The generalized difference gai sequences of fuzzy numbers defined by Orlicz functions

The concept of fuzzy sets and fuzzy set operations were first introduced by Zadeh [18] and subsequently several authors have 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. In this paper we introduce and examine the concepts of Orlicz space of entire sequence of fuzzy numbers generated by infinite matrices. Let C (R) = {A ⊂ R : Acompactandconvex} . The space C (R) has linear structure induced by the operations A + B = {a+ b : a ∈ A, b ∈ B} and λA = {λa : a ∈ A} for A,B ∈ C (R) and λ ∈ R. The Hausdorff distance between A and B of C (R) is defined as


Introduction
The concept of fuzzy sets and fuzzy set operations were first introduced by Zadeh [18] and subsequently several authors have 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.
In this paper we introduce and examine the concepts of Orlicz space of entire sequence of fuzzy numbers generated by infinite matrices.
Let C (R n ) = {A ⊂ R n : A compact and convex} .The space C (R n ) has linear structure induced by the operations A + B = {a + b : a ∈ A, b ∈ B} and λA = {λa : a ∈ A} for A, B ∈ C (R n ) and λ ∈ R. The Hausdorff distance between A and B of C (R n ) is defined as The fuzzy number is a function X from R n to [0,1] which is normal, fuzzy convex, upper semi-continuous and the closure of {x ∈ R n : X(x) > 0} is compact.These properties imply that for each 0 , and d , The additive identity in L (R n ) is denoted by 0. For simplicity in notation, we shall write throughout d instead of d q with 1 ≤ q ≤ ∞.
The fuzzy number X k denotes the value of the function at k ∈ N. We denotes by W (F ) the set of all sequences X = (X k ) of fuzzy numbers.
A complex sequence, whose k th terms is x k is denoted by {x k } or simply x.Let φ be the set of all finite sequences.Let ℓ ∞ , c, c 0 be the sequence spaces of bounded, convergent and null sequences x = (x k ) respectively.In respect of ℓ ∞ , c, c 0 we have The vector space of all analytic sequences will be denoted by Λ.A sequence x is called entire sequence if lim k→∞ |x k | 1/k = 0.The vector space of all entire sequences will be denoted by Γ.A sequence x is called gai sequence if lim k→∞ (k! |x k |) 1/k = 0.The vector space of all gai sequences will be denoted by χ.Orlicz [26] used the idea of Orlicz function to construct the space (L M ).Lindenstrauss and Tzafriri [27] investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space ℓ M contains a subspace isomorphic to ℓ p (1 ≤ p < ∞).Subsequently different classes of sequence spaces defined by Parashar and Choudhary [28], Mursaleen et al. [29], Bektas and Altin [30], Tripathy et al. [31], Rao and subramanian [32] and many others.The Orlicz sequence spaces are the special cases of Orlicz spaces studied in Ref. [33].Recall( [26], [33]) an Orlicz function is a function then this function is called modulus function, introduced by Nakano [34] and further discussed by Ruckle [35] and Maddox [36] and many others.An Orlicz function M is said to satisfy ∆ 2 − condition for all values of u, if there exists a constant K > 0, such that M (2u) ≤ KM (u)(u ≥ 0).Lindenstrauss and Tzafriri [27] used the idea of Orlicz function to construct Orlicz sequence space The space ℓ M with the norm becomes a Banach space which is called an Orlicz sequence space.For M (t) = t p , 1 ≤ p < ∞, the space ℓ M coincide with the classical sequence space ℓ p • Given a sequence x = {x k } its n th section is the sequence x (n) = {x 1 , x 2 , ..., x n , 0, 0, ...} δ (n) = (0, 0, ..., 1, 0, 0, ...) , 1 in the n th place and zero's else where.
Remark 1.1 An Orlicz function M satisfies the inequality M (λx) ≤ λM (x) for all λ with 0 < λ < 1.Let m ∈ N be fixed, then the generalized difference operation is defined by

Definitions and Prelimiaries
Let P s denotes the class of subsets of N, the natural numbers, those do not contain more than s elements.Throughout (φ n ) represents a non-decreasing sequence of real numbers such that nφ n+1 ≤ (n + 1) φ n for all n ∈ N. The sequence χ (φ) for real numbers is defined as follows: The generalized sequence space χ (∆ n , φ) of the sequence space χ (φ) for real numbers is defined as follows In this article we introduce the following classes of sequences of fuzzy numbers: Let M be an Orlicz function, then for k∈ σ ∈ P s }

Main Results
In this section we prove some results involving the classes of sequences of fuzzy numbers χ F M (∆ m , φ) , χ F M (∆ m ) and Λ F M (∆ m ) .
where λ is a scalar and |λ| Then there exist positive numbers ρ 1 and ρ 2 such that . By the equation (3.1) and since M is non-decreasing convex function, we have By the condition (3.2) and Remark, we have Proof: Let X i be a cauchy sequence in χ F M (∆ m , φ) .Then for each ǫ > 0, there exists a positive integer n 0 such that g X i , Y j < ǫ for i, j ≥ n 0 , then ) for all i,j≥ n 0 , 0 < ǫ and the fact that Let lim i→∞ X i k = X k say, for each k ∈ N. Since X i is a cauchy sequence, for each ǫ > 0, there exists n 0 = n 0 (ǫ) such that g X i , X j < ǫ for all i, j ≥ n 0 .So we have we obtain X = (X k ) ∈ χ F M .Therefore χ F M (∆ m , φ) is complete metric space.This completes the proof. 2 Proposition 3.3 The space Λ F M (∆ m ) is a complete metric space with the metric by ) .This completes the proof.
Since (φ n ) is monotonic increasing, so we have → 0 as k, s → ∞ for k ∈ σ ∈ P s .Hence the set of all fuzzy numbers.The linear structure of L (R n ) induces the addition X + Y and scalar multiplication λX, λ ∈ R, in terms of α− level 10 N.Subramanian,Ayhan Esi,U.K.Misra and M.S.Panda sets, by [

Theorem 3 . 1
If d is a translation invariant metric, then χ F M (∆ m , φ) are closed under the operations of addition and scalar multiplication Proof: Since d is a translation invariant metric implies that
defined by (see for instance Kaleva and Seikkala[42]