Weak and strong singularities problems to Liénard equation

Abstract

This paper is devoted to an investigation of the existence of a positive periodic solution for the following singular Liénard equation: x + f (x(t))x (t) + a(t)x = b(t) x α + e(t), where the external force e(t) may change sign, α is a constant and α > 0. The novelty of the present article is that for the first time we show that weak and strong singularities enables the achievement of a new existence criterion of positive periodic solution through an application of the Manásevich-Mawhin continuation theorem. Recent results in the literature are generalized and significantly improved, and we give the existence interval of periodic solution of this equation. At last, two examples and numerical solution (phase portraits and time portraits of periodic solutions of the example) are given to show applications of the theorem.