The purpose of this work is to present some new results on the existence of periodic solutions of the second-order differential Liénard equation with asymmetric nonlinearities Here, are positive constants satisfying , and is a continuous and -periodic function. Also, the limits and exist and are finite. By using some previous ideas of related works, the authors define two functions and which involve the quantities and function . Then, they prove the existence of -periodic solutions under some additional restrictions on the zeros of and
On the other hand, new nonresonant conditions are discussed if is unbounded and oscillatory and is sublinear. In this case, phase-plane analysis methods are used in the proofs.