- Research Article
- Open Access
Existence and Uniqueness of Periodic Solution for Nonlinear Second-Order Ordinary Differential Equations
© Jian Zu. 2011
- Received: 22 May 2010
- Accepted: 6 March 2011
- Published: 15 March 2011
We study periodic solutions for nonlinear second-order ordinary differential problem . 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.
- Ordinary Differential Equation
- Periodic Solution
- Degree Theory
- Periodic Problem
- Important Branch
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 [1–10]. In , 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.
Throughout this paper, we will study the existence of periodic solutions of (1.1) with the following assumptions:
The following is our main result.
Assume that and hold, then (1.1) has a unique periodic solution.
The following results will be used later.
Lemma 2.1 (see ).
and the constant is optimal.
Lemma 2.2 (see ).
then (2.4) has only the trivial -periodic solution .
where is undetermined.
we could get a contradiction.
Without loss of generality, we may assume that ; then there exists a sufficiently small such that . Since is a continuous function, there must exist a with .
which implies (2.24). By a similar argument, we have (2.25). Therefore, , a contradiction, which shows that has at least one zero in , with .
We let . If , then from a similar argument, there is a , such that and so on. So, we obtain that has at least zeros on .
we obtain , which contradicts . Therefore, has at least zeros on .
which implies for . Also . Therefore, for , a contradiction. The proof is complete.
Clearly, is a completely continuous operator in .
There exists , such that every possible periodic solution satisfies ( denote the usual normal in . If not, there exists and the solution with .
which also satisfy the condition . Notice that and are integrable on , so satisfies Lemma 2.3. Hence, we have , which contradicts . Therefore, is bounded.
So, we conclude that has at least one fixed point in , that is, (1.1) has at least one solution.
By Lemma 2.3, we have for .
with . Denote by . So, is the solution of the problem (1.1). The proof is complete.
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.
- Bereanu C, Mawhin J:Existence and multiplicity results for some nonlinear problems with singular -Laplacian. Journal of Differential Equations 2007, 243(2):536-557. 10.1016/j.jde.2007.05.014View ArticleMathSciNetGoogle Scholar
- 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 ArticleMathSciNetGoogle Scholar
- 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 ArticleMathSciNetGoogle Scholar
- Knobloch H-W: On the existence of periodic solutions for second order vector differential equations. Journal of Differential Equations 1971, 9: 67-85.View ArticleMathSciNetGoogle Scholar
- 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.MathSciNetGoogle Scholar
- 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.MathSciNetGoogle Scholar
- 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 ArticleMathSciNetGoogle Scholar
- Rachůnková I, Tvrdý M: Existence results for impulsive second-order periodic problems. Nonlinear Analysis. Theory, Methods & Applications 2004, 59(1-2):133-146.View ArticleGoogle Scholar
- 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 ArticleMathSciNetGoogle Scholar
- 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 ArticleMathSciNetGoogle Scholar
- Li Y: Boundary value problems for nonlinear ordinary differential equations. Northeastern Mathematical Journal 1990, 6(3):297-302.MathSciNetGoogle Scholar
- Mitrinović DS: Analytic Inequalities. Springer, New York, NY, USA; 1970:xii+400.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.