一类具有奇性的时滞平均曲率方程周期解的存在性问题

Existence of positive periodic solutions for a delay prescribed mean curvature equation with a singularity

摘要:

本文研究了如下具有奇性的Li\'{e}nard型时滞平均曲率方程$$(\frac{u'(t)}{\sqrt{1+(u'(t))^2}})'+f(u(t))u'(t)+g( u(t-\gamma))=e(t)$$的周期解存在性问题. 运用Mawhin重合度扩展定理, 获得了该方程至少存在一个$T$-周期正解的新结果, 最后给出一个例子来验证文章主要结论的有效性. 本文的研究丰富了时滞平均曲率方程的内容.

Abstract:

In this paper, we study the existence of periodic solutions to the following prescribed mean curvature Li\'{e}nard equation with a singularity and a deviating argument $$(\frac{u'(t)}{\sqrt{1+(u'(t))^2}})'+f(u(t))u'(t)+g( u(t-\gamma))=e(t)$$ And by applying Mawhin's continuation theorem, a new result on the existence of positive $T-$periodic solution for this equation is obtained. An example is given to illustrate the effectiveness of our results. Our research enriches the contents of prescribed mean curvature equations.