On the existence of periodic solution of perturbed generalized Liénard equations

Abstract

Under conditions of Levinson-Smith type, we prove the existence of a $τ$-periodic solution for the perturbed generalized Liénard equation u''+ϕ(u,u')u'+ψ(u)=εω(\frac{t}τ,u,u') with periodic forcing term. Also we deduce sufficient condition for existence of a periodic solution for the equation u''+\sum_{k=0}^{2s+1} p_k(u){u'}^k=εω(\frac{t}τ,u,u'). Our method can be applied also to the equation u''+[u^2+(u+u')^2-1]u'+u=εω(\frac{t}τ,u,u'). The results obtained are illustrated with numerical examples.