Structure of the set of bounded solutions and existence of pseudo almost-periodic solutions of a Liénard equation

Abstract

We study some of the properties of bounded, asymptotically almost-periodic or pseudo almost-periodic solutions of the Liénard equation, $$x'' + f(x)x' + g(x) = p(t) ,$$ where $p : \mathbb R \longrightarrow \mathbb R$ is a continuous, bounded, asymptotically almost-periodic or pseudo almost-periodic function, $f$ and $g : (a,b) \longrightarrow \mathbb R$ are continuous and $g$ is strictly decreasing. Notably, we describe the set of initial conditions of the bounded solutions on $( 0 , + \infty )$ and we state some results of existence of pseudo almost-periodic solutions.