Regularly ideal invariant convergence of double sequences

Nimet Pancaroǧlu Akın

doi:10.1186/s13660-021-02664-1

Abstract

In this paper, we introduce the notions of regularly invariant convergence, regularly strongly invariant convergence, regularly p-strongly invariant convergence, regularly (mathcal{I}_{sigma },mathcal{I}^{sigma }_{2})-convergence, regularly (mathcal{I}_{sigma }^{*},mathcal{I}^{sigma *}_{2})-convergence, regularly (mathcal{I}_{sigma },mathcal{I}^{sigma }_{2} )-Cauchy double sequence, regularly (mathcal{I}_{sigma }^{*},mathcal{I}^{sigma *}_{2})-Cauchy double sequence and investigate the relationship among them.

Highlights

1 Introduction and background Throughout the paper, N and R denote the set of all positive integers and the set of all real numbers, respectively
The concept of convergence of real sequences was extended to statistical convergence independently by Fast [18] and Schoenberg [42]
Das et al [5] introduced the concept of I-convergence of double sequences in a metric space and studied some properties of this convergence

Summary

Introduction

Introduction and backgroundThroughout the paper, N and R denote the set of all positive integers and the set of all real numbers, respectively. A double sequence (xkj) is I2∗-invariant convergent or I2σ∗-convergent to L if and only if there exists a set M2 ∈ F (I2σ ) (N × N\M2 = H ∈ I2σ ) such that, for (k, j) ∈ M2, limk,j→∞ xkj = L.

Results

Conclusion