On strongly λ-summable sequences of fuzzy numbers

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Abstract

In this paper we introduce and study the concepts of strongly λ-summable and λ-statistical convergence.

Introduction

The idea of the statistical convergence of sequence of real numbers was introduced by Fast [2]. Schoenberg [12] studied statistical convergence as a summability method and listed some of the elemantary properties of statistical convergence. Both these authors noted that if bounded sequence is statisticaly convergent to L, then it is Cesaro summable to L.

Recently, Fridy [3] has shown that k(xkxk+1)=O(1) a Tauberian condition for the statistical convergence of (xk). Statistical convergence has also been considered in locally convex spaces [5].

Existing work on statistical convergence appears to have been restricted to real or complex sequence, but in [8] Nuray and Savaş extended the idea to apply to sequences of fuzzy numbers and also introduced and discussed the concept of statistically Cauchy sequences of fuzzy numbers.

In this paper we continue the study of statistical convergence. We introduce and study the concepts of strongly λ-summable and λ-statistical convergence.

Section snippets

Preliminaries

Let C(Rn)={A⊂Rn:Acompactandconvex}. The space C(Rn) has a linear structure induced by the operations A+B={a+b:a∈A,b∈B} and λA={λa:aA} for A,B∈C(Rn) and λR. The Hausdorff distance between A and B of C(Rn) is defined as:δ(A,B)=maxsupa∈Ainfb∈B‖a−b‖,supb∈Binfa∈A‖a−b‖.It is well known that (C(Rn),δ) is a complete (not separable) metric space.

A fuzzy number is a function X from Rn to [0,1] satisfying

  • 1.

    X is normal, i.e. there exists an x0Rn such that X(x0)=1;

  • 2.

    X is fuzzy convex, i.e. for any x,y∈Rn

Strongly λ-summability and λ-statistical convergence

In this section we introduce and study the concepts of strongly λ-summability and λ-statistical convergence and find its relation with strongly λ-summability and statistical convergence.

Before giving the promised relations we will give two new definitions.

Definition 3.1

Let λ=(λn) be a non-decreasing sequence of positive numbers tending to and λn+1⩽λn+1,λ1=1 and X=(Xk) be a sequence of fuzzy numbers. The sequence X is said to be strongly λ-summable if there is a fuzzy number X0 such that

limn→∞1λnk∈IndXk,X0

For further reading

Refs. [4], [13].

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