Remarks on positive solutions of some three point boundary value problems
type: Item
instance of: scholarly article
title: Remarks on positive solutions of some three point boundary value problems
zbMATH Open document ID: 1064.34015
published in: Discrete and Continuous Dynamical Systems
publication date: 2004-02-23T00:00:00Z
review text: The author investigates the existence of positive solutions of the second-order differential equation \[ u''(t)+g(t)f(u(t))=0 \tag{1} \] subjected to one of the following three-point boundary conditions \[ u'(0)=0, \quad \alpha u(\eta)=u(1), \tag{2} \] or \[ u(0)=0, \quad \alpha u(\eta)=u(1), \tag{3} \] where \(\eta \in (0,1)\),\ \(f:[0,\infty) \to [0,\infty)\) is continuous,\ \(g\in L[0,1]\) is nonnegative and \(\int^b_ag(s)\,ds>0\) on some subinterval \([a,b]\). A solution is found as a fixed-point of a corresponding integral operator. The classical theory of fixed-point index for compact mappings in cones is used. Conditions which guarantee the existence of one or two positive solutions of (1),(2) or (1),(3) are given in terms of the quantities \[ f^\alpha = \limsup_{u\to \alpha}{f(u)\over u}, \quad f_\alpha = \liminf_{u\to \alpha}{f(u)\over u}, \quad \alpha\in \{0,\infty\}, \] \[ f_{c\rho,\rho} = \min \biggl\{ {f(u)\over \rho} :u\in [c\rho,\rho]\biggr\}, \quad f^{0,\rho} = \max \biggl\{ {f(u)\over \rho} :u\in [0,\rho]\biggr\},\quad c\in (0,1],\;\rho>0, \] \[ m(a,b)=\biggl(\max_{a\leq t \leq b} \int_a^bk(t,s)g(s)\,ds\biggr)^{-1},\quad M(a,b)=\biggl(\min_{a\leq t \leq b} \int_a^bk(t,s)g(s)\,ds\biggr)^{-1},\;[a,b]\subset [0,1]. \] Here, \(k(t,s)\) is the Green function to the boundary value problem \(u''=0\), (2) or \(u''=0\), (3). Optimal relations among the above quantities are discussed from the point of view of the solvability of problem (1), (2) or problem (1), (3).
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 47H30
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 34B10
zbMATH DE Number: 2046267
zbMATH Keywords: fixed-point index
zbMATH Keywords: cone
zbMATH Keywords: positive solutions
zbMATH Keywords: three-point boundary value problem
author: J. Webb
reviewed by: Irena Rachůnková
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2046267