On two-point boundary value problems
type: Item
instance of: scholarly article
title: On two-point boundary value problems
zbMATH Open document ID: 0723.34015
DOI: 10.1016/0022-247X(90)90274-J
author: K. N. Murty
author: P. V. S. Lakshmi
published in: Journal of Mathematical Analysis and Applications
publication date: 1990-01-01T00:00:00Z
review text: The authors study two-point boundary value problems of the form \(y'+A(t)y=f(t)\), \(My(a)+Ny(b)=\alpha\) on an interval [a,b], where A is an \(n\times n\) matrix, M and N are \(m\times n\) matrices with rank \([MN]=p\leq \min (m,n)\). Such a problem is said to be invertible if \(m=n\) and the corresponding homogeneous problem \((f=0)\) has only the trivial solution. It is known that in the invertible case the boundary value problem has a unique solution. A characteristic matrix of the boundary value problem is a matrix \(D=MY(a)+NY(b)\), where Y is a fundamental matrix of the homogeneous system \(y'+A(t)y=0\). For the non-invertible case the authors show that the boundary value problem has a unique solution if the columns of D are linearly independent and give a representation for the solution in terms of a Green's matrix.
Mathematics Subject Classification ID: 34B05
zbMATH DE Number: 4192148
zbMATH Keywords: two-point boundary value problems
zbMATH Keywords: invertible
zbMATH Keywords: unique solution
zbMATH Keywords: non- invertible case
zbMATH Keywords: representation
zbMATH Keywords: Green's matrix
reviewed by: Fred Brauer
MaRDI profile type: Publication
cites work: Q4163780
cites work: Q5606769
cites work: Q3035836
cites work: The degenerate case of boundary value problems associated with weakly nonlinear differential systems
full work available at URL: 0022-247x(90)90274-j
OpenAlex ID: W2053147688
description: scientific article; zbMATH DE number 4192148