Unbounded solutions and periodic solutions for second order differential equations with asymmetric nonlinearity
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Abstract:
In this paper we will prove the coexistence of unbounded solutions and periodic solutions for the asymmetric oscillator \[ \ddot {x}+f(\dot {x})+a x^{+}-bx^{-}=\varphi (t,x), \] where $a$ and $b$ are positive constants satisfying the nonresonant condition \[ \frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\notin \mathbb {Q} \] and $\varphi (t,x)$ is $2\pi$-periodic in the first variable and bounded.References
- José Miguel Alonso and Rafael Ortega, Roots of unity and unbounded motions of an asymmetric oscillator, J. Differential Equations 143 (1998), no. 1, 201–220. MR 1604908, DOI 10.1006/jdeq.1997.3367
- Walter Dambrosio, A note on the existence of unbounded solutions to a perturbed asymmetric oscillator, Nonlinear Anal. 50 (2002), no. 3, Ser. A: Theory Methods, 333–346. MR 1906465, DOI 10.1016/S0362-546X(01)00765-9
- E. N. Dancer, Boundary-value problems for weakly nonlinear ordinary differential equations, Bull. Austral. Math. Soc. 15 (1976), no. 3, 321–328. MR 430384, DOI 10.1017/S0004972700022747
- E. N. Dancer, On the Dirichlet problem for weakly non-linear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976/77), no. 4, 283–300. MR 499709, DOI 10.1017/S0308210500019648
- Pavel Drábek and Sergio Invernizzi, On the periodic BVP for the forced Duffing equation with jumping nonlinearity, Nonlinear Anal. 10 (1986), no. 7, 643–650. MR 849954, DOI 10.1016/0362-546X(86)90124-0
- Christian Fabry and Alessandro Fonda, Nonlinear resonance in asymmetric oscillators, J. Differential Equations 147 (1998), no. 1, 58–78. MR 1632669, DOI 10.1006/jdeq.1998.3441
- C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillator at resonance, Nonlinearity 13 (2000), no. 3, 493–505. MR 1758987, DOI 10.1088/0951-7715/13/3/302
- Svatopluk Fučík, Solvability of nonlinear equations and boundary value problems, Mathematics and its Applications, vol. 4, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1980. With a foreword by Jean Mawhin. MR 620638
- Thierry Gallouët and Otared Kavian, Resonance for jumping nonlinearities, Comm. Partial Differential Equations 7 (1982), no. 3, 325–342. MR 646710, DOI 10.1080/03605308208820225
- P. Habets, M. Ramos, and L. Sanchez, Jumping nonlinearities for Neumann BVPs with positive forcing, Nonlinear Anal. 20 (1993), no. 5, 533–549. MR 1207529, DOI 10.1016/0362-546X(93)90037-S
- Markus Kunze, Tassilo Küpper, and Bin Liu, Boundedness and unboundedness of solutions for reversible oscillators at resonance, Nonlinearity 14 (2001), no. 5, 1105–1122. MR 1862814, DOI 10.1088/0951-7715/14/5/311
- A. C. Lazer and P. J. McKenna, A semi-Fredholm principle for periodically forced systems with homogeneous nonlinearities, Proc. Amer. Math. Soc. 106 (1989), no. 1, 119–125. MR 942635, DOI 10.1090/S0002-9939-1989-0942635-9
- A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev. 32 (1990), no. 4, 537–578. MR 1084570, DOI 10.1137/1032120
- Xiong Li and Qing Ma, Boundedness of solutions for second order differential equations with asymmetric nonlinearity, J. Math. Anal. Appl. 314 (2006), no. 1, 233–253. MR 2183549, DOI 10.1016/j.jmaa.2005.03.079
- Xiong Li, Boundedness and unboundedness of solutions for reversible oscillators at resonance, Nonlinear Anal. 65 (2006), no. 3, 514–533. MR 2231069, DOI 10.1016/j.na.2005.09.025
- Bin Liu, Boundedness in asymmetric oscillations, J. Math. Anal. Appl. 231 (1999), no. 2, 355–373. MR 1669195, DOI 10.1006/jmaa.1998.6219
- Rafael Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc. (2) 53 (1996), no. 2, 325–342. MR 1373064, DOI 10.1112/jlms/53.2.325
- Rafael Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London Math. Soc. (3) 79 (1999), no. 2, 381–413. MR 1702247, DOI 10.1112/S0024611599012034
- Rafael Ortega, Invariant curves of mappings with averaged small twist, Adv. Nonlinear Stud. 1 (2001), no. 1, 14–39. MR 1850202, DOI 10.1515/ans-2001-0102
- Zaihong Wang, Irrational rotation numbers and unboundedness of solutions of the second order differential equations with asymmetric nonlinearities, Proc. Amer. Math. Soc. 131 (2003), no. 2, 523–531. MR 1933343, DOI 10.1090/S0002-9939-02-06601-7
Additional Information
- Xiong Li
- Affiliation: School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China
- Email: xli@bnu.edu.cn
- Ziheng Zhang
- Affiliation: School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China
- Received by editor(s): May 4, 2006
- Published electronically: February 9, 2007
- Additional Notes: This project was supported by the Program for New Century Excellent Talents of Ministry of Education of China and the National Natural Science Foundation of China (Grant No. 10671020 and 10301006)
- Communicated by: Carmen C. Chicone
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 2769-2777
- MSC (2000): Primary 34C11, 34C25
- DOI: https://doi.org/10.1090/S0002-9939-07-08928-9
- MathSciNet review: 2317951