In this paper, we investigate the existence of a positive periodic solution for the following \(\phi \)-Laplacian generalized Liénard equation with a singularity:

where \(\rho \) is a positive constant and m is a constant. Our proof is based on the Manásevich–Mawhin continuation theorem, and the results are applicable to weak and strong singularities of attractive type (or repulsive type, attractive-repulsive type). Besides, these results in the literature are generalized and significantly improved, and we give the existence interval of a positive periodic solution of this equation. As applications, we study the existence of positive periodic solutions for a Rayleigh–Plesset equation. Finally, three examples are given to show applications of these theorems.

在本文中，我们研究以下具有奇异性的\（\ phi \）- Laplacian广义Liénard方程的正周期解的存在：

其中\（\ rho \）是一个正常数，m是一个常数。我们的证明基于Manásevich-Mawhin连续定理，并且该结果适用于吸引型（或排斥型，吸引-排斥型）的弱奇异性和强奇异性。此外，这些结果在文献中得到了推广和显着改善，我们给出了该方程正周期解的存在区间。作为应用，我们研究了Rayleigh-Plesset方程的正周期解的存在。最后，给出了三个例子来说明这些定理的应用。