Boundedness of solutions for a class of Liénard equations with a deviating argument

Consider the Liénard equation with a deviating argument x (t)+ f (x(t))x (t)+ g1(x(t))+ g2(x(t − τ(t))) = e(t), where f, g1 and g2 are continuous functions on R = (−∞,+∞), τ (t) ≥ 0 is a bounded continuous function on R, and e(t) is a bounded continuous function on R = [0,+∞). We obtain some new sufficient conditions for all solutions and their derivatives to be bounded, which substantially extend and improve some important results from the literature. c © 2007 Elsevier Ltd. All rights reserved.