StatisticalΛ-Convergence in Probabilistic Normed Spaces

M Aldhaifallah,H M Srivastava,K S Nisar,M Mursaleen

doi:10.1155/2017/3154280

Abstract

The main objective of the study was to understand the notion of Λ-convergence and to study the notion of probabilistic normed(PN)spaces. The study has also aimed to define the statistical Λ-convergence and statistical Λ-Cauchy in PN-spaces. The concepts of these approaches have been defined by some examples, which have demonstrated the concepts of statistical Λ-convergence and statistical Λ-Cauchy in PN-spaces. Previous studies have also been used to understand similar terminologies and notations for the extraction of outcomes.

Highlights

The notion of statistical metric spaces [1,2,3], called probabilistic metric spaces, was introduced by Menger [4]; it is an important generalization of metric spaces
The concept of probabilistic normed (PN) spaces [5] is a key generalization of the concept of normed spaces
By making use of the definitions given in the preceding section, it is proposed here to systematically investigate the notion of statistical Λ-convergence and statistical Λ-Cauchy in PN-spaces and apply our findings to the problem of approximating positive linear operators

Summary

Introduction

The notion of statistical metric spaces [1,2,3], called probabilistic metric spaces, was introduced by Menger [4]; it is an important generalization of metric spaces. A number sequence y = (ym) is said to be statistically convergent to the number L if, for each ε > 0, the set In this case a sequence y = (ym)∞ m=0 is λ-convergent to the number L ∈ C, which is known as λ-limit of y, if Λ n(y) → L as n → ∞, where

Objectives

Results

Conclusion