Generalized Weighted Statistical Convergence for Double Sequences of Fuzzy Numbers and Associated Korovkin-Type Approximation Theorem

Abdullah Alotaibi

doi:10.1155/2020/9298650

Abstract

We define the notions of weighted λ,μ-statistical convergence of order γ1,γ2 and strongly weighted λ,μ-summability of γ1,γ2 for fuzzy double sequences, where 0<γ1,γ2≤1. We establish an inclusion result and a theorem presenting a connection between these concepts. Moreover, we apply our new concept of weighted λ,μ-statistical convergence of order γ1,γ2 to prove Korovkin-type approximation theorem for functions of two variables in a fuzzy sense. Finally, an illustrative example is provided with the help of q-analogue of fuzzy Bernstein operators for bivariate functions which shows the significance of our approximation theorem.

Highlights

Introduction and PreliminariesThe notion of weighted statistical convergence for sequences of real numbers has been studied by Karakaya and Chishti [1] as a generalization of the concept of statistical convergence which is related to the idea of asymptotic density of a subset of N, the set of natural numbers, according to Fast [2]
The weighted statistical convergence was further improved by Mursaleen et al [3] and later generalized by Belen and Mohiuddine [4] with a view of nondecreasing sequence of positive numbers
Çolak [11] defined the notion of statistical convergence of order αð0 < α ≤ 1Þ and strong p-Cesàro summability of order α while the order of statistical was given in [12], and it reduces to strong p-Cesàro summability for α = 1 according to Connor [13]

Summary

Introduction

Introduction and PreliminariesThe notion of weighted statistical convergence for sequences of real numbers has been studied by Karakaya and Chishti [1] as a generalization of the concept of statistical convergence which is related to the idea of asymptotic density (or natural density) of a subset of N, the set of natural numbers, according to Fast [2]. Savas and Mursaleen [26] defined statistical convergence and statistically Cauchy for fuzzy Journal of Function Spaces double sequences and obtained that these concepts are equivalent. A fuzzy double sequence ðsmnÞ is said to be weighted ðλ, μÞ -statistically convergent of order ðγ1, γ2Þ , where 0 < γ1, γ2 ≤ 1 ; in short, we shall write SN2,γðλ1,,γμ2Þ -convergent, to a fuzzy number s0 if for every ε > 0 , the set

Results

Conclusion