Antiperiodic Solutions for Liénard-Type Differential Equation with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq1_HTML.gif -Laplacian Operator

  • Taiyong Chen1Email author,

    Affiliated with

    • Wenbin Liu1 and

      Affiliated with

      • Cheng Yang1

        Affiliated with

        Boundary Value Problems20102010:194824

        DOI: 10.1155/2010/194824

        Received: 2 March 2010

        Accepted: 19 August 2010

        Published: 25 August 2010

        Abstract

        The existence of antiperiodic solutions for Liénard-type and Duffing-type differential equations with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq2_HTML.gif -Laplacian operator has been studied by using degree theory. The results obtained improve and enrich some known works to some extent.

        1. Introduction

        Antiperiodic problems arise naturally from the mathematical models of various of physical processes (see [1, 2]), and also appear in the study of partial differential equations and abstract differential equations (see [35]). For instance, electron beam focusing system in travelling-wave tube's theories is an antiperiodic problem (see [6]).

        During the past twenty years, antiperiodic problems have been studied extensively by numerous scholars. For example, for first-order ordinary differential equations, a Massera's type criterion was presented in [7] and the validity of the monotone iterative technique was shown in [8]. Moreover, for higher-order ordinary differential equations, the existence of antiperiodic solutions was considered in [912]. Recently, existence results were extended to antiperiodic boundary value problems for impulsive differential equations (see [13]), and antiperiodic wavelets were discussed in [14].

        Wang and Li (see [15]) discussed the existence of solutions of the following antiperiodic boundary value problem for second-order conservative system:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ1_HTML.gif
        (1.1)

        using of the main assumption as follows:

        ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq3_HTML.gif ) There exist constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq4_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq5_HTML.gif , such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ2_HTML.gif
        (1.2)
        The turbulent flow in a porous medium is a fundamental mechanics problem. For studying this type of problems, Leibenson (see [16]) introduced the following http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq6_HTML.gif -Laplacian equation:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ3_HTML.gif
        (1.3)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq7_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq8_HTML.gif . Obviously, the inverse operator of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq9_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq10_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq11_HTML.gif is a constant such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq12_HTML.gif .

        Notice that, when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq13_HTML.gif , the nonlinear operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq14_HTML.gif reduces to the linear operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq15_HTML.gif .

        In the past few decades, many important results relative to (1.3) with certain boundary conditions have been obtained. We refer the reader to [1720] and the references cited therein. However, to the best of our knowledge, there exist relatively few results for the existence of antiperiodic solutions of (1.3). Moreover, it is well known that the existence of antiperiodic solutions plays a key role in characterizing the behavior of nonlinear differential equations (see [21]). Thus, it is worthwhile to continue to investigate the existence of antiperiodic solutions for (1.3).

        A primary purpose of this paper is to study the existence of antiperiodic solutions for the following Liénard-type http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq16_HTML.gif -Laplacian equation:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ4_HTML.gif
        (1.4)
        and antiperiodic solutions with symmetry for Duffing-type http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq17_HTML.gif -Laplacian equation as follows:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ5_HTML.gif
        (1.5)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq19_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq20_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq21_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq22_HTML.gif . That is, we will prove that (1.4) or (1.5) has at least one solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq23_HTML.gif satisfying
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ6_HTML.gif
        (1.6)

        Note that, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq24_HTML.gif is also a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq25_HTML.gif -periodic solution of (1.4) or (1.5) if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq26_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq27_HTML.gif -antiperiodic solution of (1.4) or (1.5). Hence, from the arguments in this paper, we can also obtain the existence results of periodic solutions for above equations.

        The rest of this paper is organized as follows. Section 2 contains some necessary preliminaries. In Section 3, we establish some sufficient conditions for the existence of antiperiodic solutions of (1.4), basing on Leray-Schauder principle. Then, in Section 4, we obtain two existence results of antiperiodic solutions with symmetry for (1.5). Finally, in Section 5, some explicit examples are given to illustrate the main results. Our results are different from those of bibliographies listed above.

        2. Preliminaries

        For convenience, we introduce some notations as follows:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ7_HTML.gif
        (2.1)

        and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq28_HTML.gif denotes norm in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq29_HTML.gif .

        For each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq30_HTML.gif , there exists the following Fourier series expansion:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ8_HTML.gif
        (2.2)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq31_HTML.gif . Let us define the mapping http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq32_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ9_HTML.gif
        (2.3)
        Notice that, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq33_HTML.gif may be written as Fourier series as follows:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ10_HTML.gif
        (2.4)
        and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq34_HTML.gif may be written as the following Fourier series:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ11_HTML.gif
        (2.5)
        We define the mapping http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq35_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ12_HTML.gif
        (2.6)
        and the mapping http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq36_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ13_HTML.gif
        (2.7)

        It is easy to prove that the mappings http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq37_HTML.gif are completely continuous by using Arzelà-Ascoli theorem.

        Next, we introduce a Wirtinger inequality (see [22]) and a continuation theorem (see [23, 24]) as follows.

        Lemma 2.1 (Wirtinger inequality).

        For each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq38_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq39_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq40_HTML.gif , one has
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ14_HTML.gif
        (2.8)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ15_HTML.gif
        (2.9)

        Lemma 2.2 (Continuation theorem).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq41_HTML.gif be open-bounded in a linear normal space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq42_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq43_HTML.gif is a completely continuous field on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq44_HTML.gif . Moreover, assume that the Leray-Schauder degree
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ16_HTML.gif
        (2.10)

        Then equation http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq45_HTML.gif has at least one solution in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq46_HTML.gif .

        3. Antiperiodic Solutions for (1.4)

        In this section, an existence result of antiperiodic solutions for (1.4) will be given.

        Theorem 3.1.

        Assume that

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq47_HTML.gif there exists a nonnegative function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq48_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ17_HTML.gif
        (3.1)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ18_HTML.gif
        (3.2)

        Then (1.4) has at least one antiperiodic solution.

        Remark 3.2.

        When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq49_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq50_HTML.gif is equal to 1. It is easy to see that condition ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq51_HTML.gif ) in [15] is stronger than condition ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq52_HTML.gif ) of Theorem 3.1.

        For making use of Leray-Schauder degree theory to prove the existence of antiperiodic solutions for (1.4), we consider the homotopic equation of (1.4) as follows:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ19_HTML.gif
        (3.3)
        Define the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq53_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ20_HTML.gif
        (3.4)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ21_HTML.gif
        (3.5)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq54_HTML.gif be the Nemytski operator
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ22_HTML.gif
        (3.6)
        Obviously, the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq55_HTML.gif is invertible and the antiperiodic problem of (3.3) is equivalent to the operator equation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ23_HTML.gif
        (3.7)

        We begin with some lemmas below.

        Lemma 3.3.

        Suppose that the assumption http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq56_HTML.gif is true. Then the antiperiodic solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq57_HTML.gif of (3.3) satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ24_HTML.gif
        (3.8)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq58_HTML.gif is a positive constant only dependent of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq59_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq60_HTML.gif .

        Proof.

        Multiplying the both sides of (3.3) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq61_HTML.gif and integrating it over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq62_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ25_HTML.gif
        (3.9)
        Noting that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ26_HTML.gif
        (3.10)
        and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq63_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ27_HTML.gif
        (3.11)
        By hypothesis http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq64_HTML.gif , there exists a nonnegative constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq65_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ28_HTML.gif
        (3.12)
        Thus, from (3.11), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ29_HTML.gif
        (3.13)
        That is,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ30_HTML.gif
        (3.14)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq66_HTML.gif .

        For each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq67_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ31_HTML.gif
        (3.15)
        Similarly, we obtain that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ32_HTML.gif
        (3.16)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ33_HTML.gif
        (3.17)
        Basing on Lemma 2.1, it can be shown from (3.17) and (3.14) that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ34_HTML.gif
        (3.18)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq68_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ35_HTML.gif
        (3.19)

        The proof is complete.

        Lemma 3.4.

        Suppose that the assumption http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq69_HTML.gif is true. Then, for the possible antiperiodic solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq70_HTML.gif of (3.3), there exists a prior bounds in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq71_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq72_HTML.gif satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ36_HTML.gif
        (3.20)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq73_HTML.gif is a positive constant independent of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq74_HTML.gif .

        Proof.

        By (3.15), there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq75_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq76_HTML.gif . Hence, (3.8) yields that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ37_HTML.gif
        (3.21)
        Letting
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ38_HTML.gif
        (3.22)
        From (3.16), there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq77_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq78_HTML.gif , which implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq79_HTML.gif . Therefore, integrating the both sides of (3.3) over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq80_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ39_HTML.gif
        (3.23)
        Thus, we get from (3.8) that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ40_HTML.gif
        (3.24)
        Noting that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq81_HTML.gif , we obtain that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ41_HTML.gif
        (3.25)
        Combining (3.21) with (3.25), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ42_HTML.gif
        (3.26)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq82_HTML.gif . The proof is complete.

        Now we give the proof of Theorem 3.1.

        Proof of Theorem 3.1..

        Setting
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ43_HTML.gif
        (3.27)

        Obviously, the set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq83_HTML.gif is an open-bounded set in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq84_HTML.gif and zero element http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq85_HTML.gif .

        From the definition of operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq86_HTML.gif , it is easy to see that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ44_HTML.gif
        (3.28)
        Hence, the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq87_HTML.gif sends http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq88_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq89_HTML.gif . Let us define the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq90_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ45_HTML.gif
        (3.29)

        Obviously, the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq91_HTML.gif is completely continuous in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq92_HTML.gif and the fixed points of operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq93_HTML.gif are the antiperiodic solutions of (1.4).

        With this in mind, let us define the completely continuous field http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq94_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ46_HTML.gif
        (3.30)
        By (3.20), we get that zero element http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq95_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq96_HTML.gif . So that, the following Leray-Schauder degrees are well defined and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ47_HTML.gif
        (3.31)

        Consequently, the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq97_HTML.gif has at least one fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq98_HTML.gif by using Lemma 2.2. Namely, (1.4) has at least one antiperiodic solution. The proof is complete.

        4. Antiperiodic Solutions with Symmetry for (1.5)

        In this section, we will prove the existence of even antiperiodic solutions or odd antiperiodic solutions for (1.5).

        Theorem 4.1.

        Assume that

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq99_HTML.gif the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq101_HTML.gif are even in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq102_HTML.gif , that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ48_HTML.gif
        (4.1)
        and the assumption ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq103_HTML.gif ) is true. Then (1.5) has at least one even antiperiodic solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq104_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq105_HTML.gif satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ49_HTML.gif
        (4.2)

        Proof.

        We consider the homotopic equation of (1.5) as follows:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ50_HTML.gif
        (4.3)
        Define the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq106_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ51_HTML.gif
        (4.4)

        Obviously, the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq107_HTML.gif is continuous.

        Basing on the proof of Theorem 3.1, for the possible even antiperiodic solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq108_HTML.gif of (4.3), there exists a prior bounds in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq109_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq110_HTML.gif satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ52_HTML.gif
        (4.5)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq111_HTML.gif is a positive constant independent of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq112_HTML.gif . So that, our problem is reduced to construct one completely continuous operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq113_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq114_HTML.gif which sends http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq115_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq116_HTML.gif , such that the fixed points of operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq117_HTML.gif in some open-bounded set are the even antiperiodic solutions of (1.5).

        With this in mind, let us define the following set:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ53_HTML.gif
        (4.6)

        Obviously, the set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq118_HTML.gif is an open-bounded set in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq119_HTML.gif and zero element http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq120_HTML.gif .

        By hypothesis ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq121_HTML.gif ), it is easy to see that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ54_HTML.gif
        (4.7)
        Hence, the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq122_HTML.gif sends http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq123_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq124_HTML.gif . Let us define the completely continuous operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq125_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ55_HTML.gif
        (4.8)

        From the similar arguments in the proof of Theorem 3.1, we can prove that there exists at least one fixed point of operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq126_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq127_HTML.gif . Thus, (1.5) has at least one even antiperiodic solution. The proof is complete.

        Theorem 4.2.

        Assume that

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq128_HTML.gif the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq129_HTML.gif is odd in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq130_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq131_HTML.gif is odd in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq132_HTML.gif , that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ56_HTML.gif
        (4.9)
        and the assumption ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq133_HTML.gif ) is true. Then (1.5) has at least one odd antiperiodic solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq134_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq135_HTML.gif satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ57_HTML.gif
        (4.10)

        Proof.

        We consider the homotopic equation (4.3) of (1.5). Define the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq136_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ58_HTML.gif
        (4.11)

        Obviously, the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq137_HTML.gif is continuous.

        Based on the proof of Theorem 3.1, for the possible odd antiperiodic solutions of (4.3), there exists a prior bounds in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq138_HTML.gif . Hence, our problem is reduced to construct one completely continuous operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq139_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq140_HTML.gif which sends http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq141_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq142_HTML.gif , such that the fixed points of operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq143_HTML.gif in some open-bounded set are the odd antiperiodic solutions of (1.5).

        With this in mind, let us define the set as follows:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ59_HTML.gif
        (4.12)

        Obviously, the set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq144_HTML.gif is an open-bounded set in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq145_HTML.gif and zero element http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq146_HTML.gif .

        From the hypothesis ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq147_HTML.gif ), it is easy to see that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ60_HTML.gif
        (4.13)
        Thus, the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq148_HTML.gif sends http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq149_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq150_HTML.gif . Let us define the completely continuous operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq151_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ61_HTML.gif
        (4.14)

        By a similar way as the proof of Theorem 3.1, we can prove that there exists at least one fixed point of operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq152_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq153_HTML.gif . So that, (1.5) has at least one odd antiperiodic solution. The proof is complete.

        5. Examples

        In this section, we will give some examples to illustrate our main results.

        Consider the following second-order differential equation with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq154_HTML.gif -Laplacian operator:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ62_HTML.gif
        (5.1)

        Example 5.1.

        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ63_HTML.gif
        (5.2)

        For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq155_HTML.gif , by direct calculation, we can get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq156_HTML.gif . Choosing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq157_HTML.gif , then (5.1) satisfies the condition of Theorem 3.1. So it has at least one antiperiodic solution.

        Moreover, the conditions of Theorem 4.1 are also satisfied. Thus (5.1) has at least one even antiperiodic solution.

        Example 5.2.

        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ64_HTML.gif
        (5.3)

        We choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq158_HTML.gif . Obviously, (5.1) satisfies all the conditions of Theorem 4.2. Hence it has at least one odd antiperiodic solution.

        Declarations

        Acknowledgments

        The authors would like to thank the referees very much for their helpful comments and suggestions. This research was supported by the National Natural Science Foundation of China (10771212), the Fundamental Research Funds for the Central Universities, and the Science Foundation of China University of Mining and Technology (2008A037).

        Authors’ Affiliations

        (1)
        Department of Mathematics, China University of Mining and Technology

        References

        1. Ahn C, Rim C: Boundary flows in general coset theories. Journal of Physics 1999,32(13):2509-2525.MATHMathSciNet
        2. Kleinert H, Chervyakov A: Functional determinants from Wronski Green functions. Journal of Mathematical Physics 1999,40(11):6044-6051. 10.1063/1.533069MATHMathSciNetView Article
        3. Aizicovici S, McKibben M, Reich S: Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities. Nonlinear Analysis. Theory, Methods & Applications 2001,43(2):233-251. 10.1016/S0362-546X(99)00192-3MATHMathSciNetView Article
        4. Nakao M: Existence of an anti-periodic solution for the quasilinear wave equation with viscosity. Journal of Mathematical Analysis and Applications 1996,204(3):754-764. 10.1006/jmaa.1996.0465MATHMathSciNetView Article
        5. Souplet P: Optimal uniqueness condition for the antiperiodic solutions of some nonlinear parabolic equations. Nonlinear Analysis. Theory, Methods & Applications 1998,32(2):279-286. 10.1016/S0362-546X(97)00477-XMATHMathSciNetView Article
        6. Lu Z: Travelling Tube. Shanghai Science and Technology Press, Shanghai, China; 1962.
        7. Chen YQ: On Massera's theorem for anti-periodic solution. Advances in Mathematical Sciences and Applications 1999,9(1):125-128.MATHMathSciNet
        8. Yin Y: Monotone iterative technique and quasilinearization for some anti-periodic problems. Nonlinear World 1996,3(2):253-266.MATHMathSciNet
        9. Aftabizadeh AR, Pavel NH, Huang YK: Anti-periodic oscillations of some second-order differential equations and optimal control problems. Journal of Computational and Applied Mathematics 1994,52(1–3):3-21.MATHMathSciNetView Article
        10. Chen T, Liu W, Zhang J, Zhang M: The existence of anti-periodic solutions for Liénard equations. Journal of Mathematical Study 2007,40(2):187-195.MATHMathSciNet
        11. Liu B: Anti-periodic solutions for forced Rayleigh-type equations. Nonlinear Analysis. Real World Applications 2009,10(5):2850-2856. 10.1016/j.nonrwa.2008.08.011MATHMathSciNetView Article
        12. Liu WB, Zhang JJ, Chen TY: Anti-symmetric periodic solutions for the third order differential systems. Applied Mathematics Letters. An International Journal of Rapid Publication 2009,22(5):668-673. 10.1016/j.aml.2008.08.004MATHMathSciNet
        13. Luo Z, Shen J, Nieto JJ: Antiperiodic boundary value problem for first-order impulsive ordinary differential equations. Computers & Mathematics with Applications 2005,49(2-3):253-261. 10.1016/j.camwa.2004.08.010MATHMathSciNetView Article
        14. Chen HL: Antiperiodic wavelets. Journal of Computational Mathematics 1996,14(1):32-39.MATHMathSciNet
        15. Wang K, Li Y: A note on existence of (anti-)periodic and heteroclinic solutions for a class of second-order odes. Nonlinear Analysis. Theory, Methods & Applications 2009,70(4):1711-1724. 10.1016/j.na.2008.02.054MATHMathSciNetView Article
        16. Leibenson LS: General problem of the movement of a compressible fluid in a porous medium. Izvestiia Akademii Nauk Kirgizskoĭ SSSR 1983, 9: 7-10.MathSciNet
        17. Jiang D, Gao W: Upper and lower solution method and a singular boundary value problem for the one-dimensional p -Laplacian. Journal of Mathematical Analysis and Applications 2000,252(2):631-648. 10.1006/jmaa.2000.7012MATHMathSciNetView Article
        18. Lian LF, Ge WG: The existence of solutions of m-point p -Laplacian boundary value problems at resonance. Acta Mathematicae Applicatae Sinica 2005,28(2):288-295.MathSciNet
        19. Liu B, Yu JS: On the existence of solution for the periodic boundary value problems with p -Laplacian operator. Journal of Systems Science and Mathematical Sciences 2003,23(1):76-85.MATHMathSciNet
        20. Zhang J, Liu W, Ni J, Chen T: Multiple periodic solutions of p -Laplacian equation with one-side Nagumo condition. Journal of the Korean Mathematical Society 2008,45(6):1549-1559. 10.4134/JKMS.2008.45.6.1549MATHMathSciNetView Article
        21. Chen Y, Nieto JJ, O'Regan D: Anti-periodic solutions for fully nonlinear first-order differential equations. Mathematical and Computer Modelling 2007,46(9-10):1183-1190. 10.1016/j.mcm.2006.12.006MATHMathSciNetView Article
        22. Croce G, Dacorogna B: On a generalized Wirtinger inequality. Discrete and Continuous Dynamical Systems 2003,9(5):1329-1341.MATHMathSciNetView Article
        23. Deimling K: Nonlinear Functional Analysis. Springer, Berlin, Germany; 1985:xiv+450.MATHView Article
        24. Gaines RE, Mawhin JL: Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Mathematics, Vol. 568. Springer, Berlin, Germany; 1977:i+262.

        Copyright

        © Taiyong Chen et al. 2010

        This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.