Acta mathematica scientia,Series A ›› 2021, Vol. 41 ›› Issue (3): 686-701.

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Periodic Solutions for a Singular Liénard Equation with Sign-Changing Weight Functions

Shiping Lu*(),Shile Zhou(),Xingchen Yu()   

  1. School of Math & Statistics, Nanjing University of Information Science and Technology, Nanjing 210044
  • Received:2020-05-06 Online:2021-06-01 Published:2021-06-09
  • Contact: Shiping Lu;;
  • Supported by:
    the China Scholarship Council Project(201908320531);the Project of Innovation in Scientific Research for Graduate Students of Jiangsu Province(SJKY19_0957)


In this paper, we study the existence of positive periodic solutions for a singular Liénard equation where $f: (0, +\infty)\rightarrow \mathbb{R} $ is continuous which may have a singularity at $x=0$, $\alpha$ and $\varphi$ are $T$ -periodic functions with $\alpha, \varphi\in L([0, T], \mathbb{R})$, $\mu\in(0, +\infty)$ and $\delta\in(0, 1]$ are constants. The signs of weight functions $\alpha(t)$ and $\varphi(t)$ are allowed to change on $[0, T]$. We prove that the given equation has at least one positive $T$ -periodic solution. The method of proof relies on a continuation theorem of coincidence degree principle.

Key words: Periodic solution, Singularity, Continuation theorem, Coincidence degree principle

CLC Number: 

  • O175.2