Unboundedness of solutions for perturbed asymmetric  oscillators

Lixia Wang; Shiwang Ma

doi:10.3934/dcdsb.2011.16.409

Article Contents

2011, Volume 16, Issue 1: 409-421. Doi: 10.3934/dcdsb.2011.16.409

This issue Previous Article Local and global exponential synchronization of complex delayed dynamical networks with general topology Next Article

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

Received: February 2010

Revised: August 2010

Published: July 2011

Abstract / Introduction Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 34C25, 37B30; Secondary: 37J45.

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