Introduction to the Time Scales Calculus

Martin Bohner (), Gusein Guseinov () and Allan Peterson ()
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Martin Bohner: University of Missouri-Rolla, Department of Mathematics and Statistics
Gusein Guseinov: Atilim University, Department of Mathematics
Allan Peterson: University of Nebraska-Lincoln, Department of Mathematics and Statistics

Chapter Chapter 1 in Advances in Dynamic Equations on Time Scales, 2003, pp 1-15 from Springer

Abstract: Abstract In this chapter we introduce some basic concepts concerning the calculus on time scales that one needs to know to read this book. Most of these results will be stated without proof. Proofs can be found in the book by Bohner and Peterson [86]. A time scale is an arbitrary nonempty closed subset of the real numbers. Thus $$ \mathbb{R},\mathbb{Z},\mathbb{N},\mathbb{N}_0 , $$ i.e., the real numbers, the integers, the natural numbers, and the nonnegative integers are examples of time scales, as are $$ [0,1] \cup [2,3],[0,1] \cup \mathbb{N} $$ , and the Cantor set, while $$ \mathbb{Q},\mathbb{R}\backslash \mathbb{Q},\mathbb{C}(0,1), $$ i.e., the rational numbers, the irrational numbers, the complex numbers, and the open interval between 0 and 1, are not time scales. Throughout this book we will denote a time scale by the symbol $$ \mathbb{T} $$ . We assume throughout that a time scale $$ \mathbb{T} $$ has the topology that it inherits from the real numbers with the standard topology.

Keywords: Jump Operator; Standard Topology; Regressive Function; Forward Difference Operator; Delta Derivative (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1007/978-0-8176-8230-9_1

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