Research Article | Open Access

Erdinç Dündar, Özer Talo, Feyzi Başar, "Regularly -Convergence and Regularly -Cauchy Double Sequences of Fuzzy Numbers", *International Journal of Analysis*, vol. 2013, Article ID 749684, 7 pages, 2013. https://doi.org/10.1155/2013/749684

# Regularly -Convergence and Regularly -Cauchy Double Sequences of Fuzzy Numbers

**Academic Editor:**Malte Braack

#### Abstract

In this paper, we introduce the notions of regularly , -convergence and regularly , -Cauchy double sequence of fuzzy numbers. Also, we study some properties of these concepts.

#### 1. Introduction, Notations, and Definitions

The concept of ordinary convergence of a sequence of fuzzy numbers was firstly introduced by Matloka [1] and proved some basic theorems for sequences of fuzzy numbers. Nanda [2] studied the sequences of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers forms a complete metric space. Recently, Nuray and Savaş [3] defined the concepts of statistical convergence and statistical Cauchy for sequence of fuzzy numbers. They proved that a sequence of fuzzy numbers, is statistically convergent if and only if it is statistically Cauchy. Nuray [4] introduced Lacunary statistical convergence of sequences of fuzzy numbers whereas Savaş [5] studied some equivalent alternative conditions for a sequence of fuzzy numbers to be statistically Cauchy. A lot of developments have been made in this area after the works of [6–8].

Throughout the paper and denote the set of all positive integers and the set of all real numbers, respectively. The idea of -convergence was introduced by Kostyrko et al. [9] as a generalization of statistical convergence which is based on the structure of the ideal of subset of the set . Nuray and Ruckle [10] independently introduced the same with another name generalized statistical convergence. Das et al. [11] introduced the concept of -convergence of double sequences in a metric space and studied some properties of this convergent sequences of this type. Balcerzak et al. [12] studied on statistical convergence and ideal convergence for sequences of functions. Komisarski [13] discussed the pointwise -convergence and -convergence in measure of sequences of functions. Mursaleen and Alotaibi investigated the notion of ideal convergence in [14] for random 2-normed space and construct some interesting examples. Mursaleen and Mohiuddine defined and studied the concept of -convergence, -convergence, -limit points and -cluster points in probabilistic normed space, in [15]. Şahiner et al. introduced and investigated -convergence in 2-normed spaces, and also defined and examined some new sequence spaces using norm, in [16]. A lot of developments have been made in this area after the works of [17–21].

V. Kumar and K. Kumar studied the concepts of -convergence, -convergence, and -Cauchy sequence for sequences of fuzzy numbers, in [22]. Mursaleen et al. studied the concept of ideal convergence and ideal Cauchy for double sequences in intuitionistic fuzzy normed spaces, in [23]. Recently, Dündar and Talo have introduced the concepts of -convergence, -convergence for double sequences of fuzzy numbers and studied their some properties and relations, in [24]. Quite recently, Dündar and Talo have introduced the concepts of -Cauchy, -Cauchy double sequences of fuzzy numbers, in [25].

In this paper, we introduce the notions of regularly (), ()-convergence and regularly (), ()-Cauchy double sequence of fuzzy numbers. Also, we study some properties of those sequences.

In a possible application we can say that if we choose the statistical convergence of a special case of the ideal convergence, the results in this paper can be obtained for statistical convergence.

Now, we recall the concept of fuzzy numbers, convergence, ideal convergence of the sequences and double sequences (see [9, 11, 21, 24–30]).

A double sequence of real numbers is said to be *convergent* to in Pringsheim’s sense, if for any there exists such that whenever . In this case we write

Let . A class of subsets of is said to be an *ideal* in provided that (i),(ii) implies , (iii), implies .

is called a *nontrivial ideal* if .

Let . A non-empty class of subsets of is said to be a *filter* in provided that (i), (ii) implies , (iii), implies .

If is a nontrivial ideal in , , then the class is called a filter on associated with .

A nontrivial ideal in is called *admissible* if for each .

Throughout the paper we take as a nontrivial admissible ideal in .

Let be a nontrivial ideal, and let be a metric space. A sequence of elements of is said to be -convergent to , if for each we have .

A sequence of elements of is said to be -convergent to if and only if there exists a set (i.e., ), such that .

Throughout the paper we take as a nontrivial admissible ideal in .

A nontrivial ideal is called strongly admissible if and belong to for each . It is evident that a strongly admissible ideal is also admissible.

Let . Then is a nontrivial strongly admissible ideal and clearly an ideal is strongly admissible if and only if .

Let be a linear metric space, and let be a strongly admissible ideal. A double sequence in is said to be -convergent to , if for any we have and is written .

If is a strongly admissible ideal, then usual convergence implies -convergence.

Let be a linear metric space, and let be a strongly admissible ideal. A double sequence of elements of is said to be -convergent to , if and only if there exists a set (i.e., ) such that , for and is written .

Let be a linear metric space, and let be a strongly admissible ideal. A double sequence of elements of is said to be -Cauchy, if for every there exist such that .

Let be an ideal of and be an ideal of , then a double sequence in , which is the set of complex numbers, is said to be regularly -convergent (*r *-convergent), if it is -convergent in Pringsheim’s sense and for every , the following statements hold:
for some , for each and
for some , for each .

We say that an admissible ideal satisfies the property (AP), if for every countable family of mutually disjoint sets belonging to , there exists a countable family of sets such that is a finite set for and . (hence for each ).

We say that an admissible ideal satisfies the property (AP2), if for every countable family of mutually disjoint sets belonging to , there exists a countable family of sets such that , that is, is included in the finite union of rows and columns in for each and (hence for each ).

A fuzzy number is a fuzzy set on the real axis, that is, a mapping which satisfies the following four conditions. (i) is normal, that is, there exists an such that . (ii) is fuzzy convex, that is, for all and for all . (iii) is upper semicontinuous. (iv)The set is compact, (cf. Zadeh [31]), where denotes the closure of the set in the usual topology of .

We denote the set of all fuzzy numbers on by and called it as the space of fuzzy numbers. -level set of is defined by

The set is closed, bounded, and nonempty interval for each which is defined by . can be embedded in , since each can be regarded as a fuzzy number defined by

Theorem 1 (see [27]). *Let for and for each . Then the following statements hold.*(i)* is a bounded and nondecreasing left continuous function on . *(ii)* is a bounded and nonincreasing left continuous function on . *(iii)*The functions and are right continuous at the point . *(iv)*. **Conversely, if the pair of functions and satisfies the conditions (i)–(iv), then there exists a unique such that for each . The fuzzy number corresponding to the pair of functions and is defined by , .*

Let and . Then the operations addition, scalar multiplication, and product are defined on by where it is immediate that for all .

Let be the set of all closed bounded intervals of real numbers with endpoints and , that is, . Define the relation on by

Then it can easily be observed that is a metric on and is a complete metric space, (cf. Nanda [2]). Now, we may define the metric on by means of the Hausdorff metric as

One can see that

The partial ordering relation on is defined as follows:

Now, we may give the following.

Proposition 2 (see [6]). *Let and . Then, the following statements hold.*(i) is a complete metric space. (ii). (iii). (iv). (v).

Following Matloka [1], we give some definitions concerning the sequences of fuzzy numbers which are needed in the text.

A *sequence * of fuzzy numbers is a function from the set into the set . The fuzzy number denotes the value of the function at and is called as the general term of the sequence. By , we denote the set of all sequences of fuzzy numbers

A sequence is called *convergent* with limit , if for every there exists an such that for all .

A *double sequence * of fuzzy real numbers is defined by a function from the set into the set . The fuzzy number denotes the value of the function at .

A double sequence of fuzzy numbers is said to be *convergent* in the Pringsheim’s sense or P-convergent, if for every there exists such that for all and is denoted by . The fuzzy number is called the Pringsheim limit of .

Let be a strongly admissible ideal. (i)A double sequence of fuzzy numbers is said to be *-convergent* to a fuzzy number if for any we have and is written . (ii)A double sequence of fuzzy numbers is said to be *-convergent* to if there exists (i.e., ) such that
and is written .(iii)A double sequence of fuzzy numbers is said to be *-Cauchy*, if for each , there exist such that .(iv)A double sequence of fuzzy numbers is said to be *-Cauchy* if there exists a set (i.e., ) such that for every and for , , , that is,

Now, we begin with quoting the following five lemmas due to Dündar and Talo [24, 25] which are needed throughout the paper.

Lemma 3 (see [24, Theorem 3.3]). *Let be a strongly admissible ideal, let be a double sequence of fuzzy numbers, and let be a fuzzy number. Then, implies .*

Lemma 4 (see [24, Theorem 4.2]). *Let be a strongly admissible ideal, let be a double sequence of fuzzy numbers, and let . Then, implies .*

Lemma 5 (see [24, Theorem 4.4]). *Let be a strongly admissible ideal with property (AP2), let be a double sequence of fuzzy numbers, and let be a fuzzy real number. Then, implies .*

Lemma 6 (see [25, Theorem 3.2]). *Let be a strongly admissible ideal. A double sequence of fuzzy numbers is -convergent if and only if it is -Cauchy sequence. *

Lemma 7 (see [25, Theorem 3.4]). *Let be a strongly admissible ideal. If a double sequence of fuzzy numbers is an -Cauchy sequence, then it is -Cauchy. *

#### 2. Main Results

In this section, we study certain properties of regular convergence, regularly -convergence and regularly -Cauchy double sequences of fuzzy numbers.

*Definition 8. *A double sequence of fuzzy numbers is said to be regularly convergent, if it is convergent in Pringsheim’s sense and the limits
exist for each fixed and , respectively. Note that if is regularly convergent to a fuzzy number , then the limits
exist and are equal to . In this case we write

*Definition 9. *Let be a strongly admissible ideal, and let be an admissible ideal. A double sequence of fuzzy numbers is said to be regularly -convergent (-convergent), if it is -convergent in Pringsheim’s sense and for every , the following statements hold:
for some fuzzy numbers , for each and
for some fuzzy numbers , for each .

In the case is regularly -convergent (-convergent) to a fuzzy number , then the limits and exist and are equal to .

Theorem 10. *Let be a strongly admissible ideal of , and let be an admissible ideal of . If a double sequence of fuzzy numbers is regularly convergent, then it is -convergent. *

*Proof. *Let be regularly convergent. Then is convergent in Pringsheim’s sense and the limits and exist. By Lemma 3, is -convergent. Also, for , there exist and such that
for some fuzzy numbers and each fixed for every and
for some fuzzy numbers and each fixed for every . Then, since is admissible ideal so for , we have

Hence, is -convergent.

*Definition 11. *Let be a strongly admissible ideal of , and let be an admissible ideal of . A double sequence of fuzzy numbers is said to be -convergent, if there exist the sets (i.e., ), , and (i.e., and ) such that the limits
exist for each fixed and , respectively.

Theorem 12. *Let be a strongly admissible ideal of , and let be an admissible ideal of . If a double sequence of fuzzy numbers is -convergent, then it is -convergent. *

*Proof. *Let be -convergent. Then, it is -convergent and so, by Lemma 4, it is -convergent. Also, there exist the sets such that
for some fuzzy numbers and
for some fuzzy numbers . Hence, we have
for . Since is admissible ideal we get
and therefore . This shows that the double sequence is -convergent.

Theorem 13. *Let be a strongly admissible ideal with property (AP2), and let be an admissible ideal with property (AP). If a double sequence of fuzzy numbers is -convergent, then is -convergent. *

*Proof. *Let a double sequence of fuzzy numbers be -convergent. Then is -convergent and so is -convergent, by Lemma 5. Also, for every we have
for some fuzzy numbers , for each and
for some fuzzy numbers , for each .

Now put
for , for some fuzzy numbers and for each . It is clear that for and for each . By the property () there is a countable family of sets in such that is a finite set for each and .

We prove that
for . Let be given. Choose such that . Then, we have

Since is a finite set for , there exists such that

If and then

Thus, we have for some fuzzy numbers and for each . This implies that

Hence, we have
for some fuzzy numbers and for each .

Similarly, for the set , we have
for some fuzzy numbers and for each . Hence, a double sequence of fuzzy numbers is -convergent.

Now, we give the definitions of -Cauchy sequence and -Cauchy sequence.

*Definition 14. *Let be a strongly admissible ideal of , and let be an admissible ideal of . A double sequence of fuzzy numbers is said to be regularly -Cauchy (-Cauchy), if it is -Cauchy in Pringsheim’s sense and for every there exist and such that the following statements hold:

A double sequence is said to be regularly -Cauchy (-Cauchy), if there exist the sets , , and (i.e., , , and ) and for every there exist , , , , such that
whenever .

Theorem 15. *Let be a strongly admissible ideal of , and let be an admissible ideal of . If a double sequence of fuzzy numbers is -Cauchy, then it is -Cauchy. *

*Proof. *Since a double sequence of fuzzy numbers is -Cauchy, it is -Cauchy. We know that -Cauchy implies -Cauchy by Lemma 7. Also, since the double sequence of fuzzy numbers is -Cauchy so there exist the sets and for every there exist , such that

for and . Therefore, for we have
for and
for . Since is admissible ideal
Hence, we have and is -Cauchy.

Theorem 16. *Let be a strongly admissible ideal of , and let be an admissible ideal of . If a double sequence of fuzzy numbers is -convergent, then is -Cauchy sequence. *

*Proof. *Let be a -convergent double sequence of fuzzy numbers. Then is -convergent, and by Lemma 6, it is -Cauchy sequence of fuzzy numbers. Also for every , we have
for some fuzzy numbers , for each and
for some fuzzy numbers , for each . Since is admissible ideal, the sets
for some fuzzy numbers and
for some fuzzy numbers , are nonempty and belong to . For , ( and ) we have for some fuzzy numbers . Now, for we define the set
where . Let . Then for , ( and ) we have
for some fuzzy numbers . This shows that

Hence, we have .

Similarly, for ( and ) we have
for some fuzzy numbers . Therefore, it can be seen that

Hence, we have . This shows that is -Cauchy sequence of fuzzy numbers.

#### Acknowledgment

The authors would like to express their pleasure to the referees for making some useful remarks and helpful suggestions which improved the presentation of the paper.

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Copyright © 2013 Erdinç Dündar et al. 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.