A note on statistical convergence on time scale

Seyyidoglu M, Tan, Tan N

Description

In the present paper, we will give some new notions, such as Δ-convergence and Δ-Cauchy, by using the Δ-density and investigate their relations. It is important to say that the results presented in this work generalize some of the results mentioned in the theory of statistical convergence. MSC: 34N05. In the present paper, we will give some new notions, such as Δ-convergence and Δ-Cauchy, by using the Δ-density and investigate their relations. It is important to say that the results presented in this work generalize some of the results mentioned in the theory of statistical convergence. MSC: 34N05.

Sujets

• Time scale

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 18
• Langue: Italiano

Extrait

Seyyidoglu and TanJournal of Inequalities and Applications2012,2012:219 http://www.journalofinequalitiesandapplications.com/content/2012/1/219

R E S E A R C H

Open Access

A note on statistical convergence on time scale

* M Seyyit Seyyidoglu and N Özkan Tan

* Correspondence: [e-mail] Faculty of Sciences and Arts, Department of Mathematics, Usak University, 1 Eylul Campus, Usak, 64200, Turkey

Abstract In the present paper, we will give some new notions, such as  -convergence and  -Cauchy, by using the  -density and investigate their relations. It is important to say that the results presented in this work generalize some of the results mentioned in the theory of statistical convergence. MSC:34N05 Keywords:time scale; Lebesgue  -measure; statistical convergence

1 Introduction and background In [] Fast introduced an extension of the usual concept of sequential limits which he called statistical convergence. In [] Schoenberg gave some basic properties of statistical convergence. In [] Fridy introduced the concept of a statistically Cauchy sequence and proved that it is equivalent to statistical convergence. The theory of time scales was introduced by Hilger in his PhD thesis supervised by Auld- bach [] in . The measure theory on time scales was first constructed by Guseinov [], and then further studies were performed by Cabada-Vivero [] and Rzezuchowski []. In [] Deniz-Ufuktepe define Lebesgue-Stieltjesand-measures, and by using these measures, they define an integral adapted to a time scale, specifically Lebesgue-Steltjes -integral. A time scaleTis an arbitrary nonempty closed subset of the real numbersR. The time scaleTis a complete metric space with the usual metric. We assume throughout the paper that a time scaleThas the topology that it inherits from the real numbers with the standard topology. Fort∈T, we define theforward jump operatorσ:T→Tby

σ(t) :=inf{s∈T:s>t}.

In this definition, we putinf∅=supT. Fora,b∈Twitha≤b, we define the interval [a,b] inTby

[a,b] ={t∈T:a≤t≤b}.

Open intervals and half-open intervalsetc.are defined accordingly. LetTbe a time scale. Denote by S the family of all left-closed and right-open intervals ofTof the form [a,b) ={t∈T:a≤t<b}witha,b∈Tanda≤b. The interval [a,a) is

©2012 Seyyidoglu and Tan; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.