On generalized double statistical convergence in a random 2-normed space

Ekrem Savas

doi:10.1186/1029-242x-2012-209

Abstract

Recently, the concept of statistical convergence has been studied in 2-normed and random 2-normed spaces by various authors. In this paper, we shall introduce the concept of λ-double statistical convergence and λ-double statistical Cauchy in a random 2-normed space. We also shall prove some new results. MSC: 40A05; 40B50; 46A19; 46A45

Highlights

1 Introduction The probabilistic metric space was introduced by Menger [ ] which is an interesting and an important generalization of the notion of a metric space
Recently some new sequence spaces have been studied by Savas [ – ] by using -normed spaces
We study λ-double statistical convergence in a random -normed space which is a new and interesting idea

Summary

Introduction

The probabilistic metric space was introduced by Menger [ ] which is an interesting and an important generalization of the notion of a metric space. Definition A sequence x = (xk,l) in a random -normed space (X, F , ∗) is said to be double convergent (or F -convergent) to ∈ X with respect to F if for each ε > , η ∈ ( , ), there exists a positive integer n such that F (xk,l – , z; ε) > – η, whenever k, l ≥ n and for nonzero z ∈ X In this case we write F – limk,l xk,l = , and is called the F -limit of x = (xk,l). Definition A sequence x = (xk,l) in a random -normed space (X, F , ∗) is said to be λ-double statistically convergent or Sλ ̄ -convergent to ∈ X with respect to F if for every ε > , η ∈ ( , ) and for nonzero z ∈ X such that δλk ∈ In, l ∈ Jm : F (xk,l – , z; ε) ≤ – η =.

It follows that

If we take