On Green's functions and positive solutions for boundary value problems on time scales
type: Item
instance of: scholarly article
title: On Green's functions and positive solutions for boundary value problems on time scales
zbMATH Open document ID: 1007.34025
DOI: 10.1016/S0377-0427(01)00437-X
author: Ferhan M. Atıcı
author: Gusein Sh. Guseinov
published in: Journal of Computational and Applied Mathematics
publication date: 2002-06-24T00:00:00Z
review text: The authors investigate the second-order dynamic equation on a time scale \[ -[p(t)y^\Delta]^\nabla+q(t)y=h(t). \tag{*} \] Recall that a time scale \(\mathbf T\) is any closed subset of \(\mathbb{R}\), where the jump operators \(\sigma(t)=\inf\{s>t, s\in \mathbf T\}\), \(\rho(t)=\sup\{s<t, s\in \mathbf T\}\) are defined. Using these operators, the so-called \(\Delta\)-derivative and \(\nabla\)-derivative of a function \(f:\mathbf T\to \mathbb{R}\) are defined by \[ f^\Delta(t)=\lim_{s\to t}\frac{f(s)-f(\sigma(t))}{s-\sigma(t)},\quad f^\nabla(t)=\lim_{s\to t}\frac{f(s)-f(\rho(s))}{s-\rho(t)}. \] First, the basic properties of solutions to (*) are established (Wronskian-type identity for homogeneous equation, variation of parameters formula,\dots). Then, the attention is focused on the Sturm-Liouville boundary value problem associated with (*), in particular, properties of Green's function to this boundary value problem are investigated. In the last part of the paper, these properties of Green's function are used to study the existence of positive solutions to (*).
reviewed by: Ondřej Došlý
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34B27
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 1758470
zbMATH Keywords: time scales
zbMATH Keywords: delta and nabla derivatives
zbMATH Keywords: Green's function
zbMATH Keywords: cone
MaRDI profile type: Publication
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full work available at URL: s0377-0427(01)00437-x
OpenAlex ID: W2024813476
description: scientific article; zbMATH DE number 1758470