一类弯曲的弹性梁方程正解的存在性

数学物理学报 ›› 2024, Vol. 44 ›› Issue (6): 1476-1484.

一类弯曲的弹性梁方程正解的存在性

霍会霞(),李永祥^*()

西北师范大学数学与统计学院兰州 730070

收稿日期:2023-12-22 修回日期:2024-05-11 出版日期:2024-12-26 发布日期:2024-11-22
通讯作者: ^*李永祥,Email：liyx@nwnu.edu.cn
作者简介:霍会霞,Email：18219686720@163.com
基金资助:
国家自然科学基金(12061062);国家自然科学基金(12161080)

Existence of Positive Solutions for a Bending Elastic Beam Equation

Huo Huixia(),Li Yongxiang^*()

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070

Received:2023-12-22 Revised:2024-05-11 Online:2024-12-26 Published:2024-11-22
Supported by:
NSFC(12061062);NSFC(12161080)

摘要/Abstract

摘要：

该文讨论四阶常微分方程边值问题$\left\{\begin{array}{ll} u^{(4)}(x)=f(x,u(x),u''(x)),\quad x\in [0,\,1],\\ u'(0)=u'''(0)=u(1)=u''(1)=0 \end{array}\right.$正解的存在性, 其中, $ f:[0,\,1]\times\mathbb{R}^{+}\times\mathbb{R}^{-}\to\mathbb{R}^{+} $ 连续, 该问题是描述一类弹性梁静态形变的数学模型. 在非线性项 $ f(x,\,u,\,v) $ 满足适当的不等式条件下,应用锥上的不动点指数理论获得了正解的存在性结果.

关键词: 四阶边值问题, 正解, 锥, 不动点指数

Abstract:

This paper discusses the existence of the positive solution of the fourth-order boundary value problem$\left\{\begin{array}{ll} u^{(4)}(x)=f(x,u(x),u''(x)),\quad x\in [0,\,1],\\ u'(0)=u'''(0)=u(1)=u''(1)=0,\end{array}\right.$which models the deformations of a statically elastic beam, where $ \,f:[0,\,1]\times\mathbb{R}^{+}\times\mathbb{R}^{-}\to\mathbb{R}^{+} $ is continuous. Under that the nonlinearity $ f(x,\,u,\,v) $ satisfies some inequality conditions, the existence results of positive solutions of this problem are obtained by applying the fixed point index theory in cones.

Key words: Fourth-order boundary value problem, Positive solution, Cone, Fixed point index

中图分类号:

0175.8

霍会霞, 李永祥. 一类弯曲的弹性梁方程正解的存在性[J]. 数学物理学报, 2024, 44(6): 1476-1484.

Huo Huixia, Li Yongxiang. Existence of Positive Solutions for a Bending Elastic Beam Equation[J]. Acta mathematica scientia,Series A, 2024, 44(6): 1476-1484.

图/表 1

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