Existence of positive solutions for second-order ODEs with reverse Carathéodory functions
type: Item
instance of: scholarly article
title: Existence of positive solutions for second-order ODEs with reverse Carathéodory functions
zbMATH Open document ID: 1032.34004
DOI: 10.1016/S0362-546X(03)00057-9
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-06-17T00:00:00Z
review text: Given real intervals \([x_0,L] \) and \([u_0,R] \), the nonnegative function \[ f:[x_0,L] \times [u_0,R] \to \mathbb{R} \] is defined as a reverse Carathéodory function if it satisfies: (A) for all \(x\in [x_0,L] \), \(f(x,.)\) is measurable; (B) almost everywhere (a.e.) for \(u\in [u_0,R] \), \(f(.,u) \) is continuous; (C) there exists \(m\in L^1(u_0,R)\) such that, for all \(x\in [x_0,L] \) and a.e. \(u\in [u_0,R]\), \(0<m(u)\leq f(x,u)\). With such an \(f\), the existence of a solution is established for the second-order initial value problem \[ \frac {d}{dx}(l(x)k(x,u(x))u'(x))=f(x,u(x))u'(x),\text{ a.e. for }x\in I:= [0,L], \tag{1} \] \[ u(0)=0, \quad l(0)k(0,u(0))u'(0)=0. \] This result is obtained by defining a convenient operator in a closed convex subset of the space of continuous functions on \(I\), equipped with the maximum norm, and showing that the operator satisfies the hypotheses in Schauder's fixed-point theorem. The fixed-point is then a solution to (1). Here, necessary and sufficient conditions are given that ensure existence and uniqueness of a solution for a simplified version of (1), where both \(k\) and \(f\) only depend on \(u\).
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 1929511
zbMATH Keywords: second-order problems
zbMATH Keywords: Carathéodory conditions
zbMATH Keywords: Schauder's fixed-point theorem
zbMATH Keywords: singular differential equations
author: J. Ángel Cid
reviewed by: Juan Pedro Milaszewicz
MaRDI profile type: Publication
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full work available at URL: s0362-546x(03)00057-9
OpenAlex ID: W2154579899
description: scientific article; zbMATH DE number 1929511