On certain classes of mild solutions of the scalar Liénard equation revisited

Abstract

In this work we revisit the existence, uniqueness and expo- nential decay of some classes of mild solutions which are almost periodic (AP-), asymptotically almost periodic (AAP-) and pseudo almost peri- odic (PAP-) of the scalar Lin ́eard equation by employing the notion of Green function and Massera-type principle. First, by changing variable we convert this equation to a system of first order differential equations. Then, we transform the problem into a framework of an abstract parabolic evolution equation which associates with an evolution family equipped an exponential dichtonomy and the corresponding Green function is exponen- tially almost periodic.