On multi-point boundary value problems for linear ordinary differential equations with singularities
type: Item
instance of: scholarly article
title: On multi-point boundary value problems for linear ordinary differential equations with singularities
zbMATH Open document ID: 1058.34012
author: Ravi P. Agarwal
author: Ivan Kiguradze
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-10-01T00:00:00Z
review text: The authors investigate the singular linear differential equation \[ u^{(n)}= \sum_{i=1}^n p_i(t)u^{(i-1)}+q(t) \tag{1} \] on \([a,b]\subset \mathbb{R}\), where the functions \(p_i\) and \(q\) can have singularities at \(t=a, t=b\) and \(t=t_0\in (a,b)\). This means that \(p_i\) and \(q\) are not integrable on \([a,b]\). Equation (1) is studied with the boundary conditions \[ u^{(i-1)}(t_0)=0 \text{ for } 1\leq i\leq n-1,\;\sum_{j=1}^{n-n_1}\alpha_{1j}u^{(j-1)}(t_{1j}) + \sum_{j=1}^{n-n_2}\alpha_{2j}u^{(j-1)}(t_{2j})=0, \tag{2} \] or \[ u^{(i-1)}(a)=0\text{ for }1\leq i\leq n-1,\;\sum_{j=1}^{n-n_0}\alpha_{j}u^{(j-1)}(t_{j})=0, \tag{3} \] where \(t_{1j}, t_{2j}, t_j\) are certain interior points in \((a,b)\). The authors introduce the Fredholm property for these problems which means that the unique solvability of the corresponding homogeneous problem implies the unique solvability of the nonhomogeneous problem for every \(q\) which is weith-integrable on \([a,b]\). Then, for the solvability of a problem having the Fredholm property, it sufficies to show that the corresponding homogeneous problem has only the trivial solution. In this way, the authors prove main theorems on the existence of a unique solution of (1),(2) and of (1),(3). Examples verifying the optimality of the conditions in various corollaries are shown as well.
Mathematics Subject Classification ID: 34B10
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 34B16
zbMATH DE Number: 2105432
zbMATH Keywords: multipoint boundary value problem
zbMATH Keywords: linear differential equation
zbMATH Keywords: higher order
zbMATH Keywords: singular
zbMATH Keywords: sharp conditions
reviewed by: Irena Rachůnková
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.05.002
OpenAlex ID: W2067652448
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DOI: 10.1016/J.JMAA.2004.05.002
description: scientific article; zbMATH DE number 2105432