On the Logarithmic Summability of Sequences in Intuitionistic Fuzzy Normed Spaces

Abstract

We introduce logarithmic summability in intuitionistic fuzzy normed spaces(IFNS) and give some Tauberian conditions for which logarithmic summability yields convergence in IFNS. Besides, we define the concept of slow oscillation with respect to logarithmic summability in IFNS, investigate its relation with the concept of q-boundedness and give Tauberian theorems by means of q-boundedness and slow oscillation with respect to logarithmic summability. A comparison theorem between Cesàro summability method and logarithmic summability method in IFNS is also proved in the paper. Definition 1.1. [6] The triplicate (N, µ, ν) is said to be an IFNS if N is a real vector space, and µ, ν are fuzzy sets on N × R satisfying the following conditions for every u, w ∈ N and t, s ∈ R:

Tauberian theorems for weighted means of double sequences in intuitionistic fuzzy normed spaces

“…The concept of intuitionistic fuzzy norm was studied by Saadati and Park [11], and by Lael [7]. convergence of sequences in IF N S was studied and some convergence and summation methods were introduced to recover the convergence where ordinary convergence of sequences in IF N S fails [6,8,12,13]. In this study, we extend the results of [2] to intuitionistic fuzzy normed spaces.…”

On $\mathcal{I}_{\theta }$-convergence in Neutrosophic Normed Spaces

“…Statistical convergence in IFNS was given by Karakuş et al [11]. Notable results on this topic can be found in [12]- [17]. By a lacunary sequence we mean an increasing integer sequence θ = {k r } such that k 0 = 0 and h r = k r − k r−1 → ∞ as r → ∞.…”

Tauberian theorems for statistical Cesàro and statistical logarithmic summability of sequences in intuitionistic fuzzy normed spaces

“…Convergence of sequences with respect to IF −norm is defined and various convergence methods in the statistical sense are proposed to achieve a limit where ordinary convergence fails [6][7][8][9][10]. Also, there are recent studies applying some weighted mean summation methods in IF N S to achieve limit of sequences and providing Tauberian conditions which guarantee ordinary convergence [11,12].…”

“…Definition 1.9. [12] A sequence (u k ) is slowly oscillating with respect to logarithmic summability if and only if for all t > 0 and for all ε ∈ (0, 1) there exist λ > 1 and m 0 ∈ N, depending on t and ε, such that…”

“…Theorem 1.13. [12] Let sequence (u k ) be in (N, µ, ν). If (u k ) is logarithmic summable to some a ∈ N and slowly oscillating with respect to logarithmic summability, then (u k ) converges to a.…”