On a generalization of statistical cluster and limit points

Neşet Özkan Tan,Mustafa Seyyit Seyyidog̃Lu

doi:10.1186/s13662-015-0395-9

Abstract

This paper is concerned with the notions of statistical limit and cluster points defined by Fridy. Following the concept of a Δ-density for a subset of a time scale, we established a generalization of these notions which are called Δ-limit and Δ-cluster points for a function defined on a time scale $\mathbb{T}$ .

Highlights

1 Introduction The theory of time scales was first constructed by Hilger in his PhD thesis in [ ]
Deniz and Ufuktepe defined the Lebesgue-Stieltjes and ∇-measures and by using these measures, they defined an integral which is adaptable to a time scale, the Lebesgue-Stieltjes -integral, in [ ]
Throughout this paper we consider a time scale T with the topology inherited from the real numbers with the standard topology

Summary

Introduction

The theory of time scales was first constructed by Hilger in his PhD thesis in [ ]. Let A be a -measurable subset of T and a = min T, the -density of A in T is defined by μ (A(s)) δ (A) = lim s→∞ σ (s) – a (if this limit exists) where A(s) = {t ∈ A : t ≤ s}. ( -Limit point) A real number L is called a -limit point of a function f : T → R if there exists a subset K of T with a non-zero -density or if it does not have a. Example (i) Let the time scale be T = N This case is called the discrete case and it is easy to see that all definitions above coincide with the definition of a limit point and cluster point in the classical statistically convergence theory. If f (K) is not closed we can consider f (K) by adding limit points βi (i ∈ I) of f (K) and we have f (K) ⊂ N f (t) ∪ N (βi)

We have

Then we have n