Positive solution for a class of nonlinear fourth-order boundary value problem

Yanhong Zhang; Li Chen; Yanhong Zhang; Li Chen

doi:10.3934/math.2023049

AIMS Mathematics

2023, Volume 8, Issue 1: 1014-1021. doi: 10.3934/math.2023049

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Research article

Positive solution for a class of nonlinear fourth-order boundary value problem

Yanhong Zhang ,
Li Chen ^,

School of Information Engineering, Lanzou City University, Lanzhou 730070, China

Received: 21 July 2022 Revised: 08 September 2022 Accepted: 13 September 2022 Published: 14 October 2022
MSC : 34A08, 34B15, 35J05

In this paper, we are concerned with the existence of positive solutions for boundary value problems of nonlinear fourth-order differential equations

$ \begin{eqnarray*} &&u^{(4)}+c(x)u = \lambda a(x)f(u), \; \; x\in (0,1),\\ &&u(0) = u(1) = u''(0) = u''(1) = 0, \end{eqnarray*} $

where $ a(x) $ may change signs. The proof of main results is based on Leray-Schauder's fixed point theorem and the properties of Green's function of the fourth-order differential operator $ L_cu = u^{(4)}+c(x)u $.
- fourth-order differential operator,
- positive solution,
- boundary value problem,
- Leray-Schauder's fixed point theorem
Citation: Yanhong Zhang, Li Chen. Positive solution for a class of nonlinear fourth-order boundary value problem[J]. AIMS Mathematics, 2023, 8(1): 1014-1021. doi: 10.3934/math.2023049

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Abstract

In this paper, we are concerned with the existence of positive solutions for boundary value problems of nonlinear fourth-order differential equations

$ \begin{eqnarray*} &&u^{(4)}+c(x)u = \lambda a(x)f(u), \; \; x\in (0,1),\\ &&u(0) = u(1) = u''(0) = u''(1) = 0, \end{eqnarray*} $

where $ a(x) $ may change signs. The proof of main results is based on Leray-Schauder's fixed point theorem and the properties of Green's function of the fourth-order differential operator $ L_cu = u^{(4)}+c(x)u $.

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