# American Institute of Mathematical Sciences

April  2013, 33(4): 1563-1581. doi: 10.3934/dcds.2013.33.1563

## Periodic solutions of Liénard equations with resonant isochronous potentials

 1 School of Mathematical Sciences, Capital Normal University, Beijing 100048, China, China

Received  September 2011 Revised  September 2012 Published  October 2012

In this paper, we study the existence and multiplicity of periodic solutions of Liénard equations $$x''+f(x)x'+V'(x)+g(x)=p(t),$$ where $V$ is a $2\pi/n$-isochronous potential. When $F(F(x)=\int_0^xf(s)ds)$ and $g$ are bounded, we provide new sufficient conditions to ensure the existence of periodic solutions of this equations. Moreover, we prove the multiplicity of periodic solutions of the given equations under certain bounded conditions by using topological degree method. When $F, g$ satisfy a certain class of unbounded conditions, we also give sufficient conditions to ensure the existence of $2\pi$-periodic solutions of the given equations.
Citation: Tiantian Ma, Zaihong Wang. Periodic solutions of Liénard equations with resonant isochronous potentials. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1563-1581. doi: 10.3934/dcds.2013.33.1563
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