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

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], Subrahmanyamdefined theCesàro 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].


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 Savas ¸ [4], Diamond and Kloeden [5], Matloka [2], Esi [6], Kaleva [7], Nanda [8], [3], Dubois and Prade [9], Altınok, C ¸olak, 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 C ¸akan [15], Altın, Mursaleen, and Altınok [16], and Yavuz [17], [18].
(ii) V is fuzzy convex; i.e., V[ + (1 − )] ≥ min{V(), V()} for all ,  ∈ R and for all  ∈ [0 We denote the set of all fuzzy numbers on R by  1 and called it the space of fuzzy numbers.-level set The set [V]  is a closed, bounded, and nonempty interval for each  ∈ [0, 1] which is defined by R can be embedded in  1 , since each  ∈ R can be regarded as a fuzzy number  defined as 2

Advances in Fuzzy Systems
Definition 2 (Talo and Bas ¸ar [20]).Let , ,  ∈  1 and  ∈ R. Then the operations addition, scalar multiplication, and product are defined on  1 by and where it is immediate that and for all  ∈ [0, 1].
By W  we denote the set of all single sequences of fuzzy numbers on R.
Matloka [2] introduced bounded and convergent sequences of fuzzy numbers and studied their properties.We now quote the following definitions given by Talo and Bas ¸ar [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 (  ) ∈ W  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  ∈ [0, 1].The fact that the boundedness of the sequence (  ) ∈ W  is equivalent to the uniform boundedness of the functions  −  () and  +  () on [0, 1].Therefore, one can say that the boundedness of the sequence Definition 8. Consider the sequence of fuzzy numbers (  ) ∈ W  .If for every  > 0 there exists  0 =  0 () ∈  and  ∈  1 such that D(  , ) <  for all  >  0 , then we say that the sequence is said to be convergent to the limit  and write and we have the sets ℓ  ∞ , C  , C  0 consisting of the bounded, convergent, and convergent to 0 sequences of fuzzy numbers (Talo and Bas ¸ar [20]) as follows: Throughout the text, the summations without limits run from 0 to ∞; for example, Definition 9 (Talo and Bas ¸ar [20]).Let (  ) ∈ W  .Then the expression is 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  → ∞ that and and converge uniformly in  ∈ [0, 1], then  = {( − (),  + () :  ∈ [0, 1]} defines a fuzzy number such that  = ∑ ∞ =0   .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 Bas ¸ar [20]).Let   be a space of convergent sequences of fuzzy numbers.The sum of a series with respect to this rule is defined by Definition 11.Following Khan and Rahman [22], we define the Cesàro sequence space [, (  )]  as follows: where Following Maddox [23], throughout the paper we use the following inequality.
The classical analogy of ()  was introduced and studied by Lim [24].
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.

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.

Computation of the Köthe-Toeplitz Dual
Definition 13 (Talo and Bas ¸ar [20]).The Köthe-Toeplitz dual or the −dual of a set   ⊂ W  , denoted by {  }  , is defined as follows: where ℓ  1 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.

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 Bas ¸ar [20]).Let  and we have  = {()  } ∈N .A sequence of fuzzy numbers  = (  ) is said to be  −  to  if  converges to  which is called the −limit of .Also by  ∈ (  1 :   2 ; ) we denote that  preserves the limit; that is, −limit of  is equal to the limit of  for all  = (  ) ∈    1 .We write where, for each  ∈ N, the maximum is taken with respect to  ∈ [2  , 2 +1 ).
1is a nonnegative fuzzy number if and only if ( 0 ) = 0 for all  0 < 0. It is immediate that  ≽ 0 if  is a nonnegative fuzzy number.