Existence of asymptotically stable periodic solutions of a Rayleigh type equation

In this work, we investigate the following Rayleigh type equation: x″(t)+f(x′(t))+g(t,x(t))=e(t).x″(t)+f(x′(t))+g(t,x(t))=e(t). Some criteria for guaranteeing the existence of asymptotically stable (in the sense of Lyapunov) periodic solutions of this equation is presented by using Mawhin’s continuation theorem, Floquet theory, Lyapunov stability theory and some analysis techniques. Moreover, an example is provided to illustrate the validity of our results.