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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 670287, 6 pages
Periodic Solutions for Second-Order Ordinary Differential Equations with Linear Nonlinearity
1School of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
2Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China
Received 17 February 2013; Accepted 29 October 2013; Published 30 January 2014
Academic Editor: Pei Yu
Copyright © 2014 Xiaohong Hu et al. 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.
By using minimax methods in critical point theory, we obtain the existence of periodic solutions for second-order ordinary differential equations with linear nonlinearity.
1. Introduction and Main Results
Consider the second-order ordinary differential systems where , , is a nonnegative integer; and satisfies the following assumption: is measurable in for every and continuously differentiable in for , and there exist and such that for all and , where is the set of all nonnegative real numbers.
In the case of , the existence of periodic solutions for problem (1) is obtained in articles [1–17] with many solvability conditions, such as the coercive type potential condition (see ), the convex type potential condition (see ), the periodic type potential conditions (see ), the even type potential condition (see ), the subquadratic potential condition in Rabinowitz’s sense (see ), the bounded nonlinearity condition (see ), the subadditive condition (see ), the sublinear nonlinearity condition (see [9, 15]), and the linear nonlinearity condition (see [13, 14, 16, 17]).
In this paper, motivated by the results mentioned above, we investigate the existence of periodic solutions of problem (1) in the case of .
Let be a Hilbert space defined by with the norm for .
Let then (). For all , we have , where , , and . It is easy to obtain
Furthermore, we have for some and all (see, [6, Proposition 1.3]).
Our main results are the following theorems.
2. Proof of Main Results
Let for any . It follows from assumption that the functional on is continuously differentiable; moreover we obtain for any . It is well known that the solutions of problem (1) correspond to the critical points of (see ).
For the sake of convenience, we denote
Proof of Theorem 1. Firstly, we assert that the functional satisfies (PS) condition. Let be a sequence in such that is bounded and as . By the proof of  Proposition 4.1, we only need to prove that is bounded. On one hand, we have
So where .
Since (14), so . Then where .
On the other hand, we have
So where .
Then where .
From (22) and (25), we have where .
By (8), (26) we get It follows from (26), (27), and the boundedness of that
The above inequality and (15) imply that is bounded. Hence is bounded by (26).
Secondly, we assert that as in , which implies that ; as in ,for all ; that is, ; then by (8) and (12) we have
for in if and only if or . So, by , (14), and (15), we obtain as in .
Let ; then by (8) and (13), we have
So, by (14), is bounded from below on .
Hence, by Rabinowitz’s Saddle point Theorem (see [19, Theorem 4.6]), we obtain that the problem (1) has at least one solution.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank Professor Zhang S. Q. for his guidance, Editor Pei Yu for his hard work, and the anonymous referees for their comments and suggestions. This paper is supported partially by NSF of China and by Dr. Start-up fund of Chongqing University of Posts and Telecommunications.
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