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

Unboundedness of solutions for perturbed asymmetric oscillators

1. 

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

Received  February 2010 Revised  August 2010 Published  April 2011

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.

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.

show all references

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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