Variational Method to the Impulsive Equation with Neumann Boundary Conditions

  • Juntao Sun1Email author and

    Affiliated with

    • Haibo Chen1

      Affiliated with

      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

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

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

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

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq15_HTML.gif is continuous, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq16_HTML.gif , are continuous, and http://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:

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

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq18_HTML.gif is continuous, and http://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:

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

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq20_HTML.gif is continuous, and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq22_HTML.gif such that for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq23_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq24_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq25_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ5_HTML.gif
      (1.5)

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

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

      (H3)There exist numbers http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq30_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq31_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ6_HTML.gif
      (1.6)
      (H4)There exist numbers http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq32_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq33_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ7_HTML.gif
      (1.7)
      (H5)There exist numbers http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq34_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq35_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ8_HTML.gif
      (1.8)
      (H6)There exist numbers http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq36_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq37_HTML.gif such that
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq38_HTML.gif . Then http://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:

      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq41_HTML.gif . We say that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq43_HTML.gif for which http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq44_HTML.gif is bounded and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq45_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq46_HTML.gif possesses a convergent subsequence in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq47_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq48_HTML.gif be the open ball in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq49_HTML.gif with the radius http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq50_HTML.gif and centered at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq51_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq53_HTML.gif with http://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) http://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) http://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) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq57_HTML.gif is weakly sequentially lower semicontinuous on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq59_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq60_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq61_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq62_HTML.gif , one has http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq66_HTML.gif and a bounded open neighborhood http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq67_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq68_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq69_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ11_HTML.gif
      (2.2)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ12_HTML.gif
      (2.3)

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

      Theorem 2.3 ([23]).

      Let http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq78_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq79_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq80_HTML.gif is finite dimensional, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq81_HTML.gif satisfies that

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

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

      Then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq90_HTML.gif the Sobolev space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq91_HTML.gif , and consider the inner product

      http://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

      http://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

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

      and the norm

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

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

      For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq99_HTML.gif , we have that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq100_HTML.gif are absolutely continuous, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq101_HTML.gif , hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq102_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq103_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq104_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq105_HTML.gif is absolutely continuous and http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq109_HTML.gif , the limits http://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, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq111_HTML.gif exist and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq112_HTML.gif . Moreover, for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq113_HTML.gif satisfy http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq114_HTML.gif .

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

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

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

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

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

      Lemma 2.4.

      If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq123_HTML.gif , then http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq126_HTML.gif . It shows that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ19_HTML.gif
      (2.10)
      holds for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq127_HTML.gif . Choose any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq128_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq129_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq130_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq131_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq132_HTML.gif . Equation (2.10) implies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ20_HTML.gif
      (2.11)
      This means, for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq133_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ21_HTML.gif
      (2.12)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq134_HTML.gif . Thus http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ22_HTML.gif
      (2.13)
      and therefore http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq136_HTML.gif Let http://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:
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ24_HTML.gif
      (2.15)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq138_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq139_HTML.gif are two constants. Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq140_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq141_HTML.gif . Therefore, http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq145_HTML.gif and http://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
      http://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
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq148_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ27_HTML.gif
      (2.18)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ28_HTML.gif
      (2.19)
      Obviously, http://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
      http://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 http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ30_HTML.gif
      (2.21)

      for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq151_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq152_HTML.gif are arbitrary, (2.21) shows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq153_HTML.gif and it implies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq154_HTML.gif . Therefore, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq156_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq157_HTML.gif , where
      http://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, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq158_HTML.gif and the impulsive functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq159_HTML.gif are odd about http://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, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq161_HTML.gif is an even functional and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq163_HTML.gif satisfies the Palais-Smale condition. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq164_HTML.gif be a bounded sequence such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq165_HTML.gif . Then there exists constants http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq166_HTML.gif such that
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ33_HTML.gif
      (3.2)
      It follows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq167_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq168_HTML.gif . From the reflexivity of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq170_HTML.gif in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq172_HTML.gif strongly converges to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq173_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq174_HTML.gif . By (2.9) we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ34_HTML.gif
      (3.3)
      By http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq175_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq176_HTML.gif , we see that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq177_HTML.gif uniformly converges to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq178_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq179_HTML.gif . So
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq180_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq181_HTML.gif . That is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq182_HTML.gif strongly converges to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq183_HTML.gif in http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq186_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq187_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq188_HTML.gif . In view of (H2), take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq189_HTML.gif , there exists an http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq190_HTML.gif such that for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq191_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq192_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ36_HTML.gif
      (3.5)
      Hence, for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq193_HTML.gif with http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ37_HTML.gif
      (3.6)

      Take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq195_HTML.gif , then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq197_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq198_HTML.gif we have that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ38_HTML.gif
      (3.7)
      Hence
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ39_HTML.gif
      (3.8)
      for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq199_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq200_HTML.gif . This implies that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq201_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq202_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq204_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq205_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq206_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq207_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq208_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq209_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq210_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq211_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq212_HTML.gif . Thus, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ40_HTML.gif
      (3.9)

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

      Similarly, there exist constants http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq214_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ41_HTML.gif
      (3.10)
      For every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq215_HTML.gif and http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ42_HTML.gif
      (3.11)

      holds. Take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq217_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq218_HTML.gif , since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq219_HTML.gif , (3.11) implies that there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq220_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq221_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq222_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq223_HTML.gif . Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq225_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq226_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq227_HTML.gif . By Theorem 2.3, http://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, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq229_HTML.gif is odd about http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq230_HTML.gif and the impulsive functions http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq233_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq234_HTML.gif . So we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq235_HTML.gif . Take http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq237_HTML.gif and the impulsive functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq238_HTML.gif are odd about http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq240_HTML.gif such that

      http://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, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq241_HTML.gif is an even functional and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq242_HTML.gif . Firstly, we will show that http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ44_HTML.gif
      (3.13)

      It follows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq244_HTML.gif is bounded in http://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 http://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 http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq248_HTML.gif , then we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ46_HTML.gif
      (3.15)

      Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq249_HTML.gif By Theorem 2.3, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq251_HTML.gif such that
      http://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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ48_HTML.gif
      (3.17)

      It follows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq253_HTML.gif is bounded in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq255_HTML.gif , which will be determined later. Set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq256_HTML.gif , then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq258_HTML.gif , we can easily obtain that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq260_HTML.gif is weakly lower semicontinuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq261_HTML.gif . Let
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ49_HTML.gif
      (3.18)
      Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq262_HTML.gif . By http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq263_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq264_HTML.gif we see that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq265_HTML.gif uniformly converges to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq266_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq267_HTML.gif . So http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq268_HTML.gif is weakly continuous. Clearly, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq270_HTML.gif , implies that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq271_HTML.gif is weakly lower semicontinuous. Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq272_HTML.gif is weakly lower semi-continuous on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq274_HTML.gif . Now we will show that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ50_HTML.gif
      (3.19)
      For any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq275_HTML.gif , by (H5) and (H6), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ51_HTML.gif
      (3.20)
      Hence
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq276_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq277_HTML.gif , for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq278_HTML.gif . So http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq280_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq281_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq282_HTML.gif . Let http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ53_HTML.gif
      (3.22)

      Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq284_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq285_HTML.gif . Therefore, there exists a sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq286_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq287_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq288_HTML.gif . Set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq289_HTML.gif , then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq291_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq292_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq293_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq294_HTML.gif are two critical points of http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq297_HTML.gif and impulsive functions http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ54_HTML.gif
      (4.1)

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

      Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq300_HTML.gif are odd on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq301_HTML.gif . Compared to IBVP (1.1), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq302_HTML.gif . By simple calculations, we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq303_HTML.gif . Let http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ55_HTML.gif
      (4.2)

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

      Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq306_HTML.gif are odd on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq307_HTML.gif . Compared to IBVP (1.1), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq308_HTML.gif . By simple calculations, we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq309_HTML.gif . Let http://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 http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ56_HTML.gif
      (4.3)

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

      Compared to IBVP (1.1), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq313_HTML.gif . By simple calculations, we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_IEq314_HTML.gif . Let http://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 http://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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