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

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 https://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:
https://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:

( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq3_HTML.gif ) There exist constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq4_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq5_HTML.gif , such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq6_HTML.gif -Laplacian equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ3_HTML.gif
(1.3)

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

Notice that, when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq13_HTML.gif , the nonlinear operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq14_HTML.gif reduces to the linear operator https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq16_HTML.gif -Laplacian equation:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq17_HTML.gif -Laplacian equation as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ5_HTML.gif
(1.5)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq18_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq19_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq20_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq21_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq23_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ6_HTML.gif
(1.6)

Note that, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq24_HTML.gif is also a https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq26_HTML.gif is a https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ7_HTML.gif
(2.1)

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

For each https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ8_HTML.gif
(2.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq31_HTML.gif . Let us define the mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq32_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ9_HTML.gif
(2.3)
Notice that, https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ10_HTML.gif
(2.4)
and https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ11_HTML.gif
(2.5)
We define the mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq35_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ12_HTML.gif
(2.6)
and the mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq36_HTML.gif by
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq38_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq39_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq40_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ14_HTML.gif
(2.8)
where
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq42_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq43_HTML.gif is a completely continuous field on https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ16_HTML.gif
(2.10)

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

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq47_HTML.gif there exists a nonnegative function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq48_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ17_HTML.gif
(3.1)
where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq49_HTML.gif , https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq51_HTML.gif ) in [15] is stronger than condition ( https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ19_HTML.gif
(3.3)
Define the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq53_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ20_HTML.gif
(3.4)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ21_HTML.gif
(3.5)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq54_HTML.gif be the Nemytski operator
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ22_HTML.gif
(3.6)
Obviously, the operator https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq56_HTML.gif is true. Then the antiperiodic solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq57_HTML.gif of (3.3) satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ24_HTML.gif
(3.8)

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq59_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq61_HTML.gif and integrating it over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq62_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ25_HTML.gif
(3.9)
Noting that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ26_HTML.gif
(3.10)
and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq63_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ27_HTML.gif
(3.11)
By hypothesis https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq64_HTML.gif , there exists a nonnegative constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq65_HTML.gif such that
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ29_HTML.gif
(3.13)
That is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ30_HTML.gif
(3.14)

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

For each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq67_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ31_HTML.gif
(3.15)
Similarly, we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ32_HTML.gif
(3.16)
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ34_HTML.gif
(3.18)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq68_HTML.gif , then
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq71_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq72_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ36_HTML.gif
(3.20)

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

Proof.

By (3.15), there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq75_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq76_HTML.gif . Hence, (3.8) yields that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ37_HTML.gif
(3.21)
Letting
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq77_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq78_HTML.gif , which implies that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq80_HTML.gif , we have
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ40_HTML.gif
(3.24)
Noting that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq81_HTML.gif , we obtain that
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ42_HTML.gif
(3.26)

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

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

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

Obviously, the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq91_HTML.gif is completely continuous in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq92_HTML.gif and the fixed points of operator https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq94_HTML.gif by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq95_HTML.gif for all https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ47_HTML.gif
(3.31)

Consequently, the operator https://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 https://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

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq99_HTML.gif the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq100_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq101_HTML.gif are even in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq102_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ48_HTML.gif
(4.1)
and the assumption ( https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq104_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq105_HTML.gif satisfies
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ50_HTML.gif
(4.3)
Define the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq106_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ51_HTML.gif
(4.4)

Obviously, the operator https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq109_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq110_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ52_HTML.gif
(4.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq111_HTML.gif is a positive constant independent of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq113_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq114_HTML.gif which sends https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq115_HTML.gif into https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ53_HTML.gif
(4.6)

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

By hypothesis ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq121_HTML.gif ), it is easy to see that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ54_HTML.gif
(4.7)
Hence, the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq122_HTML.gif sends https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq123_HTML.gif into https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq125_HTML.gif by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq126_HTML.gif in https://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

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq128_HTML.gif the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq129_HTML.gif is odd in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq130_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq131_HTML.gif is odd in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq132_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ56_HTML.gif
(4.9)
and the assumption ( https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq134_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq135_HTML.gif satisfies
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq136_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ58_HTML.gif
(4.11)

Obviously, the operator https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq139_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq140_HTML.gif which sends https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq141_HTML.gif into https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ59_HTML.gif
(4.12)

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

From the hypothesis ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq147_HTML.gif ), it is easy to see that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ60_HTML.gif
(4.13)
Thus, the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq148_HTML.gif sends https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq149_HTML.gif into https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq151_HTML.gif by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq152_HTML.gif in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq154_HTML.gif -Laplacian operator:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ62_HTML.gif
(5.1)

Example 5.1.

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

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq155_HTML.gif , by direct calculation, we can get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_IEq156_HTML.gif . Choosing https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F194824/MediaObjects/13661_2010_Article_901_Equ64_HTML.gif
(5.3)

We choose https://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

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Copyright

© Taiyong Chen et al. 2010

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