Some new results for singular problems with sign changing nonlinearities
type: Item
instance of: scholarly article
title: Some new results for singular problems with sign changing nonlinearities
zbMATH Open document ID: 0943.34014
DOI: 10.1016/S0377-0427(99)00239-3
author: Ravi P. Agarwal
author: Donal O'Regan
published in: Journal of Computational and Applied Mathematics
publication date: 2000-02-03T00:00:00Z
review text: The authors deal with second-order differential equations of the form \[ y''= q(t)f(t,y),\quad 0< t< 1,\tag{1} \] where \(q\in C(0,1)\) can have singularities at \(0\) and \(1\) and \(f:[0,1]\times (0,\infty)\to \mathbb{R}\) is continuous with a possible singularity at \(y= 0\). \(f\) is allowed to change sign. The authors prove the existence of a solution to (1) which satisfies either the Dirichlet conditions \(y(0)= y(1)= 0\) or the nonlinear conditions \(y(0)= 0\), \(\theta(y'(1)+ y(1))= 0\) with \(\theta: \mathbb{R}\to\mathbb{R}\) continuous, nondecreasing and satisfying \(\theta(0)= 0\). The first existence result, concerning the Dirichlet problem, is formulated in terms of the lower and upper solutions and its proof is based on the Schauder fixed point theorem. The existence of a solution to the above nonlinear boundary value problem is proved via the nonlinear alternative of Leray-Schauder type. In this case some growth restrictions are imposed on \(f\). These results generalize and complete earlier results obtained by the authors.
Mathematics Subject Classification ID: 34B16
zbMATH DE Number: 1398391
zbMATH Keywords: singular boundary value problem
zbMATH Keywords: growth restriction
zbMATH Keywords: second-order differential equations
zbMATH Keywords: lower and upper solutions
zbMATH Keywords: nonlinear boundary value problem
zbMATH Keywords: nonlinear alternative of Leray-Schauder type
reviewed by: Irena Rachůnková
MaRDI profile type: Publication
cites work: Positive solutions to superlinear singular boundary value problems
cites work: Q3941926
cites work: Singular nonlinear boundary value problems for higher order ordinary differential equations
cites work: Upper and lower solutions for a generalized Emden-Fowler equation
cites work: Iterative solution of negative exponent Emden-Fowler problems
cites work: Singular differential equations with linear and nonlinear boundary conditions
cites work: A nonlinear singular boundary value problem
cites work: Existence theorems for a singular two-point dirichlet problem
full work available at URL: s0377-0427(99)00239-3
OpenAlex ID: W2033992093
description: scientific article; zbMATH DE number 1398391