Journal of East China Normal University(Natural Science) ›› 2020, Vol. 2020 ›› Issue (6): 38-45.doi: 10.3969/j.issn.1000-5641.201911026

• Mathematics • Previous Articles     Next Articles

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

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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