Higher order boundary value problems on time scales
type: Item
instance of: scholarly article
title: Higher order boundary value problems on time scales
zbMATH Open document ID: 1124.34009
author: S. Gulsan Topal
author: Erbil Çetin
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-07-19T00:00:00Z
review text: This paper deals with the existence of positive solutions to the Lidstone boundary value problem \[ (-1)^n y^{\Delta^{2n}}(t) = f(t, y(\sigma(t))), \quad t \in [0,1], \] \[ y^{\Delta^{2i}}(0) = y^{\Delta^{2i}}(\sigma(1)) = 0, \quad 0 \leq i \leq n-1, \] where \(n \geq 1\) and \(f: [0,\sigma(1)] \times \mathbb{R} \to \mathbb{R}\) is continuous. The problem is considered on a time scale with the right-dense \(\sigma(1)\). The paper develops as follows. The introductory Section 1 provides an overview of related recent results and estimates on the Green function associated with the problem. Section 2 contains an existence result under the assumption of a bounded inhomogeneous term, which follows from the Schauder fixed point theorem. Another existence result is based on the assumption that lower and upper solutions \(u\), \(v\) exist and are in the well order, that is, \(u \leq v\). Sections 3 is devoted to the existence of a positive solutions. The assumptions made here are the sub- and super-linearity of \(f \geq 0\). The results of the last section follow from the Krasnosel'skiĭ fixed point theorem of cone compression-expansion type.
reviewed by: Nickolai Kosmatov
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34B60
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 5172992
zbMATH Keywords: Lidstone boundary value problem
zbMATH Keywords: positive solutions
zbMATH Keywords: upper and lower solutions
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.08.078
OpenAlex ID: W2046762849
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DOI: 10.1016/J.JMAA.2006.08.078
description: scientific article; zbMATH DE number 5172992