华东师范大学学报(自然科学版) ›› 2009, Vol. 2009 ›› Issue (1): 32-36.

• 应用数学与基础数学 • 上一篇    下一篇

非线性Sturm-Liouville问题的一个正解存在定理

姚庆六   

  1. 南京财经大学 应用数学系, 南京 210003
  • 收稿日期:2008-03-07 修回日期:2008-04-13 出版日期:2009-01-25 发布日期:2009-01-25
  • 通讯作者: 姚庆六

Existence theorem of positive solution to a nonlinear Sturm-Liouville problem (Chinese)

YAO Qing-liu   

  1. Department of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210003 China
  • Received:2008-03-07 Revised:2008-04-13 Online:2009-01-25 Published:2009-01-25
  • Contact: YAO Qing-liu

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摘要： 研究了非线性~Sturm-Liouville~边值问题的正解存在性,~%
其中非线性项~$f(t,u)$~可以在~$t = 0,\,t = 1$~处奇异.~%
通过引入非线性项在有界集合上的高度函数的积分来描述非线性项的增长变化.~%
在极限函数~$\mathop {\lim }\limits_{u \to + 0} f(t,u) / u$,$\mathop
{\lim }\limits_{u \to + \infty } f(t,u) /
u$~存在的情况下利用度理论中的~Krasnosel'skii~不动点定理和实变函数论中的控制收敛定理证明了一个正解存在定理.

关键词: 非线性常微分方程, 边值问题, 正解, 存在性, 非线性常微分方程, 边值问题, 正解, 存在性

Abstract:

The existence of positive solution was studied for the
nonlinear Sturm-Liouville boundary value problem, where the
nonlinear term $f(t,u)$ may be singular at $t = 0,\,t = 1$. By
introducing the integrations of height functions of nonlinear term
on bounded set the growths of nonlinear term were described. By
applying the Krasnoselskii fixed point theorem in degree theory and
the dominated convergence theorems in real variable, an existence
theorem of positive solution was proved when there are limit
functions $\mathop {\lim }\limits_{u \to + 0} f(t,u) / u$ and
$\mathop {\lim }\limits_{u \to + \infty } f(t,u) / u$.

Key words: boundary value problem, positive solution, existence, nonlinear ordinary differential equation, boundary value problem, positive solution, existence

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