Open Access

Existence and Lyapunov Stability of Periodic Solutions for Generalized Higher-Order Neutral Differential Equations

Boundary Value Problems20102011:635767

DOI: 10.1155/2011/635767

Received: 17 May 2010

Accepted: 23 June 2010

Published: 19 July 2010

Abstract

Existence and Lyapunov stability of periodic solutions for a generalized higher-order neutral differential equation are established.

1. Introduction

In recent years, there is a good amount of work on periodic solutions for neutral differential equations (see [111] and the references cited therein). For example, the following neutral differential equations
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ1_HTML.gif
(1.1)
have been studied in [1, 3, 8], respectively, and existence criteria of periodic solutions were established for these equations. Afterwards, along with intensive research on the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq1_HTML.gif -Laplacian, some authors [4, 11] start to consider the following https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq2_HTML.gif -Laplacian neutral functional differential equations:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ2_HTML.gif
(1.2)

and by using topological degree theory and some analysis skills, existence results of periodic solutions for (1.2) have been presented.

In general, most of the existing results are concentrated on lower-order neutral functional differential equations, while studies on higher-order neutral functional differential equations are rather infrequent, especially on higher-order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq3_HTML.gif -Laplacian neutral functional differential equations. In this paper, we consider the following generalized higher-order neutral functional differential equation:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ3_HTML.gif
(1.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq4_HTML.gif is given by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq5_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq6_HTML.gif being a constant, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq7_HTML.gif is a continuous function defined on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq8_HTML.gif and is periodic with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq9_HTML.gif with period https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq10_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq11_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq12_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq13_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq14_HTML.gif are constants.

Since the neutral operator is divided into two cases https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq15_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq16_HTML.gif , it is natural to study the neutral differential equation separately according to these two cases. The case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq17_HTML.gif has been studied in [5]. Now we consider (1.3) for the case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq18_HTML.gif . So throughout this paper, we always assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq19_HTML.gif , and the paper is organized as follows. We first transform (1.3) into a system of first-order differential equations, and then by applying Mawhin's continuation theory and some new inequalities, we obtain sufficient conditions for the existence of periodic solutions for (1.3). The Lyapunov stability of periodic solutions for the equation will then be established. Finally, an example is given to illustrate our results.

2. Preparation

First, we recall two lemmas. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq20_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq21_HTML.gif be real Banach spaces and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq22_HTML.gif be a Fredholm operator with index zero; here https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq23_HTML.gif denotes the domain of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq24_HTML.gif . This means that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq25_HTML.gif is closed in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq26_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq27_HTML.gif . Consider supplementary subspaces https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq28_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq29_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq30_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq31_HTML.gif , respectively, such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq32_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq33_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq34_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq35_HTML.gif denote the natural projections. Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq36_HTML.gif and so the restriction https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq37_HTML.gif is invertible. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq38_HTML.gif denote the inverse of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq39_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq40_HTML.gif be an open bounded subset of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq41_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq42_HTML.gif . A map https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq43_HTML.gif is said to be https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq44_HTML.gif -compact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq45_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq46_HTML.gif is bounded and the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq47_HTML.gif is compact.

Lemma 2.1 (see [12]).

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq48_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq49_HTML.gif are two Banach spaces, and suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq50_HTML.gif is a Fredholm operator with index zero. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq51_HTML.gif be an open bounded set and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq52_HTML.gif be https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq53_HTML.gif -compact on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq54_HTML.gif . Assume that the following conditions hold:

(1) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq55_HTML.gif

(2) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq56_HTML.gif

(3) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq57_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq58_HTML.gif is an isomorphism.

Then, the equation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq59_HTML.gif has a solution in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq60_HTML.gif .

Lemma 2.2 (see [13]).

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq61_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq62_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ4_HTML.gif
(2.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq63_HTML.gif is a fixed real number with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq64_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ5_HTML.gif
(2.2)
For the sake of convenience, throughout this paper we denote by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq65_HTML.gif a positive real number, and for any continuous function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq66_HTML.gif , we write
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ6_HTML.gif
(2.3)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq67_HTML.gif be the operator on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq68_HTML.gif given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ7_HTML.gif
(2.4)

Lemma 2.3.

The operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq69_HTML.gif has a continuous inverse https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq70_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq71_HTML.gif satisfying the following:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ8_HTML.gif
(2.5)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ9_HTML.gif
(2.6)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ10_HTML.gif
(2.7)

Remark 2.4.

This lemma is basically proved in [3, 10]. For the convenience of the readers, we present a detailed proof here as follows.

Proof.

We split it into the following two cases.

Case 1 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq72_HTML.gif ).

Define an operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq73_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ11_HTML.gif
(2.8)
Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq74_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq75_HTML.gif . Note also that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq76_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq77_HTML.gif has a continuous inverse https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq78_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq79_HTML.gif ; here https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq80_HTML.gif . Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ12_HTML.gif
(2.9)
and so
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ13_HTML.gif
(2.10)

Case 2 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq81_HTML.gif ).

Define operators
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ14_HTML.gif
(2.11)
From the definition of the linear operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq82_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ15_HTML.gif
(2.12)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq83_HTML.gif , the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq84_HTML.gif has a bounded inverse https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq85_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ16_HTML.gif
(2.13)
and so, for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq86_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ17_HTML.gif
(2.14)
On the other hand, from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq87_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ18_HTML.gif
(2.15)
That is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ19_HTML.gif
(2.16)
Now, for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq88_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq89_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ20_HTML.gif
(2.17)
then we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ21_HTML.gif
(2.18)
or
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ22_HTML.gif
(2.19)
So, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ23_HTML.gif
(2.20)
So, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq90_HTML.gif exists and satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ24_HTML.gif
(2.21)

This proves (1) and (2) of Lemma 2.3. Finally, (3) is easily verified.

By Hale's terminology [14], a solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq91_HTML.gif of (1.3) is that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq92_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq93_HTML.gif and (1.3) is satisfied on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq94_HTML.gif . In general, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq95_HTML.gif does not belong to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq96_HTML.gif But we can see easily from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq97_HTML.gif that a solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq98_HTML.gif of (1.3) must belong to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq99_HTML.gif . Equation (1.3) is transformed into
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ25_HTML.gif
(2.22)

Lemma 2.5 (see [4]).

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq100_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ26_HTML.gif
(2.23)
Now we consider (2.22). Define the conjugate index https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq101_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq102_HTML.gif . Introducing new variables
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ27_HTML.gif
(2.24)
Using the fact that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq103_HTML.gif and by Lemma 2.3, (1.3) can be rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ28_HTML.gif
(2.25)

It is clear that, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq104_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq105_HTML.gif -periodic solution to (2.25), then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq106_HTML.gif must be a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq107_HTML.gif -periodic solution to (1.3). Thus, the problem of finding a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq108_HTML.gif -periodic solution for (1.3) reduces to finding one for (2.25).

Define the linear spaces
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ29_HTML.gif
(2.26)
with norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq109_HTML.gif . Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq110_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq111_HTML.gif are Banach spaces. Define
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ30_HTML.gif
(2.27)
by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ31_HTML.gif
(2.28)
Moreover, define
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ32_HTML.gif
(2.29)
by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ33_HTML.gif
(2.30)
Then, (2.25) can be rewritten as the abstract equation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq112_HTML.gif . From the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq113_HTML.gif , one can easily see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq114_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq115_HTML.gif . So, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq116_HTML.gif is a Fredholm operator with index zero. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq117_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq118_HTML.gif be defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ34_HTML.gif
(2.31)
It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq119_HTML.gif . Moreover, for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq120_HTML.gif , if we write https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq121_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq122_HTML.gif and so https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq123_HTML.gif . This is to say https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq124_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq125_HTML.gif So, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq126_HTML.gif is a Fredholm operator with index zero. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq127_HTML.gif denote the inverse of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq128_HTML.gif , then we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ35_HTML.gif
(2.32)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ36_HTML.gif
(2.33)

From (2.30) and (2.33), it is clear that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq129_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq130_HTML.gif are continuous, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq131_HTML.gif is bounded, and so https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq132_HTML.gif is compact for any open bounded https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq133_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq134_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq135_HTML.gif -compact on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq136_HTML.gif . For the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq137_HTML.gif defined as (2.24), we have the following.

Lemma 2.6.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq138_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq139_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ37_HTML.gif
(2.34)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq140_HTML.gif

Proof.

From https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq141_HTML.gif , there is a point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq142_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq143_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq144_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq145_HTML.gif . From https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq146_HTML.gif , there is a point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq147_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq148_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq149_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq150_HTML.gif Continuing this way, we get from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq151_HTML.gif a point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq152_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq153_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq154_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq155_HTML.gif From https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq156_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq157_HTML.gif , so there is a point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq158_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq159_HTML.gif ; hence, we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq160_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq161_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq162_HTML.gif Continuing this way, we get from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq163_HTML.gif that there is a point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq164_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq165_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq166_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq167_HTML.gif By Lemma 2.2, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ38_HTML.gif
(2.35)
By Lemma 2.5 and Lemma 2.2, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ39_HTML.gif
(2.36)
Combining (2.35) and (2.36), we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ40_HTML.gif
(2.37)
Similarly, we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ41_HTML.gif
(2.38)

This completes the proof of Lemma 2.6.

Remark 2.7.

In particular, if we take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq168_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq169_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ42_HTML.gif
(2.39)
In this case, (2.34) is transformed into
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ43_HTML.gif
(2.40)

3. Main Results

For the sake of convenience, we list the following assumptions which will be used repeatedly in the sequel.

(H1)There exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq171_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ44_HTML.gif
(3.1)
H2There exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq173_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ45_HTML.gif
(3.2)
H3 There exist nonnegative constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq175_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ46_HTML.gif
(3.3)
H4 There exist nonnegative constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq177_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ47_HTML.gif
(3.4)

for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq178_HTML.gif .

Theorem 3.1.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq179_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq180_HTML.gif hold, then (1.3) has at least one nonconstant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq181_HTML.gif -periodic solution.

Proof.

Consider the equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ48_HTML.gif
(3.5)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq182_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq183_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ49_HTML.gif
(3.6)
We first claim that there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq184_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ50_HTML.gif
(3.7)
Integrating the last equation of (3.6) over https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq185_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ51_HTML.gif
(3.8)
By the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq186_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq187_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ52_HTML.gif
(3.9)
From assumption https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq188_HTML.gif , we get (3.7). As a consequence, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ53_HTML.gif
(3.10)
On the other hand, multiplying both sides of the last equation of (3.6) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq189_HTML.gif and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq190_HTML.gif , using assumption https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq191_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ54_HTML.gif
(3.11)
It is easy to see that there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq192_HTML.gif (independent of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq193_HTML.gif ) such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ55_HTML.gif
(3.12)
From https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq194_HTML.gif , there exists a point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq195_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq196_HTML.gif . By Hölder's inequality, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ56_HTML.gif
(3.13)
From https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq197_HTML.gif , there exists a point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq198_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq199_HTML.gif , and we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ57_HTML.gif
(3.14)
Continuing this way for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq200_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ58_HTML.gif
(3.15)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ59_HTML.gif
(3.16)
Meanwhile, from (3.10), we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ60_HTML.gif
(3.17)

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq201_HTML.gif . Then, obviously https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq202_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq203_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq204_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq205_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq206_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq207_HTML.gif , which means that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq208_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq209_HTML.gif . We see that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ61_HTML.gif
(3.18)
So,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ62_HTML.gif
(3.19)

Now take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq210_HTML.gif . By the analysis above, it is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq211_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq212_HTML.gif , and conditions (1) and (2) of Lemma 2.1 are satisfied.

Next we show that condition (3) of Lemma 2.1 is also satisfied. Define an isomorphism https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq213_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ63_HTML.gif
(3.20)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq214_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq215_HTML.gif . Then, for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq216_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ64_HTML.gif
(3.21)
From https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq217_HTML.gif , it is obvious that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq218_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq219_HTML.gif . Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ65_HTML.gif
(3.22)

which means that condition (3) of Lemma 2.1 is also satisfied. By applying Lemma 2.1, we conclude that equation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq220_HTML.gif has a solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq221_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq222_HTML.gif ; that is, (1.3) has a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq223_HTML.gif -periodic solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq224_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq225_HTML.gif .

Finally, observe that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq226_HTML.gif is not constant. For, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq227_HTML.gif (constant), then from (1.3) we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq228_HTML.gif , which contradicts the assumption that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq229_HTML.gif . The proof is complete.

Theorem 3.2.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq230_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq231_HTML.gif hold, then (1.3) has at least one nonconstant   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq232_HTML.gif -periodic solution if one of the following conditions holds:

(1) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq233_HTML.gif ,

(2) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq234_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq235_HTML.gif

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq236_HTML.gif be defined as in Theorem 3.1. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq237_HTML.gif then from the proof of Theorem 3.1 we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ66_HTML.gif
(3.23)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ67_HTML.gif
(3.24)

We claim that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq238_HTML.gif is bounded.

Multiplying both sides of (3.23) by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq239_HTML.gif and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq240_HTML.gif , by using assumption https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq241_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ68_HTML.gif
(3.25)
Applying Hölder's inequality, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ69_HTML.gif
(3.26)
Applying Lemma 2.6 and (3.26), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ70_HTML.gif
(3.27)

Case 1.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq242_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq243_HTML.gif , then it is easy to see that there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq244_HTML.gif (independent of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq245_HTML.gif ) such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ71_HTML.gif
(3.28)

Case 2.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq246_HTML.gif , then it is easy to see that there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq247_HTML.gif (independent of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq248_HTML.gif ) such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ72_HTML.gif
(3.29)
From https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq249_HTML.gif , there exists a point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq250_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq251_HTML.gif . By Hölder's inequality, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ73_HTML.gif
(3.30)

This proves the claim, and the rest of the proof of the theorem is identical to that of Theorem 3.1.

Remark 3.3.

If (1.3) takes the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ74_HTML.gif
(3.31)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq252_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq253_HTML.gif , then the results of Theorems 3.1 and 3.2 still hold.

Remark 3.4.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq254_HTML.gif , then (1.3) is transformed into
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ75_HTML.gif
(3.32)

and the results of Theorems 3.1 and 3.2 still hold.

Next, we study the Lyapunov stability of the periodic solutions of (3.32).

Theorem 3.5.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq255_HTML.gif holds. Then every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq256_HTML.gif -periodic solution of (3.32) is Lyapunov stable.

Proof.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ76_HTML.gif
(3.33)
Then, system (3.32) is transformed into
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ77_HTML.gif
(3.34)
Suppose now that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq257_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq258_HTML.gif -periodic solution of (3.34). Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq259_HTML.gif be any arbitrary solution of (3.34). For any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq260_HTML.gif , write https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq261_HTML.gif . Then, it follows from (3.34) that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ78_HTML.gif
(3.35)
and so
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ79_HTML.gif
(3.36)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq262_HTML.gif . Then,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ80_HTML.gif
(3.37)
Take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq263_HTML.gif , and define a function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq264_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ81_HTML.gif
(3.38)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq265_HTML.gif . It is obvious that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq266_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq267_HTML.gif . From https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq268_HTML.gif and Lemma 2.3, we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ82_HTML.gif
(3.39)

Hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq269_HTML.gif is a Lyapunov function for nonautonomous (3.32) (see [15, page 50]), and so the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq270_HTML.gif -periodic solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq271_HTML.gif of (3.32) is Lyapunov stable.

Finally, we present an example to illustrate our result.

Example 3.6.

Consider the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq272_HTML.gif -order delay differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ83_HTML.gif
(3.40)
Here https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq273_HTML.gif is a constant with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq274_HTML.gif . Comparing with (1.3), we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq275_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ84_HTML.gif
(3.41)
Observe that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq276_HTML.gif has period https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq277_HTML.gif and satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ85_HTML.gif
(3.42)
Pick https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq278_HTML.gif . Then,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ86_HTML.gif
(3.43)
for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq279_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq280_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq281_HTML.gif holds. On the other hand, since
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ87_HTML.gif
(3.44)

assumption https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq282_HTML.gif holds with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq283_HTML.gif .

Case 1.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq284_HTML.gif , then by (1) of Theorem 3.2, (3.40) has at least one nonconstant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq285_HTML.gif -periodic solution.

Case 2.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq286_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_Equ88_HTML.gif
(3.45)

So by (2) of Theorem 3.2, (3.40) has at least one nonconstant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F635767/MediaObjects/13661_2010_Article_50_IEq287_HTML.gif -periodic solution.

Declarations

Acknowledgments

This paper is partially supported by the National Natural Science Foundation of China (10971202), and the Research Grant Council of Hong Kong SAR, China (project no. HKU7016/07P).

Authors’ Affiliations

(1)
Department of Mathematics, Zhengzhou University
(2)
Department of Mathematics, The University of Hong Kong

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© Jingli Ren et al. 2011

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