Unboundedness of solutions for perturbed asymmetric oscillators

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July 2011, 16(1): 409-421. doi: 10.3934/dcdsb.2011.16.409

Unboundedness of solutions for perturbed asymmetric oscillators

Lixia Wang ^1, and Shiwang Ma ^1,

1.	School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

Received February 2010 Revised August 2010 Published April 2011

Abstract
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In this paper, we consider the existence of unbounded solutions and periodic solutions for the perturbed asymmetric oscillator with damping

$x'' + f(x )x' + ax^+ - bx^-$ $+ g(x)=p(t), $

where $x^+ =\max\{x,0\}, x^-$ $=\max\{-x,0\}$, $a$ and $b$ are two positive constants, $f(x)$ is a continuous function and $ p(t)$ is a $2\pi $-periodic continuous function, $g(x)$ is locally Lipschitz continuous and bounded. We discuss the existence of periodic solutions and unbounded solutions under two classes of conditions: the resonance case $\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}\in Q$ and the nonresonance case $\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}} \notin Q$. Unlike many existing results in the literature where the function $g(x)$ is required to have asymptotic limits at infinity, our main results here allow $g(x)$ be oscillatory without asymptotic limits.

Keywords: resonance, periodic solution, Unbounded solution, nonresonance..

Mathematics Subject Classification: Primary: 34C25, 37B30; Secondary: 37J4.

Citation: Lixia Wang, Shiwang Ma. Unboundedness of solutions for perturbed asymmetric oscillators. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 409-421. doi: 10.3934/dcdsb.2011.16.409

References:

[1]	J. M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance, Nonlinearity, 9 (1996), 1099-1111. doi: 10.1088/0951-7715/9/5/003.
[2]	J. M. Alonso and R. Ortega, Roots of unity and unbounded motions of an asymmetric oscillator, J. Differential Equations, 143 (1998), 201-220. doi: 10.1006/jdeq.1997.3367.
[3]	W. Dambrosio, A note on the existence of unbounded solutions to a perturbed asymmetric oscillator, Nonlinear Anal., 50 (2002), 333-346. doi: 10.1016/S0362-546X(01)00765-9.
[4]	E. N. Dancer, Boundary-value problems for weakly nonlinear ordinary differential equations, Bull. Austral. Math. Soc., 15 (1976), 321-328. doi: 10.1017/S0004972700022747.
[5]	E. N. Dancer, On the Dirichlet problem for weakly nonlinear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh Sect.A, 76 (1976), 283-300.
[6]	S. Fučik, "Sovability of Nonlinear Equations and Boundary Value Problems," D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1980.
[7]	M. Kunze, T. Küpper and B. Liu, Boundedness and unboundedness solutions of reversible oscillators at resonance, Nonlinearity, 14 (2001), 1105-1122. doi: 10.1088/0951-7715/14/5/311.
[8]	B. Liu, Boundedness in asymmetric oscillations, J. Math. Anal. Appl., 231 (1999), 355-373. doi: 10.1006/jmaa.1998.6219.
[9]	X. Li and Z. H. Zhang, Unbounded solutions and periodic solutions for second order differential equations with asymmetric nonlinearity, Proc. Amer. Math. Soc., 135 (2007), 2769-2777. doi: 10.1090/S0002-9939-07-08928-9.
[10]	N. J. Lloyd, "Degree Theory," Cambridge University Press, Cambridge-New York-Melbourne, 1978.
[11]	S. W. Ma and J. H. Wu, A small twist theorem and boundedness of solutions for semilinear Duffing equations at resonance, Nonlinear Anal., 67 (2007), 200-237. doi: 10.1016/j.na.2006.04.023.
[12]	L. X. Wang and S. W. Ma, Boundedness and unboundedness of solutions for asymmetric oscillators at resonance, Preprint.
[13]	Z. H. Wang, Coexistence of unbounded solutions and periodic solutions of Liénard equations with asymmetric nonlinearities at resonance, Sci. China Ser. A, 50 (2007), 1205-1216. doi: 10.1007/s11425-007-0070-z.
[14]	Z. H. Wang, Irrational rotation numbers and unboundedness of solutions of the second order differential equations with asymmetric nonlinearities, Proc. Amer. Math. Soc., 131 (2003), 523-531. doi: 10.1090/S0002-9939-02-06601-7.

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References:

[1]	J. M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance, Nonlinearity, 9 (1996), 1099-1111. doi: 10.1088/0951-7715/9/5/003.
[2]	J. M. Alonso and R. Ortega, Roots of unity and unbounded motions of an asymmetric oscillator, J. Differential Equations, 143 (1998), 201-220. doi: 10.1006/jdeq.1997.3367.
[3]	W. Dambrosio, A note on the existence of unbounded solutions to a perturbed asymmetric oscillator, Nonlinear Anal., 50 (2002), 333-346. doi: 10.1016/S0362-546X(01)00765-9.
[4]	E. N. Dancer, Boundary-value problems for weakly nonlinear ordinary differential equations, Bull. Austral. Math. Soc., 15 (1976), 321-328. doi: 10.1017/S0004972700022747.
[5]	E. N. Dancer, On the Dirichlet problem for weakly nonlinear elliptic partial differential equations, Proc. Roy. Soc. Edinburgh Sect.A, 76 (1976), 283-300.
[6]	S. Fučik, "Sovability of Nonlinear Equations and Boundary Value Problems," D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1980.
[7]	M. Kunze, T. Küpper and B. Liu, Boundedness and unboundedness solutions of reversible oscillators at resonance, Nonlinearity, 14 (2001), 1105-1122. doi: 10.1088/0951-7715/14/5/311.
[8]	B. Liu, Boundedness in asymmetric oscillations, J. Math. Anal. Appl., 231 (1999), 355-373. doi: 10.1006/jmaa.1998.6219.
[9]	X. Li and Z. H. Zhang, Unbounded solutions and periodic solutions for second order differential equations with asymmetric nonlinearity, Proc. Amer. Math. Soc., 135 (2007), 2769-2777. doi: 10.1090/S0002-9939-07-08928-9.
[10]	N. J. Lloyd, "Degree Theory," Cambridge University Press, Cambridge-New York-Melbourne, 1978.
[11]	S. W. Ma and J. H. Wu, A small twist theorem and boundedness of solutions for semilinear Duffing equations at resonance, Nonlinear Anal., 67 (2007), 200-237. doi: 10.1016/j.na.2006.04.023.
[12]	L. X. Wang and S. W. Ma, Boundedness and unboundedness of solutions for asymmetric oscillators at resonance, Preprint.
[13]	Z. H. Wang, Coexistence of unbounded solutions and periodic solutions of Liénard equations with asymmetric nonlinearities at resonance, Sci. China Ser. A, 50 (2007), 1205-1216. doi: 10.1007/s11425-007-0070-z.
[14]	Z. H. Wang, Irrational rotation numbers and unboundedness of solutions of the second order differential equations with asymmetric nonlinearities, Proc. Amer. Math. Soc., 131 (2003), 523-531. doi: 10.1090/S0002-9939-02-06601-7.

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