On the structure of the set of bounded solutions on an almost periodic Liénard equation (Q1883103)

The Liénard equation \[ x''+f(x)x'+g(x)=p(t) \] is studied, where \(p:{\mathbb R}\rightarrow {\mathbb R}\) is continuous and \(f,g:(a,b)\rightarrow {\mathbb R}\) are locally Lipschitz. Additionally, \(p\) is an almost-periodic function, and \(g\) is strictly decreasing. A set of initial conditions of the bounded solutions on \((0,\infty )\) is described, and it is shown that these solutions are asymptotically almost-periodic. A model example of this equation is the equation \[ x''+cx'+(1/x^{\alpha})=p(t), \quad c\geq 0 \text{ and }\alpha >0, \] which has a singular nonlinear restoring force.