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E. Savaş, H. Sevli, M. Cancan (2010)
On Asymptotically -Statistical Equivalent Sequences of Fuzzy NumbersJournal of Inequalities and Applications, 2010
R. Patterson (2003)
ON ASYMPTOTICALLY STATISTICALLY EQUIVALENT SEQUENCES, 36
(2010)
On asymptotically (3bb,3c3)-statistical equivalent sequences of fuzzy numbers, 2010
E. Savaş (2014)
On convergent double sequence spaces of fuzzy numbers defined by ideal and Orlicz functionJ. Intell. Fuzzy Syst., 26
(2011)
-double sequence spaces of fuzzy numbers via Orlicz function, 8
R. Çolak, Y. Altın (2013)
Statistical Convergence of Double Sequences of OrderJournal of Function Spaces and Applications, 2013
E. Savaş, R. Patterson (2006)
Lacunary statistical convergence of multiple sequencesAppl. Math. Lett., 19
(1900)
Zur theorie der zweifach unendlichen zahlen folgen
(1980)
Asymptotic equivalence of some linear transformation defined a nonnegative matrix and reduced to generalized equivalence in the sense of Cesaro and Abel
E. Savaş (2017)
ℐ$$ \mathrm{\mathcal{I}} $$λ-Double Statistical Convergence of Order α in Topological GroupsUkrainian Mathematical Journal, 68
R. Patterson (2003)
Rates of Convergence for Double SequencesSoutheast Asian Bulletin of Mathematics, 26
E. Savaş, M. Gürdal (2014)
Generalized statistically convergent sequences of functions in fuzzy 2-normed spacesJ. Intell. Fuzzy Syst., 27
Mursaleen, O. Edely (2003)
Statistical Convergence of Double SequencesJournal of Mathematical Analysis and Applications, 288
M. Marouf (1993)
Asymptotic equivalence and summabilityInternational Journal of Mathematics and Mathematical Sciences, 16
E. Savaş (2013)
Some I-Convergent Sequence Spaces of Fuzzy Numbers Defined by Infinite MatrixMathematical & Computational Applications, 18
J. Fridy (1985)
ON STATISTICAL CONVERGENCE, 5
E. Savaş (2011)
$(A)_ {Delta}$ - double Sequence Spaces of fuzzy numbers via Orlicz FunctionIranian Journal of Fuzzy Systems, 8
(2016)
double statistical convergence of order α in topological groups, 68
E. Savaş (2015)
On some summability methods using ideals and fuzzy numbersJ. Intell. Fuzzy Syst., 28
FUZZY INFORMATION AND ENGINEERING 2022, VOL. 14, NO. 2, 143–151 https://doi.org/10.1080/16168658.2022.2120225 Multidimensional Asymptotically Lacunary Statistical Equivalent of Order α for Sequences of Fuzzy Numbers Rabia Savaş Department of Mathematics and Science Education, Istanbul Medeniyet University, Istanbul, Turkey ABSTRACT KEYWORDS Asymptotically Statistical The main goal of this article is to present the notion of double asymp- Equivalent; Double Lacunary totically lacunary statistical equivalent of order α for sequences of Sequences; Statistical Limit fuzzy numbers by considering fuzzy numbers and Pringsheim limit. Points; Fuzzy Numbers To accomplish this goal, we mainly investigate some fundamental properties of the newly introduced notion. Additionally, it should be 2000 MATHEMATICS note that some interesting inclusion theorems are examined and also SUBJECT new variations are presented. CLASSIFICATIONS Primary 40A99; Secondary 40A05 1. 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 equiva- lent 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 ) has Pringsheim limit ∈ R (symbolized by r,s P − lim y = ) if given ε> 0 there exists M ∈ N such that |y − | <ε, whenever r,s r,s r,s→∞ r, s > M. We shall show such an y shortly as ‘P−convergent’. Recently, Mursaleen and Edely [6] extended Pringsheim’s definition of statistical con- vergence for double sequences, and Patterson [7] also defined double asymptotically equivalent as follows: Definition 1.2: Two nonnegative double sequences (y ) and (z ) are said to be asymp- r,s r,s totically equivalent provided that r,s P − lim = 1. r,s r,s If this condition is met, it is symbolised by y ∼ z. CONTACT Rabia Savaş rabiasavass@hotmail.com © 2022 The Authors. Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/ licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 144 RABIA SAVAŞ Later on the following definition was given by Savaş and Patterson [8]. Definition 1.3: The double sequence ={(r , s )} is named double lacunary provided ξ,η ξ η that there exist two increasing of integers in such a way that r = 0, γ = r − r →∞ 0 ξ ξ ξ −1 whenever ξ →∞ and s = 0, γ = s − s →∞ as η →∞. Also, r = r s , γ = 0 η η−1 ξ,η ξ η ξ,η γ γ , is determined by ξ ξ,η J = (r, s) : r < r ≤ r &s < s ≤ s , ξ,η ξ −1 ξ η−1 η r s ξ η ζ = , ζ = ,and ζ = ζ ζ . ξ ξ,η ξ η η r s ξ −1 η−1 Also, in Ref. [9,10], a different direction was presented to the concept of statistical con- vergence for double sequences in which the notion of statistical convergence of order α, 0 <α ≤ 1 was introduced by shifting (mn) by (mn) in the denominator in the definition of statistical convergence. Please note that the notion of metric space can define as an arbi- trary 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 pref- erential metric among these metrics is the Hausdorff distance for fuzzy numbers depended on the classical Hausdorff distance between compact convex subsets of R . Recently, fuzzy numbers were studied and generalised more general concepts via statis- tical convergence in Refs. [11–16]. p p p We now consider C(R ) ={K ⊂ R : K compact and convex}. The spaces C(R ) has a linear structure induced by the operations K + H = {k+h, k∈ K, h ∈ H} and ςA = {ςa, ς ∈ E} p p for K, H ∈ C(R ) and ς ∈ R. The Hausdorff distance between K and H of C(R ) is defined as δ (K, H) = max sup inf k−h , sup inf k−h h∈H h∈H k∈E k∈K It is obvious that (C(R ), δ ) is a complete (not separable) metric space. A fuzzy number is a function Y from R to [0, 1] satisfying (1) Y is normal, in other words, there exists an τ ∈ R such that Y(τ ) = 1; 0 0 (2) Y is fuzzy convex, in other words, for any τ, υ ∈ R and 0 ≤ ς ≤ 1, { } Y (ςτ + (1 − ς ) υ) ≥ min Y(τ ) ,Y(υ) ; (3) Y is upper semi-continuous; p 0 (4) The closure of {υ∈ R : Y(τ ) > 0}; denoted by Y , is compact. These features imply that for each 0 <β ≤ 1, the β−level set β p Y = τ ∈ R : Y (τ ) >β p 0 p is a nonempty compact convex, subset of R , as is the support Y .Let L(R ) denote the set of all fuzzy numbers. The linear structure of L(R ) shows addition Y + Z and scalar FUZZY INFORMATION AND ENGINEERING 145 multiplication ςY, ς ∈ R, in the sense of β−level sets, by β β β Y+Z = Y + Z [ ] [ ] [ ] and β β [ςY] = ς [Y] for each 0 ≤ β ≤ 1. Define for each 1 ≤ q < ∞ 1 q β β d (Y, Z) = δ Y , Z dβ q ∞ β β and d = sup δ (Y ,Z ). Clearly d (Y, Z) = lim d (Y, Z) with d ≤ d if q ≤ r. ∞ ∞ ∞ q→∞ q q r 0≤β≤1 Moreover, d is a complete, separable and locally compact metric space [11]. During the paper, d will denote d with 1 ≤ q ≤∞. Also, a metric d on L(R) is said to be a translation invariant if d(Y + W, Z + W) = d(Y, Z) for Y, Z, W ∈ L(R). Provided that d is a translation invariant metric on L(R), then it is clear to see that d(Y + Z,0) ≤ d(Y,0) + d(Z,0). 2. Main Results In this section, we will present new concepts and examine the relationship among these concepts. Definition 2.1: Let ={(r , s )} be a double lacunary sequence; and p = (p ) be a ξ,η ξ η u,v 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 u,v 1 Y u,v P − lim d , = 0 ξ,η→∞ γ Z u,v ξ,η (u,v)∈J ξ,η (p) (denoted by Y ∼ Z), and simply double lacunary strongly asymptotically equivalent of p (p) N N order α if = 1. If we take p = p for all u and v, then we write Y ∼ Z instead of Y ∼ u,v Z. Definition 2.2: Let p = (p ) be a sequence of positive real numbers, and we consider u,v 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 m,n u,v 1 Y u,v P − lim d , = 0 m,n (mn) Z u,v u,v=1,1 σ (p) (denoted by Y ∼ Z) and in simple terms double strongly Cesáro asymptotically equiva- lent of order α if = 1. 146 RABIA SAVAŞ Definition 2.3: Let ={(r , s )} be a double lacunary sequence; the two nonnegative ξ,η ξ η sequences Y and Z are called as double asymptotically lacunary statistical equivalent of order α to , where 0 <α ≤ 1 if for every > 0, 1 Y u,v P − lim (u, v) ∈ J : d , ≥ = 0 ξ,η ξ,η→∞ γ Z u,v ξ,η (denoted by Y ∼ Z) and simply asymptotically lacunary statistical equivalent, if = 1. Let us prove the following theorems. Theorem 2.1: Let be a double lacunary sequence. Then ξ,η N S (1) If Y ∼ Z then Y ∼ Z. S N 2 ∗ (2) If Y, Z ∈ l (F ) and Y ∼ Z then Y ∼ Z. Proof: (1) If > 0and Y ∼ Z then p p u,v u,v 1 Y 1 Y u,v u,v d , ≥ d , α α γ Z γ Z u,v u,v ξ,η ξ,η (u,v)∈J Y ξ,η u,v (u,v)∈J &d − ≥ ξ,η u,v 1 Y u,v ≥ (u, v) ∈ J : d , ≥ . ξ,η γ Z u,v ξ,η α α S S Therefore, Y ∼ Z. (2) Suppose that Y and Z are in l (F) and Y ∼ Z, then we can presume u,v that d( , ) ≤ M for u and v.Let > 0 be given and N be such that u,v 1 Y u,v p (u, v) ∈ J : d , ≥ ≤ ξ,η γ Z 2 2M u,v ξ,η for all ξ, η> N and let u,v p L := (u, v) ∈ J : d , ≥ . u,v ξ,η Z 2 u,v Now for all ξ, η> N we are granted p p u,v u,v 1 Y 1 Y u,v u,v d − = d , α α γ Z γ Z u,v u,v ξ,η ξ,η (u,v)∈J u,v∈L ξ,η u,v u,v 1 Y u,v + d , γ Z u,v ξ,η u,v∈L u,v 1 1 ξ,η p α ≥ M + γ . α α ξ,η γ 2M γ 2 ξ,η ξ,η Hence, we obtain Y ∼ Z. FUZZY INFORMATION AND ENGINEERING 147 Theorem 2.2: Let be a double lacunary sequence and sup p = H then Y ∼ Z ξ,η u,v u,v implies Y ∼ Z. Proof: Let Y ∼ Z and > 0 be given. Then p p u,v u,v 1 Y 1 Y u,v u,v d , = d , α α γ Z γ Z u,v u,v ξ,η ξ,η (u,v)∈J Y u,v ξ,η (u,v)∈J &d , ≥ ξ,η u,v u,v 1 Y u,v + d , γ Z u,v ξ,η u,v (u,v)∈J &d , < ξ,η u,v u,v 1 Y u,v ≥ d , γ Z u,v ξ,η u,v (u,v)∈J &d , ≥ ξ,η u,v u,v ≥ () ξ,η u,v (u,v)∈J &d , ≥ ξ,η u,v inf p H u,v ≥ min{() , () } ξ,η u,v (u,v)∈J &d , ≥ ξ,η u,v 1 Y u,v inf p H u,v ≥ (u, v) ∈ J : d , ≥ min () , () . ξ,η γ Z u,v ξ,η Hence, Y ∼ Z Theorem 2.3: Let Y and Z be bounded and 0 < h = inf p ≤ sup p = H < ∞. Then u,v u,v u,v u,v S N Y ∼ Z implies Y ∼ Z. Proof: Suppose that Y and Z be bounded and > 0. Since Y and Z are bounded, there is u,v an integer K such that d( , ) ≤ K for all u and v; then u,v u,v p u,v u,v 1 Y 1 Y u,v u,v d , = d , α α γ Z γ Z u,v u,v ξ,η ξ,η (u,v)∈J Y ξ,η u,v (u,v)∈J &d , ≥ ξ,η u,v u,v u,v 1 Y u,v + d , γ Z u,v ξ,η u,v (u,v)∈J &d , < ξ,η u,v h H ≤ max{K , K } ξ,η u,v (u,v)∈J &d , ≥ ξ,η u,v 148 RABIA SAVAŞ u,v + max{} ξ,η u,v (u,v)∈J &d , < ξ,η u,v 1 Y u,v h H ≤ max{K , K } (u, v) ∈ J : d , ≥ ξ,η γ Z u,v ξ,η h H + max{ , }. Hence, Y ∼ Z. Theorem 2.4: Let be a double lacunary sequence with lim inf ζ > 1 and lim inf ζ > ξ,η ξ ξ η σ (p) 1, then Y ∼ Z implies Y ∼ Z. Proof: Suppose lim inf ζ > 1 and lim inf ζ > 1; then there is aδ> 0suchthat ζ > 1 + ξ ξ η ξ σ (p) γ γ ξ η δ δ δ and ζ > 1 + δ. This applies ≥ and ≥ . Then for Y ∼ Z, r 1+δ s 1+δ ξ η u,v 1 Y u,v A = d , ξ,η γ Z u,v ξ,η (ξ,η)∈J ξ,η r s ξ η u,v 1 Y u,v = d , γ Z u,v ξ,η r=1 s=1 r s ξ −1 η−1 u,v 1 Y u,v − d , γ Z u,v ξ,η r=1 s=1 r s ξ η−1 u,v 1 Y u,v − d , γ Z u,v ξ,η r=r +1 s=1 ξ −1 r s ξ −1 η u,v 1 Y u,v − d , γ Z u,v ξ,η r=1 s=s +1 η−1 r s α ξ η u,v r s 1 Y ξ η u,v = d , γ Z r s u,v ξ η ξ,η r=1 s=1 r s α ξ −1 η−1 u,v r s 1 Y ξ −1 η−1 u,v − d , γ Z r s ξ −1 η−1 u,v r=1 s=1 r s ξ α η−1 u,v 1 1 Y η−1 u,v − d , α α α γ γ Z u,v ξ η η−1 r=r +1 s=1 ξ −1 s r η ξ −1 u,v 1 r 1 Y ξ −1 u,v − d , α α γ γ r Z u,v s ξ −1 s=s +1 r=1 η−1 FUZZY INFORMATION AND ENGINEERING 149 Since Y ∼ Z the last two terms tends to zero in the Pringsheim sense, thus α r s ξ η u,v r s 1 Y ξ η u,v A = d , ξ,η γ Z r s u,v ξ η ξ,η r=1 s=1 r s α ξ −1 η−1 u,v r s 1 Y ξ −1 η−1 u,v − d , + o(1). γ r s α Z u,v ξ −1 η−1 ξ,η r=1 s=1 α α α Since γ = (r s ) − (r s ) we are granted the following: ξ η ξ −1 η−1 ξ,η α α r s r s 1 + δ 1 ξ η ξ −1 η−1 ≤ and ≤ . α α γ δ γ δ ξ,η ξ,η The terms r s ξ η u,v 1 Y u,v d , r s u,v ξ η r=1 s=1 and r s ξ −1 η−1 u,v 1 Y u,v d , r s u,v ξ −1 η−1 r=1 s=1 are both Pringsheim null sequences. Hence, A is a Pringsheim null sequence. Conse- ξ,η quently, Y ∼ Z. Theorem 2.5: Let be a double lacunary sequence with lim sup ζ < ∞ and ξ,η ξ σ (p) lim sup ζ < ∞, then Y ∼ Z implies Y ∼ Z. η η r s r ξ η ξ Proof: Since lim sup < ∞ and lim sup < ∞ there exists H > 0suchthat < ξ η r s r ξ −1 η−1 ξ −1 H and < H for all ξ and η.Let Y ∼ Z and > 0 and there exist ξ > 0and η > 0such 0 0 η−1 that for every i ≥ ξ and j ≥ η 0 0 u,v 1 Y u,v A = d , <. i,j γ Z u,v ξ,η (u,v)∈J ξ,η 150 RABIA SAVAŞ Let T = max{A :1 ≤ ξ ≤ ξ and 1 ≤ η ≤ η },and q and w be such that r < q ≤ r and i,j 0 0 ξ −1 ξ s < w ≤ s .Thus, η−1 η m,n u,v 1 Y u,v d , (qw) Z u,v r,s=1,1 r s ξ η u,v 1 Y u,v ≤ d , r s u,v ξ −1 η−1 r,s=1,1 ⎛ ⎞ ξ,η u,v 1 Y u,v ⎝ ⎠ ≤ d , r s Z u,v ξ −1 η−1 t,w=1,1 (u,v)∈J t,w ξ ,η 0 0 1 1 = γ A + γ A t,w t,u t,w t,w α α r s r s ξ −1 η−1 ξ −1 η−1 t,w=1,1 (ξ <t≤ξ)∪(η <w≤η) 0 0 ξ ,η 0 0 T 1 ≤ γ + γ A t,u t,w t,w α α r s r s ξ −1 η−1 ξ −1 η−1 t,w=1,1 (ξ <t≤ξ)∪(η <w≤η) 0 0 Tr s ξ η 1 ξ η 0 0 0 0 ≤ + A γ t,w t,w α α r s r s ξ −1 η−1 ξ −1 η−1 (ξ <t≤ξ)∪(η <w≤η) 0 0 Tr s ξ η 1 ξ η 0 0 0 0 ≤ + sup A γ t,w t,w α α r s r s ξ −1 η−1 t≥ξ ∪w≥η ξ −1 η−1 0 0 (ξ <t≤ξ)∪(η <w≤η) 0 0 Tr s ξ η ξ η 0 0 0 0 ≤ + A t,w r s ξ −1 η−1 (ξ <t≤ξ)∪(η <w≤η) 0 0 Tr s ξ η ξ η 0 0 0 0 2 ≤ + H . r s ξ −1 η−1 Since r and s both approach infinity as both q and w approach infinity. Therefore ξ η q,w u,v 1 Y u,v d , → 0. (qw) Z u,v u,v=1,1 σ (p) Therefore, Y ∼ Z. Disclosure statement The Author declares that there is no conflict of interest. Notes on contributor Rabia Savaş received her BSc (Mathematics) (2012) and MSc (Mathematics) (2014) degree at the Istan- bul Commerce University and Ph.D. (Mathematics) (2019) degree at Sakarya University in Turkey. She also completed her Ph.D. researches at the University of North Florida in the USA (2017-2019). Her research interests are in the areas of pure mathematics, probability, mathematical statistics, FUZZY INFORMATION AND ENGINEERING 151 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 work- ing at Istanbul Medeniyet University in the Department of Mathematics and Science Education as an associate professor. References [1] Fridy JA. On statistical convergence. 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Fuzzy Information and Engineering – Taylor & Francis
Published: Apr 3, 2022
Keywords: Asymptotically Statistical Equivalent; Double Lacunary Sequences; Statistical Limit Points; Fuzzy Numbers; Primary 40A99; Secondary 40A05
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