Multidimensional Asymptotically Lacunary Statistical Equivalent of Order α for Sequences of Fuzzy Numbers

The main goal of this article is to present the notion of double asymptotically lacunary statistical equivalent of order α for sequences of fuzzy numbers by considering fuzzy numbers and Pringsheim limit. To accomplish this goal, we mainly investigate some fundamental properties of the newly introduced notion. Additionally, it should be note that some interesting inclusion theorems are examined and also new variations are presented.


Introduction
Fridy [1] put forward the idea of statistical convergence, which is a significant part of the Summability Theory. In 1980, Pobyvancts [2] presented the idea of asymptotically regular matrices. In 2003, Patterson [3] introduced the notions of asymptotically statistically equivalent sequences by combining the notion of asymptotically equivalent introduced by Marouf [4] and statistical convergence.
On the other hand, the notion of convergence for double sequences was presented by Pringsheim in [5]. Definition 1.1: A double sequence y = (y r,s ) has Pringsheim limit ∈ R (symbolized by P − lim r,s→∞ y r,s = ) if given ε > 0 there exists M ∈ N such that |y r,s − | < ε, whenever r, s > M. We shall show such an y shortly as 'P−convergent'.
Recently, Mursaleen and Edely [6] extended Pringsheim's definition of statistical convergence for double sequences, and Patterson [7] also defined double asymptotically equivalent as follows: Definition 1.2: Two nonnegative double sequences (y r,s ) and (z r,s ) are said to be asymptotically equivalent provided that P − lim r,s y r,s z r,s = 1.
If this condition is met, it is symbolised by y Later on the following definition was given by Savaş and Patterson [8].

Definition 1.3:
The double sequence ξ ,η = {(r ξ , s η )} is named double lacunary provided that there exist two increasing of integers in such a way that r 0 = 0, γ ξ = r ξ − r ξ −1 → ∞ whenever ξ → ∞ and s 0 = 0, γ η = s η − s η−1 → ∞ as η → ∞. Also, r ξ ,η = r ξ s η , γ ξ ,η = γ ξ γ η , ξ ,η is determined by Also, in Ref. [9,10], a different direction was presented to the concept of statistical convergence for double sequences in which the notion of statistical convergence of order α, 0 < α ≤ 1 was introduced by shifting (mn) by (mn) α in the denominator in the definition of statistical convergence. Please note that the notion of metric space can define as an arbitrary fuzzy set that a distance between all elements of the set are described. It is possible to define many different metrics on the space of fuzzy numbers; nonetheless, the most preferential metric among these metrics is the Hausdorff distance for fuzzy numbers depended on the classical Hausdorff distance between compact convex subsets of R p .
We now consider C(R p ) = {K ⊂ R p : K compact and convex}. The spaces C(R p ) has a linear structure induced by the operations for K, H ∈ C(R p ) and ς ∈ R. The Hausdorff distance between K and H of C(R p ) is defined as (2) Y is fuzzy convex, in other words, for any τ , υ ∈ R p and 0 ≤ ς ≤ 1, These features imply that for each 0 < β ≤ 1, the β−level set is a nonempty compact convex, subset of R p , as is the support Y 0 . Let L(R p ) denote the set of all fuzzy numbers. The linear structure of L(R p ) shows addition Y + Z and scalar multiplication ςY, ς ∈ R, in the sense of β−level sets, by Moreover, d q is a complete, separable and locally compact metric space [11]. During the paper, d will denote Provided that d is a translation invariant metric on L(R), then it is clear to see

Main Results
In this section, we will present new concepts and examine the relationship among these concepts. Definition 2.1: Let ξ ,η = {(r ξ , s η )} be a double lacunary sequence; and p = (p u,v ) be a sequence of positive real numbers; two sequences Y and Z of fuzzy numbers are double strongly asymptotically equivalent of order α to multiple , where 0 < α ≤ 1 provided that ∼ Z), and simply double lacunary strongly asymptotically equivalent of

Definition 2.2:
Let p = (p u,v ) be a sequence of positive real numbers, and we consider two sequences Y and Z of fuzzy numbers. The two sequences Y and Z are called as double strongly Cesáro summable of order α to , where 0 < α ≤ 1 provided that ∼ Z) and in simple terms double strongly Cesáro asymptotically equivalent of order α if = 1.
Let us prove the following theorems.
Theorem 2.1: Let ξ ,η be a double lacunary sequence. Then (2) Suppose that Y and Z are in l 2 ∞ (F) and Y S α ∼ Z, then we can presume that d( Y u,v Z u,v , ) ≤ M for u and v. Let > 0 be given and N be such that for all ξ , η > N and let ∼ Z and > 0 be given. Then

Theorem 2.3: Let Y and Z be bounded and
Proof: Suppose that Y and Z be bounded and > 0. Since Y and Z are bounded, there is an integer K such that d( Since Y fuzzy logic, soft computing, matrix theory. She has published research articles in many different high-quality journals. She is also a referee and editor of some mathematical journals. She is working at Istanbul Medeniyet University in the Department of Mathematics and Science Education as an associate professor.