电竞博彩-电子竞技博彩

Existence and multiplicity of positive periodic solutions to a class of Liénard equations with repulsive singularities

Jinrong Wang, Ben Wang, Yongwei Miao & Xingchen Yu 

Journal of Fixed Point Theory and Applications volume 24, Article number: 64 (2022) 

Abstract

The purpose of this paper is to study the non-existence, existence and multiplicity of positive T-periodic solutions to the following parameter-dependent Liénard equations with repulsive singularities \begin{aligned} x''+f(x)x'-\frac{a(t)}{x^\mu }+\varphi (t)x^\delta =s, \end{aligned} where f:(0,+\infty )\rightarrow \mathbb {R}, a\in C[0,T] is a nonnegative function with \overline{a}>0, \varphi \in C[0,T] is a sign-changing function with \overline{\varphi }<0, constants \mu >0,~\delta \in (0,1) are fixed, and s\in \mathbb {R} is a parameter. Using a continuation theorem of coincidence degree theory and certain properties of Leray–Schauder degree, we prove the existence of critical point s_*\in \mathbb {R} such that the equation has at least two, at least one or no positive T-periodic solution, according to s<s_*, s=s_* or s>s_*, respectively.