# Generalized Weighted Statistical Convergence for Double Sequences of Fuzzy Numbers and Associated Korovkin-Type Approximation Theorem

Authors: Abdullah Alotaibi
Journal: Journal of Function Spaces
 (2020)

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## Abstract

We define the notions of weightedλ,μ-statistical convergence of order γ1,γ2 and strongly weighted λ,μ-summability of γ1,γ2 for fuzzy double sequences, where 0<γ1,γ2≤1. We establish an inclusion result and a theorem presenting a connection between these concepts. Moreover, we apply our new concept of weighted λ,μ-statistical convergence of order γ1,γ2 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 q-analogue of fuzzy Bernstein operators for bivariate functions which shows the significance of our approximation theorem.


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## Body

## 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 ℕ, 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 α0<α≤1 and strong p-Cesàro summability of order α while the order of statistical was also given in [12], and it reduces to strong p-Cesàro summability for α=1 according to Connor [13].The idea convergence for double sequencey=ykl was presented by Pringsheim [14]; that is, ykl is convergent to ξ in the Pringsheim sense if for every ε>0, there is N∈ℕ such that ykl−ξ<ε whenever k,l>N. The idea of statistical convergence for y=yklk,l∈ℕ×ℕ was introduced by Mursaleen and Edely [15] as follows: ykl is statistically convergent to ξ if for every ε>0,
(1)limk1,l1→∞1k1l1k,l,k≤k1,l≤l1:ykl−ξ≥∈=0,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 λ,μ-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 p-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, u:ℝ⟶0,1 which is normal, fuzzy convex, upper semicontinuous and the closure of the set y∈ℝ:uy>0 is compact. By using the symbol Lℝ, we denote the set of all fuzzy numbers on ℝ. α-Level set uα of u is given as follows:
(2)ua=y∈ℝ:uy≥αif0<α≤1,y∈ℝ:uy>0¯ifα=0.For eachα∈0,1, uα is closed and bounded and nonempty interval for each α∈0,1 defined by uα=uα−,uα+, where uα−≤uα+ and uα−,uα+∈ℝ. For any u1,u2∈Lℝ, the partial ordering is defined by
(3)u1⪯u2⇔u1α−≤u2α−,u1α+≤u2α+,for all 0≤α≤1. For any u1,u2∈Lℝ and α∈0,1, define the following:
(4)u1⊕u2α=u1α⊕u2α,μ⊙μ1α=μu1αμ∈ℝ.By means of the Hausdorff metricD, the metric d:Lℝ×Lℝ⟶ℝ+ is defined by
(5)du1,u2=supα∈0,1Du1α,u2α=supα∈0,1maxu1α−−u2α−,u1α+−u2α+.It is clear from [33] that Lℝ,d is a complete metric space.Supposeg1,g2:a,b×a,b⟶Lℝ are two fuzzy number-valued functions. Then, the distance between g1 and g2 is defined by
(6)d∗g1,g2=supy∈a,b×a,bsupα∈0,1maxg1α−−g2α−,g1α+−g2α+.


## 2. Generalized Weighted Statistical Convergence of Orderγ1,γ2

Suppose two nondecreasing sequenceλ=λk and μ=μl of positive integers such that both λ and μ tending to infinity; that is, λk⟶∞ and μl⟶∞ as k⟶∞ and l⟶∞, respectively, such that
(7)λk+1≤λk+1,λk=1,μl+1≤μl+1,μl=1.Consider two sequencesu=um and v=vn of nonnegative numbers such that liminfmum>0, liminfnvn>0 and
(8)Uλk=∑m∈k−λk+1,kum⟶∞k⟶∞,Vμ1∑n∈l−μl+1,lvn⟶∞l⟶∞.Assume thatM⊆ℕ×ℕ and let 0<γ1,γ2≤1. We define the weighted λ,μ density of order γ1,γ2, denoted by δ2,γ1,γ2N¯λ,μ, of M by
(9)δ2,γ1,γ2N¯λ,μM=limk,l→∞AUλk,VμlUλkγ1Vμlγ2,provided the limit in the last relation (9) exists, where
(10)AUλk,Vμl=m≤Uλk,n≤Vμl:m,n∈M.Definition 1.
A fuzzy double sequencesmn is said to be weighted λ,μ -statistically convergent of order γ1,γ2 , where 0<γ1,γ2≤1 ; in short, we shall write S2,γ1,γ2N¯λ,μ -convergent, to a fuzzy number s0 if for every ε>0 , the set
(11)B=m,n∈ℕ×ℕ::umvndsmn,s0≥∈has δ2,γ1,γ2N¯λ,μ density zero; that is,
(12)S2,γ1,γ2N¯λ,μB=0.
Equivalently, we can write(13)limk,l→∞1Uλkγ1Vμlγ2m,n,m≤Uλk,n≤Vμl:umvndsmn,s0≥∈=0.We denote this convergence bysmn⟶S2,γ1,γ2N¯λ,μs0 or S2,γ1,γ2N¯λ,μ−limsmn=s0. We denote the set of all S2,γ1,γ2N¯λ,μ-convergence by S2,γ1,γ2N¯λ,μ.The choice ofγ1=γ2=1 in the notion of S2,γ1,γ2N¯λ,μ-convergence gives weighted λ,μ-statistically convergence for fuzzy double sequences. In addition, if we choose λk=k, μl=l, um=1, and vn=1 for all m,n, then S2,γ1,γ2N¯λ,μ-convergence gives the notion of λ,μ-statistically convergence in a fuzzy sense [26].For0<γ1,γ2≤1, the notion of weighted λ,μ-statistically convergent of order γ1,γ2 is well but not well defined for γ1,γ2>1. To show this assertion, we consider the following example.Example 1.
Assume thatγ1,γ2>1 , and define the fuzzy double sequence smn by
(14)smn=1¯ifm+neven,0¯ifm+nodd.
Letε>0 be given, and let um=1 and vn=1∀m,n∈ℕ. Then,
(15)limk,l→∞1Uλkγ1Vμlγ2m,n,m≤Uλk,n≤Vμl:umvndsmn,1¯≥∈≤limk,l→∞λkμl2λkγ1μlγ2=0.
On the other hand,(16)limk,l→∞1Uλkγ1Vμlγ2m,n,m≤Uλk,n≤Vμl:umvndsmn,0¯≥∈≤limk,l→∞λkμl2λkγ1μlγ2=0.
We conclude from (15) and (16) that
(17)S2,γ1,γ2N¯λ,μ−limsmn=1¯,S2,γ1,γ2N¯λ,μ−limsmn=0¯,which is impossible.Example 2.
Definesmn as
(18)smn=bifm=k2,n=l2,0¯otherwise,fori,j∈ℕ, where b stands for a fixed fuzzy number. Let γ1=γ2=1, and let um=1 and vn=1∀m,n. Let λk=k and μl=l. Suppose bα+ is the upper bound of the α-cut. Then, for ε>0, 0≤α≤1 and i,j∈ℕ, one writes
(19)dsmn,0¯=sup0≤α≤1bα+ifm=k2,n=l2,0¯otherwise.Thus,(20)limk,l→∞1Uλkγ1Vμlγ2m,n,m≤Uλk,n≤Vμl:umvndsmn,0¯≥ε≤limk,l→∞k2l2kl=0.Hence,smn⟶S2,γ1,γ2N¯λ,μ0¯ but smn is not convergent.Theorem 2.
Suppose0<γ1,γ2,η1,η2≤1 such that γ1≤η1 and γ2≤η2. Then, S2,γ1,γ2N¯λ,μ⊆S2,η1,η2N¯λ,μ.Proof.
Letε>0 be given, and let γ1,γ2,η1,η2∈0,1. If γ1≤η1 and γ2≤η2, then, for every ε>0, we can write
(21)1Uλkγ1Vμlγ2m,n,m≤Uλk,n≤Vμl:umvndsmn,s0≥ε,≤1Uλkη1Vμlη2m,n,m≤Uλk,n≤Vμl:umvndsmn,s0≥ε,which yields that S2,γ1,γ2N¯λ,μ⊆S2,η1,η2N¯λ,μ.Theorem 3.
Suppose two fuzzy double sequencesmn and tmn such that S2,γ1,γ2N¯λ,μ−limsmn=s0 and S2,γ1,γ2N¯λ,μ−limtmn=t0. Then,
(i)
S2,γ1,γ2N¯λ,μ−limcsmn=cs0c∈ℝ(ii)
S2,γ1,γ2N¯λ,μ−limsmn+tmn=s0+t0Proof.
(i)
LetS2,γ1,γ2N¯λ,μ−limsmn=s0. It is clear for c=0. Let c≠0. For ε>0, we write
(22)1Uλkγ1Vμlγ2m,n,m≤Uλk,n≤Vμl:umvndcsmn,cs0≥ε=1Uλkγ1Vμlγ2m,n,m≤Uλk,n≤Vμl:umvncdsmn,s0≥ε=1Uλkγ1Vμlγ2m,n,m≤Uλk,n≤Vμl:umvndsmn,s0≥εc
Thus, (i) holds.
Suppose thatS2,γ1,γ2N¯λ,μ−limsmn=s0 and S2,γ1,γ2N¯λ,μ−limtmn=t0. Let ε>0 be given. Then,
(23)1Uλkγ1Vμlγ2m,n,m≤Uλk,n≤Vμl:umvndsmn+tmn,s0+t0≥ε≤1Uλkγ1Vμlγ2m,n,m≤Uλk,n≤Vμl:umvndsmn,s0≥ε2+1Uλkγ1Vμlγ2m,n,m≤Uλk,n≤Vμl:umvndtmn,t0≥ε2
Hence, (ii) holds.Definition 4.
Assume thatγ1,γ2∈0,1. Then, a fuzzy double sequence smn is said to be strongly weighted λ,μ-summable of order γ1,γ2; in short, we shall write S2,γ1,γ2N¯λ,μ-summable, to fuzzy number s0, denoted by
(24)smn⟶M2,γ1,γ2N¯λ,μs0,such that
(25)limk,l→∞1Uλkγ1Vμlγ2∑m∈k−λk+1,k∑n∈l−μl+1,lumvndsmn,s0=0.Theorem 5.
Consider a fuzzy double sequences=smn, and let 0<γ1,γ2≤1. Then,
(i)
smn⟶M2,γ1,γ2N¯λ,μs0 implies smn⟶S2,γ1,γ2N¯λ,μs0(ii)
smn is a bounded double sequence and smn⟶S2,γ1,γ2N¯λ,μs0 imply smn⟶M2,γ1,γ2N¯λ,μs0Proof.
(i)
Letsmn⟶M2,γ1,γ2N¯λ,μs0, and let ε>0 be given. Then,(26)∑m∈k−λk+1,k∑n∈l−μl+1,lumvndsmn,s0=∑m∈k−λk+1,kumvndsmn,s0≥ε∑n∈l−μl+1,lumvndsmn,s0≥εumvndsmn,s0+∑m∈k−λk+1,kumvndsmn,s0<ε∑n∈l−μl+1,lumvndsmn,s0<εumvndsmn,s0≥∑m∈k−λk+1,kumvndsmn,s0≥ε∑n∈l−μl+1,lumvndsmn,s0≥εumvndsmn,s0Therefore, we get(27)1Uλkγ1Vμlγ2∑m∈k−λk+1,k∑n∈l−μl+1,lumvndsmn,s0≥1Uλkγ1Vμlγ2∑m∈k−λk+1,kumvndsmn,s0≥ε∑n∈l−μl+1,lumvndsmn,s0≥εε≥1Uλkγ1Vμlγ2m,n,m≤Uλk,n≤Vμl:umvndsmn,s0≥εε.Lettingk,l⟶∞ on both sides of the above relation, we get smn⟶S2,γ1,γ2N¯λ,μs0.Letsmn⟶S2,γ1,γ2N¯λ,μs0. Since s=smn∈L2,∞, dmn,s0≤C for all m,n∈ℕ. For a given ε>0, one writes
(28)1Uλkγ1Vμlγ2∑m∈k−λk+1,k∑n∈l−μl+1,lumvndsmn,s0=1Uλkγ1Vμlγ2∑m∈k−λk+1,kumvndsmn,s0≥ε∑n∈l−μl+1,lumvndsmn,s0≥εumvndsmn,s0+1Uλkγ1Vμlγ2∑m∈k−λk+1,kumvndsmn,s0<ε∑n∈l−μl+1,lumvndsmn,s0<εumvndsmn,s0≤1Uλkγ1Vμlγ2∑m∈k−λk+1,kumvndsmn,s0<ε∑n∈l−μl+1,lumvndsmn,s0<εC+1Uλkγ1Vμlγ2∑m∈k−λk+1,kumvndsmn,s0<ε∑n∈l−μl+1,lumvndsmn,s0<εε≤1Uλkγ1Vμlγ2m,n,m≤Uλk,n≤Vμl:umvndsmn,s0≥εC+ε.It follows thatsmn⟶M2,γ1,γ2N¯λ,μs0.


## 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 Karakus [45] with the help of A-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 S2,γ1,γ2N¯λ,μ-convergence.Assume thatI=a,ba,b∈ℝ and g:I×I=I2⟶Lℝ is a fuzzy number-valued function. Then, g is fuzzy continuous at a point S of I×I if Dgs,gS⟶0 whenever s=smn is Pringsheim’s convergent to S in a fuzzy sense. We denote the set of fuzzy continuous function on I2=a,b×a,bby CLI2. Here, CLI2 is only a cone but not a vector space and let T:CLI2⟶CLI2 be an operator. Further, T is said to be a fuzzy linear operator if
(29)Tβ1·g1⊕β2·g2;x,y=β1·Tg1;x,y⊕β2·Tg2;x,y,for all β1,β2∈ℝ, g1,g2∈CLI2, and x,y∈I2. In addition, if Tg1;x,y°Tg2;x,y holds for all g1,g2∈CLI2 and x,y∈I2 with g1x,y°g2x,y, then T is called fuzzy position linear operator.Theorem 6.
LetAnrn,r∈ℕ be a double sequence such that Anr:CLI2⟶CLI2. Let A~nrn,r∈ℕ be corresponding positive linear operators acting from CI2 into itself having the relation
(30)Anrg;x,yα±=A~nrgα±;x,y∀α∈0,1,x,y∈I2,g∈CLI2,n,r∈ℕ.
Assume further that(31)S2,γ1,γ2N¯λ,μ−limA~nrgi−gi=0i=0,1,2,3,where g0t1,t2=1, g1t1,t2=t1, g2t1,t2=t2, and g3t1,t2=t1+t2. Then,
(32)S2,γ1,γ2N¯λ,μ−limd∗Anrg,g=0,
for alln,r∈ℕ and g∈CLI2.Proof.
Suppose that (31) holds and n,r∈ℕ×ℕ. Suppose also that x,y∈I2, g∈CLI2, and α∈0,1. Since gα±∈CI2, it follows that, for a given ε>0, there exists δ>0 such that for all t1,t2∈I2, we have gα±t1,t2−gα±x,y<εwhenevert1−x<δandt2−y<δ.
By the fuzzy boundedness ofg, we may write gα±x,y≤Cα± for all x,y∈I2, where Cα±=gα± for all a<y<b. We then obtain for all t1,t2,x,y∈I2 that
(33)gα±t1,t2−gα±x,y≤2Cα±.With the help of last relations, for allt1−x<δ and t2−y<δ, we obtain
(34)gα±t1,t2−gα±x,y<ε+2Cα±δ2t1−x2+t2−y2.SinceA~nr is a positive linear operator, by applying this operator to (34), we get
(35)A~nrgα±t1,t2;x,y−gα±x,y=A~nrgα±t1,t2−gα±x,y;x,y+gα±x,yA~nrg0;x,y−g0x,y≤A~nrgα±t1,t2−gα±x,y;x,y+Cα±A~nrg0;x,y−g0x,y≤A~nrε+2Cα±δ2t1−x2+t2−y2;x,y+Cα±A~nrg0;x,y−g0x,y≤ε+ε+Cα±+4Cα±E2δ2A~nrg0;x,y−g0x,y+4Cα±Eδ2A~nrg1;x,y−g1x,y+4Cα±Eδ2A~nrg2;x,y−g2x,y+2Cα±δ2A~nrg3;x,y−g3x,y,where E=maxx,y. We thus obtain by letting
(36)Ωα±ε=maxε+Cα±+4Cα±E2δ2,4Cα±Eδ2,2Cα±Eδ2and taking supx,y∈I2 that
(37)A~nrgα±−gα±≤ε+Ωα±εA~nrg0−g0+A~nrg1−g1+A~nrg2−g2+A~nrg3−g3.From (30), we can write
(38)d∗Amrg,g=supx,y∈I2dAmrg;x,y,gx,y=supx,y∈I2supα∈0,1maxA~mrgα−;x,y−gα−x,y,A~mrgα+;x,y−gα+x,y,and so
(39)d∗Anrg,g=supα∈0,1maxA~nrgα−−gα−,A~nrgα+−gα+.With the help of (37) and (39), we get
(40)d∗Anrg,g≤ε+ΩεA~nrg0−g0+A~nrg1−g1+A~nrg2−g2+A~nrg3−g3,where
(41)Ωε=supα∈0,1maxΩα−ε,Ωα+ε.Multiplying by the product of two nonnegative sequencesun and vr as details given in the previous section, one gets
(42)unvrd∗Anrg,g≤unvrε+ΩεunvrA~nrg0−g0+unvrA~nrg1−g1+unvrA~nrg2−g2+unvrA~nrg3−g3.For a givenϱ>0, choose ε>0 such that εunvr<ϱ. Then, upon setting,
(43)F=n,r:unvrd∗Anrg,g≥ϱ,F0=n,r:unvrA~nrg0−g0≥ϱ−εunvr4Ωε,F1=n,r:unvrA~nrg1−g1≥ϱ−εunvr4Ωε,F2=n,r:unvrA~nrg2−g2≥ϱ−εunvr4Ωε,F3=n,r:unvrA~nrg3−g3≥ϱ−εunvr4Ωε.It follows that(44)F⊂F0∪F1∪F2∪F3.Consequently, by taking weightedλ,μ density of order γ1,γ2, we obtain
(45)δ2,γ1,γ2N¯λ,μF≤δ2,γ1,γ2N¯λ,μF0+δ2,γ1,γ2N¯λ,μF1+δ2,γ1,γ2N¯λ,μF2+δ2,γ1,γ2N¯λ,μF3.Using the hypothesis (31), we conclude that
(46)S2,γ1,γ2N¯λ,μ−limd∗Anrg,g=0holds for all n,r∈ℕ×ℕ and g∈CLI2.Recall that, for any nonnegative integerj, the q-integer jqq>0 is given by
(47)jq=1−qj1−qifq≠1,jifq=1,and the q-factorial jq! by
(48)jq!=jq!j−1q!⋯1q!,0q!=1.Theq-binomial coefficients, for 0≤m≤j, is defined by
(49)jmq=jq!mq!j−mq!.The bivariate case of classicalq-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 bivariateq-Bernstein operators, defined by
(50)BnrL,q1,q2g;x,y=⨁k=0n⨁m=0rgkq1nq1,mq2rq2⊙bk,m,n,rq1,q2x,yn,r∈ℕ,for all g∈CLI2I2=0,1×0,1, x,y∈0,1×0,1, and 0<q1,q2<1, where
(51)bk,m,n,rq1,q2x,y=nkq1rmq2xk1−xq1n−kym1−yq2r−m.Note that(52)1−xqn=∏k=0n−11−qkx.The above operatorsBnrL,q1,q2g;x,y are positive and linear, and one writes
(53)BnrL,q1,q2g;x,yα±=Bnrq1,q2gα±;x,y=∑k=0n∑m=0rbk,m,n,rq1,q2x,ygα±kq1nq1,mq2rq2,for all 0≤α≤1, gα±∈C0,1. Suppose that q1=q1n and q2=q2r such that q1n⟶1 as s⟶∞ and q2r⟶1 as r⟶∞. In view of this assumption, one can easily find that
(54)nq1⟶∞n⟶∞,rq2⟶∞r⟶∞.We define the sequence of positive linear operators by(55)Φ~nrgα±;x,y=1+snrBnrq1,q2gα±;x,y.In this case,snr is same as defined in Example 2 with γ1=γ2=1, λk=k, μl=l, un=1, and vr=1 for all n,r∈ℕ. Therefore, we get
(56)Φ~nrg0;x,y=1+snr,Φ~nrg1;x,y=1+snrx,Φ~nrg2;x,y=1+snry,Φ~nrg3;x,y=1+snrx2+y2+x−x2nq1+y−y2rq2.Since(57)S2,γ1,γ2N¯λ,μ−limsnr=0,we observe that
(58)S2,γ1,γ2N¯λ,μ−limΦ~nrgi−gi=0i=0,1,2,3.Consequently, by Theorem6, we obtain
(59)S2,γ1,γ2N¯λ,μ−limd∗Φnrg,g=0,for all g∈CLI2. Hence, all the conditions of Theorem 6 hold true for our operators Φ~nr but Theorem 2.2 obtained in [45] for convergence in Pringsheim’s sense does not work for Φ~nr since the fuzzy double sequence snr 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.



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