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 http://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
http://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:

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq2_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq3_HTML.gif are continuous in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq4_HTML.gif , and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ2_HTML.gif
(1.2)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq5_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ3_HTML.gif
(1.3)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq6_HTML.gif is some positive integer,
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq7_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq10_HTML.gif with
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ5_HTML.gif
(2.1)
then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ6_HTML.gif
(2.2)

and the constant http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq12_HTML.gif with the boundary value conditions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq13_HTML.gif , then
http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq14_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq15_HTML.gif -integrable http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq16_HTML.gif periodic function, where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq17_HTML.gif satisfy the condition (H2), with
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq18_HTML.gif -periodic solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq20_HTML.gif -periodic solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq21_HTML.gif , then using (2.4), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ10_HTML.gif
(2.6)

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

Firstly, we prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq23_HTML.gif has at least one zero in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq24_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq25_HTML.gif , we may assume http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq26_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq27_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq28_HTML.gif -periodic solution, there exists a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq29_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq30_HTML.gif . Then,
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq31_HTML.gif ; then there exists a sufficiently small http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq32_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq33_HTML.gif . Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq35_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq36_HTML.gif .

Secondly, we prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq37_HTML.gif has at least http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq38_HTML.gif zeros on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq39_HTML.gif . Considering the initial value problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ12_HTML.gif
(2.8)
Obviously,
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ14_HTML.gif
(2.10)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq40_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq41_HTML.gif . Since
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq42_HTML.gif , there is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq43_HTML.gif , such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ16_HTML.gif
(2.12)
Now, let http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ17_HTML.gif
(2.13)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ18_HTML.gif
(2.14)
Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq45_HTML.gif is decreasing in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq46_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq47_HTML.gif . Therefore,
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ20_HTML.gif
(2.16)
Clearly,
http://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 http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ22_HTML.gif
(2.18)
Hence, there exists a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq49_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq50_HTML.gif , such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ23_HTML.gif
(2.19)
Then,
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ25_HTML.gif
(2.21)
By http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq51_HTML.gif and (2.21), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ26_HTML.gif
(2.22)
Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq52_HTML.gif is decreasing on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq53_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq54_HTML.gif , and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ27_HTML.gif
(2.23)
We now prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq55_HTML.gif has a zero point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq56_HTML.gif . If on the contrary http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq57_HTML.gif for http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ28_HTML.gif
(2.24)
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ30_HTML.gif
(2.26)
with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq59_HTML.gif . Setting http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq60_HTML.gif , and since
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ31_HTML.gif
(2.27)
we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ32_HTML.gif
(2.28)
Notice that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq61_HTML.gif , which implies
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ33_HTML.gif
(2.29)
So, we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ34_HTML.gif
(2.30)
Integrating from 0 to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq62_HTML.gif , we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ35_HTML.gif
(2.31)
Therefore,
http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq63_HTML.gif , a contradiction, which shows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq64_HTML.gif has at least one zero in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq65_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq66_HTML.gif .

We let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq67_HTML.gif . If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq69_HTML.gif , such that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq71_HTML.gif has at least http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq72_HTML.gif zeros on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq73_HTML.gif .

Thirdly, we prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq74_HTML.gif has at least http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq75_HTML.gif zeros on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq77_HTML.gif only has http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq78_HTML.gif zeros on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq79_HTML.gif , we write them as
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ37_HTML.gif
(2.33)
Obviously,
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq80_HTML.gif . Since
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ39_HTML.gif
(2.35)

we obtain http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq81_HTML.gif , which contradicts http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq82_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq83_HTML.gif has at least http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq84_HTML.gif zeros on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq86_HTML.gif has at least http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq87_HTML.gif zeros on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq88_HTML.gif , there are two zeros http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq89_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq90_HTML.gif with http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ40_HTML.gif
(2.36)
From http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq92_HTML.gif , it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ41_HTML.gif
(2.37)
Hence,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ42_HTML.gif
(2.38)

which implies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq93_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq94_HTML.gif . Also http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq95_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq96_HTML.gif for http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ43_HTML.gif
(3.1)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq98_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq99_HTML.gif . When http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq101_HTML.gif is the fundamental solution matrix of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq102_HTML.gif with http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ44_HTML.gif
(3.2)
From http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq104_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq105_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq106_HTML.gif periodic solution of (3.2), then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ45_HTML.gif
(3.3)
For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq107_HTML.gif is invertible,
http://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),
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ47_HTML.gif
(3.5)
Define an operator
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ48_HTML.gif
(3.6)
such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ49_HTML.gif
(3.7)

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

There exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq110_HTML.gif , such that every possible periodic solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq111_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq112_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq113_HTML.gif denote the usual normal in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq114_HTML.gif . If not, there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq115_HTML.gif and the solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq116_HTML.gif with http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ50_HTML.gif
(3.8)
Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq118_HTML.gif , obviously http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq119_HTML.gif . It satisfies the following problem:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ51_HTML.gif
(3.9)
in which we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ52_HTML.gif
(3.10)
Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq120_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq122_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq123_HTML.gif and a subsequence of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq124_HTML.gif (denote it again by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq125_HTML.gif ), such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq126_HTML.gif ,  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq127_HTML.gif uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq128_HTML.gif . Using http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq129_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq130_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq131_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq133_HTML.gif integrable function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq134_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq135_HTML.gif , and a subsequence of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq136_HTML.gif (denote it again by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq137_HTML.gif ), such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ53_HTML.gif
(3.11)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq138_HTML.gif denotes "weakly converges to" in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq139_HTML.gif . As a consequence, we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ54_HTML.gif
(3.12)
that is,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ55_HTML.gif
(3.13)
Denote that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq140_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq141_HTML.gif , then we get
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq142_HTML.gif . Notice that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq143_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq144_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq145_HTML.gif integrable on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq146_HTML.gif , so http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq148_HTML.gif , which contradicts http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq149_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq150_HTML.gif is bounded.

Denote
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ57_HTML.gif
(3.15)
Because http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq151_HTML.gif for http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ58_HTML.gif
(3.16)

So, we conclude that http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq155_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq156_HTML.gif holds. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq157_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq158_HTML.gif be two http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq159_HTML.gif -periodic solutions of the problem. Denote http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq160_HTML.gif , then http://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:
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq162_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq163_HTML.gif .

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

with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq165_HTML.gif . Denote http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq166_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq167_HTML.gif . So, http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_Equ61_HTML.gif
(4.1)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq169_HTML.gif is a continuous function. Obviously,
http://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 http://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

References

  1. Bereanu C, Mawhin J:Existence and multiplicity results for some nonlinear problems with singular http://static-content.springer.com/image/art%3A10.1155%2F2011%2F192156/MediaObjects/13661_2010_Article_28_IEq171_HTML.gif -Laplacian. Journal of Differential Equations 2007, 243(2):536-557. 10.1016/j.jde.2007.05.014View ArticleMathSciNetMATH
  2. Ehme J, Eloe PW, Henderson J: Upper and lower solution methods for fully nonlinear boundary value problems. Journal of Differential Equations 2002, 180(1):51-64. 10.1006/jdeq.2001.4056View ArticleMathSciNetMATH
  3. Kannan R, Lakshmikantham V: Existence of periodic solutions of nonlinear boundary value problems and the method of upper and lower solutions. Applicable Analysis 1983, 17(2):103-113.View ArticleMathSciNet
  4. Knobloch H-W: On the existence of periodic solutions for second order vector differential equations. Journal of Differential Equations 1971, 9: 67-85.View ArticleMathSciNetMATH
  5. Knobloch HW, Schmitt K: Non-linear boundary value problems for systems of differential equations. Proceedings of the Royal Society of Edinburgh. Section A 1977, 78(1-2):139-159.MathSciNetMATH
  6. Liu Y, Ge W: Positive periodic solutions of nonlinear Duffing equations with delay and variable coefficients. Tamsui Oxford Journal of Mathematical Sciences 2004, 20(2):235-255.MathSciNetMATH
  7. Ortega R, Tarallo M: Almost periodic upper and lower solutions. Journal of Differential Equations 2003, 193(2):343-358. 10.1016/S0022-0396(03)00130-XView ArticleMathSciNetMATH
  8. Rachůnková I, Tvrdý M: Existence results for impulsive second-order periodic problems. Nonlinear Analysis. Theory, Methods & Applications 2004, 59(1-2):133-146.View ArticleMATH
  9. Schmitt K: Periodic solutions of linear second order differential equations with deviating argument. Proceedings of the American Mathematical Society 1970, 26: 282-285. 10.1090/S0002-9939-1970-0265722-5View ArticleMathSciNetMATH
  10. Sędziwy S: Nonlinear periodic boundary value problem for a second order ordinary differential equation. Nonlinear Analysis. Theory, Methods & Applications 1998, 32(7):881-890. 10.1016/S0362-546X(97)00533-6View ArticleMathSciNetMATH
  11. Li Y: Boundary value problems for nonlinear ordinary differential equations. Northeastern Mathematical Journal 1990, 6(3):297-302.MathSciNetMATH
  12. Mitrinović DS: Analytic Inequalities. Springer, New York, NY, USA; 1970:xii+400.View ArticleMATH

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© Jian Zu. 2011

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