具有不确定奇性的Liénard方程周期正解的存在性

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

该文讨论了下述具有奇性的Liénard方程x''(t)+f(x)x'−φ(t)x^(δ)(t)+α(t)xμ(t)=0周期正解的存在性,其中f:(0,+∞)→R为连续函数,且允许其在原点处具有奇性,函数α,φ∈L([0,T],R)都是T-周期的,μ∈(0,+∞),δ∈(0,1]为常数.函数α(t),φ(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.

In this paper,we study the existence of positive periodic solutions for a singular Liénard equation x''(t)+f(x(t))x'(t)−φ(t)x^(δ)(t)+α(t)xμ(t)=0,where f:(0,+∞)→R is continuous which may have a singularity at x=0,αandφare T-periodic functions withα,φ∈L([0,T],R),μ∈(0,+∞)andδ∈(0,1]are constants.The signs of weight functionsα(t)andφ(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.