif
(2.20)
which contradicts (2.18). So https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq150_HTML.gif satisfies the impulsive conditions in IBVP (2.1). Thus, (2.17) becomes the following form:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ30_HTML.gif
(2.21)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq151_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq152_HTML.gif are arbitrary, (2.21) shows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq153_HTML.gif and it implies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq154_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq155_HTML.gif is a classical solution of IBVP (2.1).

Lemma 2.5.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq156_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq157_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ31_HTML.gif
(2.22)

Proof.

By using the same methods of [15, Lemma 2.6], we easily obtain the above result, and we omit it here.

3. Main Results

In this section, we will show our main results and prove them.

Theorem 3.1.

Assume that (H1) and (H2) hold. Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq158_HTML.gif and the impulsive functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq159_HTML.gif are odd about https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq160_HTML.gif , then IBVP (1.1) has infinitely many classical solutions.

Proof.

Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq161_HTML.gif is an even functional and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq162_HTML.gif . We divide our proof into three parts in order to show Theorem 3.1.

Firstly, We will show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq163_HTML.gif satisfies the Palais-Smale condition. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq164_HTML.gif be a bounded sequence such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq165_HTML.gif . Then there exists constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq166_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ32_HTML.gif
(3.1)
By (2.8), (2.9), (3.1), and (H1), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ33_HTML.gif
(3.2)
It follows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq167_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq168_HTML.gif . From the reflexivity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq169_HTML.gif , we may extract a weakly convergent subsequence that, for simplicity, we call https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq170_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq171_HTML.gif . In the following we will verify that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq172_HTML.gif strongly converges to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq173_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq174_HTML.gif . By (2.9) we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ34_HTML.gif
(3.3)
By https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq175_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq176_HTML.gif , we see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq177_HTML.gif uniformly converges to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq178_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq179_HTML.gif . So
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ35_HTML.gif
(3.4)

By (3.3), (3.4), we obtain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq180_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq181_HTML.gif . That is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq182_HTML.gif strongly converges to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq183_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq184_HTML.gif , which means the that P. S. condition holds for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq185_HTML.gif .

Secondly, we verify the condition (A1) in Theorem 2.3. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq186_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq187_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq188_HTML.gif . In view of (H2), take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq189_HTML.gif , there exists an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq190_HTML.gif such that for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq191_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq192_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ36_HTML.gif
(3.5)
Hence, for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq193_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq194_HTML.gif , by (2.8) and (3.5) , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ37_HTML.gif
(3.6)

Take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq195_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq196_HTML.gif

Finally, we verify condition (A2) in Theorem 2.3. According to (H1), for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq197_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq198_HTML.gif we have that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ38_HTML.gif
(3.7)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ39_HTML.gif
(3.8)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq199_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq200_HTML.gif . This implies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq201_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq202_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq203_HTML.gif . Similarly, we can prove that there is a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq204_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq205_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq206_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq207_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq208_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq209_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq210_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq211_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq212_HTML.gif . Thus, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ40_HTML.gif
(3.9)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq213_HTML.gif .

Similarly, there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq214_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ41_HTML.gif
(3.10)
For every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq215_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq216_HTML.gif , by (2.8), (3.9), and (3.10), we have that the following inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ42_HTML.gif
(3.11)

holds. Take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq217_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq218_HTML.gif , since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq219_HTML.gif , (3.11) implies that there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq220_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq221_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq222_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq223_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq224_HTML.gif is a finite dimensional subspace, there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq225_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq226_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq227_HTML.gif . By Theorem 2.3, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq228_HTML.gif possesses infinite many critical points; that is, IBVP (1.1) has infinite many classical solutions.

Theorem 3.2.

Assume that (H1) and the first equality in (H2) hold. Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq229_HTML.gif is odd about https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq230_HTML.gif and the impulsive functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq231_HTML.gif are odd and nonincreasing. Then IBVP (1.1) has infinitely many classical solutions.

Proof.

We only verify (A1) in Theorem 2.3. Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq232_HTML.gif are odd and nonincreasing continuous functions, then for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq233_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq234_HTML.gif . So we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq235_HTML.gif . Take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq236_HTML.gif , like in (3.6) we can obtain the result.

Theorem 3.3.

Suppose that the first inequalities in (H1), (H3), and (H4) hold. Furthermore, one assumes that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq237_HTML.gif and the impulsive functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq238_HTML.gif are odd about https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq239_HTML.gif and we have the following.

(H7)There exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq240_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ43_HTML.gif
(3.12)

Then IBVP (1.1) has infinitely many classical solutions.

Proof.

Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq241_HTML.gif is an even functional and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq242_HTML.gif . Firstly, we will show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq243_HTML.gif satisfies the Palais-Smale condition. As in the proof of Theorem 3.1, by (2.8), (2.9), (3.1), the first inequalities in (H1) and (H4), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ44_HTML.gif
(3.13)

It follows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq244_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq245_HTML.gif . In the following, the proof of P. S. condition is the same as that in Theorem 3.1, and we omit it here.

Secondly, as in Theorem 3.1, we can obtain that condition (A2) in Theorem 2.1 is satisfied.

Take the same direct sum decomposition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq246_HTML.gif as in Theorem 3.1. For any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq247_HTML.gif , by (2.8), (H3), and (H4), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ45_HTML.gif
(3.14)
In view of (H7), set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq248_HTML.gif , then we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ46_HTML.gif
(3.15)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq249_HTML.gif By Theorem 2.3, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq250_HTML.gif possesses infinite many critical points, that is, IBVP (1.1) has infinite many classical solutions.

Theorem 3.4.

Assume that the second inequalities in (H1), (H5), and (H6) hold, moreover, one assumes the following.

(H8) There exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq251_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ47_HTML.gif
(3.16)

Then IBVP (1.1) has at least two classical solutions.

Proof.

We will use Theorems 2.1 and 2.2 to prove the main results. Firstly, we will show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq252_HTML.gif satisfies the Palais-Smale condition. Similarly, as in the proof of Theorem 3.1, by (2.8), (2.9), (3.1), the second inequalities in (H1) and (H5), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ48_HTML.gif
(3.17)

It follows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq253_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq254_HTML.gif . In the following, the proof of P. S. condition is the same as that in Theorem 3.1, and we omit it here.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq255_HTML.gif , which will be determined later. Set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq256_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq257_HTML.gif is a closed ball. From the reflexivity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq258_HTML.gif , we can easily obtain that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq259_HTML.gif is bounded and weakly sequentially closed. We will show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq260_HTML.gif is weakly lower semicontinuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq261_HTML.gif . Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ49_HTML.gif
(3.18)
Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq262_HTML.gif . By https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq263_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq264_HTML.gif we see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq265_HTML.gif uniformly converges to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq266_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq267_HTML.gif . So https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq268_HTML.gif is weakly continuous. Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq269_HTML.gif is continuous, which, together with the convexity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq270_HTML.gif , implies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq271_HTML.gif is weakly lower semicontinuous. Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq272_HTML.gif is weakly lower semi-continuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq273_HTML.gif . So by Theorem 2.1, without loss of generality, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq274_HTML.gif . Now we will show that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ50_HTML.gif
(3.19)
For any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq275_HTML.gif , by (H5) and (H6), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ51_HTML.gif
(3.20)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ52_HTML.gif
(3.21)

In view of (H8), take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq276_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq277_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq278_HTML.gif . So https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq279_HTML.gif .

Next we will verify that there exists a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq280_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq281_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq282_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq283_HTML.gif . Then by (3.10) and (H5), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ53_HTML.gif
(3.22)

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq284_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq285_HTML.gif . Therefore, there exists a sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq286_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq287_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq288_HTML.gif . Set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq289_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq290_HTML.gif . So by Theorem 2.2, there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq291_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq292_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq293_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq294_HTML.gif are two critical points of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq295_HTML.gif , and they are classical solutions of IBVP (1.1).

Remark 3.5.

Obviously, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq296_HTML.gif is a bounded function, in view of Theorem 3.4, we can obtain the same result.

Theorem 3.6.

Suppose that (H4) and (H5) hold. Then IBVP (1.1) has at least one solution.

Proof.

The proof is similar to that in [19], and we omit it here.

Corollary 3.7.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq297_HTML.gif and impulsive functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq298_HTML.gif are bounded, then IBVP (1.1) has at least one solution.

4. Some Examples

Example 4.1.

Consider the following problem:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ54_HTML.gif
(4.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq299_HTML.gif

Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq300_HTML.gif are odd on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq301_HTML.gif . Compared to IBVP (1.1), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq302_HTML.gif . By simple calculations, we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq303_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq304_HTML.gif . Clearly, (H1), (H2) are satisfied. Applying Theorem 3.1, IBVP (4.1) has infinitely many classical solutions.

Example 4.2.

Consider the following problem:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ55_HTML.gif
(4.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq305_HTML.gif .

Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq306_HTML.gif are odd on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq307_HTML.gif . Compared to IBVP (1.1), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq308_HTML.gif . By simple calculations, we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq309_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq310_HTML.gif . Clearly, the first inequalities in (H1), (H3), and (H4) are satisfied. Take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq311_HTML.gif , then (H7) is also satisfied. Applying Theorem 3.3, IBVP (4.2) has infinitely many classical solutions.

Example 4.3.

Consider the following problem:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ56_HTML.gif
(4.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq312_HTML.gif .

Compared to IBVP (1.1), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq313_HTML.gif . By simple calculations, we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq314_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq315_HTML.gif . Clearly, the second inequalities in (H1), (H5), and (H6) are satisfied. Take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq316_HTML.gif , then (H8) is also satisfied. Applying Theorem 3.4, IBVP (4.3) has at least two classical solutions.

Declarations

Acknowledgment

This project was supported by the National Natural Science Foundation of China (10871206).

Authors’ Affiliations

(1)
Department of Mathematics, Central South University

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© J. Sun and H. Chen. 2009

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