一类非线性微分方程的非平凡解

ON Nontrivial Solution to Some Nonlinear Ordinary Differential Equations

本文讨论了一类非线性二阶常微分方程 (um+ 1(x) )″+(x- 1) p(x) u′(x) =0 ;x∈ (0 ,1],m>0 ,非平凡解的存在问题 ,用非线性算子方法证明了非平凡解的存在性以及惟一性。从而使W.OKRASINSKI发表于《数学分析与应用》的论文“一类非线性常微分方程的非平凡解”中的部分结果成为本文的 1个推论。

In this paper, an initial value problem for a nonlinear ordinary differential equation of the second order(u m+1(x))″+(x-1)p(x)u′(x)=0 x∈(0,1], m>0 is considered. Some nonlinear operator methods are used to prove the existence and uniqueness of nontrivial solutions of this kind of equations. Moreover, the results improve and generalize the results in 《On Nontrival Solution to Some Nonlinear Ordinary Differential Equations》(W.OKRASINSKI(Poland)1995).