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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 670287, 6 pages
http://dx.doi.org/10.1155/2014/670287
Research Article

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.

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 [117] 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.

References

  1. M. S. Berger and M. Schechter, “On the solvability of semilinear gradient operator equations,” Advances in Mathematics, vol. 25, no. 2, pp. 97–132, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. J. Mawhin, “Semicoercive monotone variational problems,” Bulletin de la Classe des Sciences. Académie Royale de Belgique, vol. 73, no. 3-4, pp. 118–130, 1987. View at MathSciNet
  3. M. Willem, “Oscillations forces de systmes hamiltoniens,” in Public. Smin. Analyse Non Linaire, Université de Besançon, 1981.
  4. Y. M. Long, “Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials,” Nonlinear Analysis: Theory, Methods & Applications, vol. 24, no. 12, pp. 1665–1671, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. P. H. Rabinowitz, “On subharmonic solutions of Hamiltonian systems,” Communications on Pure and Applied Mathematics, vol. 33, no. 5, pp. 609–633, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989. View at MathSciNet
  7. C. Tang, “Periodic solutions of non-autonomous second order systems with γ-quasisubadditive potential,” Journal of Mathematical Analysis and Applications, vol. 189, no. 3, pp. 671–675, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  8. C. Tang, “Periodic solutions of non-autonomous second order systems,” Journal of Mathematical Analysis and Applications, vol. 202, no. 2, pp. 465–469, 1996. View at Publisher · View at Google Scholar · View at Scopus
  9. C.-L. Tang, “Periodic solutions for nonautonomous second order systems with sublinear nonlinearity,” Proceedings of the American Mathematical Society, vol. 126, no. 11, pp. 3263–3270, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. C.-L. Tang and X.-P. Wu, “Periodic solutions for second order systems with not uniformly coercive potential,” Journal of Mathematical Analysis and Applications, vol. 259, no. 2, pp. 386–397, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. X. Wu, “Saddle point characterization and multiplicity of periodic solutions of non-autonomous second-order systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 58, no. 7-8, pp. 899–907, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. X.-P. Wu and C.-L. Tang, “Periodic solutions of a class of non-autonomous second-order systems,” Journal of Mathematical Analysis and Applications, vol. 236, no. 2, pp. 227–235, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. F. Zhao and X. Wu, “Periodic solutions for a class of non-autonomous second order systems,” Journal of Mathematical Analysis and Applications, vol. 296, no. 2, pp. 422–434, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  14. F. Zhao and X. Wu, “Existence and multiplicity of periodic solution for non-autonomous second-order systems with linear nonlinearity,” Nonlinear Analysis: Theory, Methods & Applications, vol. 60, no. 2, pp. 325–335, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Z. Q. Han, “2π-periodic solutions to n-Duffng systems,” in Nonlinear Analysis and Its Aplications, D. J. Guo, Ed., pp. 182–191, Beijng Scientific and Technical, Beijing, China, 1994, (Chinese).
  16. Q. Meng and X. H. Tang, “Solutions of a second-order Hamiltonian system with periodic boundary conditions,” Communications on Pure and Applied Analysis, vol. 9, no. 4, pp. 1053–1067, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. X. H. Tang and Q. Meng, “Solutions of a second-order Hamiltonian system with periodic boundary conditions,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 3722–3733, 2010. View at Publisher · View at Google Scholar · View at Scopus
  18. Z. Q. Han, “2π-periodic solutions to ordinary differential systems at resonance,” Acta Mathematica Sinica. Chinese Series, vol. 43, no. 4, pp. 639–644, 2000 (Chinese). View at Zentralblatt MATH · View at MathSciNet
  19. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Washington, DC, USA, 1986. View at MathSciNet