Existence of Periodic Solutions of a Class of Planar Systems

Abstract

In this paper, we consider the existence of periodic solutions for the following planar system:

\[ J u'=\D H(u)+ G(u)+h(t)\,, \]

where the function $H(u)\in C^3({\mathbb{R}}^2\backslash \{0\},\mathbb{R})$ is positive for $u\ne 0$ and positively $(q,p)$-quasi-homogeneous of quasi-degree $pq,G:{\mathbb{R}}^2\to {\mathbb{R}}^2$ is local Lipschitz and bounded, $h\in L^{\infty }(0,2\pi )$ is $2\pi$-periodic and $J$ is the standard symplectic matrix.