Positive solutions for nonlinear \(m\)-point boundary value problems of Dirichlet type via fixed-point index theory
type: Item
instance of: scholarly article
title: Positive solutions for nonlinear \(m\)-point boundary value problems of Dirichlet type via fixed-point index theory
zbMATH Open document ID: 1070.34039
DOI: 10.1016/S0893-9659(03)90009-7
author: Lishun Ren
author: Ruyun Ma
published in: Applied Mathematics Letters
publication date: 2004-06-11T00:00:00Z
review text: This paper deals with the \(m\)-point boundary value problem \[ u''(t)+a(t)u'(t)+b(t)u(t)+g(t)f(u(t))=0, \quad t \in [0,1], \] \[ u(0)=0, u(1)=\sum^{m-2}_{i=1}\alpha_i u(\xi_i). \] It is assumed that \(a, b : [0,1] \to \mathbb R\) are continuous, with \(b(t)\leq 0 \), and \(\xi_i\in \;]0,1[\), \(\alpha_i \in \; ]0,+\infty[\) are such that \(0<\sum^{m-2}_{i=1}\alpha_i\phi_1(\xi_i)<1\), where \(\phi_1 \) is the solution of \[ u''(t)+a(t)u'(t)+b(t)u(t) =0, \quad t \in [0,1], \quad u(0)=0, u(1)=1. \] Further, \(g : [0,1] \to [0,+\infty[\) is continuous, with \(\int_0^1 g >0\), and \(f : [0,+\infty[ \; \to [0,+\infty[\) is continuous too. Under suitable conditions on the quotient \(f(u)/u\), the authors prove the existence of at least one or at least two positive solutions. The proof is based on some results on Hammerstein integral equations obtained by \textit{K. Lan} and \textit{J. R. L. Webb} [J. Differ. Equations 148, 407--421 (1998; Zbl 0909.34013)] and \textit{K. Q. Lan} [J. Lond. Math. Soc., II. Ser. 63, No.3, 690--704 (2001; Zbl 1032.34019)].
reviewed by: Pierpaolo Omari
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B10
zbMATH DE Number: 2073697
zbMATH Keywords: multipoint boundary value problem
zbMATH Keywords: positive solutions
zbMATH Keywords: cone
zbMATH Keywords: fixed-point index
MaRDI profile type: Publication
cites work: Q3820939
cites work: Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation
cites work: A sharper condition for the solvability of a three-point second order boundary value problem
cites work: Solvability of three point boundary value problems at resonance
cites work: Q4264050
cites work: Multiple Positive Solutions of Semilinear Differential Equations with Singularities
cites work: Positive solutions of semilinear differential equations with singularities
cites work: Existence of positive solutions for semilinear elliptic equations in annular domains
cites work: Multiplicity of positive solutions for a nonlinear differential equation with nonlinear boundary conditions
cites work: Q3994493
full work available at URL: s0893-9659(03)90009-7
OpenAlex ID: W2163181269
Recommended article: Q3516567
Recommended article: Positive solutions for nonhomogeneous \(m\)-point boundary value problems
Recommended article: Positive solutions of a nonlinear \(m\)-point boundary value problem
Recommended article: Q5432825
Recommended article: Q3092911
Recommended article: Q3416255
Recommended article: Q3417252
Recommended article: Q4827149
Recommended article: Q5453370
Recommended article: On positive solutions of an \(m\)-point nonhomogeneous singular boundary value problem
description: scientific article; zbMATH DE number 2073697