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),x″(t)+f(x(t))x′(t)+g1(x(t))+g2(x(t−τ(t)))=e(t), where f,g1f,g1 and g2g2 are continuous functions on R=(−∞,+∞),τ(t)≥0R=(−∞,+∞),τ(t)≥0 is a bounded continuous function on RR, and e(t)e(t) is a bounded continuous function on R+=[0,+∞)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.