具有不确定奇性的Liénard方程周期正解的存在性

数学物理学报 ›› 2021, Vol. 41 ›› Issue (3): 686-701.

具有不确定奇性的Liénard方程周期正解的存在性

鲁世平*(),周诗乐(),余星辰()

^{南京信息工程大学数学与统计学院南京 210044}

收稿日期:2020-05-06 出版日期:2021-06-01 发布日期:2021-06-09
通讯作者: 鲁世平 E-mail:lushiping88@sohu.com;zhoushile96@163.com;yuxingchen0@yeah.net
作者简介:周诗乐, E-mail: zhoushile96@163.com|余星辰, E-mail: yuxingchen0@yeah.net
基金资助:
国家留学基金(201908320531);江苏省研究生科研创新项目(SJKY19_0957)

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

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

^{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 E-mail:lushiping88@sohu.com;zhoushile96@163.com;yuxingchen0@yeah.net
Supported by:
the China Scholarship Council Project(201908320531);the Project of Innovation in Scientific Research for Graduate Students of Jiangsu Province(SJKY19_0957)

摘要/Abstract

摘要：

该文讨论了下述具有奇性的Liénard方程 $x''(t)+f(x)x'-\varphi(t)x^\delta(t)+\frac{\alpha(t)}{x^{\mu}}(t)=0$ 周期正解的存在性，其中 $f: (0，+\infty)\rightarrow\mathbb{R}$ 为连续函数，且允许其在原点处具有奇性，函数 $\alpha，\varphi\in L([0，T]，\mathbb{R})$ 都是 $T$ -周期的， $\mu\in(0，+\infty)$ ， $\delta\in(0，1]$ 为常数.函数 $\alpha(t)，\varphi(t)$ 在 $[0，T]$ 上可变号.利用重合度拓展定理证明了上述方程至少存在一个 $T$ -周期正解.

关键词: 周期解, 奇性, 拓展定理, 重合度理论

Abstract:

In this paper, we study the existence of positive periodic solutions for a singular Liénard equation $x''(t)+f(x(t))x'(t)-\varphi(t)x^\delta(t)+\frac{\alpha(t)}{x^{\mu}(t)}=0,$ 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

中图分类号:

O175.2

鲁世平,周诗乐,余星辰. 具有不确定奇性的Liénard方程周期正解的存在性[J]. 数学物理学报, 2021, 41(3): 686-701.

Shiping Lu,Shile Zhou,Xingchen Yu. Periodic Solutions for a Singular Liénard Equation with Sign-Changing Weight Functions[J]. Acta mathematica scientia,Series A, 2021, 41(3): 686-701.

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