On Some Further Generalizations of Strong Convergence in Probabilistic Metric Spaces Using Ideals

Pratulananda Das,Sanjoy Ghosal,Vatan Karakaya,Kaustubh Dutta

doi:10.1155/2013/765060

Pratulananda Das, Sanjoy Ghosal + Show 2 more

Open Access

https://doi.org/10.1155/2013/765060

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Abstract

Following the line of (Das et al., 2011, Savas and Das, 2011), we make a new approach in this paper to extend the notion of strong convergence and more general strong statistical convergence (Şençimen and Pehlivan, 2008) using ideals and introduce the notion of strongℐ- andℐ*-statistical convergence and two related concepts, namely, strongℐ-lacunary statistical convergence and strongℐ-λ-statistical convergence in a probabilistic metric space endowed with strong topology. We mainly investigate their interrelationship and study some of their important properties.

Highlights

Introduction and BackgroundThe usual idea of convergence is not enough to understand behaviours of those sequences which are not convergent
We introduce the definition of strong I-statistical convergence in PM space
We introduce the main definitions for this section

Summary

Introduction and Background

The usual idea of convergence is not enough to understand behaviours of those sequences which are not convergent. In [8], Karakaya et al defined statistical convergence of sequences of functions in intuitionistic fuzzy normed spaces and used the concepts of lacunary and λ-statistical convergence for sequences of functions in these spaces in [9, 10] It must be mentioned in this context that some of the above mentioned convergence methods have applications in number theory, measure theory, fourier analysis, optimization, and many branches of mathematics. First of all, we introduce the notion of strong I-statistical convergence in probabilistic metric spaces which happens to be more general than strong statistical convergence. We introduce the concepts of strong Ilacunary statistical convergence and strong I-λ-statistical convergence in probabilistic metric spaces and investigate some of their important properties, in particular their relations with strong I-statistical convergence. As these concepts are more general than the concept of strong statistical convergence, we believe that they can extend the general framework introduced by [25] which in the future may enhance the applicability of strong convergence in probabilistic metric spaces

Preliminaries

Strong I-Lacunary Statistical Convergence in PM Space

Strong I-λ-Statistical Convergence in PM Space