Positive periodic solution to indefinite singular Lienard equation

In this paper, we investigate the existence of a positive periodic solution for the following Lienard equation with a indefinite singularity x ''+f(x)x '+b(t)x=p(t), is a T-periodic sign-changing function. The novelty of the present article is that for the first time we show that a indefinite singularity enables the achievement of a new existence criterion of positive periodic solutions through a application of a topological degree theorem by Mawhin. Recent results in the literature are generalized and significantly improved, and we give the existence interval of a positive periodic solution of this equation. At last, an example is given to show applications of the theorems.