Existence of positive solutions of nonlinear discrete Sturm-Liouville problems
type: Item
instance of: scholarly article
title: Existence of positive solutions of nonlinear discrete Sturm-Liouville problems
zbMATH Open document ID: 0965.39009
DOI: 10.1016/S0895-7177(00)00156-4
author: Ferhan M. Atıcı
published in: Mathematical and Computer Modelling
publication date: 2001-08-02T00:00:00Z
review text: Consider the nonlinear boundary-value problem \[ -\Delta[p(t- 1)\Delta y(t-1)]+ q(t) y(t)= f(t,y(t)),\quad t\in [1,b+1], \] \[ \alpha y(0)- \beta p(0)\Delta y(0)= 0,\quad \gamma y(b+ 1)+\delta p(b+1)\Delta y(b+ 1)= 0, \] where \(b\geq 2\) is a fixed integer in the discrete interval \([1,b+1]\). The author gives the Green function \(G(s,t)\) for the problem under consideration. Let \(A\) be the operator defined by \[ Ay(t):= \sum^{b+1}_{s=1} G(t,s) f(s,y(s)). \] It is shown that under some assumptions the operator \(A\) is differentiable along a certain cone in the Banach space of all nonnegative functions \(y: [0,b+2]\to \mathbb{R}\) satisfying the given boundary conditions. On the basis of this result, the author proves that the discrete Sturm-Liouville boundary-value problem as above but with boundary conditions \[ y(0)= y(b+ 2)= 0, \] has at least one positive solution.
reviewed by: Dobiesław Bobrowski
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 34B24
zbMATH DE Number: 1545779
zbMATH Keywords: nonlinear discrete Sturm-Liouville problems
zbMATH Keywords: differentiable operator along a cone
zbMATH Keywords: nonlinear boundary-value problem
zbMATH Keywords: Green's function
zbMATH Keywords: positive solution
MaRDI profile type: Publication
cites work: Q5732050
cites work: Q5185568
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Wikidata QID: Q126815468
description: scientific article; zbMATH DE number 1545779