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`Advances in Fuzzy SystemsVolume 2018, Article ID 8636121, 9 pageshttps://doi.org/10.1155/2018/8636121`
Research Article

## Duals and Matrix Classes Involving Cesàro Type Classes of Sequences of Fuzzy Numbers

Department of Mathematics, Gauhati University, Guwahati 781014, India

Correspondence should be addressed to Hemen Dutta; moc.liamffider@80attud_nemeh

Received 31 May 2018; Accepted 26 July 2018; Published 8 August 2018

Copyright © 2018 Hemen Dutta and Jyotishmaan Gogoi. 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 first define Cesàro type classes of sequences of fuzzy numbers and equip the set with a complete metric. Then we compute the Köthe-Toeplitz dual and characterize some related matrix classes involving such classes of sequences of fuzzy numbers.

#### 1. Introduction

In 1965, Zadeh [1] introduced the concept of fuzzy sets and fuzzy set operations as an extension of the classical notion of the set theory. Later on several authors have discussed different aspects of the theory of fuzzy sets and applied it in various areas of science and engineering such as fuzzy topological spaces, similarity relations and fuzzy orderings, fuzzy possibility theory, fuzzy measures of fuzzy events, and fuzzy mathematical programming. Nowadays, fuzzy set theory is used as a powerful mathematical tool in solving complex real life problems which yields a notion of uncertainty and vagueness. Matloka [2] introduced bounded and convergent sequences of fuzzy numbers and studied their properties. In [3], Nanda studied sequences of fuzzy numbers and proved that every Cauchy sequence of fuzzy numbers is convergent. Since then, different classes of sequences of fuzzy numbers were introduced and studied by various authors. For the works on convergence of fuzzy sequences and series, we refer to Nuray and Savaş [4], Diamond and Kloeden [5], Matloka [2], Esi [6], Kaleva [7], Nanda [8],[3], Dubois and Prade [9], Altınok, Çolak, and Altın [10], Stojaković and Stojaković [11],[12], and Mursaleen, Srivastava, and Sharma [13]. In [14], Subrahmanyam defined the Cesàro summability of sequences of fuzzy numbers and proved some related Tauberian theorems. Some interesting results related to Cesàro summability method of sequences of fuzzy numbers and the Tauberian conditions which guarantee the convergence of summable sequences of fuzzy numbers can be found in Subrahmanyam [14], Talo and Çakan [15], Altın, Mursaleen, and Altınok [16], and Yavuz [17],[18].

Definition 1 (Goetschel and Voxman [19]). A fuzzy number is a fuzzy set on the real axis, i.e., a mapping , which satisfies the following four conditions:(i) is normal; i.e., there exists an such that (ii) is fuzzy convex; i.e., 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 called it the space of fuzzy numbers. -level set of is defined by
The set is a 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 as

Definition 2 (Talo and Başar [20]). Let and . Then the operations addition, scalar multiplication, and product are defined on byand where it is immediate thatandfor all

Definition 3 (Talo and Başar [20]). Let be the set of all closed bounded intervals of real numbers such that . Define the relation on as follows:where . Then is a complete metric space (see Diamond and Kloeden [5], Nanda [8]). Talo and Başar [20] defined the metric on by means of Hausdorff metric asThe partial ordering relation on is defined as follows:

Definition 4 (Talo and Başar [20]). is a nonnegative fuzzy number if and only if for all It is immediate that if is a nonnegative fuzzy number.
One can see that

Lemma 5 (Bede and Gal [21]). and. Then (i) is a complete metric space.(ii)(iii)(iv)(v)

Lemma 6 (Talo and Başar [20]). The following statements hold:(i)(ii)If

By we denote the set of all single sequences of fuzzy numbers on

Matloka [2] introduced bounded and convergent sequences of fuzzy numbers and studied their properties. We now quote the following definitions given by Talo and Başar [20] which we will use in a later part of this paper.

Definition 7. A sequence of fuzzy numbers is said to be bounded if the set of fuzzy numbers consisting of the terms of the sequence is a bounded set. That is to say that a sequence is said to be bounded if and only if there exist two fuzzy numbers and such that for all This means that and for all
The fact that the boundedness of the sequence is equivalent to the uniform boundedness of the functions and on Therefore, one can say that the boundedness of the sequence is equivalent to the fact that

Definition 8. Consider the sequence of fuzzy numbers . If for every there exists and for all , then we say that the sequence is said to be convergent to the limit and writeand we have the sets consisting of the bounded, convergent, and convergent to sequences of fuzzy numbers (Talo and Başar [20]) as follows:Throughout the text, the summations without limits run from to ; for example,means that

Definition 9 (Talo and Başar [20]). Let Then the expressionis called a series corresponding to the sequence of fuzzy number. We denote If the sequence () converges to a fuzzy number , then we say that the series converges to and write which implies as thatuniformly in Conversely, if the fuzzy numbers , and converge uniformly in , then defines a fuzzy number such that
Otherwise, we say the series of fuzzy numbers diverges. Additionally, if the sequence is bounded then we say that the series of fuzzy numbers is bounded.

Definition 10 (Talo and Başar [20]). Let be a space of convergent sequences of fuzzy numbers. The sum of a serieswith respect to this rule is defined by

Definition 11. Following Khan and Rahman [22], we define the Cesàro sequence space as follows:
If is a positive sequence of real numbers, then, for ,where and denotes summation over the range If for all , then reduces to defined byFollowing Maddox [23], throughout the paper we use the following inequality.
For any and we havewhere and
The classical analogy of was introduced and studied by Lim [24].
The classical Cesàro sequence space and its algebraic dual and related matrix transformations were introduced and studied by various authors like Shiue [25], Leibowitz [26], Lim [24], Khan and Khan [27],[28], Khan and Rahman [22], Johnson and Mohapatra [29], Rahman and Karim [30], etc.
The main purpose of this paper is to define and study the Cesàro sequence space and determine the Köthe-Toeplitz dual and give some related matrix transformations.

#### 2. Complete Metric Structure

We equip with a metric and show that this set is complete with respect to the metric defined in the following theorem.

Theorem 12. is complete with the metric defined bywhere

Proof. We first show that is a metric space.
It is obvious that and
Now we prove the triangle inequality.
Suppose .Then, using Minkowski’s inequality,Next, to show that is complete under , let us consider that is a Cauchy sequence in Then, for given there exists such thatwhich implies that ; that is, is a Cauchy sequence in So converges to a limit, say ; i.e.,
Suppose For given , there exists such that, for any ,Letting , we obtainSince is arbitrary, letting , we obtainwhich implies that
Next we show that
We have that is bounded in ; i.e., there exists such that Now, for any ,Now,Letting , we obtainThis implies that
This step completes the proof.

#### 3. Computation of the Köthe-Toeplitz Dual

Definition 13 (Talo and Başar [20]). The Köthe-Toeplitz dual or the dual of a set , denoted by , is defined as follows:where denotes the absolutely summable sequences of fuzzy numbers defined as follows:We now give the following theorem by which the Köthe-Toeplitz dual of will be determined.

Theorem 14. If and , then

Proof. Let and
We define We want to show that
Let and Then, using inequality (26) together with Lemma 6, we getwhich implies that Thus
Conversely, suppose that for all but Then, for every ,So, following Khan and Rahman [22], we can define a sequence such that and we haveNow we define a sequence as follows:and , where is such thatandwhere the maximum is taken with respect to in .
Therefore, It follows that is divergent. MoreoverNowThus, So, , which is a contradiction to our assumption. Hence
That is,
Then, combining the two results, we obtain
This step completes the proof.

#### 4. Characterization of Matrix Classes

An infinite matrix is one of the most general linear operators between two sequence spaces. The study of theory of matrix transformations has always been of great interest to mathematicians in the study of sequence spaces, which is motivated by special results in summability theory.

Definition 15 (Talo and Başar [20]). Let and be any two-dimensional infinite matrix of fuzzy numbers. Then we say that defines a mapping from into and denote it by if, for every sequence , the transform of , , given byexists for each each and is in .
if and only if the series on the right hand side of (53) converges for each and every and we have . A sequence of fuzzy numbers is said to be to if converges to which is called the limit of . Also by we denote that preserves the limit; that is, limit of is equal to the limit of for all .
We writewhere, for each , the maximum is taken with respect to

Theorem 16. Let be an infinite matrix of fuzzy numbers and . Then if there exists an integer such that whereand

Proof. Suppose there exists an integer such that . Let ThenUsing inequality (26), we obtainTherefore,
The necessity of the above theorem is still open; i.e., we do not know if then the condition (55) holds or not.

Theorem 17. Let be an infinite matrix of fuzzy numbers and . Then if (55) holds and, for fixed , we have

Proof. Suppose conditions (55) and (58) hold. ThenFrom (59), using similar argument as in Theorem 14, it is easy to verify that the seriesThis implies that Thus, exists.
Now, for each and , we can choose an integer such thatThen It follows thatas . This shows that which proves the theorem.

Corollary 18. Let be an infinite matrix of fuzzy numbers and . Then if (55) holds and, for fixed , we have

Proof. The proof is obvious.

#### Data Availability

The paper is theoretical in nature and all necessary references are included in the References with proper citation within the main text.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The authors would like to express their gratitude to the University Grants Commission, New Delhi, India, for offering Fellowship to the author J. Gogoi via Award Letter no F./2015-16/NFO-2015-17-OBC-ASS-36722/(SA-III/Website).

#### References

1. L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, pp. 338–353, 1965.
2. H. Matloka, “Sequences of fuzzy numbers,” BUSEFAL, vol. 28, pp. 28–37, 1986.
3. S. Nanda, “On sequences of fuzzy numbers,” Fuzzy Sets and Systems, vol. 33, no. 1, pp. 123–126, 1989.
4. F. Nuray and E. Savaş, “Statistical convergence of sequences of fuzzy numbers,” Mathematica Slovaca, vol. 45, no. 3, pp. 269–273, 1995.
5. P. Diamond and P. Kloeden, “Metric spaces of fuzzy sets,” Fuzzy Sets and Systems, vol. 35, no. 2, pp. 241–249, 1990.
6. A. Esi, “On some new paranormed sequence spaces of fuzzy numbers defined by Orlicz functions and statistical convergence,” Mathematical Modelling and Analysis, vol. 11, no. 4, pp. 379–388, 2006.
7. O. Kaleva, “On the convergence of fuzzy sets,” Fuzzy Sets and Systems, vol. 17, no. 1, pp. 53–65, 1985.
8. S. Nanda, Matrix transformations and sequence spaces, vol. 74, ICTP, Trieste, Italy, 1983.
9. D. Dubois and H. Prade, “Operations on fuzzy numbers,” International Journal of Systems Science, vol. 9, no. 6, pp. 613–626, 1978.
10. H. Altinok, R. Çolak, and Y. Altin, “On the class of λ-statistically convergent difference sequences of fuzzy numbers,” Soft Computing, vol. 16, no. 6, pp. 1029–1034, 2012.
11. M. Stojakovic and Z. Stojakovic, “Addition and series of fuzzy sets,” Fuzzy Sets and Systems, vol. 83, pp. 341–346, 1996.
12. M. Stojaković and Z. Stojaković, “Series of fuzzy sets,” Fuzzy Sets and Systems, vol. 160, no. 21, pp. 3115–3127, 2009.
13. M. Mursaleen, H. M. Srivastava, and S. K. Sharma, “Generalized statistically convergent sequences of fuzzy numbers,” Journal of Intelligent and Fuzzy Systems, vol. 30, no. 3, pp. 1511–1518, 2016.
14. P. V. Subrahmanyam, “Cesàro summability of fuzzy real numbers,” Journal of Analysis, vol. 7, pp. 159–168, 1999.
15. Ö. Talo and C. Çakan, “On the Cesàro convergence of sequences of fuzzy numbers,” Applied Mathematics Letters, vol. 25, pp. 676–681, 2012.
16. Y. Altin, M. Mursaleen, and H. Altinok, “Statistical summability (C, 1) for sequences of fuzzy real numbers and a Tauberian theorem,” Journal of Intelligent and Fuzzy Systems, vol. 21, no. 6, pp. 379–384, 2010.
17. E. Yavuz, “Comparison theorems for summability methods of sequences of fuzzy numbers,” Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology, pp. 1–8, 2018.
18. E. Yavuz, “Euler summability method of sequences of fuzzy numbers and a Tauberian theorem,” Journal of Intelligent and Fuzzy Systems, p. 10, 2016.
19. J. Goetschel and W. Voxman, “Elementary fuzzy calculus,” Fuzzy Sets and Systems, vol. 18, no. 1, pp. 31–43, 1986.
20. Ö. Talo and F. Başar, “Determination of duals of classical sets of sequences of fuzzy numbers and related matrix transformations,” Computers & Mathematics with Applications, vol. 58, pp. 717–733, 2009.
21. B. Bede and S. G. Gal, “Almost periodic fuzzy-number-valued functions,” Fuzzy Sets and Systems, vol. 147, no. 3, pp. 385–403, 2004.
22. F. M. Khan and M. F. Rahman, “Infinite matrices and Cesàro sequence spaces,” Analysis Mathematica, vol. 23, pp. 3–11, 1997.
23. I. J. Maddox, “Continuous and Köthe-Toeplitz dual of certain sequence spaces,” Proceedings of the Cambridge Philosophical Society, vol. 65, pp. 431–435, 1969.
24. K. P. Lim, “Matrix transformation in the Cesàro sequence spaces,” Kyungpook Mathematical Journal, vol. 14, pp. 221–227, 1974.
25. J. S. Shiue, “On the Cesàro sequence spaces,” Tamkang Journal of Mathematics, vol. 1, pp. 19–25, 1970.
26. G. M. Leibowitz, “A note on the Cesàro sequence spaces,” Tamkang Journal of Mathematics, vol. 2, pp. 151–157, 1971.
27. F. M. Khan and M. A. Khan, “The sequence space ces(p, s) and related matrix transformations,” Review of Research Mathematics Series, vol. 21, pp. 95–104, 1991.
28. F. M. Khan and M. A. Khan, “Matrix transformations between Cesàro sequence spaces,” Indian Journal of Pure and Applied Mathematics, vol. 25, no. 6, pp. 641–645, 1994.
29. P. D. Johnson and R. N. Mohapatra, “Density of finitely non-zero sequences in some sequence spaces,” Mathematica Japonica, vol. 24, no. 3, pp. 253–262, 1979.
30. M. F. Rahman and A. B. M. Karim, “Dual spaces of generalized Cesàro sequence space and related matrix mapping,” International Journal of Mathematics and Statistics Invention, vol. 4, no. 4, pp. 44–50, 2016.