华东师范大学学报（自然科学版） ›› 2020, Vol. 2020 ›› Issue (6): 38-45.doi: 10.3969/j.issn.1000-5641.201911026

• 数学 • 上一篇    下一篇

带有完全非线性项的四阶边值问题的多正解性

姚燕燕, 李杰梅   

  1. 兰州交通大学 数理学院, 兰州 730070
  • 收稿日期:2019-06-03 发布日期:2020-12-01
  • 通讯作者: 李杰梅, 女, 副教授, 主要研究方向为分歧理论及常微分方程边值问题. E-mail: lijiemei81@126.com E-mail:lijiemei81@126.com
  • 基金资助:
    国家自然科学基金(11801243, 61863022); 甘肃省高等学校创新能力提升项目(2019B-054); 兰州交通大学青年科学基金(2017012)

Existence and multiplicity of positive solutions for fourth-order boundary value problems with a fully nonlinear term

YAO Yanyan, LI Jiemei   

  1. School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China
  • Received:2019-06-03 Published:2020-12-01

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摘要： 本文讨论四阶两点边值问题 $\left\{ {\begin{array}{*{20}{l}} {{u^{(4)}}(t) = f(t,u(t),u'(t),u''(t),u'''(t)), t \in (0,1), }\\ {u(0) = u'(0) = u''(1) = u'''(1){\rm{ = 0}}. } \end{array}} \right.$这里非线性项$f$中含有项$u'$, $u''$和$u'''$, 因而该问题为带有完全非线性项的四阶边值问题. 运用Leggett-Williams型的两个不动点定理, 在$f$满足一定条件的情况下, 获得了该问题至少存在两个或者三个正解的结果. 最后举例验证了所获定理的有效性.

关键词: 完全非线性项, 多正解, Leggett-Williams不动点定理

Abstract: In this paper, we discuss the fourth-order two-point boundary value problem $\left\{ {\begin{array}{*{20}{l}} {{u^{(4)}}(t) = f(t,u(t),u'(t),u''(t),u'''(t)),t \in (0,1), }\\ {u(0) = u'(0) = u''(1) = u'''(1){\rm{ = 0}}. } \end{array}} \right.$ Here, the nonlinear term $f$ contains $u'$, $u''$ and $u'''$; therefore, the problem is a fourth-order boundary value problem with a fully nonlinear term. By using the two fixed point theorems of Leggett-Williams type, the existence of at least two or at least three positive solutions are obtained under the term $f$ that satisfies certain conditions. Finally, two examples are given to verify the theorems.

Key words: fully nonlinear term, multiple positive solutions, Leggett-Williams fixed point theorem

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