Publication detail

Some generalizations in theory of rapid variation on time scales and its applications in dynamic equations

VÍTOVEC, J.

Original Title

Some generalizations in theory of rapid variation on time scales and its applications in dynamic equations

English Title

Some generalizations in theory of rapid variation on time scales and its applications in dynamic equations

Type

journal article

Language

en

Original Abstract

In this paper we introduce a new definition of rapidly varying function on time scales. Unlike the recently studied concept of rapid variation, this new concept is more general and naturally extends and complements the already established class of rapidly varying functions. We prove some of its properties and show the relation between this new type of definition and recently introduced classical Karamata type of definition of rapid variation on time scales. Note that the theory of rapid variation on time scales unifies the existing theories from continuous and discrete cases. As an application, we establish necessary and sufficient conditions for all positive solutions of the second order half-linear dynamic equations on time scales to be rapidly varying.

English abstract

In this paper we introduce a new definition of rapidly varying function on time scales. Unlike the recently studied concept of rapid variation, this new concept is more general and naturally extends and complements the already established class of rapidly varying functions. We prove some of its properties and show the relation between this new type of definition and recently introduced classical Karamata type of definition of rapid variation on time scales. Note that the theory of rapid variation on time scales unifies the existing theories from continuous and discrete cases. As an application, we establish necessary and sufficient conditions for all positive solutions of the second order half-linear dynamic equations on time scales to be rapidly varying.

Keywords

Rapidly varying function, regularly varying function, regularly bounded function, time scale, half-linear dynamic equation.

RIV year

2013

Released

15.01.2013

Pages from

139

Pages to

146

Pages count

8

BibTex


@article{BUT97750,
  author="Jiří {Vítovec}",
  title="Some generalizations in theory of rapid variation on time scales and its applications in dynamic equations",
  annote="In this paper we introduce a new definition of rapidly varying function on time scales. Unlike the recently studied concept of rapid variation, this new concept is more general and naturally extends and complements the already established class of rapidly varying functions. We prove some of its properties and show the relation between this new type of definition and recently introduced classical Karamata type of definition of rapid variation on time scales. Note that the theory of rapid variation on time scales unifies the existing theories from continuous and discrete cases. As an application, we establish necessary and sufficient conditions for all positive solutions of the second order half-linear dynamic equations on time scales to be rapidly varying.",
  chapter="97750",
  number="2",
  volume="5 (2012)",
  year="2013",
  month="january",
  pages="139--146",
  type="journal article"
}