# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - B, 2011, 16 (1) : 409-421. doi: 10.3934/dcdsb.2011.16.409
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##### References:
 [1] Anna Capietto, Walter Dambrosio, Tiantian Ma, Zaihong Wang. Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1835-1856. doi: 10.3934/dcds.2013.33.1835 [2] José F. Caicedo, Alfonso Castro. A semilinear wave equation with smooth data and no resonance having no continuous solution. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 653-658. doi: 10.3934/dcds.2009.24.653 [3] Francisco Odair de Paiva, Humberto Ramos Quoirin. Resonance and nonresonance for p-Laplacian problems with weighted eigenvalues conditions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1219-1227. doi: 10.3934/dcds.2009.25.1219 [4] Dominique Blanchard, Olivier Guibé, Hicham Redwane. Existence and uniqueness of a solution for a class of parabolic equations with two unbounded nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (1) : 197-217. doi: 10.3934/cpaa.2016.15.197 [5] Claudianor O. Alves. Existence of periodic solution for a class of systems involving nonlinear wave equations. Communications on Pure & Applied Analysis, 2005, 4 (3) : 487-498. doi: 10.3934/cpaa.2005.4.487 [6] Jingli Ren, Zhibo Cheng, Stefan Siegmund. Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 385-392. doi: 10.3934/dcdsb.2011.16.385 [7] Kaifa Wang, Aijun Fan. Uniform persistence and periodic solution of chemostat-type model with antibiotic. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 789-795. doi: 10.3934/dcdsb.2004.4.789 [8] Mi-Young Kim. Uniqueness and stability of positive periodic numerical solution of an epidemic model. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 365-375. doi: 10.3934/dcdsb.2007.7.365 [9] Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121 [10] Nikolaos S. Papageorgiou, Patrick Winkert. Double resonance for Robin problems with indefinite and unbounded potential. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 323-344. doi: 10.3934/dcdss.2018018 [11] Yongkun Li, Pan Wang. Almost periodic solution for neutral functional dynamic equations with Stepanov-almost periodic terms on time scales. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 463-473. doi: 10.3934/dcdss.2017022 [12] Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995 [13] Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012 [14] Yingte Sun, Xiaoping Yuan. Quasi-periodic solution of quasi-linear fifth-order KdV equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6241-6285. doi: 10.3934/dcds.2018268 [15] Benjamin B. Kennedy. A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 47-66. doi: 10.3934/dcdss.2020003 [16] Shouchuan Hu, Nikolaos S. Papageorgiou. Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2005-2021. doi: 10.3934/cpaa.2012.11.2005 [17] D. Ruiz, J. R. Ward. Some notes on periodic systems with linear part at resonance. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 337-350. doi: 10.3934/dcds.2004.11.337 [18] Natalia Ptitsyna, Stephen P. Shipman. A lattice model for resonance in open periodic waveguides. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 989-1020. doi: 10.3934/dcdss.2012.5.989 [19] D. Bonheure, C. Fabry, D. Smets. Periodic solutions of forced isochronous oscillators at resonance. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 907-930. doi: 10.3934/dcds.2002.8.907 [20] V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277

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