Dynamic equations on time scales: A survey
type: Item
instance of: scholarly article
title: Dynamic equations on time scales: A survey
zbMATH Open document ID: 1020.39008
DOI: 10.1016/S0377-0427(01)00432-0
author: Ravi P. Agarwal
author: Donal O'Regan
author: Martin Bohner
author: Allan Peterson
published in: Journal of Computational and Applied Mathematics
publication date: 2002-06-24T00:00:00Z
review text: The authors present a survey of some basic results concerning dynamic equations on time scales. The study of such objects goes back to \textit{S. Hilger} [Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Diss. (1988; Zbl 0695.34001)], who created the calculus of time scales (or, more generally, the calculus of measure chains) in order to unify continuous and discrete analysis. This calculus enables an investigation of dynamic equations that cover as special cases differential equations, difference equations, as well as many other equations, where the domain of an unknown function is a closed subset of reals. The authors give an introduction to the time scale calculus and present some basic properties of elementary functions on time scales, such as exponential, hyperbolic and trigonometric functions. Those are used to solve certain linear dynamic equations of first and second order. They give further basic results (as a variation of constants) for higher order linear equations, and also several examples and applications are considered. Finally, they mention results on the positivity of quadratic functionals and the solvability of Riccati dynamic equations which correspond to self-adjoint equations and, more generally, symplectic dynamic systems. The paper contains an extensive list of related publications.
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 39-02
Mathematics Subject Classification ID: 37J10
zbMATH DE Number: 1758465
zbMATH Keywords: time scales
zbMATH Keywords: measure chains
zbMATH Keywords: dynamic equations
zbMATH Keywords: research survey
zbMATH Keywords: elementary functions
zbMATH Keywords: Riccati dynamic equations
zbMATH Keywords: symplectic dynamic systems
reviewed by: Pavel Rehak
MaRDI profile type: Publication
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full work available at URL: s0377-0427(01)00432-0
OpenAlex ID: W2137369682
Wikidata QID: Q127571645
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description: scientific article; zbMATH DE number 1758465