Lacunary Statistical Convergence in Measure for Double Sequences of Fuzzy Valued Functions

Kişi, Ömer

doi:https://doi.org/10.1155/2021/6655630

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Research Article | Open Access

Volume 2021 | Article ID 6655630 | https://doi.org/10.1155/2021/6655630

Lacunary Statistical Convergence in Measure for Double Sequences of Fuzzy Valued Functions

Ömer Kişi¹

Academic Editor: Hijaz Ahmad

Received10 Dec 2020

Revised25 Jan 2021

Accepted29 Jan 2021

Published04 Mar 2021

Abstract

Based on the concept of lacunary statistical convergence of sequences of fuzzy numbers, the lacunary statistical convergence, uniformly lacunary statistical convergence, and equi-lacunary statistical convergence of double sequences of fuzzy-valued functions are defined and investigated in this paper. The relationship among lacunary statistical convergence, uniformly lacunary statistical convergence, equi-lacunary statistical convergence of double sequences of fuzzy-valued functions, and their representations of sequences of -level cuts are discussed. In addition, we obtain the lacunary statistical form of Egorov’s theorem for double sequences of fuzzy-valued measurable functions in a finite measurable space. Finally, the lacunary statistical convergence in measure for double sequences of fuzzy-valued measurable functions is examined, and it is proved that the inner and outer lacunary statistical convergence in measure are equivalent in a finite measure set for a double sequence of fuzzy-valued measurable functions.

1. Introduction

Fast [1] initiated statistical convergence for a real sequence. After the work of Fridy [2] and Šalát [3], it became a noteworthy topic in summability theory. Mursaleen and Edely [4] examined the statistical convergence via double sequences.

Various types of convergence for sequences of functions, such as pointwise, equi-statistical (or ideal), and uniform convergence, were originated by Balcerzak et al. [5]. Pointwise and uniform statistical convergence of double sequences was studied by Gökhan et al. [6].

Additionally, Duman and Orhan [7] studied convergence in -density and -statistical convergence of sequences of functions and presented the notions of -statistical pointwise convergence and -statistical uniform convergence.

Lacunary statistical convergence was firstly studied by Fridy and Orhan [8]. In [9], lacunary statistical convergence via double sequence was investigated.

Zadeh [10] initiated the notion of fuzzy sets. The publication of the paper affected deeply all the scientific fields. This notion is significant for real-life conditions, but has no adequate solution to some problems. Such problems lead to new quests. Matloka [11] identified ordinary convergence of a sequence of fuzzy numbers. Nanda [12] examined the seqeuences with fuzzy numbers. Negoita [13] gave the Hausdorff distance between two fuzzy numbers. Statistical convergence by utilizing fuzzy numbers was given by Nuray and Savaş [14]. Savaş and Mursaleen [15] investigated statistical convergent double sequences via fuzzy numbers. By using fuzzy numbers, Aytar and Pehlivan [16] defined the statistical convergence of sequences. Pointwise statistical convergence sequences of fuzzy mappings studied by Altin et al. [17]. Some beneficial results on this topic can be found in [18–32].

In recent times, Gong et al. studied statistical convergence, equi-statistical convergence, and uniformly statistical convergence for sequences of fuzzy-valued functions. These concepts were extended to the double sequences by Hazarika et al. Kişi and Dündar examined lacunary statistical convergence in measure for sequences of fuzzy-valued functions and established noteworthy results.

According to Zadeh [10], a fuzzy subset of is a nonempty subset of for some function . A function is called a fuzzy number if the function holds the following properties:(i) is convex, i.e., , where (ii) is normal, i.e., there exists an such that (iii) is upper semicontinuous, i.e., for every , , for all is open in the usual topology of (iv) is compact, where is the closure operator

We indicate the set of all fuzzy numbers by . The set of real numbers can be included in if we take by

For , -cut of is given by which is a closed bounded interval of . As in [13], the Hausdorff distance between two fuzzy numbers and is denoted by :where is the Hausdorff metric.

For and , is named th partial density of if

Ifexists, it is named the natural density of . is denoted the zero density set.

A sequence of fuzzy numbers is called to be statistically convergent to a fuzzy number if for every , , i.e., . It is demonstrated by or , .

Let and . is named the th partial lacunary density of ifwhere .

The number is denoted as the lacunary density (-density) of ifexists. Also, is called to be zero density set.

A SFVF is pointwise lacunary statistically convergent to FVF on , denoted by or ; if for each and all there exists such that, for all , we get . It is obvious that if for every and all , . Here, is named the lacunary statistical limit function of .

Some definitions and significant results about lacunary statistical convergence in measure for sequences of fuzzy-valued functions were given in.

A double sequence is named lacunary sequence if there exist two increasing sequences of integers and such that

We use the following symbols in the sequel:

In the paper, by , we will indicate a double lacunary sequence of positive real numbers.

In this article, we proposed the concepts of lacunary statistical convergence, uniformly lacunary statistical convergence, and equi-lacunary statistical convergence for double sequences of fuzzy-valued functions and proved some classical results in this new setting and their representations of sequences of -level cuts. We proved lacunary statistical form of Egorov’s theorem for double sequences of fuzzy-valued measurable functions defined on a finite measure space . Finally, we define the notion of lacunary statistical convergence in measure for double sequences of fuzzy-valued measurable functions and prove some interesting results. Our results were emphasized with examples.

2. Main Results

In the section, we presume that and are the fuzzy-valued function and a double sequence of fuzzy-valued functions for all . We indicate FVF and DSFVF in place of fuzzy-valued function and double sequence of fuzzy-valued functions.

Definition 1. Let and . is called the th partial lacunary density of ifThe number is named the lacunary density or -density of ifexists. Additionally,is named to be zero density set.

Definition 2. We call that a DSFVF is pointwise lacunary statistically convergent to FVF on , indicated byif for every and for all there exists such that, for all , we haveIt is obvious that if for every and for each

Definition 3. We call that a DSFVF is uniformly lacunary statistically convergent to FVF on , denoted by on , or , if for each there exists such that, for all , we obtainwhich provides for all . It is obvious that if, for all ,for all .

Remark 1. If , then .

Remark 2. iff .

Proof. Assume that . Then, for each and for every , we have so thatHence, .
Next, suppose that on . For every , we writeIf , then , which impliesas is arbitrary. This gives that . Therefore, we obtainwhich gives . Hence, .

Theorem 1. Presume that the sequence of FVF on , where is equi-continuous on ; then, is continuous and on .

Proof. First, we demonstrate that is continuous. Let and . By the equi-continuity of , then there exists such thatfor any and . For some , since , the setSo, there exists such thatWe obtainTherefore, we proved the continuity of .
At the moment, we will show that on . Let . Since is continuous on , it provides that is uniformly continuous and is equi-uniformly continuous on . So, pick such that, for any and , we haveUsing the finite covering theorem, select finite open coveringsfrom the cover of . Using , there exists a set such thatfor all and . Let and . Hence, for some . As a resultwhich gives that on .

Definition 4. We call that a DSFVF is equi-lacunary statistically convergent to FVF , denoted by , if, for given ,with regards to is uniformly convergent to zero function. Thus, iff for all , , for each , for all ,Note that, by monotonicity of , we can also utilize .

Remark 3. It is obvious that iff for every and for every , for all ,In this instance, we may take . Obviously, implies . Furthermore, we see that indicates .

Corollary 1. Let be a DSFVF and be a FVF on . Then, . The converse implications do not hold in general.

Proof. Proving the above result, we can consider the examples as given below.

Example 1. Consider DSFVF is defined by for . Then, we get for each and . Therefore, converges pointwise to and so . However, for each , we consider . Therefore, for all , we haveThus, for all , one obtainsHence, is not equi-lacunary statistically convergent to on .

Example 2. Let us define a DSFVF byThen, for every , we haveTherefore, for every ,This yields that is pointwise lacunary statistically convergent to . However,for all and is not uniformly lacunary statistically convergent to on .

Theorem 2. A of SFVMF is uniformly lacunary statistically convergent to FVMF iff is uniformly lacunary statistically convergent to uniformly with regards to and .

Proof. Select . Given , there exists such that , for any and , i.e.,That is, there arefor any and . Moreover,Consequently, we get is uniformly lacunary statistically convergent to uniformly with regards to and .
Conversely, for any and for any , is uniformly lacunary statistically convergent to with regards to and . Thus, for given , there exists such thatfor all , , and any . Additionally, we see that, for given , there exists such thatfor all , and any . Take . We getfor all , , and any . As a result, we havethat is,This concludes the proof.

Theorem 3. Let FVMF and SFVMF. Fix if on and all are continuous on , then is continuous on .

Proof. Let . , and we can identify numbers such that, for all ,Let , . Therefore, for all . Via the continuity of at , there is a neighborhood of such thatfor all and . Fix . Since and , we find . Thus,Thus, we obtainfor all , i.e., is continuous on .

Theorem 4. A of SFVMF is equi-lacunary statistically convergent to FVMF iff is equi-lacunary statistically convergent to uniformly for any and any .

Proof. shows that, for any and , there exists , for all , and any such thatSo, for any , we obtainTherefore, for any , we acquireThen, emphasize thatHence, is uniformly lacunary statistically convergent to for any and any .
Conversely, let and , and there exists such thatfor all , and any and for any . Similarly, there exists such thatfor all and and any and for any . Then, select and . We obtainfor all and and any and for any . Thus, we havethat is,This concludes the proof.
The next result is the lacunary statistical form of Egorov’s theorem for the DSFVF.

Theorem 5. Take measurable space . Presume that and are measurable and identified almost everywhere on . Suppose also that almost everywhere on . Then, for every , there exists such that and on .

Proof. We presume that and are defined everywhere on and also suppose that for all . Now, for any fix , consider that the setis measurable. Then, the function , , is measurable. Let . For every , we have iffSince the functionis measurable, so we have . For , one writesFrom the previous observation, we conclude that is measurable. Also, we obtainAs a result, . Take . For each , select such that . SetThen, we haveLetThus, . Hence, we get ,This gives that on .

Now, we give the concepts of inner and outer lacunary statistical convergence in measure of DSFVF and demonstrate the equivalence of these concepts. For our convenience, we use FVMF and DSFVMF in place of fuzzy-valued function and double sequence of fuzzy-valued measurable function, respectively.

Definition 5. Take measurable space . Presume that the set of all FVMF defined almost everywhere on . Take and in . The outer lacunary statistical convergence in measure of a DSFVMF to a FVMF is defined byfor , . It is denoted by . If we take (68) as follows, we get the inner statistical convergence in measure of a DSFVMF to a FVMF :It is demonstrated by .

Theorem 6. Take measurable space . Presume and in :(i)If , then (ii)If , then , provided

Proof. Since is a probability measure, we can think the product measure on the product algebra of subsets of . For fix , we obtainDefine a function asis -measurable. Therefore, we get . For any , one writes(i)In order to acquire this, we have to show that Fix and . Since , one can find such that , one get the following: Assume that Then, we have from situation (74) that Hence, for all , one acquires Take . Therefore, we have To acquire relation (73), it is enough to demonstrate that For the set and for every fix , we can utilize the Fubini theorem for the characteristic function of of the finite measure . Actually, where Thus, Suppose and such that , one obtains which gives that strict inequality (80) is valid.(ii)Presume that . Fix . To verify our result, we need to denote thatLet and be given. Since , one can find such that, for all and , we haveBy considering the Fubini theorem for the function of , we obtainSupposing and such that, for all , we haveThis concludes the proof.

Theorem 6 indicates that both kinds of convergence (in Definition 5) in measure are equivalent if is a finite measurable set. By thinking the set , we give the following definition.

Definition 6. DSFVMF is called to be lacunary statistical convergent in measure (shortly, LSCM) to a FVMF , in symbol, , if is lacunary statistically convergent to zero for any and all . This concept can be denoted by the following formula:Here, we can take or , .

Proposition 1. Take measurable space . Suppose that and in . Then, .

Proof. We presume that . Take . In this way, there is a set such thatThus, we obtainThis indicates that .

Theorem 7. Let be a FVMF such that, for each , . Then, given and , there is a measurable set with and integers such that

Proof. Letand setWe have that , and for each , there is some to which does not belong, since . Thus, we get , so we have . Hence, given , so that , i.e.,If we write for this , then and

Theorem 8. If DSFVMF pointwise lacunary statistically convergent to a FVMF almost everywhere on , then .

Proof. Assume that almost everywhere on . We have from Theorem 6 that is the same as . So, to demonstrate our result, we have to show that . Presume that and . It is clear from Theorem 5 that such that and . Select indexes and such thatfor every , and . Thus, for all and , we obtainTherefore, for every and ,as desired.

Theorem 9. If almost everywhere on , then .

Proof. Take . By considering Theorem 5, there is an such that and . Consider such thatwhich yieldsTherefore, one obtains

Corollary 2. If , then a subsequence of such that almost everywhere on .

Proof. Presume that , so any subsequence of also lacunary statistically convergent in measure to . In this way, possesses a subsequence that lacunary statistically convergent in measure to almost everywhere on . This gives that almost everywhere on .

Definition 7. Take finite measurable space . Assume that , , , and . The double sequence is uniformly lacunary statistically convergent in measure (briefly, we can write ULSCM) to with regards to ifboth lacunary statistically convergent measure to zero for each . Notice that this concept is given by the following formula:In that case, we can write or , .

Theorem 10. DSFVMF is LSCM to FVMF iff is ULSCM to with regards to .

Proof. Suppose that is LSCM to . Then,is lacunary statistically convergent to zero for each , i.e.,Thus,For every , one obtainswhich gives thatHence, is ULSCM to with regards to .
Next, we presume that is ULSCM to with regards to . Then, for every , one hasThus,From the last two relations, we obtainwhich gives thatIt means that is LSCM to FVMF .

3. Conclusion 1

In this study, we propose the notions of pointwise lacunary statistical convergence, uniformly lacunary statistical convergence, and equi-lacunary statistical convergence of the double sequence of fuzzy-valued functions and discuss the reationship between various kinds of lacunary statistical convergence for the double sequence of fuzzy-valued functions and the sequence of -level cut functions. In addition, we obtain the Egorov theorem for the double sequence of fuzzy-valued functions and investigate the lacunary statistical convergence in measure for the double sequence of fuzzy-valued functions. Note that, in contrast to the case for single sequences, a convergent double sequence need not be bounded, so the studies on lacunary statistical convergence of double sequences have a rapid growth and an emerging area in mathematical research. We conclude that our results are more general than the one proved earlier for single sequences by Kişi and Dündar. As an application, researchers who linked two theories such as the theory of approximation and the theory of lacunary statistical summability may prove fuzzy analogue of Korovkin’s type approximation theorem for several test functions by using our convergence methods.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

H. Fast, “Sur la convergence statistique,” Colloquium Mathematicum, vol. 2, no. 3-4, pp. 241–244, 1951.
View at: Publisher Site | Google Scholar
J. A. Fridy, “On statistical convergence,” Analysis, vol. 5, pp. 301–313, 1985.
View at: Publisher Site | Google Scholar
T. S̆alát, “On statistically convergent sequences of real numbers,” Mathematica Slovaca, vol. 30, pp. 139–150, 1980.
View at: Google Scholar
M. Mursaleen and O. H. H. Edely, “Statistical convergence of double sequences,” Journal of Mathematical Analysis and Applications, vol. 288, no. 1, pp. 223–231, 2003.
View at: Publisher Site | Google Scholar
M. Balcerzak, K. Dems, and A. Komisarski, “Statistical convergence and ideal convergence for sequences of functions,” Journal of Mathematical Analysis and Applications, vol. 328, no. 1, pp. 715–729, 2007.
View at: Publisher Site | Google Scholar
A. Gökhan, M. Güngör, and M. Et, “Statistical convergence of double sequences of real-valued functions,” International Mathematical Forum, vol. 2, no. 5–8, pp. 365–374, 2007.
View at: Publisher Site | Google Scholar
O. Duman and C. Orhan, “-Statistically convergent function sequences,” Czechoslovak Mathematical Journal, vol. 54, no. 129, pp. 413–422, 2004.
View at: Publisher Site | Google Scholar
J. Fridy and C. Orhan, “Lacunary statistical convergence,” Pacific Journal of Mathematics, vol. 160, no. 1, pp. 43–51, 1993.
View at: Publisher Site | Google Scholar
C. Çakan, B. Altay, and H. Çoşkun, “Double lacunary density and lacunary statistical convergence of double sequences,” Studia Scientiarum Mathematicarum Hungarica, vol. 47, no. 1, pp. 35–45, 2010.
View at: Publisher Site | Google Scholar
L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.
View at: Publisher Site | Google Scholar
M. Matloka, “Sequences of fuzzy numbers,” Busefal, vol. 28, pp. 28–37, 1986.
View at: Google Scholar
S. Nanda, “On sequences of fuzzy numbers,” Fuzzy Sets and Systems, vol. 33, no. 1, pp. 123–126, 1989.
View at: Publisher Site | Google Scholar
C. V. Negoita and D. Ralescu, Applications of Fuzzy Sets to Systems Analysis, Wiley, New York, NY, USA, 1975.
F. Nuray and E. Savaş, “Statistical convergence of sequences of fuzzy numbers,” Mathematica Slovaca, vol. 45, no. 3, pp. 269–273, 1995.
View at: Google Scholar
E. Savaş and M. Mursaleen, “On statistically convergent double sequences of fuzzy numbers,” Information Sciences, vol. 162, no. 3-4, pp. 183–192, 2004.
View at: Google Scholar
S. Aytar and S. Pehlivan, “Statistical convergence of sequences of fuzzy numbers and sequences of α−cuts,” International Journal of General Systems, vol. 37, no. 2, pp. 231–237, 2008.
View at: Publisher Site | Google Scholar
Y. Altin, M. Et, and B. C. Tripathy, “On pointwise statistical convergence sequences of fuzzy mappings,” Kuwait Journal of Science, vol. 15, no. 2, pp. 425–433, 2007.
View at: Google Scholar
H. Altinok, Y. Altin, and M. Işık, “Statistical convergence and strong-Cesàro summability of order in sequences of fuzzy numbers,” Iranian Journal of Fuzzy Systems, vol. 9, no. 2, pp. 65–75, 2012.
View at: Google Scholar
H. Altinok and M. Et, “Statistical convergence of order β, γ for sequences of fuzzy numbers,” Soft Computing, vol. 23, no. 15, pp. 6017–6022, 2019.
View at: Publisher Site | Google Scholar
M. Çınar, M. Karakaş, and M. Et, “On pointwise and uniform statistical convergence of order for sequences of functions,” Fixed Point Theory and Applications, vol. 33, 2020.
View at: Google Scholar
S. A. Mohiuddine, A. Asiri, and B. Hazarika, “Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems,” International Journal of General Systems, vol. 48, no. 5, pp. 492–506, 2019.
View at: Publisher Site | Google Scholar
S. A. Mohiuddine, B. Hazarika, and A. Alotaibi, “On statistical convergence of double sequences of fuzzy valued functions,” Journal of Intelligent & Fuzzy Systems, vol. 32, no. 6, pp. 4331–4342, 2017.
View at: Publisher Site | Google Scholar
F. Nuray, “Lacunary statistical convergence of sequences of fuzzy numbers,” Fuzzy Sets and Systems, vol. 99, no. 3, pp. 353–355, 1998.
View at: Publisher Site | Google Scholar
E. Savaş and M. Gürdal, “Certain summability methods in intuitionistic fuzzy normed spaces,” Journal of Intelligent and Fuzzy Systems, vol. 27, no. 4, pp. 1621–1629, 2014.
View at: Google Scholar
E. Savaş and M. Gürdal, “Generalized statistically convergent sequences of functions in fuzzy 2-normed spaces,” Journal of Intelligent and Fuzzy Systems, vol. 27, no. 4, pp. 2067–2075, 2014.
View at: Google Scholar
E. Savaş and M. Gürdal, “A generalized statistical convergence in intuitionistic fuzzy normed spaces,” Science Asia, vol. 41, pp. 289–294, 2015.
View at: Google Scholar
E. Savaş, “A note on double sequences of fuzzy numbers,” Turkish Journal of Mathematics, vol. 20, no. 2, pp. 175–178, 1996.
View at: Google Scholar
E. Savaş, “On statistically convergent sequences of fuzzy numbers,” Information Sciences, vol. 137, no. 1–4, pp. 277–282, 2001.
View at: Google Scholar
C. Şençimen and S. Pehlivan, “Statistical convergence in fuzzy normed linear spaces,” Fuzzy Sets and Systems, vol. 159, pp. 361–370, 2008.
View at: Google Scholar
Z. Gong, L. Zhang, and X. Zhu, “The statistical convergence for sequences of fuzzy-number-valued functions,” Information Sciences, vol. 295, pp. 182–195, 2015.
View at: Publisher Site | Google Scholar
B. Hazarika, A. Alotaibi, and S. A. Mohiuddine, “Statistical convergence in measure for double sequences of fuzzy-valued functions,” Soft Computing, vol. 24, no. 9, pp. 6613–6622, 2020.
View at: Publisher Site | Google Scholar
Ö. Kişi and E. Dündar, “Lacunary statistical convergence in measure for sequences of fuzzy valued functions,” Konuralp Journal of Mathematics, vol. 8, no. 2, pp. 252–262, 2020.
View at: Google Scholar

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Copyright © 2021 Ömer Kişi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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