华东师范大学学报（自然科学版） >

2020 , Vol. 2020 >Issue 6: 38 - 45

DOI: https://doi.org/10.3969/j.issn.1000-5641.201911026

数学

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
展开
  • 兰州交通大学 数理学院, 兰州 730070

收稿日期: 2019-06-03

  网络出版日期: 2020-12-01

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)

收起

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

  • YAO Yanyan ,
  • LI Jiemei
Expand
  • School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China

Received date: 2019-06-03

  Online published: 2020-12-01

Fold

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

本文引用格式

姚燕燕 , 李杰梅 . 带有完全非线性项的四阶边值问题的多正解性[J]. 华东师范大学学报（自然科学版）, 2020 , 2020(6) : 38 -45 . DOI: 10.3969/j.issn.1000-5641.201911026

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

参考文献

[1] BAI Z B, WANG H Y. On positive solutions of some nonlinear fourth-order beam equations [J]. J Math Anal Appl, 2002, 270(2): 357-368
[2] 席进华. 四阶两点常微分方程边值问题解的存在性 [J]. 山东大学学报(理学版), 2009, 44(1): 67-73
[3] CABADA A, SAAVEDRA L. Existence of solutions for nth-order nonlinear differential boundary value problems by means of fixed point theorems [J]. Nonlinear Anal: Real World Appl, 2018, 42: 180-206
[4] LI Y X. Existence of positive solutions for the cantilever beam equations with fully nonlinear terms [J]. Nonlinear Anal: Real World Appl, 2016, 27: 221-237
[5] DANG Q A, NGO T K Q. Existence results and iterative method for solving the cantilever beam equation with fully nonlinear term [J]. Nonlinear Anal: Real World Appl, 2017, 36: 56-68
[6] FENG H Y, JI D H, GE W G. Existence and uniqueness of solutions for a fourth-order boundary value problem [J]. Nonlinear Anal: Real World Appl, 2009, 70(10): 3561-3566
[7] 王云杰, 朱江. 一类四阶边值问题的正解的存在性与多解性 [J]. 中山大学学报(自然科学版), 2012, 51(1): 25-29
[8] BAI Z B. The upper and lower solution method for some fourth-order boundary value problems [J]. Nonlinear Anal: Theory Methods and App, 2007, 67(15): 1704-1709
[9] 卢整智, 韩晓玲. 一类四阶两点边值问题多个正解的存在性 [J]. 西南大学学报(自然科学版), 2013, 35(4): 79-83
[10] 马如云. 一类四阶两点边值问题的可解性 [J]. 应用数学, 1997, 10(2): 60-62
[11] 达举霞, 韩晓玲. 奇异四阶三点边值问题正解的存在性 [J]. 四川大学学报(自然科学版), 2017, 54(3): 441-446
[12] 崔玉军, 赵聪. 四阶微分方程奇异边值问题解的唯一性 [J]. 山东大学学报(理学版), 2017, 52(2): 73-76
Options
文章导航

/

〈 〉