About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 218346, 6 pages
http://dx.doi.org/10.1155/2013/218346
Research Article

Unbounded Solutions of Asymmetric Oscillator

College of Science, Hebei University of Engineering, Handan 056038, China

Received 4 July 2013; Revised 26 September 2013; Accepted 1 October 2013

Academic Editor: Shaoyong Lai

Copyright © 2013 Tieguo Ji and Zhenhui Zhang. 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

We obtain sufficient conditions for the existence of unbounded solutions of the following nonlinear differential equation , where are positive constants, and is continuous, bounded, and -periodic in for some .

1. Introduction

In 1960s, Lazer and Leach [1] considered the existence of -periodic solution of the following differential equation: where ,   is continuous and -periodic, and is bounded and continuous. If exist, they showed that (1) has at least one -periodic solution provided: Assume are positive constants satisfying Fabry and Mawhin [2] considered the following asymmetric oscillator: where are continuous and bounded, is continuous and -periodic, , and . Moreover, the limits exist; has a sublinear primitive; that is, They introduced the function where is the solution of the following initial value problem: and gave approximations of solutions of (4) with large initial values. Moreover, they proved that (4) has unbounded solutions if the function has only simple zeros.

Later, Kunze et al. [3] considered the following resonant oscillator: where , is bounded, is -periodic, and . They showed that if where is bounded, , then every solution of (8) is unbounded.

Wang [4] considered the unboundedness of solutions of the equation: where and is continuous and -periodic. By using the well-known Birkhoff Ergodic theorem [5], he showed that if the function is bounded and the limits exist, then is a sufficient condition for the existence of unbounded solutions of (10).

Liu [6] discussed the boundedness of all solutions of the following nonlinear equation: where with ,  . Let ; he defined a -periodic function as where exist, , , and is the solution of with the initial value . Under some smoothness conditions, he showed that all solutions of (11) are bounded if the function has no zero for all .

Recently, Li and Zhang [7] studied the unboundedness of solutions of the following asymmetric oscillator: where are positive constants satisfying nonresonance condition: and are bounded and is -periodic in . By using the similar method used in [4], they obtained some sufficient conditions for the existence of unbounded solutions of (15). For more recent results on the boundedness or unboundedness of solutions of differential equations of second order, we refer to [6, 818] and the references therein.

Assume are positive constants. Let be the solution of the following initial value problem: then it is well-known that is -periodic with

In this paper, we consider the following asymmetric oscillator: where ,  , is continuous, bounded, and -periodic in , and the limits exist. We assume that satisfies the following resonance condition: A solution of (19) is called to have large initial value if . We will give some sufficient conditions for the unboundedness of solutions of (19). Especially, if , we will obtain higher order approximation and corresponding sufficient conditions for the unboundedness of solutions of (19) with large initial values. The results of this paper are new which improve some relative results on the literature in some sense. Throughout this paper, we assume with .

The main results of this paper are the following.

Theorem 1. Introduce a -periodic function where Then all solutions of (19) with large initial values are unbounded provided that the function has only finite number of zeros in and all its zeros are simple.

Corollary 2. Assume is -periodic and define a -periodic function as If has only finite number of zeros in and all its zeros are simple, then all solutions of (19) with large initial values are unbounded.

Theorem 3. Consider (19) with is -periodic. Let be given in Corollary 2 and assume . Define -periodic functions and as Then all solutions of (19) with large initial values are unbounded provided one of the following conditions holds: (i) and has only finite number of zeros in and all of them are simple;(ii) and the function has no zero in .

2. Generalized Polar Coordinate Transformation

Let be the solution of the following initial value problem: and define a function as then it is easy to verify that ; is -periodic and satisfies the identity:

Moreover, one can verify that is the solution of (17).

For ,   , define the generalized polar coordinate transformation by

Under the mapping and by using (27), it is not difficult to verify that (19) can be transformed into the following planar system: where are given by (28).

Lemma 4. Let be the solution of (29) satisfying initial condition , then for ; the Poincaré mapping satisfies where is given by (21).

Proof. For , , by the boundedness of and , we obtain from (29) which implies Substituting (32) into (29) and integrating it over , we obtain Substituting (32) and (33) into (29) and integrating it over , we obtain where , and are given by (21). Moreover, it is easy to verify from (21) and (35) that for all ,

Lemma 5. Let and be -periodic. Assume , where is given in Corollary 2 and is the Poincaré mapping of the solution of (29). Then for , there exists the following higher order approximation: where and are given in Theorem 3.

Proof. Since in (29), we integrate the first equation of (29) from to . For , we obtain Substituting (38) into the second equation of (29) and integrating it over , we get where Substituting (38) and (40) into (29) and integrating over we get where By (41) we get the following approximation Substituting (43) into (29) and integrating it over , we get where Substituting (40) and (42) into (45) and then integrating it over , we obtain where Let then (46) reduces to (37).

Lemma 6. If , then

Proof. It follows from the second equation of (48) that which, together with ,   ,   implies (50).

3. Unboundedness Motions of Planar Mappings

In this section, we adopt the notations used in [9]. Given , let the set be the exterior of the open ball centered at the origin with radius ; that is, then .

Define ; then the points and the group distance in can be described, respectively, by Let be a one to one and continuous mapping. We denote its lift by in the form where is -periodic, is a constant, and are continuous, -periodic and satisfy .

Given a point , let be the unique solution of the initial value problem for the differential equation: This solution is defined in a maximal interval where are certain numbers in the set satisfying The solution is said to be defined in the future if and is said to be defined in the past if .

Let be the ordered sequence of zeros of in such that

Lemma 7. If has a simple zero ; that is, , then there exists orbits of (54) which are to be defined in the future and satisfy or they are defined in the past and satisfy Moreover, if and all zeros of are simple, then every orbit of (54) with large initial value is either to be defined in the future satisfying or is defined in the past satisfying The proof of the above lemma is similar to that of Proposition  3.1 in [9].

Lemma 8. Assume that for ,  ,   has the following expression: where ,   , and are continuous, -periodic in and satisfy ,   uniformly in . Moreover If ,   for  all  , then all orbits of (63) with large initial values are defined in the future and satisfy . If ,   for  all  , then all orbits of (63) with large initial values are defined in the past and satisfy .

Proof. Let ,  for  all  , then and it follows from (63) that for , By induction, we get from (65) and replacing by by we get from (64) Obviously, (66) implies that the orbit is defined in the future. Next we claim that the orbit is unbounded in the future. In fact, (67) implies that is monotone increasing, hence the limit exists. If , taking the limit on both sides of (67) we obtain which is a contradiction. Hence the orbit of (63) is defined in the future and satisfies . Similarly, if ,   for  all  , we can prove that the orbit of (63) is defined in the past and satisfies

4. Proofs of Theorems and an Example

Proof of Theorem 1. Since , by Lemma 4, for , the mapping has the approximate expression of (30) with , where is given by Theorem 1 and . Now Lemma 7 implies that all solutions of (30) with go to infinity either in the future or in the past, which implies that the solutions of (19) with large initial values are unbounded.

Proof of Theorem 3. By Lemma 5, for ,   , the mapping has the form (37) with and and are given by Theorem 3. If , then it follows from Lemma 6 that Now, Lemma 7 implies that all orbits of (37) with go to infinity either in the future or in the past, which means that all solutions of (19) with large initial values are unbounded.
If , then by assumption of Theorem 3, and the function has no sign-changing. Lemma 8 implies that all orbits of (37) with go to infinity either in the future or in the past, which implies that all solutions of (19) with large initial values are unbounded.

Example 1. For (19), we assume ,   ,   ,  , and for some . We write in Fourier series form as where It follows from Corollary 2 that and it is not difficult to show that . Then has only finite number of zeros in and all of them are simple. Corollary 2 implies that all solutions of (19) with large initial values are unbounded.

Acknowledgments

This paper is supported by Chinese NSF (11371117) and Hebei Province NSF (A2012402036).

References

  1. A. C. Lazer and D. E. Leach, “Bounded perturbations of forced harmonic oscillators at resonance,” Annali di Matematica Pura ed Applicata, vol. 82, pp. 49–68, 1969. View at MathSciNet
  2. C. Fabry and J. Mawhin, “Oscillations of a forced asymmetric oscillator at resonance,” Nonlinearity, vol. 13, no. 3, pp. 493–505, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  3. M. Kunze, T. Küpper, and B. Liu, “Boundedness and unboundedness of solutions for reversible oscillators at resonance,” Nonlinearity, vol. 14, no. 5, pp. 1105–1122, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  4. Z. Wang, “Irrational rotation numbers and unboundedness of solutions of the second order differential equations with asymmetric nonlinearities,” Proceedings of the American Mathematical Society, vol. 131, no. 2, pp. 523–531, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  5. P. Walters, An Introduction to Ergodic Theory, vol. 79, Springer, New York, NY, USA, 1982. View at MathSciNet
  6. B. Liu, “Boundedness of solutions for equations with p-Laplacian and an asymmetric nonlinear term,” Journal of Differential Equations, vol. 207, no. 1, pp. 73–92, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  7. X. Li and Z. Zhang, “Unbounded solutions and periodic solutions for second order differential equations with asymmetric nonlinearity,” Proceedings of the American Mathematical Society, vol. 135, no. 9, pp. 2769–2777, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  8. J. M. Alonso and R. Ortega, “Unbounded solutions of semilinear equations at resonance,” Nonlinearity, vol. 9, no. 5, pp. 1099–1111, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  9. J. M. Alonso and R. Ortega, “Roots of unity and unbounded motions of an asymmetric oscillator,” Journal of Differential Equations, vol. 143, no. 1, pp. 201–220, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  10. C. Fabry, “Landesman-Lazer conditions for periodic boundary value problems with asymmetric nonlinearities,” Journal of Differential Equations, vol. 116, no. 2, pp. 405–418, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  11. C. Fabry and A. Fonda, “Nonlinear resonance in asymmetric oscillators,” Journal of Differential Equations, vol. 147, no. 1, pp. 58–78, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  12. C. Fabry and A. Fonda, “Unbounded motions of perturbed isochronous Hamiltonian systems at resonance,” Advanced Nonlinear Studies, vol. 5, no. 3, pp. 351–373, 2005. View at MathSciNet
  13. B. Liu, “Boundedness in nonlinear oscillations at resonance,” Journal of Differential Equations, vol. 153, no. 1, pp. 142–174, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  14. R. Ortega, “Asymmetric oscillators and twist mappings,” Journal of the London Mathematical Society, vol. 53, no. 2, pp. 325–342, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  15. R. Ortega, “Boundedness in a piecewise linear oscillator and a variant of the small twist theorem,” Proceedings of the London Mathematical Society, vol. 79, no. 2, pp. 381–413, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  16. X. Yang, “Boundedness in nonlinear asymmetric oscillations,” Journal of Differential Equations, vol. 183, no. 1, pp. 108–131, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  17. X. Yang, “Unbounded solutions of asymmetric oscillator,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 137, no. 2, pp. 487–494, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  18. X. Yang, “Unboundedness of the large solutions of an asymmetric oscillator,” Journal of Mathematical Analysis and Applications, vol. 303, no. 1, pp. 304–314, 2005. View at Publisher · View at Google Scholar · View at MathSciNet