Positive periodic solution for p-Laplacian Rayleigh equation with weak and strong singularities of repulsive type

Abstract

In this paper, we investigate the existence of a positive periodic solution for the following p-Laplacian generalized Rayleigh equation with a singularity: $$\begin{aligned} (\phi _p(x'(t)))'+f(t,x'(t))+g(x(t))=e(t), \end{aligned}$$where g has a singularity of repulsive type at the origin. The novelty of the present article is that for the first time, we show that a weakly singularity enables the achievement of a new existence criterion of positive periodic solutions through a application of the Manasevich–Mawhin continuation theorem. Recent results in the literature are generalized and significantly improved, the result is applicable to the case of a strong singularity as well as the case of a weak singularity, and we give the existence interval of a positive periodic solution of this equation. At last, example and numerical solution (phase portraits and time portraits of periodic solutions of the example) are given to show applications of the theorem.