Existence criteria for singular boundary value problems with sign changing nonlinearities
type: Item
instance of: scholarly article
title: Existence criteria for singular boundary value problems with sign changing nonlinearities
zbMATH Open document ID: 1032.34018
author: Ravi P. Agarwal
author: Donal O'Regan
published in: Journal of Differential Equations
publication date: 2002-11-28T00:00:00Z
review text: In this interesting paper, a general theory on the existence of solutions to the singular Dirichlet problem \[ y''(t)+q(t)f(t,y)=0, \quad 0<t<1,\qquad y(0)=y(1)=0,\tag \(*\) \] is presented. The nonlinearity \(f(t,y)\) might be singular with respect to the variable \(y\) at 0 and it may change sign. The authors show the existence of a solution \(y\in C[0,1]\cap{C^2(0,1)}\) to \((*)\) such that \(y(t)>0,\) for \(0<t<1.\) To do this, they approximate problem \((*)\) with an appropriate sequence of (regular) Dirichlet problems \((*_n)\), say, and by making use of an earlier existence result [the authors, J. Differ. Equations 175, 393-414 (2001; Zbl 0999.34018)], they get the existence of a solution \((y_n)\) to \((*_n).\) The basic assumption is the existence of a lower solution \(\alpha_n\) (generated from a positive given function \(\alpha\) which it turns to be a lower solution to \((*)\)) and of an upper solution \(\beta_n\). Then, the result follows by applying the classical Ascoli theorem. Several comments on the assumptions are presented and, in particular, some general criteria are given which help to verify whether a particular equation has a lower solution. An application of the result is given. On the other hand, two other theorems give existence results by using conditions which do not require any knowledge of upper and lower solutions.
Mathematics Subject Classification ID: 34B16
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 1836801
zbMATH Keywords: singular nonlinear boundary value problems
reviewed by: George Karakostas
MaRDI profile type: Publication
full work available at URL: jdeq.2001.4127
OpenAlex ID: W2095302171
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DOI: 10.1006/JDEQ.2001.4127
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description: scientific article; zbMATH DE number 1836801