Certain spaces of sequences of fuzzy numbers defined by a modulus function

Abstract The main purpose of the present paper is to introduce the spaces ℓ∞ (F, f), c(F, f), c0(F, f) and ℓp(F, f, s) of sequences of fuzzy numbers defined by a modulus function. Furthermore, some inclusion theorems related to these sets are given and shown that ℓ∞ (F, f), c0(F, f) and ℓp(F, f, s) are solid.

We denote the set of all fuzzy numbers on R by L(R) and called it as the space of fuzzy numbers. α-level set [u] α of u ∈ L(R) is defined by It is known that (L(R), d) is a complete metric space, (cf. [8]). Following Matloka [5], we give some definitions below, which are needed in the text: . The fuzzy number x k denotes the value of the function at k ∈ N and is called the k th term of the sequence. By w(F ), we denote the set of all sequences of fuzzy numbers. Definition 1.2. A sequence x = (x k ) of fuzzy numbers is said to be convergent to a fuzzy number l, if for every ε > 0 there exists a positive integer n 0 such that d(x k , l) < ε for all k > n 0 .
By c(F ) and c 0 (F ), we denote the set of all convergent sequences and the set of all sequences converging to 0 of fuzzy numbers; respectively. Definition 1.3. A sequence x = (x k ) of fuzzy numbers is said to be Cauchy if for every ε > 0 there exists a positive integer n 0 such that d(x k , x m ) < ε for all k, m > n 0 .
By C(F ), we denote the set of all Cauchy sequences of fuzzy numbers. Definition 1.4. A sequence x = (x k ) of fuzzy numbers is said to be bounded if the set of fuzzy numbers consisting of the terms of the sequence (x k ) is a bounded set. That is to say that a sequence x = (x k ) of fuzzy numbers is said to be bounded if there exist two fuzzy numbers l and u such that l ≤ x n ≤ u for any n ∈ N. By ℓ ∞ (F ), we denote the set of all bounded sequences of fuzzy numbers.
Definition 1.5. Let x = (x k ) be a sequence of fuzzy numbers. Then the expression ∞ k=0 x k is called a series of fuzzy numbers. Denote s n = n k=0 x k for all n ∈ N, if the sequence (s n ) converges to a fuzzy number s then we say that the series ∞ k=0 x k of fuzzy numbers converges to s and write ∞ k=0 x k = s. We say otherwise the series of fuzzy numbers is divergent. The notion of modulus function was introduced by Nakano [7], as follows; is called a modulus if the following conditions hold: (d) f is continuous from the right at 0.
Hence, f is continuous on the interval [0, ∞). Now, we may give the concept of solidity of a space of sequences of fuzzy numbers defined by Sarma [11].
Zadeh introduced the concept of fuzzy sets and define the fuzzy set operations, in his significant article [15]. Subsequently several authors discussed various aspects of the theory and applications of fuzzy sets such as fuzzy topological spaces, similarity relations and fuzzy orderings, fuzzy measures of fuzzy events, fuzzy mathematical programming. Especially, El Naschie [9] studied the E infinity theory which has very important applications in quantum particle physics. In [8], it was shown that c(F ) and ℓ ∞ (F ) are complete metric spaces with the Haussdorff metric D ∞ defined by is also a complete metric space with respect to the Haussdorff metric D ∞ . Further, Nanda [8] has introduced and proved that the space ℓ p (F ) of all absolutely p-summable sequences of fuzzy numbers defined by is a complete metric space with the Haussdorff metric D p defined by We assume here and in what follows 1 ≤ p < ∞ and for simplicity in notation, the summation without limits runs from 0 to ∞. Nuray and Savaş [10] have recently shown that the space ℓ(p, F ) of sequences of fuzzy numbers is a complete metric space with the metric ̺ defined by is a bounded sequence of strictly positive real numbers and M = max{1, sup k∈N p k }, and x = (x k ), y = (y k ) are the points of the space ℓ(p, F ). Mursaleen and Başarir [6] have introduced some new spaces of sequences of fuzzy numbers generated by a non-negative regular matrix A some of which reduced to the Maddox spaces ℓ ∞ (p, F ), c(p, F ), c 0 (p, F ) and ℓ(p, F ) of sequences of fuzzy numbers for the special cases of that matrix A. Altın, Et and Çolak [1] have recently introduced the concepts of lacunary statistical convergence and lacunary strongly convergence of generalized difference sequences of fuzzy numbers, and gave some relations related to these concepts. Talo and Başar [12] have extended the main results of Başar and Altay [2] to the fuzzy numbers.
In [13], Talo and Başar have recently studied the corresponding sets ℓ ∞ (F ), c(F ), c 0 (F ) and ℓ p (F ) of sequences of fuzzy numbers to the classical spaces ℓ ∞ , c, c 0 and ℓ p of sequences with real or complex terms. After determining the α-, β-and γ-duals of the sets ℓ ∞ (F ), c(F ), c 0 (F ) and ℓ p (F ), they characterize some classes of matrix transformations between the classical sets of sequences of fuzzy numbers. Furthermore, they also emphasize the solidness of the sets ℓ ∞ (F ), c 0 (F ) and ℓ p (F ). Quite recently; Talo and Başar [14] have worked the quasilinearity of the classical sets ℓ ∞ (F ), c(F ), c 0 (F ) and ℓ p (F ) of sequences of fuzzy numbers and obtained the β−, α−duals of the set ℓ 1 (F ), and characterized the class of infinite matrices of fuzzy numbers from ℓ 1 (F ) to ℓ p (F ). Additionally, they proved that ℓ ∞ (F ) and c(F ) are normed quasialgebras and an operator defined by an infinite matrix belonging to the class (ℓ ∞ (F ) : ℓ ∞ (F )) is bounded and quasilinear.
In the present paper, we essentially deal with the metric spaces ℓ ∞ (F, f ), c(F, f ), c 0 (F, f ) and ℓ p (F, f, s) of sequences of fuzzy numbers defined by a modulus function which are the generalization of the metric spaces ℓ ∞ (F ), c(F ), c 0 (F ) and ℓ p (F ) of sequences of fuzzy numbers. Additionally, we state and prove some inclusion theorems related to those sets. Finally, we establish that the sets ℓ ∞ (F, f ), c 0 (F, f ) and ℓ p (F, f, s) are solid as a consequence of the fact that the sets ℓ ∞ (F ), c 0 (F ) and ℓ p (F ) are solid.

s) of sequences of fuzzy numbers defined by a modulus function
Let f be a modulus function. We introduce the sets ℓ ∞ (F, f ), c(F, f ), c 0 (F, f ) and ℓ p (F, f, s) of sequences of fuzzy numbers defined by a modulus function by Now, we may begin with the following theorem which is essential in the text: Theorem 2.1. The sets ℓ ∞ (F, f ), c(F, f ), c 0 (F, f ) and ℓ p (F, f, s) of sequences of fuzzy numbers defined by a modulus function are closed under the coordinatewise addition and scalar multiplication.
Proof. Since it is not hard to show that the sets ℓ ∞ (F, f ), c(F, f ), c 0 (F, f ) and ℓ p (F, f, s) are closed with respect to the coordinatewise addition and scalar multiplication, we omit the detail. Proof. Since the proof is analogue for the spaces ℓ ∞ (F, f ), c(F, f ) and ℓ p (F, f, s), we consider only the space c 0 (F, f ). One can easily establish that D ∞ defines a metric on c 0 (F, f ) which is a routine verification, so we leave it to the reader. It remains to prove the completeness of the space c 0 (F, f ). Let {x i } be any Cauchy sequence in the space Then, for a given ε > 0 there exists a positive integer n 0 (ε) such that for all i, j ≥ n 0 (ε). We obtain for each fixed k ∈ N from (2.1) that for every i, j ≥ n 0 (ε). (2.2) means that Therefore, since f is a modulus function one can derive by (2.4) that k } is a Cauchy sequence in L(R) for every fixed k ∈ N. Since L(R) is complete, it converges, say x (i) k → x k as i → ∞. Using these infinitely many limits, we define the sequence x = (x 0 , x 1 , x 2 , . . .). Let us pass to limit firstly as j → ∞ and nextly taking supremum over k ∈ N in (2.2) to obtain k , 0 ≤ ε 2 for every k ≥ k 0 (ε) and for each fixed i ∈ N. Therefore, since This shows that x ∈ c 0 (F, f ). Since {x i } was an arbitrary Cauchy sequence, the space c 0 (F, f ) is complete.
This step concludes the proof.
Proof. We give the proof for λ = c 0 . Since the proof can also be given for λ = c or λ = ℓ ∞ , we leave the detail to the reader. (a) Let x = (x k ) ∈ λ(F, f 1 ) ∩ λ(F, f 2 ). Since one can see by passing to limit as k → ∞ and taking supremum over k ∈ N from (2.6) that x ∈ λ(F, f 1 + f 2 ), where λ ∈ {ℓ ∞ , c, c 0 }.