Tauberian theorems for statistically (C,1,1) summable double sequences of fuzzy numbers

Abstract In this paper, we prove that a bounded double sequence of fuzzy numbers which is statistically convergent is also statistically (C, 1, 1) summable to the same number. We construct an example that the converse of this statement is not true in general. We obtain that the statistically (C, 1, 1) summable double sequence of fuzzy numbers is convergent and statistically convergent to the same number under the slowly oscillating and statistically slowly oscillating conditions in certain senses, respectively.


Introduction
Developed based on the concept of fuzzy sets which was discovered and introduced by Zadeh [1] almost fifty years ago, fuzzy set theory have received more and more attention from researchers in a wide range of disciplines in the past few years. Intending to apply the concept of fuzziness to individual works with different aspects from theoretical to practical in almost all sciences, technology and industry, researchers have arrived uncountable and varied applications of this theory in fields such as artificial intelligence, decision theory, computer science, pattern recognition, finance and stock market, weather prediction, nuclear science, robotics, biomedicine, handwriting analysis, engineering, agriculture, geography, statistics etc. In addition to these, one of the areas which the concept of fuzziness was practised is pure mathematics, as well as there have been many studies in this field as in other areas. Dubois and Prade [2] have introduced the notion of fuzzy numbers and defined the basic operations of addition, subtraction, multiplication and division. In [3], Goetschel and Voxman have presented a less restrictive definition of fuzzy numbers. Matloka [4] has introduced the concepts of bounded and convergent sequences of fuzzy numbers, studied some of their properties and showed that every convergent sequence of fuzzy numbers is bounded. In [5], Nanda has studied the spaces of bounded and convergent sequences of fuzzy numbers and proved that these are complete metric spaces.
In recent years, there has been an increasing interest on the summability methods of sequences of fuzzy numbers. Subrahmanyam [6] has defined the Cesàro summability method for sequences of fuzzy numbers and obtained fuzzy analogues of some classical Tauberian theorems. Talo and Çakan [7] have presented some Tauberian theorems for sequences of fuzzy numbers that convergence follows from Cesàro convergence under some Tauberian conditions.
There are also some interesting results related to Tauberian theorems in which Cesàro summability method of sequences of fuzzy numbers is used (see, [8][9][10]). After the concept of the statistical convergence which is a natural generalization of the concept of ordinary convergence was introduced by Fast [11] and Schoenberg [12] independently, Nuray and Savaş [13] extended this concept to sequences of fuzzy numbers. Talo and Başar [14] have established some Tauberian theorems for sequences of fuzzy numbers which convergence follows from statistical convergence and Cesàro summability, respectively. Moreover, Talo and Çakan [15], Altın et al. [16] have obtained some Tauberian theorems for the statistically Cesàro summable sequences of fuzzy numbers.
After the concept of double sequences of fuzzy numbers was introduced by Savaş [17], Tripathy and Dutta [18] have studied some spaces of double sequences of fuzzy numbers and proved that every P -convergent and bounded double sequence of fuzzy numbers is .C; 1; 1/ summable to its P -convergence. Moreover, Tripathy and Dutta [19] have defined the space of .C; 1; 1/ summable double sequences of fuzzy numbers and obtained some results regarding it. Finally, Çanak et al. [20] have defined the slow oscillation of double sequences of fuzzy numbers in different senses and proved that some Tauberian theorems for .C; 1; 1/ summability of double sequences of fuzzy numbers.
In this paper, we prove that a bounded double sequence of fuzzy numbers which is statistically convergent is also statistically .C; 1; 1/ summable to the same number. We construct an example that the converse of this statement is not true in general. We obtain that the statistically .C; 1; 1/ summable double sequence of fuzzy numbers is convergent and statistically convergent to the same number under the slowly oscillating and statistically slowly oscillating conditions in certain senses, respectively.
We begin by basic definitions and some notations with respect to fuzzy numbers, its linear structure and its algebraic properties. In [3], Goetschel and Voxman introduced the concept of fuzzy numbers as follows: Definition 1.1. Consider a fuzzy subset of the real line u W R ! OE0; 1. Then the mapping u is a fuzzy number if it satisfies the following additional properties: (i) u is normal, i.e., there exists a t 0 2 R such that u.t 0 / D 1: (ii) u is fuzzy convex, i.e., for any t 0 ; t 1 2 R and for any˛2 OE0; 1, u.˛t 0 C .1 ˛/t 1 / minfu.t 0 /; u.t 1 /g: (iii) u is upper semicontinuous on R.
(iv) The support of u, OEu 0 WD ft 2 R W u.t / > 0g is compact, where ft 2 R W u.t / > 0g denotes the closure of the set ft 2 R W u.t / > 0g in the usual topology of R.
We denote the set of all fuzzy numbers on R by E 1 and call it the space of fuzzy numbers. We recall the linear structure of E 1 as follows. For u 2 E 1 , the˛-level set of u is defined by Then, it is easily established (see [21]) that u is a fuzzy number if and only if OEu˛is a closed, bounded and nonempty interval for each˛2 OE0; 1 with OEuˇÂ OEu˛if 0 Ä˛ÄˇÄ 1: From this characterization of fuzzy numbers, it follows that a fuzzy number u is completely determined by the end points of the intervals OEu˛D OEu .˛/; u C .˛/ where u .˛/ Ä u C .˛/ and u .˛/; u C .˛/ 2 R for each˛2 OE0; 1. Immediately after, Goetschel and Voxman [3] presented another representation of a fuzzy number as a pair of functions that satisfy some properties. Theorem 1.2 (Representation Theorem, [3]). Let u 2 E 1 and let OEu˛D OEu .˛/; u C .˛/. Then the functions u ; u C W OE0; 1 ! R, defining the endpoints of the˛-level sets, satisfy the following conditions: (i) u .˛/ 2 R is a bounded, non-decreasing and left continuous function on .0; 1.
(ii) u C .˛/ 2 R is a bounded, non-increasing and left continuous function on .0; 1.
(iii) The functions u .˛/ and u C .˛/ are right continuous at˛D 0.
Conversely, if the pair of functions f and g satisfies the above conditions (i)-(iv), then there exists a unique fuzzy number u such that OEu˛WD OEf .˛/; g.˛/ for each˛2 OE0; 1 and u.x/ WD sup 2OE0;1 f˛W f .˛/ Ä x Ä g.˛/g : Suppose that u; v 2 E 1 are represented by OEu .˛/; u C .˛/ and OEv .˛/; v C .˛/ for each˛2 OE0; 1, respectively. Then, the operations addition and scalar multiplication on the set of fuzzy numbers are defined as follows: OEku˛D kOEu˛D ( ku .˛/; ku C .˛/ ; k 0; ku C .˛/; ku .˛/ ; k < 0: The set of all real numbers can be embedded in 0; x ¤ r: The following lemma deals with the algebraic properties of fuzzy numbers. 22]). (i) The addition of fuzzy numbers is associative and commutative, i.e., (iv) For any a; b 2 R with ab 0 and any u 2 E 1 , we have .a C b/u D au C bu. For general a; b 2 R, this property does not hold.
(v) For any a 2 R and u; v 2 E 1 , we have a.u C v/ D au C av.
(vi) For any a; b 2 R and u 2 E 1 , we have .ab/u D a.bu/.
As a consequence of Lemma 1.3, we attain that the space of fuzzy numbers is not a linear space.
The concept of metric space may be defined as an arbitrary fuzzy set in which the distance between all elements of the set are described. It is possible to define several different metrics on the space of fuzzy numbers; however, the most well known and preferential metric among these metrics is the Hausdorff distance for fuzzy numbers based on the classical Hausdorff distance between compact convex subsets of R n . Let W denote the set of all closed and bounded intervals. For the particular case when A D OEa ; a C , B D OEb ; b C are two intervals, the Hausdorff distance on W is defined by It is known that W is a complete separable metric space in consideration of the Hausdorf distance (cf. Nanda [5]). At the moment, we may define the metric D on the space of fuzzy numbers with the help of the Hausdorff metric d .
Then D is called the Hausdorff distance between fuzzy numbers u and v.
The following proposition presents some properties of the Hausdorff distance between fuzzy numbers.

Preliminaries
In this section, we recall some notations and basic definitions with respect to double sequences of fuzzy numbers which are used throughout this paper. In [17], Savaş introduced the following definitions for double sequences of fuzzy numbers which we need in the sequel: Definition 2.1. A double sequence u D .u mn / of fuzzy numbers is a function u from N N (N is the set of all natural numbers) into the set E 1 . The fuzzy number u mn denotes the value of the function at a point .m; n/ 2 N N and is called the .m; n/-term of the double sequence.
We denote the set of all double sequences of fuzzy numbers by w 2 .F /.
Definition 2.2. A double sequence u D .u mn / of fuzzy numbers is said to be convergent in Pringsheim's sense (or P-convergent) to the fuzzy number 0 , written as lim m;n!1 u mn D 0 , if for every > 0 there exists a positive integer n 0 . / such that D.u mn ; 0 / < whenever m; n n 0 , and we denote by P -lim u D 0 . The number 0 is called the Pringsheim limit of u.
More exactly, we say that a double sequence .u mn / converges to a fuzzy number 0 if .u mn / tends to 0 as both m and n tend to infinity independently of one another. We denote the space of all P -convergent double sequences of fuzzy numbers by c 2 .F /. Note that throughout this paper, we always mean convergence in Pringsheim's sense. We denote the set of all bounded double sequences of fuzzy numbers by`2 1 .F /.
Note that unlike single sequences of fuzzy numbers, every P -convergent double sequences of fuzzy numbers need not be bounded.
It is clear that .u mn / is P -convergent to 0. One can check that the endpoints of the˛-level set of the sequence .u mn / are For a double sequence .u mn / of fuzzy numbers, its .C; 1; 1/ means are defined by mn WD for all nonnegative integers m and n (see [23]).
As in the single sequence of fuzzy numbers, we give the definition of natural density of K N N and we present the statistically convergent double sequence of fuzzy numbers by using this concept. Let K N N be a two dimensional set of positive integers and let K mn D f.j; k/ 2 K W j Ä m; k Ä ng: We say that K has a double natural density if the sequence has a limit in Pringsheim's sense. In this case, we write where the vertical bars denote the cardinality of the enclosed set.
In [23], Savaş and Mursaleen introduced the concept of statistical convergence for double sequences of fuzzy numbers as follows: We denote the set of all statistically convergent double sequences of fuzzy numbers by st 2 .F /. We note that if a double sequence of fuzzy numbers is P -convergent to the fuzzy number 0 , then it is also statistically convergent to same number (see [23]). However, the converse is not necessarily true. In other words, a double sequence of fuzzy numbers which is statistically convergent need not be P -convergent. Now, we construct an example of a double sequence of fuzzy numbers which is statistically convergent, but not P -convergent as follows.
Example 2.7. Consider the double sequence u D .u mn / of fuzzy numbers defined by if n D k 2 ; k 2 N and for all m 2 N 0 otherwise.
It is clear that .u mn / is divergent. On the other hand, it is statistically convergent to 0, since lim m;n!1 for every > 0: We say that the double sequence .u mn / of fuzzy numbers is called statistically .C; 1; 1/ summable to 0 if st 2 lim m;n!1 mn D 0 . One of the main theorems of this paper shows that if a double sequence of fuzzy numbers is statistically convergent, then it is also statistically .C; 1; 1/ summable provided that it is bounded.
We now define the concepts of slow oscillation for the double sequences .u mn / of fuzzy numbers in certain senses as follows.
We say that a double sequence .u mn / of fuzzy numbers is slowly oscillating in sense or equivalently, if for every > 0 there exist n 0 D n 0 . / and D . / > 1 such that Here, by n we denote the integral part of the product n. It easily follows from (3) that every P -convergent double sequence of fuzzy numbers is slowly oscillating in sense .1; 1/, but the converse is not true in general. An example indicating that the converse is not true was constructed by Çanak et al. [20].
We say that a double sequence .u mn / of fuzzy numbers is said to be slowly oscillating in sense We say that a double sequence .u mn / of fuzzy numbers is said to be slowly oscillating in the strong sense We say that a double sequence .u mn / of fuzzy numbers satisfies the two-sided Tauberian condition of Hardy type in sense .1; 0/ if there exist positive constants n 0 and H such that jD.u j n ; u j 1;n / Ä H whenever j; n > n 0 : It is clear that if (6) holds, then .u mn / is slowly oscillating in both sense .1; 0/ and the strong sense .1; 0/. Similarly, the slow oscillation of the double sequence .u mn / of fuzzy numbers in sense .0; 1/ and the strong sense .0; 1/ can be analogously defined. In addition to these, a double sequence .u mn / of fuzzy numbers which satisfies the two-sided Tauberian condition of Hardy type in sense .0; 1/ can be defined and it is slowly oscillating in both sense .0; 1/ and strong sense .0; 1/.
As a matter of fact, without loss of generality we suppose that the double sequence .u mn / is slowly oscillating in senses .1; 0/, .0; 1/ and in the strong sense .1; 0/. For every large enough m and n, that is, m; n n 0 and > 1, Taking the lim sup and the limit of both sides of this inequality as m; n ! 1 and ! 1 C respectively, we obtain that the terms on the right-hand side of this inequality tends to 0. Therefore, we obtain that .u mn / is slowly oscillating in sense .1; 1/.
On the other hand, a double sequence .u mn / of fuzzy numbers is said to be statistically slowly oscillating in A double sequence .u mn / of fuzzy numbers is statistically slowly oscillating in the strong sense .1; 0/ if, for every > 0, (7) is satisfied with (5). Similarly, the statistical slow oscillation of the double sequence .u mn / of fuzzy numbers in sense .0; 1/ and the strong sense .0; 1/ can be analogously defined.
We note that if the double sequence .u mn / of fuzzy numbers is slowly oscillating in sense .1; 0/, then .u mn / is statistically slowly oscillating in sense .1; 0/.
As a matter of fact, suppose that the double sequence .u mn / is slowly oscillating in sense .1; 0/. Given any > 0. For every large enough m and n, that is, m; n n 0 . /, we have Taking the lim sup of both sides of the inequality (8) as M; N ! 1, we obtain that the term on the right-hand side of the inequality (8) tends to 0. Then taking the limit of both sides of the last inequality as ! 1 C , we conclude that .u mn / is statistically slowly oscillating in sense .1; 0/. Note that there is a similar relation between the concepts of the slow oscillation and the statistical slow oscillation in sense .0; 1/. In a similar way, we can also indicate that the double sequence .u mn / of fuzzy numbers which is slowly oscillating in the strong sense .1; 0/ (or .0; 1/) is statistically slowly oscillating in the strong sense .1; 0/ (or .0; 1/).

Lemmas
In this part of the paper, we state and prove the following assertions which will be used in the proofs of our main theorems. The following lemma is the decomposition theorem for statistically convergent double sequences of fuzzy numbers. (i) The double sequence .u mn / of fuzzy numbers is statistically convergent to a fuzzy number 0 .
(ii) There exists a P -convergent double sequence .w mn / of fuzzy numbers such that In the following lemma, we present two representations for the distance between the general terms of the double sequences .u mn / and . mn /.
ii) If 0 < < 1, m < m, and n < n, then Proof. i) For > 1, we obtain by the definition of the .C; 1; 1/ means of .u mn / that which completes the proof of Lemma 3.2 (i). ii) For 0 < < 1, following a procedure which is similar to the proof of Lemma 3.2 (i), we can reach the conclusion in Lemma 3.2 (ii).
In the following two lemmas, we state the fuzzy analogues of the lemmas given for single sequences of complex numbers by Móricz (see [24], Lemma 9) for double sequences of fuzzy numbers. These lemmas play a crucial role in the proofs of subsequent lemmas which are required for the proofs of our main theorems. Proof. Assume that n 0 is large enough to satisfy the condition without loss of generality. Let n 0 < j . We define the subsequence j 0 WD j and j p WD 1 C Ä j p 1 ; p D 1; 2; :::; r where r is determined by the condition j rC1 Ä n 0 < j r . It follows from the definition of the subsequence .j p / that we find j p < j p 1 < j p ; p D 1; 2; :::; r C 1: Choose m such that 1 Ä m Ä j . We examine chosen m in two cases such that n 0 Ä m Ä j and 1 Ä m < n 0 . We firstly consider the case n 0 Ä m Ä j . Then, we define p such that j pC1 Ä m < j p for some 1 Ä p Ä r: Taking into account the assumption above, we obtain for n 0 Ä n D.u j n ; u mn / Ä D.u j n ; u j 1 n / C D.u j 1 n ; u mn / Ä D.u j n ; u j 1 n / C D.u j 1 n ; u j 2 n / C D.u j 2 n ; u mn / Ä D.u j n ; u j 1 n / C D.u j 1 n ; u j 2 n / C ::: C D.u j p 1 n ; u j p n / C D.u j p n ; u mn / Ä p C 1: By the definition of the subsequence .j p /, we find :::; From this point of view, we arrive by using (9) and (10). It follows from the fact that we find If we combine (11) and (12), then we conclude for n 0 < n that On the other hand, we consider the case 1 Ä m < n 0 . Then going through similar process as above and taking the assumption into account, for n 0 < n we get D.u j n ; u mn / Ä D.u j n ; u j 1 n / C D.u j 1 n ; u mn / Ä D.u j n ; u j 1 n / C D.u j 1 n ; u j 2 n / C D.u j 2 n ; u mn / Ä D.u j n ; u j 1 n / C D.u j 1 n ; u j 2 n / C ::: C D.u j r n ; u n 0 n / C D.u n 0 n ; u mn / where c WD max 1Äm<n 0 D.u n 0 n ; u mn /. If we follow a similar process to (12), then we find r Ä 1 log log 2j m whenever 1 Ä m < n 0 : If we combine (14) and (15), then we conclude for n 0 < n that D.u j n ; u mn / Ä 1 C c C 1 log log 2j m ; whenever 1 Ä m < n 0 : It follows from (13) and (16) that we have for n 0 < n D.u j n ; Proof. Following a procedure which is similar to the proof of Lemma 3.3, we can prove Lemma 3.4.
In [25], Armitage and Maddox proved that a single sequence .t n / D . P n kD0 .s n s k // of real numbers is bounded below provided that the single sequence .s n / of real numbers satisfies the slowly decreasing condition which is less restrictive than slowly oscillating condition. Based on this statement, we also demonstrate that the double sequences of fuzzy numbers given in Lemma 3.5, Lemma 3.6 and Lemma 3.7 are bounded under the slowly oscillating conditions in certain senses. Proof. Assume that a double sequence .u mn / of fuzzy numbers is slowly oscillating in sense .1; 0/. Then there exist a positive integer n 0 and > 1 such that D.u j n ; u mn / Ä 1 whenever n 0 Ä m < j < m and n 0 Ä n: It follows from Lemma 3.3 that there exists a constant H such that D.u j n ; u mn / Ä H log Â j m Ã whenever 1 Ä m Ä j and n 0 Ä n: If we especially take m D 1, we have D.u j n ; u 1n / Ä H log j for Ä j and n 0 Ä n. Therefore, we get that the sequence Â 1 j C 1 D.u j n ; u 0n / Ã is bounded. In order to accomplish the proof, it is enough to show that 0 @ 1 j C 1 j X mD1 D.u j n ; u mn / 1 A is bounded. Using (17) and (18), we obtain that for n 0 Ä j and n 0 Ä n j X mD1 D.u j n ; u mn / D

<
: where OE: denotes the integer part of a real number. Therefore, we conclude that the sequence is bounded. Proof. Following a procedure which is similar to the proof of Lemma 3.5, we can prove Lemma 3.6. Lemma 3.7. If the double sequence .u mn / of fuzzy numbers is slowly oscillating in senses .1; 0/, .0; 1/ and slowly oscillating in the strong sense .1; 0/ or .0; 1/, then the sequence Proof. Assume that a double sequence .u mn / of fuzzy numbers is slowly oscillating in senses .1; 0/, .0; 1/ and slowly oscillating in the strong sense .1; 0/ without loss of generality. Then there exist positive integers n 0 , n 1 and > 1 such that D.u j n ; u mn / Ä 1 whenever n 0 Ä m < j < m and n 0 Ä n and D.u mk ; u mn / Ä 1 whenever n 1 Ä n < k < n and n 1 Ä m; respectively. It follows from Lemma 3.3 and Lemma 3.4 that there exist constants H 0 and H 1 such that and D.u mk ; u mn / Ä H 1 log Â k n Ã whenever 1 Ä n Ä k and n 1 Ä m; respectively. If we especially take m; n D 1, we have D.u j k ; u 1k / Ä H 0 log j for Ä j , n 0 Ä k and D.u 1k ; u 11 / Ä H 1 log k for n 0 < Ä k. Using these inequalities, Lemma 3.5 and Lemma 3.6, if we consider that the sequences is bounded, which were obtained in proofs of Lemma 3.5 and Lemma 3.6 respectively, then we can find Therefore, the sequence In order to accomplish the proof, it is enough to show that 1 .j C1/.kC1/ P j mD1 P k nD1 D.u j k ; u mn / Á is bounded. If we consider n 2 D maxfn 0 ; n 1 g and combine (19), (20), (21) and (22), we obtain that for n 2 Ä j and where H D maxfH 0 ; H 1 g and OE: denotes the integer part of a real number. Therefore, we conclude that the sequence is bounded.
At once, we present a lemma for statistical convergence which is studied as a summability method. For each .m; n/, we write m and n as m C 1 D a n .m/ C b n .m/ and n C 1 D a m .n/ C b m .n/ where a n .m/ D maxfp Ä m W u pn D w pn g and a m .n/ D maxfq Ä n W u mq D w mq g. If these sets are empty, we can take a n .m/ D a m .n/ D 1. Furthermore, this can take place for at most finite numbers of m and n. To show that lim m;n!1 b n .m/ a n .m/ D lim assume that the contrary of this statement holds, i.e., b n .m/ a n .m/ > > 0; b m .n/ a m .n/ > > 0 (24) for infinitely many m and n. In that case, It follows that .u mn / is not statistically convergent. However, this contradicts the hypothesis. Hence, (23) must be true. Starting from this point of view, we can also write that 1 Ä a n .m/ C b n .m/ a n .m/ ! 1; 1 Ä a m .n/ C b m .n/ a m .n/ ! 1 (25) as m; n ! 1. Now, consider the distance between the double sequences w a n .m/;a m .n/ and u mn of fuzzy numbers. Then, we have D w a n .m/;a m .n/ ; u mn D D u a n .m/;a m .n/ ; u a n .m/Cb n .m/;a m .n/Cb m .n/ : If we consider that .u mn / is slowly oscillating in sense .1; 1/, the distance between the double sequences w a n .m/;a m .n/ and u mn tends to 0 as m; n ! 1. Since lim m;n!1 w mn D 0 , we conclude lim We note that another proof of Lemma 3.8 was given by Talo and Bayazit [26].
Since .u mn / is statistically .C; 1; 1/ summable to a fuzzy number 0 , the last term on the right-hand side of the inequality (26) as M; N ! 1 equal to 0. For this reason, we have st 2 lim m;n!1 m ; n D 0 : Let 0 < < 1. Firstly, we must prove that the same term mn cannot occur more than or equivalently, m Ä k < .k C 1/ < ::: < .k C r 1/ < m C 1 Ä .k C r/; n Ä `< .`C 1/ < ::: < .`C s 1/ < n C 1 Ä .`C s/: From this point of view, we can write that m C .r 1/ Ä .k C r 1/ < m C 1; n C .s 1/ Ä .`C s 1/ < n C 1; that is, r < 1 C 1 and s < 1 C 1 . By virtue of M C 1 M C 1 < 2 , we obtain that jfm Ä M and n Ä N W D . mn ; 0 / j gj : Since .u mn / is statistically .C; 1; 1/ summable to a fuzzy number 0 , the last term on the right-hand side of the inequality (27) as M; N ! 1 equal to 0. Therefore, we also obtain that st 2 lim m;n!1 m ; n D 0 in this case.

The statistical (C,1,1) summability
In this section, we prove that the statistical convergent double sequence of fuzzy numbers is statistically .C; 1; 1/ summable to the same number under the boundedness condition of the double sequence, as well. Immediately after, we give an example that the converse of this statement is not true in general. Finally, we indicate that the conditions under which P -convergence and statistical convergence follow from the statistical .C; 1; 1/ summability. Theorem 4.1. Let the double sequence .u mn / of fuzzy numbers be bounded. If .u mn / is statistically convergent to a fuzzy number 0 , then .u mn / is statistically .C; 1; 1/ summable to the same number.
Proof. Assume that the double sequence .u mn / of fuzzy numbers is bounded and statistically convergent to a fuzzy number 0 . Then, there exists a positive constant C such that sup m;n D.u mn ; 0 / < C for all m; n 2 N. Given an > 0, by the definition of statistical convergence, we have lim m;n!1 where K mn D fj Ä m and k Ä n W D.u j k ; 0 / g: By the arithmetic mean . mn / of .u mn /, we may write that for any given > 0 there exists a n 0 D n 0 . / 0 such that for m; n > n 0 . / in other words, . mn / is statistically convergent to 0 . Therefore, we conclude that .u mn / is statistically .C; 1; 1/ summable to 0 .
By the following example we indicate that the converse of Theorem 4.1 does not hold in general. if m is odd,n is even. Therefore, the double sequences . mn .˛// and . C mn .˛// converge to .˛ 2/=12 and .2 ˛/=12 as m; n ! 1, respectively. If we take 0 D .! 0 C 0 /=4, we obtain lim m;n!1 D. mn ; 0 / D 0, that is, . mn / is convergent to 0 . In addition to this, we can say that . mn / is statistically convergent to 0 because the every convergent double sequence of fuzzy number is statistically convergent to same number. In other words, .u mn / is statistically .C; 1; 1/ summable to 0 . On the other hand, we can easily check that D .u mn ; 0 / for every m; n 2 N and 0 < < 1=6. In this case, we obtain that The main focus of this work is to find some suitable Tauberian conditions which ensure that the converse of Theorem 4.1 is true.
Proof. Assume that a double sequence .u mn / of fuzzy numbers which is statistically .C; 1; 1/ summable to a fuzzy number 0 is slowly oscillating in senses .1; 0/; .0; 1/ and slowly oscillating in the strong sense .1; 0/ without loss of generality. In order to prove that .u mn / is P -convergent to the same number, we firstly prove that if .u mn / is slowly oscillating in senses .1; 0/, .0; 1/ and the strong sense .1; 0/, then . mn /, which is the arithmetic means of .u mn /, is slowly oscillating in sense .1; 1/. Given an > 0, from the assumptions, there exist positive integers n 0 D n 0 . /, n 1 D n 1 . / and D . / > 1 such that D.u j n ; u mn / Ä whenever n 0 Ä m < j Ä m and n 0 Ä n and D.u mk ; u mn / Ä whenever n 1 Ä n < k Ä n and n 1 Ä m; respectively. In addition to this, it is known that if .u mn / is slowly oscillating in senses .1; 0/, .0; 1/ and slowly oscillating in the strong sense .1; 0/, then it is also slowly oscillating in sense .1; 1/. In this case, there exist positive integer n 2 D maxfn 0 ; n 1 g and D . / > 1 such that D.u j k ; u mn / Ä whenever n 2 Ä m < j Ä m and n 2 Ä n < k Ä n : Let n 2 Ä m < j Ä m and n 2 Ä n < k Ä n . By the definition of the .C; 1; 1/ means of .u mn /, we obtain that .j m/.k n/ .j C 1/.k C 1/.m C 1/.n C 1/ .j m/.k n/ .j C 1/.k C 1/.m C 1/.
.j m/.k n/ .j C 1/.k C 1/.m C 1/.n C 1/ m X pD0 n X qD0 D .u mn ; u pq / C .j m/ .j C 1/.k C 1/.m C 1/ m X pD0 n X qD0 D .u mq ; u pq / C .k n/ .j C 1/.k C 1/.n C 1/ m X pD0 n X qD0 D .u pn ; u pq / : (27) By Lemma 3.5, Lemma 3.6 and Lemma 3.7, there exist constants H 0 ; H 1 and H 2 such that for all non-negative integers m and n. Using these inequalities and the definition of slow oscillation in certain senses, we find that since we have that for > 1 and whenever m < j Ä m and n < k Ä n . If we choose maxfH 0 ; H 1 ; H 2 g D H and 1 < Ä r C H C 1, then we arrive D j k ; mn Ä whenever n 2 Ä m < j Ä m and n 2 Ä n < k Ä n . Therefore, we reach that . mn / is slowly oscillating in sense .1; 1/. Since . mn / is slowly oscillating in sense .1; 1/ and statistically convergent to 0 , we obtain that . mn / is convergent to 0 by Lemma 3.8. If we consider that the condition of slowly oscillating in sense .1; 1/ is Tauberian condition for .C; 1; 1/ means of sequence as a result of Lemma 3.9, then we conclude that .u mn / is P -convergent to 0 .

Remark 4.4.
Because it is known that if the two-sided Tauberian conditions of Hardy type in senses .1; 0/ and .0; 1/ hold, then the double sequence of fuzzy numbers is slowly oscillating in senses .1; 0/ and .0; 1/ and also slowly oscillating in the strong senses .1; 0/ and .0; 1/ respectively, we can say that the double sequence .u mn / of fuzzy numbers which is statistically .C; 1; 1/ summable to a fuzzy number is P-convergent to same number under the two-sided Tauberian conditions of Hardy type in senses .1; 0/ and .0; 1/.
If we replace the conditions of slow oscillation by the conditions of statistically slow oscillation in Theorem 4.3, then we may not attain P -convergence of the double sequence .u mn / of fuzzy numbers. In this case, we reach its statistical convergence instead of its P -convergence. Proof. Assume that a double sequence .u mn / of fuzzy numbers which is statistically .C; 1; 1/ summable to a fuzzy number 0 is statistically slowly oscillating in senses .1; 0/; .0; 1/ and statistically slowly oscillating in the strong sense .1; 0/ without loss of generality. In order to prove that .u mn / is statistically convergent to the same number, it is enough to prove that Therefore, we conclude that .u mn / is statistically convergent to 0 . Corollary 4.6. Let the double sequence .u mn / of fuzzy numbers be statistically .C; 1; 1/ summable to a fuzzy number 0 . If .u mn / satisfies the two-sided Tauberian conditions of Hardy type in senses .1; 0/ and .0; 1/, then .u mn / is also statistically convergent to 0 .
Proof. Assume that a double sequence .u mn / of fuzzy numbers which is statistically .C; 1; 1/ summable to a fuzzy number 0 satisfies the two-sided Tauberian conditions of Hardy type in senses .1; 0/ and .0; 1/, that is, there exist positive constants n 0 D n 0 . / and C 1 ; C 2 such that mD.u mn ; u m 1;n / Ä C 1 and nD.u mn ; u m;n 1 / Ä C 2 (34) whenever m; n > n 0 . Using the condition in sense .1; 0/, we obtain that for every > 0 and m; n > n 0 max mC1Äj Ä m nC1ÄkÄ n D u j k ; u mk Ä max 1/ Ä whenever 1 < Ä 1 C =C 1 . Therefore, the set fn 0 < m Ä M and n 0 < n Ä N W max mC1Äj Ä m nC1ÄkÄ n D u j k ; u mk g is empty. This implies that if the two-sided Tauberian condition of Hardy type in sense .1; 0/ holds, then .u mn / is statistically slowly oscillating in the strong sense .1; 0/. Similarly, we can indicate that .u mn / which satisfies the two-sided Tauberian condition of Hardy type in sense .0; 1/ is statistically slowly oscillating in the strong sense .0; 1/. In addition to this, if we follow a similar procedure to the above, we also attain that .u mn / is statistically slowly oscillating in senses .1; 0/ and .0; 1/ by taking advantage of these conditions. As a result, it follows from Theorem 4.5 that .u mn / is statistically convergent to 0 .