Open Access

Existence and Uniqueness of Periodic Solution for Nonlinear Second-Order Ordinary Differential Equations

Boundary Value Problems20112011:192156

DOI: 10.1155/2011/192156

Received: 22 May 2010

Accepted: 6 March 2011

Published: 15 March 2011

Abstract

We study periodic solutions for nonlinear second-order ordinary differential problem https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq1_HTML.gif . By constructing upper and lower boundaries and using Leray-Schauder degree theory, we present a result about the existence and uniqueness of a periodic solution for second-order ordinary differential equations with some assumption.

1. Introduction

The study on periodic solutions for ordinary differential equations is a very important branch in the differential equation theory. Many results about the existence of periodic solutions for second-order differential equations have been obtained by combining the classical method of lower and upper solutions and the method of alternative problems (The Lyapunov-Schmidt method) as discussed by many authors [110]. In [11], the author gives a simple method to discuss the existence and uniqueness of nonlinear two-point boundary value problems. In this paper, we will extend this method to the periodic problem.

We consider the second-order ordinary differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ1_HTML.gif
(1.1)

Throughout this paper, we will study the existence of periodic solutions of (1.1) with the following assumptions:

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq2_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq3_HTML.gif are continuous in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq4_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ2_HTML.gif
(1.2)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq5_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ3_HTML.gif
(1.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq6_HTML.gif is some positive integer,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ4_HTML.gif
(1.4)

The following is our main result.

Theorem 1.1.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq7_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq8_HTML.gif hold, then (1.1) has a unique https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq9_HTML.gif periodic solution.

2. Basic Lemmas

The following results will be used later.

Lemma 2.1 (see [12]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq10_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ5_HTML.gif
(2.1)
then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ6_HTML.gif
(2.2)

and the constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq11_HTML.gif is optimal.

Lemma 2.2 (see [12]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq12_HTML.gif with the boundary value conditions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq13_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ7_HTML.gif
(2.3)
Consider the periodic boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ8_HTML.gif
(2.4)

Lemma 2.3.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq14_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq15_HTML.gif -integrable https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq16_HTML.gif periodic function, where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq17_HTML.gif satisfy the condition (H2), with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ9_HTML.gif
(2.5)

then (2.4) has only the trivial https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq18_HTML.gif -periodic solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq19_HTML.gif .

Proof.

If on the contrary, (2.4) has a nonzero https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq20_HTML.gif -periodic solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq21_HTML.gif , then using (2.4), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ10_HTML.gif
(2.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq22_HTML.gif is undetermined.

Firstly, we prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq23_HTML.gif has at least one zero in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq24_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq25_HTML.gif , we may assume https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq26_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq27_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq28_HTML.gif -periodic solution, there exists a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq29_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq30_HTML.gif . Then,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ11_HTML.gif
(2.7)

we could get a contradiction.

Without loss of generality, we may assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq31_HTML.gif ; then there exists a sufficiently small https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq32_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq33_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq34_HTML.gif is a continuous function, there must exist a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq35_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq36_HTML.gif .

Secondly, we prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq37_HTML.gif has at least https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq38_HTML.gif zeros on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq39_HTML.gif . Considering the initial value problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ12_HTML.gif
(2.8)
Obviously,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ13_HTML.gif
(2.9)
is the solution of (2.8) and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ14_HTML.gif
(2.10)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq40_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq41_HTML.gif . Since
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ15_HTML.gif
(2.11)
holds under the assumptions of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq42_HTML.gif , there is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq43_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ16_HTML.gif
(2.12)
Now, let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq44_HTML.gif . By the conditions (H2), (2.11), and (2.12), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ17_HTML.gif
(2.13)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ18_HTML.gif
(2.14)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq45_HTML.gif is decreasing in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq46_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq47_HTML.gif . Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ19_HTML.gif
(2.15)
We also consider the initial value problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ20_HTML.gif
(2.16)
Clearly,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ21_HTML.gif
(2.17)
is the solution of (2.16), where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq48_HTML.gif is the same as the previous one, and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ22_HTML.gif
(2.18)
Hence, there exists a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq49_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq50_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ23_HTML.gif
(2.19)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ24_HTML.gif
(2.20)
From (2.12) and (2.19), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ25_HTML.gif
(2.21)
By https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq51_HTML.gif and (2.21), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ26_HTML.gif
(2.22)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq52_HTML.gif is decreasing on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq53_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq54_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ27_HTML.gif
(2.23)
We now prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq55_HTML.gif has a zero point in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq56_HTML.gif . If on the contrary https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq57_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq58_HTML.gif , then we would have the following inequalities:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ28_HTML.gif
(2.24)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ29_HTML.gif
(2.25)
In fact, from(2.4), (2.8), and (2.15), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ30_HTML.gif
(2.26)
with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq59_HTML.gif . Setting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq60_HTML.gif , and since
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ31_HTML.gif
(2.27)
we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ32_HTML.gif
(2.28)
Notice that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq61_HTML.gif , which implies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ33_HTML.gif
(2.29)
So, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ34_HTML.gif
(2.30)
Integrating from 0 to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq62_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ35_HTML.gif
(2.31)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ36_HTML.gif
(2.32)

which implies (2.24). By a similar argument, we have (2.25). Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq63_HTML.gif , a contradiction, which shows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq64_HTML.gif has at least one zero in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq65_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq66_HTML.gif .

We let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq67_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq68_HTML.gif , then from a similar argument, there is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq69_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq70_HTML.gif and so on. So, we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq71_HTML.gif has at least https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq72_HTML.gif zeros on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq73_HTML.gif .

Thirdly, we prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq74_HTML.gif has at least https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq75_HTML.gif zeros on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq76_HTML.gif . If, on the contrary, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq77_HTML.gif only has https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq78_HTML.gif zeros on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq79_HTML.gif , we write them as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ37_HTML.gif
(2.33)
Obviously,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ38_HTML.gif
(2.34)
Without loss of generality, we may assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq80_HTML.gif . Since
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ39_HTML.gif
(2.35)

we obtain https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq81_HTML.gif , which contradicts https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq82_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq83_HTML.gif has at least https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq84_HTML.gif zeros on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq85_HTML.gif .

Finally, we prove Lemma 2.3. Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq86_HTML.gif has at least https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq87_HTML.gif zeros on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq88_HTML.gif , there are two zeros https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq89_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq90_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq91_HTML.gif . By Lemmas 2.1 and 2.2, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ40_HTML.gif
(2.36)
From https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq92_HTML.gif , it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ41_HTML.gif
(2.37)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ42_HTML.gif
(2.38)

which implies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq93_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq94_HTML.gif . Also https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq95_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq96_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq97_HTML.gif , a contradiction. The proof is complete.

3. Proof of Theorem 1.1

Firstly, we prove the existence of the solution. Consider the homotopy equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ43_HTML.gif
(3.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq98_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq99_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq100_HTML.gif , it holds (1.1). We assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq101_HTML.gif is the fundamental solution matrix of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq102_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq103_HTML.gif . Equation (3.1) can be transformed into the integral equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ44_HTML.gif
(3.2)
From https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq104_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq105_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq106_HTML.gif periodic solution of (3.2), then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ45_HTML.gif
(3.3)
For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq107_HTML.gif is invertible,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ46_HTML.gif
(3.4)
We substitute (3.4) into (3.2),
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ47_HTML.gif
(3.5)
Define an operator
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ48_HTML.gif
(3.6)
such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ49_HTML.gif
(3.7)

Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq108_HTML.gif is a completely continuous operator in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq109_HTML.gif .

There exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq110_HTML.gif , such that every possible periodic solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq111_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq112_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq113_HTML.gif denote the usual normal in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq114_HTML.gif . If not, there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq115_HTML.gif and the solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq116_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq117_HTML.gif .

We can rewrite (3.1) in the following form:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ50_HTML.gif
(3.8)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq118_HTML.gif , obviously https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq119_HTML.gif . It satisfies the following problem:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ51_HTML.gif
(3.9)
in which we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ52_HTML.gif
(3.10)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq120_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq121_HTML.gif are uniformly bounded and equicontinuous, there exists continuous function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq122_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq123_HTML.gif and a subsequence of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq124_HTML.gif (denote it again by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq125_HTML.gif ), such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq126_HTML.gif ,  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq127_HTML.gif uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq128_HTML.gif . Using https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq129_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq130_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq131_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq132_HTML.gif are uniformly bounded. By the Hahn-Banach theorem, there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq133_HTML.gif integrable function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq134_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq135_HTML.gif , and a subsequence of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq136_HTML.gif (denote it again by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq137_HTML.gif ), such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ53_HTML.gif
(3.11)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq138_HTML.gif denotes "weakly converges to" in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq139_HTML.gif . As a consequence, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ54_HTML.gif
(3.12)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ55_HTML.gif
(3.13)
Denote that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq140_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq141_HTML.gif , then we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ56_HTML.gif
(3.14)

which also satisfy the condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq142_HTML.gif . Notice that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq143_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq144_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq145_HTML.gif integrable on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq146_HTML.gif , so https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq147_HTML.gif satisfies Lemma 2.3. Hence, we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq148_HTML.gif , which contradicts https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq149_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq150_HTML.gif is bounded.

Denote
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ57_HTML.gif
(3.15)
Because https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq151_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq152_HTML.gif , by Leray-Schauder degree theory, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ58_HTML.gif
(3.16)

So, we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq153_HTML.gif has at least one fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq154_HTML.gif , that is, (1.1) has at least one solution.

Finally, we prove the uniqueness of the equation when the condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq155_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq156_HTML.gif holds. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq157_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq158_HTML.gif be two https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq159_HTML.gif -periodic solutions of the problem. Denote https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq160_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq161_HTML.gif is a solution of the following problem:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ59_HTML.gif
(3.17)

By Lemma 2.3, we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq162_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq163_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq164_HTML.gif . We have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ60_HTML.gif
(3.18)

with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq165_HTML.gif . Denote https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq166_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq167_HTML.gif . So, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq168_HTML.gif is the solution of the problem (1.1). The proof is complete.

4. An Example

Consider the system
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ61_HTML.gif
(4.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq169_HTML.gif is a continuous function. Obviously,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ62_HTML.gif
(4.2)

satisfy Theorem 1.1, then there is a unique https://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq170_HTML.gif -periodic solution in this system.

Declarations

Acknowledgments

The author expresses sincere thanks to Professor Yong Li for useful discussion. He would like to thank the reviewers for helpful comments on an earlier draft of this paper.

Authors’ Affiliations

(1)
College of Mathematics, Jilin University

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© Jian Zu. 2011

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