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Regular variation on measure chains

✍ Scribed by Pavel Řehák; Jiří Vítovec

Book ID
103850178
Publisher
Elsevier Science
Year
2010
Tongue
English
Weight
606 KB
Volume
72
Category
Article
ISSN
0362-546X

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✦ Synopsis

In this paper we show how the recently introduced concept of regular variation on time scales (or measure chains) is related to a Karamata type definition. We also present characterization theorems and an embedding theorem for regularly varying functions defined on suitable subsets of reals. We demonstrate that for a ''reasonable'' theory of regular variation on time scales, certain additional condition on a graininess is needed, which cannot be omitted. We establish a number of elementary properties of regularly varying functions. As an application, we study the asymptotic properties of solution to second order dynamic equations.

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Mimetic methods on measure chains
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✍ M. Bohner; J.E. Castillo 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 273 KB

We introduce the divergence and the gradient for functions defined on a measure chain, and this includes as special cases both continuous derivatives and discrete forward differences. It is shown that in one dimension, subject to Dirichlet boundary conditions, the divergence and the gradient are neg