Boundedness of solutions for a class of retarded Liénard equation ✩

Consider the retarded Liénard equation ¨x +f1(x) ˙x + f2(x) ˙x2 + g x(t − h) = e(t ), where h is a nonnegative constant, f1, f2, and g are continuous functions on R = (−∞,+∞), and e(t ) is a continuous function on R+ = [0,+∞). We obtain some new sufficient conditions, as well as some new necessary and sufficient conditions for all solutions and their derivatives to be bounded, which substantially extend and improve some important results in the literature.  2003 Elsevier Inc. All rights reserved