Abstract and Applied Analysis

Volume 2014 (2014), Article ID 670287, 6 pages

http://dx.doi.org/10.1155/2014/670287

## Periodic Solutions for Second-Order Ordinary Differential Equations with Linear Nonlinearity

^{1}School of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China^{2}Department 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.

#### Abstract

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 [1]), the convex type potential condition (see [2]), the periodic type potential conditions (see [3]), the even type potential condition (see [4]), the subquadratic potential condition in Rabinowitz’s sense (see [5]), the bounded nonlinearity condition (see [6]), the subadditive condition (see [7]), the sublinear nonlinearity condition (see [9, 15]), and the linear nonlinearity condition (see [13, 14, 16, 17]).

In the case of , Mawhin and Willem [6] prove that problem (1) has at least one solution under the bounded nonlinearity condition; that is, for some , each , and when or

Under the sublinear nonlinearity condition, that is, there exist and , such that for and . , Han [18] proves that problem (1) has at least one solution when or

Recently, when , Zhao and Wu [13, 14] and Meng and Tang [16, 17] also prove the existence of solutions for problem (1) under linear nonlinearity condition; that is, there exist such that

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 ([6]). 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.

Theorem 1. *Suppose that (A) and (8) hold and**
where is a parameter and satisfies ;**
Then problem (1) has at least one solution.*

Theorem 2. *Suppose that (A), (8) and (i) hold and**
Then problem (1) has at least one solution.*

*Remark 3. *(i) It is worth noting that, in the case of , one solution was obtained by Tang [9] and Han [15] under the sublinear nonlinearity condition.

(ii) It is also worth noting that the sublinear nonlinearity condition in [15, 18] is different from that of [9].

#### 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 [6]).

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 [6] 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.

*Proof of Theorem 2. *The proof of Theorem 2 is similar to the proof of Theorem 1, so we omit it here.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

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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