Several kinds of oscillations in forced Liénard equations

Abstract

Near an equilibrium we study the existence of asymptotically a.p. (almost periodic), asymptotically a.a. (almost automorphic), pseudo a.p., pseudo a.a., weighed pseudo a.p. and weighed pseudo a.a. solutions of Liénard differential equations in the form x(t) + f (x(t), p) x(t) + g(x(t), p) = ep(t), where the forcing term possesses a similar nature, and where p is a parameter in a Banach space. We use a perturbation method around an equilibrium. We also study two special cases of the previous family of equations that are x(t) + f (x(t)) x(t) + g(x(t)) = e(t) and x(t) + f (x(t), q) x(t) + g(x(t), q) = e(t). © 2013 Gao and Gao; licensee Springer.