Existence and non-existence of periodic solutions of generalized Lineard equations with damping of no lower bound

Abstract

This paper establishes criteria for the existence and non-existence of nonzero periodic solutions of the generalized Lienard equationx +f(x,x)x +g(x)=0. The main goal is to study to what extent the dampingf can be small so as to guarantee the existence of nonzero periodic solutions of such a system. With some standard additional assumptions we prove that if for a small ¦x¦, ∫±∞ ¦f(x,y)¦−1dy=±∞, then the system has at least one nonzero periodic solution, otherwise, the system has no nonzero periodic solution. Many classical and well-known results can be proved as corollaries to ours.