SEQUENCE SPACES OF FUZZY NUMBERS DEFINED BY A SEQUENCE OF MODULI

In this paper, we introduce to certain class of sequence spaces of fuzzy numbers defined by a sequence of modulii and study the topology that arises on the said spaces.


Introduction
The fuzzy theory has emmerged as the most active area of research in many branches of science and engineering.Among various developments of the theory of fuzzy sets [16] a progressive development has been made to find the fuzzy analogues of the classical set theory.In fact the fuzzy theory has become an area of active research for the last 50 years.It has a wide electrical engineering, nonlinear dynamical system, population dynamics and biological engineering etc.
In [9], Nanda studied the spaces of bounded and convergent sequences of fuzzy numbers and shown that they are complete metric spaces with the metric By using the metric , many spaces of fuzzy sequences have been built and published in famous math journals.By reviewing the literature, one can reach them easily, (e. g., see, [1][2][3][4][5][6], [9], [10], [12][13][14][15][16] and the references there in.)Here we give the preliminaries.Let D denote the set of all closed bounded intervals A=[A , A] on the real line R.A fuzzy number is a fuzzy subset of the real line R which is bounded, convex and normal.Let L(R) denote the set of all fuzzy numbers which are upper semicontinuous and have compact support.In other words, if X∈L(R) then for any α ∈[0,1] X α is compact where Now we quote the following definitions which will be needed in the sequel Definition 1.1.A sequence X={X k } of fuzzy numbers is a function X from the set N of all positive integers into L(R).The fuzzy number {X k } denotes the value of the function at k ∈ N and is called the k th term of the sequence.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 .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.
Let C(R n ) = {A ⊂ R n : A is compact and convex}.The space C(R n ) has a linear structure induced by the operations The Hausdorff distance between A and B is defined as A fuzzy number is a fuzzy subset of the real line R, i.e., a mapping which is bounded, convex, normal, upper semicontinuous and have compact support.Let L(R n ) denote the the set of all fuzzy numbers.The linear structure of L(R n ) induces addition X+Y and scalar multiplication µX, µ ∈ R in terms of α-level sets, by By ω(F), we denote the set of all sequences of fuzzy numbers.
By ∞ (F), c(F), c 0 (F) and p (F) we denote the set of all bounded, convergent, null and absolutely p-summable sequences of fuzzy numbers respectively.
Here we give a brief account of the recent developments in this direction.
Talo and Basar [12][13][14] used the idea of modulus function f to define the following fuzzy sequence spaces Hazarika [6] used the Orlicz function M and Λ = (γ k ) a sequence of non zero scalars to define the following fuzzy sequence spaces Recently Mursaleen and Noman [7][8] introduced the notion of λ -convergent sequences.Let ω be the set of all complex sequences x = (x k ) and λ = (λ k ) ∞ k=1 be a strictly increasing sequence of positive real numbers tending to infinity.
It is well known that if lim m x m = a in the ordinary sense of convergence, then Let C denotes the space whose elements are the sets of distinct positive integers.Given any element σ of C, we denote by c(σ ) the sequence {c n (σ )} such that Sargent [11] defined the sequence space where {ϕ k } is a real sequence.
Alotaibi, Mursaleen, Sharma and Mohiuddine [1] used the Musielak-Orlicz function M= (M k ) , p = (p k ) a bounded sequence of positive real numbers and σ one-to-one mapping from the set of positive integers into it self such that σ k (n) = σ (σ k−1 (n)) to define the following fuzzy sequence spaces

Main results
In this article we introduce the following classes of sequences of fuzzy numbers using the When σ (n) = σ (n + 1) we obtain the following classes of sequences of fuzzy numbers If p = (p k ) = 1 then we have the following classes of sequences of fuzzy numbers Theorem 2.1.F ∞ (F, Λ, σ , p), F 1 (F, Λ, σ , p) and m F (F, Λ, ϕ, σ , p) are linear spaces over the field C of complex numbers.
Since F = ( f k ) is non decreasing and continuous, we have . Hence m F (F, Λ, ϕ, σ , p) is a linear space.Similarly we can prove that F ∞ (F, Λ, σ , p) and F 1 (F, Λ, σ , p) are linear spaces.
Theorem 2.2.Let F = ( f k ) be a sequence of modulii and p = (p k ) be a bounded sequence of positive real numbers, then the sequence space m F (F, Λ, ϕ, σ , p) is a complete metric space, with the metric defined by This implies that g(X i , X) < ε for all i ≥ n 0 i.e X i → X as i → ∞.
Theorem 2.3.Let F = ( f k ) be a sequence of modulii and p = (p k ) be a bounded sequence of positive real numbers, then the sequence space (a) F 1 (F, Λ, σ , p) is a complete metric space, with the metric defined by ) is a complete metric space, with the metric defined by Proof.The proof is analogous to Theorem 2.2, so we omit the details.

Therefore we have
For A, B ∈ D define A≤ B iff A ≤ B and A ≤ B, d(A, B) = max(|A − B|, |A − B|)It is easy to see that d defines a metric on D and (D, d) is a complete metric space.Also ≤ is a partial order on D.
where the ||.|| denotes the usual Euclidean norm in R n .It is known that (