On triple difference sequences of real numbers in probabilistic normed spaces

Binod Chandra Tripathy,Rupanjali Goswami

doi:10.4067/s0716-09172014000200003

Binod Chandra Tripathy, Rupanjali Goswami

Open Access

https://doi.org/10.4067/s0716-09172014000200003

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Abstract

In this paper we define concept of triple Δ-statistical convergent sequences in probabilistic normed space and give some results. Also we introduce the notions of Δ-statistical limit point and Δ-statistical cluster point and investigate their different properties.

Highlights

The notion of probabilistic normed space (P N -space) is a generalization of normed linear space
Development and applications in different subjects of the notion of probabilistic normed spaces, one may refer to Alotaibi [1], Alsina etal ([2], [3]), Constantin etal [5], Esi [6], Karakus [11], Menger [18], Lafuerza etal ([14], [15]), Lafuerza etal [16], Schweizer and Sklar ([23], [24]), Tripathy etal [34]
Statistical convergence has been studied in abstract spaces such as the fuzzy number space by Esi ([6], [8]), Fast [9]), locally convex spaces by Maddox [17]

Summary

Introduction

The notion of probabilistic normed space (P N -space) is a generalization of normed linear space. In an ordinary normed linear space norm of vectors are represented by a positive number. As a generalization of ordinary convergence for sequences of real numbers, the notion of statistical convergence was first introduced by Fast [9]. After it was studied by many researchers like Connor [4], Fridy [10], Karakus [11], Karakus and Demirci [12], Salat [20], Tripathy ([26], [27]), Tripathy and Baruah [28], Tripathy etal [29], Tripathy and Dutta [32], Tripathy and Sarma [33]. In this paper we introduce the concept of statistical convergence of triple difference sequence in probabilistic normed spaces and establish some basic properties in P N -spaces

Definitions and Preliminaries

Main results