一类具有奇性Rayleigh方程周期正解的存在性

Existence of periodic solutions for a kind of Rayleigh equation

研究Rayleigh方程x″(t)+f(x′(t))+g(t,x)=0周期正解的存在性问题,其中f:R→R连续,g:R×(0,+∞)→R连续,关于t为T周期,且在x=0处具有奇性,即limx→0+g(x)=+∞.利用Mawhin重合度延拓定理,证明了上述方程至少存在一个T周期正解.

By using Mawhin’s continuation theorem,this paper studies the existence of positive periodic solutions for the Rayleigh equation：x″（t）＋f（x′（t））＋g（t,x）=0,where f：R→R,g：R×（0,＋∞）→Rare continuous,g（t,x）is T-periodic function with t,and with singularity at x=0,i.e.limx→0＋g（x）= ＋ ∞.It is proven that the above equation has at least one positive T-periodic solution.