Basic existence, uniqueness and approximation results for positive solutions to nonlinear dynamic equations on time scales
type: Item
instance of: scholarly article
title: Basic existence, uniqueness and approximation results for positive solutions to nonlinear dynamic equations on time scales
zbMATH Open document ID: 1158.34025
author: Christopher C. Tisdell
author: Lynn Erbe
author: Allan Peterson
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2008-09-10T00:00:00Z
review text: This paper studies the existence and uniqueness of solutions of the second order nonlinear dynamic equations \[ \big(r(t)\,x^\Delta\big)^\Delta+F(t,x^\sigma)=0, \quad t\in[a,\infty)\cap {\mathbb T}, \] \[ \big(r(t)\,x^\Delta\big)^\Delta+F(t,x^\sigma,x^{\Delta\sigma})=0, \quad t\in[a,\infty)\cap {\mathbb T}, \] where \({\mathbb T}\) is a time scale (i.e., a nonempty closed subset of \({\mathbb R}\)). These equations generalize and unify the corresponding second order differential and difference equations. The authors use the Banach fixed point theorem to prove existence and uniqueness of solutions with an asymptote, i.e., \(\lim_{t\to\infty}x(t)=M>0\) (the so-called terminal value problem). The function \(F\) is assumed to be nonnegative and \(r\) is positive. In the time scale environment one also needs to assume that \(r(t)\) is bounded away from zero, i.e., \(\inf\{r(t),\;t\in[a,b]\cap{\mathbb T}\}>0\) for all \(b\in(a,\infty)\cap{\mathbb T}\) (so that the function \(1/r(\cdot)\) is rd-continuous) as e.g. in [\textit{R.~Hilscher} and \textit{C.~C.~Tisdell}, ``Terminal value problems for first and second order nonlinear equations on time scales'', Electron. J. Differ. Equ. 2008, Paper No.~68, 21 pp. (2008)]. Note also that the set \(\chi_M\) is a closed subset of the Banach space \(C_{\mathbb T}^1\) (and not a Banach space itself).
reviewed by: Roman Šimon Hilscher
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 39A12
zbMATH DE Number: 5324201
zbMATH Keywords: positive solution
zbMATH Keywords: time scale
zbMATH Keywords: terminal value problem
zbMATH Keywords: second order dynamic equation
MaRDI profile type: Publication
full work available at URL: j.na.2007.08.010
OpenAlex ID: W2096811056
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DOI: 10.1016/J.NA.2007.08.010
description: scientific article; zbMATH DE number 5324201