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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 891986, 7 pages
http://dx.doi.org/10.1155/2013/891986
Research Article

On the Slowly Decreasing Sequences of Fuzzy Numbers

1Department of Mathematics, Faculty of Arts and Sciences, Celal Bayar University, 45040 Manisa, Turkey
2Department of Mathematics, Faculty of Arts and Sciences, Fatih University, Büyükçekmece Campus, 34500 İstanbul, Turkey

Received 11 March 2013; Accepted 11 April 2013

Academic Editor: Ljubisa Kocinac

Copyright © 2013 Özer Talo and Feyzi Başar. 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.

Abstract

We introduce the slowly decreasing condition for sequences of fuzzy numbers. We prove that this is a Tauberian condition for the statistical convergence and the Cesáro convergence of a sequence of fuzzy numbers.

1. Introduction

The concept of statistical convergence was introduced by Fast [1]. A sequence of real numbers is said to be statistically convergent to some number if for every we have where by and , we denote the number of the elements in the set and the set of natural numbers, respectively. In this case, we write .

A sequence of real numbers is said to be -convergent to if its Cesàro transform of order one converges to as , where In this case, we write .

We recall that a sequence of real numbers is said to be slowly decreasing according to Schmidt [2] if where we denote by the integral part of the product , in symbol .

It is easy to see that (3) is satisfied if and only if for every there exist and , as close to 1 as we wish, such that

Lemma 1 (see [3, Lemma 1]). Let be a sequence of real numbers. Condition (3) is equivalent to the following relation:

A sequence of real numbers is said to be slowly increasing if Clearly, it is trivial that is slowly increasing if and only if the sequence is slowly decreasing.

Furthermore, if a sequence of real numbers satisfies Landau’s one-sided Tauberian condition (see [4, page 121]) then is slowly decreasing.

Móricz [3, Lemma 6] proved that if a sequence is slowly decreasing, then Also, Hardy [4, Theorem 68] proved that if a sequence is slowly decreasing, then

Maddox [5] defined a slowly decreasing sequence in an ordered linear space and proved implication (9) for slowly decreasing sequences in an ordered linear space.

We recall in this section the basic definitions dealing with fuzzy numbers. In 1972, Chang and Zadeh [6] introduced the concept of fuzzy number which is commonly used in fuzzy analysis and in many applications.

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, where denotes the closure of the set in the usual topology of .

We denote the set of all fuzzy numbers on by and call it 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

Let and . Then the operations addition and scalar multiplication are defined on by (cf. Bede and Gal [7]).

Lemma 2 (see [7]). The following statements hold. (i) is neutral element with respect to +, that is, for all . (ii)With respect to , none of , has opposite in .(iii)For any with or and any , we have . For general , the above property does not hold. (iv)For any and any , we have .(v)For any and any , we have .

Notice that is not a linear space over .

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 be easily observed that is a metric on and is a complete metric space (cf. Nanda [8]). Now, we may define the metric on by means of the Hausdorff metric as follows: One can see that Now, we may give the following.

Proposition 3 (see [7]). Let and . Then, the following statements hold. (i)  is a complete metric space.(ii).(iii).(iv).(v).

One can extend the natural order relation on the real line to intervals as follows: Also, the partial ordering relation on is defined as follows: We say that if and there exists such that or (cf. Aytar et al. [9]).

Lemma 4 (see [9, Lemma 6]). Let and . The following statements are equivalent. (i). (ii).

Lemma 5 (see [10, Lemma 5]). Let . If for every , then .

Lemma 6 (see [11, Lemma 3.4]). Let . Then, the following statements hold. (i)If and , then . (ii)If and , then .

Theorem 7 (see [11, Teorem 4.9]). Let . Then, the following statements hold (i)If and , then . (ii)If and , then .

Following Matloka [12], we give some definitions concerning sequences of fuzzy numbers. Nanda [8] introduced the concept of Cauchy sequence of fuzzy numbers and showed that every convergent sequence of fuzzy numbers is Cauchy.

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 the th term of the sequence. We denote by , the set of all sequences of fuzzy numbers.

A sequence is called convergent to the limit if and only if for every there exists an such that We denote by , the set of all convergent sequences of fuzzy numbers.

A sequence of fuzzy numbers is said to be Cauchy if for every there exists a positive integer such that We denote by , the set of all Cauchy sequences of fuzzy numbers.

If for every , then is said to be a monotone increasing sequence.

Statistical convergence of a sequence of fuzzy numbers was introduced by Nuray and Savaş [13]. A sequence of fuzzy numbers is said to be statistically convergent to some number if for every we have Nuray and Savaş [13] proved that if a sequence is convergent, then is statistically convergent. However, the converse is false, in general.

Lemma 8 (see [14, Remark 3.7]). If is statistically convergent to some , then there exists a sequence which is convergent (in the ordinary sense) to and

Basic results on statistical convergence of sequences of fuzzy numbers can be found in [10, 1517].

The Cesàro convergence of a sequence of fuzzy numbers is defined in [18] as follows. The sequence is said to be Cesàro convergent (written -convergent) to a fuzzy number if Talo and Çakan [19, Theorem 2.1] have recently proved that if a sequence of fuzzy numbers is convergent, then is -convergent. However, the converse is false, in general.

Definition 9 (see [14]). A sequence of fuzzy numbers is said to be slowly oscillating if It is easy to see that (23) is satisfied if and only if for every there exist and , as close to 1 as wished, such that whenever .

Talo and Çakan [19, Corollary 2.7] proved that if a sequence of fuzzy numbers is slowly oscillating, then the implication (9) holds.

In this paper, we define the slowly decreasing sequence over which is partially ordered and is not a linear space. Also, we prove that if is slowly decreasing, then the implications (8) and (9) hold.

2. The Main Results

Definition 10. A sequence of fuzzy numbers is said to be slowly decreasing if for every there exist and , as close to 1 as wished, such that for every Similarly, is said to be slowly increasing if for every there exist and , as close to 1 as wished, such that for every

Remark 11. Each slowly oscillating sequence of fuzzy numbers is slowly decreasing. On the other hand, we define the sequence , where Then, for each , since is increasing. Therefore, is slowly decreasing. However, it is not slowly oscillating because for each and we get for and the statements and hold.

Lemma 12. Let be a sequence of fuzzy numbers. If is slowly decreasing, then for every there exist and , as close to 1 as wished, such that for every

Proof. We prove the lemma by an indirect way. Assume that the sequence is slowly decreasing and there exists some such that for all and there exist integers and for which Therefore, there exists such that For the sake of definiteness, we only consider the case . Clearly, (5) is not satisfied by . That is, is not slowly decreasing. This contradicts the hypothesis that is slowly decreasing.

Theorem 13. Let be a sequence of fuzzy number. If is statistically convergent to some and slowly decreasing, then is convergent to .

Proof. Let us start by setting in (21), where is a subsequence of those indices for which . Therefore, we have Consequently, it follows that By the definition of the subsequence , we have Since is slowly decreasing for every there exist and , as close to 1 as we wish, such that for every For every large enough By (33), we have for every large enough , whence it follows that By (34) and Lemma 4, for every large enough we have Combining (37) and (38) we can see that On the other hand, by virtue of Lemma 12, for every there exist and such that for every For every large enough By (33), we have for every large enough , whence it follows that By (34) and Lemma 4, for every large enough we have Therefore, (42) and (43) lead us to the consequence that which yields with (39) for each and Lemma 4 that Therefore, (45) gives together with (34) that the whole sequence is convergent to .

Lemma 14. Let . If , then .

Proof. Let . If , then for all . Therefore, we have and for all . This means that .

Theorem 15. Let . If is -convergent to some and slowly decreasing, then is convergent to .

Proof. Assume that is satisfied (22) and is slowly decreasing. Then for every there exist and , as close to 1 as we wish, such that for every If is large enough in the sense that , then For every large enough , since we have By Lemma 4, we obtain for large enough that By (22), for large enough we obtain Since is slowly decreasing, we have Combining (51), (52), and (53) we obtain by (48) for each that By Lemma 14, we have On the other hand, by virtue of Lemma 12, for every there exist and such that for every If is large enough in the sense that , then For large enough , since we have Using the similar argument above, we conclude that Therefore, combining (55) and (60) for each and large enough , it is obtained that . This completes the proof.

Now, we define the Landau’s one-sided Tauberian condition for sequences of fuzzy numbers.

Lemma 16. If a sequence satisfies the one-sided Tauberian condition then is slowly decreasing.

Proof. A sequence of fuzzy numbers satisfies for , where is suitably chosen. Therefore, for all we have For all and , we obtain Hence, for each and we get for all Similarly, for all and we have Combining (65) and (66), one can see that which proves that is slowly decreasing.

By Theorems 13, 15 and Lemma 16, we derive the following two consequences.

Corollary 17. Let be a sequence of fuzzy numbers which is statistically convergent to a fuzzy number . If (61) is satisfied, then .

Corollary 18. Let be a sequence of fuzzy numbers which is -convergent to a fuzzy number . If (61) is satisfied, then .

Lemma 19. If the sequence satisfies (61), then

Proof. Assume that the sequence satisfies (61), then for all we have By the proof of Theorem  2.3 in [20], we obtain This means that , as desired.

Corollary 20. If the sequence satisfies (61), then

Proof. By Lemma 19, which is a Tauberian condition for statistical convergence by Corollary 17. Therefore, implies that . Then, Corollary 18 yields that .

3. Conclusion

In the present paper, we introduce the slowly decreasing condition for a sequence of fuzzy numbers. This is a Tauberian condition from to and from to .

Since we are not able to prove the fact that “-statistical convergence can be replaced by -convergence as a weaker condition, if it is proved that    is slowly decreasing while    is slowly decreasing,” this problem is still open. So, it is meaningful to solve this problem.

Finally, we note that our results can be extended to Riesz means of sequences of fuzzy numbers which are introduced by Tripathy and Baruah in [21].

Acknowledgment

The authors would like to express their pleasure to the anonymous referees for many helpful suggestions and interesting comments on the main results of the earlier version of the paper which improved the presentation of the paper.

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