Singular two point boundary value problems for second order differential systems
type: Item
instance of: scholarly article
title: Singular two point boundary value problems for second order differential systems
zbMATH Open document ID: 0930.34012
author: Zheyan Zhou
author: Qinde Zhou
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-02-16T00:00:00Z
review text: The authors consider singular two-point boundary value problems \[ A(t,x,x') x''= f(t,x,x'),\quad x(a)- Px'(a)= 0,\quad x(b)+ Qx'(b)= 0,\tag{1} \] with \(x: [a,b]\to \mathbb{R}^n\), \(A\in C([a,b]\times \mathbb{R}^n\times \mathbb{R}^n,\mathbb{R}^{n\times n})\) is symmetric and positive definite on \([a,b)\times \mathbb{R}^n\times \mathbb{R}^n\), with \(\text{rank }A(b,-Qy, y)< n\) for \(y\in \mathbb{R}^n\), \(P\) and \(Q\) are constant \(n\times n\)-matrices, \(f\in C([a, b]\times \mathbb{R}^n\times \mathbb{R}^n,\mathbb{R}^n)\). The authors give some existence results for (1) provided \(f\) fulfils a growth condition and the eigenvalues of \(A\) are bounded below by a continuous function \(\lambda_0\), positive on \([a,b)\), with \(\int^b_a (ds/\lambda_0(s))< +\infty\). In the proofs, the method of a priori estimates, guiding functions and Leray-Schauder topological degree arguments are used. An application of these existence results to some examples of semilinear boundary value problems is shown.
Mathematics Subject Classification ID: 34B16
zbMATH DE Number: 1295261
zbMATH Keywords: existence
zbMATH Keywords: Leray-Schauder topological degree
zbMATH Keywords: singular two-point boundary value problems
reviewed by: Irena Rachůnková
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6304
OpenAlex ID: W2051084656
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DOI: 10.1006/JMAA.1999.6304
description: scientific article; zbMATH DE number 1295261