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 Manásevich–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.

中文翻译：

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具有排斥型的弱奇点和强奇点的p -Laplacian Rayleigh方程的正周期解

在本文中，我们研究以下具有奇点的p -Laplacian广义Rayleigh方程的正周期解的存在：$$ \ begin {aligned}（\ phi _p（x'（t）））'+ f（t ，x'（t））+ g（x（t））= e（t），\ end {aligned} $$，其中g在原点具有排斥型的奇异性。本文的新颖之处在于，我们首次证明了弱奇点通过应用Manásevich-Mawhin连续定理可以实现正周期解的新存在准则。对文献中的最新结果进行了概括和显着改进，该结果适用于强奇异性情况和弱奇异性情况，并且给出了该方程正周期解的存在区间。最后，给出了实例和数值解（实例的周期解的相图和时间图）以展示该定理的应用。