Existence of periodic solution for perturbed generalized Lienard equations

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