Abstract
We define the notions of weighted -statistical convergence of order and strongly weighted -summability of for fuzzy double sequences, where . We establish an inclusion result and a theorem presenting a connection between these concepts. Moreover, we apply our new concept of weighted -statistical convergence of order to prove Korovkin-type approximation theorem for functions of two variables in a fuzzy sense. Finally, an illustrative example is provided with the help of -analogue of fuzzy Bernstein operators for bivariate functions which shows the significance of our approximation theorem.
1. Introduction and Preliminaries
The notion of weighted statistical convergence for sequences of real numbers has been studied by Karakaya and Chishti [1] as a generalization of the concept of statistical convergence which is related to the idea of asymptotic density (or natural density) of a subset of , the set of natural numbers, according to Fast [2]. The weighted statistical convergence was further improved by Mursaleen et al. [3] and later generalized by Belen and Mohiuddine [4] with a view of nondecreasing sequence of positive numbers. Ghosal [5] added constraints to these ideas. For some other related work, we refer the interested reader to [6–10]. Recently, Çolak [11] defined the notion of statistical convergence of order and strong -Cesàro summability of order while the order of statistical was also given in [12], and it reduces to strong -Cesàro summability for according to Connor [13].
The idea convergence for double sequence was presented by Pringsheim [14]; that is, is convergent to in the Pringsheim sense if for every , there is such that whenever . The idea of statistical convergence for was introduced by Mursaleen and Edely [15] as follows: is statistically convergent to if for every , where the limit is taken in Pringsheim’s sense while the order of notion was studied in [16]. This idea was further generalized by Mursaleen et al. [17] and called it -statistical convergence, and for single sequence in [18] and its weighted variant were presented by Cinar and Et [19], which were later on studied by various authors in various setups [20–22]. Throughout this paper, limit of the double sequences means limit in Pringsheim’s sense.
In the very recent past, for sequence of fuzzy numbers, Mohiuddine et al. [23] weighted statistical convergence and strong weighted summability by using the idea of difference operator and established a connection between these notions while under certain conditions, these notions reduce to statistical convergence and strong -Cesáro summability, respectively, in [24, 25]. Savas and Mursaleen [26] defined statistical convergence and statistically Cauchy for fuzzy double sequences and obtained that these concepts are equivalent. The statistical analogue of pointwise and uniformly convergent double sequences for fuzzy-valued functions were discussed by Mohiuddine et al. [21]. These concepts were studied and extended by several authors; for instance, see [27–31].
Recall that, in [32], a fuzzy number is a fuzzy set on the real axis; that is, which is normal, fuzzy convex, upper semicontinuous and the closure of the set is compact. By using the symbol , we denote the set of all fuzzy numbers on . -Level set of is given as follows:
For each , is closed and bounded and nonempty interval for each defined by , where and . For any , the partial ordering is defined by for all . For any and , define the following:
By means of the Hausdorff metric , the metric is defined by
It is clear from [33] that is a complete metric space.
Suppose are two fuzzy number-valued functions. Then, the distance between and is defined by
2. Generalized Weighted Statistical Convergence of Order
Suppose two nondecreasing sequence and of positive integers such that both and tending to infinity; that is, and as and , respectively, such that
Consider two sequences and of nonnegative numbers such that , and
Assume that and let . We define the weighted density of order , denoted by , of by provided the limit in the last relation (9) exists, where
Definition 1. A fuzzy double sequence is said to be weighted -statistically convergent of order , where ; in short, we shall write -convergent, to a fuzzy number if for every , the set has density zero; that is, Equivalently, we can write
We denote this convergence by or . We denote the set of all -convergence by .
The choice of in the notion of -convergence gives weighted -statistically convergence for fuzzy double sequences. In addition, if we choose , , , and for all , then -convergence gives the notion of -statistically convergence in a fuzzy sense [26].
For , the notion of weighted -statistically convergent of order is well but not well defined for . To show this assertion, we consider the following example.
Example 1. Assume that , and define the fuzzy double sequence by Let be given, and let and . Then, On the other hand, We conclude from (15) and (16) that which is impossible.
Example 2. Define as
for , where stands for a fixed fuzzy number. Let , and let and . Let and . Suppose is the upper bound of the -cut. Then, for , and , one writes
Thus,
Hence, but is not convergent.
Theorem 2. Suppose such that and . Then, .
Proof. Let be given, and let . If and , then, for every , we can write which yields that .
Theorem 3. Suppose two fuzzy double sequence and such that and . Then, (i) (ii)
Proof. (i)Let . It is clear for . Let . For , we write
Thus, (i) holds.
Suppose that and . Let be given. Then,
Hence, (ii) holds.
Definition 4. Assume that . Then, a fuzzy double sequence is said to be strongly weighted -summable of order ; in short, we shall write -summable, to fuzzy number , denoted by such that
Theorem 5. Consider a fuzzy double sequence , and let . Then, (i) implies (ii) is a bounded double sequence and imply
Proof. (i)Let , and let be given. Then,
Therefore, we get
Letting on both sides of the above relation, we get .
Let . Since , for all . For a given , one writes
It follows that .
3. Application to Fuzzy Korovkin-Type Theorems
The fuzzy version of Korovkin theorem has been obtained by Anastassiou [34] while the classification of this result has been established by Korovkin in [35] (also see [36–42]), and then Anastassiou and Duman [43] and Karaisa and Kadak [44] studied this result in a statistical sense. In the recent past, Mohiuddine et al. [23] investigated the fuzzy Korovkin-type approximation theorem through weighted statistical convergence of fuzzy sequence based on difference operators. The fuzzy Korovkin theorem for function of two variables was discussed by Demirci and Karaku [45] with the help of -statistical convergence for a sequence of fuzzy numbers. Here, we prove the fuzzy Korovkin-type theorem for the function of two variables with the help of -convergence.
Assume that and is a fuzzy number-valued function. Then, is fuzzy continuous at a point of if whenever is Pringsheim’s convergent to in a fuzzy sense. We denote the set of fuzzy continuous function on by . Here, is only a cone but not a vector space and let be an operator. Further, is said to be a fuzzy linear operator if for all , , and . In addition, if holds for all and with , then is called fuzzy position linear operator.
Theorem 6. Let be a double sequence such that . Let be corresponding positive linear operators acting from into itself having the relation Assume further that where , , , and . Then, for all and .
Proof. Suppose that (31) holds and . Suppose also that , , and . Since , it follows that, for a given , there exists such that for all , we have
By the fuzzy boundedness of , we may write for all , where for all . We then obtain for all that
With the help of last relations, for all and , we obtain
Since is a positive linear operator, by applying this operator to (34), we get where . We thus obtain by letting and taking that
From (30), we can write and so
With the help of (37) and (39), we get where
Multiplying by the product of two nonnegative sequences and as details given in the previous section, one gets
For a given , choose such that . Then, upon setting,
It follows that
Consequently, by taking weighted density of order , we obtain
Using the hypothesis (31), we conclude that holds for all and .
Recall that, for any nonnegative integer , the -integer is given by and the -factorial by
The -binomial coefficients, for , is defined by
The bivariate case of classical -Bernstein operators were introduced and studied by Barbosu [46]. We are now considering the fuzzy analogue of these operators to construct an example to illustrate our last theorem.
Example 3. Consider the fuzzy analogue of bivariate -Bernstein operators, defined by for all , , and , where
Note that
The above operators are positive and linear, and one writes for all , . Suppose that and such that as and as . In view of this assumption, one can easily find that
We define the sequence of positive linear operators by
In this case, is same as defined in Example 2 with , , , , and for all . Therefore, we get
Since we observe that
Consequently, by Theorem 6, we obtain for all . Hence, all the conditions of Theorem 6 hold true for our operators but Theorem 2.2 obtained in [45] for convergence in Pringsheim’s sense does not work for since the fuzzy double sequence is not convergent in Pringsheim’s sense. Thus, we conclude that Theorem 6 is stronger than the one proved for convergence in Pringsheim’s sense.
Data Availability
Not applicable.
Conflicts of Interest
The author declares there are no conflicts of interest.
Acknowledgments
This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. (RG-36-130-38). The author, therefore, acknowledges with thank DSR for technical and financial support.