Boundedness of solutions to a retarded Liénard equation

This paper is concerned with the following retarded Lienard equation x '' (t) + f1(x(t))(x ' (t)) 2 + f2(x(t))x ' (t) + g1(x(t)) + g2(x(t − (t))) = e(t). We prove a new theorem which ensures that all solutions of the above Lienard equation satisfying given initial conditions are bounded. As one will see, our results improve some earlier results even in the case of f1(x) ≡ 0.