Solvability of an \(m\)-point boundary value problem for second order ordinary differential equations
type: Item
instance of: scholarly article
title: Solvability of an \(m\)-point boundary value problem for second order ordinary differential equations
zbMATH Open document ID: 0819.34012
author: Chaitan P. Gupta
author: Sotiris K. Ntouyas
author: Panagiotis Ch. Tsamatos
published in: Journal of Mathematical Analysis and Applications
publication date: 1995-08-27T00:00:00Z
review text: The paper deals with the \(m\)-point boundary value problem (1) \(x''(t) = f(t,x(t), x'(t)) + e(t)\), \(t \in (0,1)\), \(x(1) = \sum^{m-2}_{i=1} a_ i x(\xi_ i)\), \(x'(0) = 0\), where \(f\) is a Carathéodory function, \(e \in L(0,1)\), \(a_ i \in \mathbb{R}\) have the same sign, \(0 < \xi_ 1 < \cdots < \xi_{m-2} < 1\). The authors transform the above problem onto the auxiliary three-point problem with the same equation and the boundary conditions (2) \(x'(0) = 0\), \(x(1) = \alpha x (\eta)\), where \(\alpha = \sum^{m-2}_{i=1} a_ i\), \(\eta \in [\xi_ 1, \xi_{m-2}]\). Then they find a common a priori bound for all solutions of \(x''(t) = \lambda f(t,x(t), x'(t)) + \lambda e(t)\), (2), which is independent on \(\eta\) and \(\lambda \in [0,1]\). Finally, using the Mawhin's continuation theorem, they prove the existence of solutions of the auxiliary three-point problem which implies the existence of a solution of problem (1). The existence results are proved provided \(f\) satisfies a linear growth conditions with respect to the phase variables with sufficiently small coefficients.
Mathematics Subject Classification ID: 34B10
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 743918
zbMATH Keywords: \(m\)-point boundary value problem
zbMATH Keywords: a priori bound
zbMATH Keywords: Mawhin's continuation theorem
reviewed by: Irena Rachůnková
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1036
OpenAlex ID: W2009919704
DOI: 10.1006/JMAA.1995.1036
description: scientific article; zbMATH DE number 743918