Some Classes of Difference Sequence Spaces of Fuzzy Real Numbers Defined by Orlicz Function

The idea of Kizmaz [8] was applied to introduce different type of difference sequence spaces and study their different properties by Et et al. [9], Tripathy et al. [10], Tripathy and Baruah [2], Tripathy and Borgohain [3], Tripathy and Esi [11], Tripathy et al. [12], Tripathy and Mahanta [13], and many others. Tripathy and Esi [11] introduced a new type of difference sequence spaces as follows. Let m ∈ N be fixed, then Z(Δm) = {x = (xk) : (Δmxk) ∈ Z}, for Z = ∞, c, c0, where Δmx = (Δmxk) = (xk − xk+m). (3)

Tripathy and Esi [11] introduced a new type of difference sequence spaces as follows.Let m ∈ N be fixed, then where Δ m x = (Δ m x k ) = (x k − x k+m ). ( The above sequence spaces are Banach spaces, normed by Tripathy et al. [12] further generalized this notion and introduced the following.For m ≥ 1 and n ≥ 1, This generalized difference has the following binomial representation: An Orlicz function is a function M : [0, ∞) → [0, ∞), which is continuous, nondecreasing, and convex with M(0) = 0, M(x) > 0, for x > 0 and M(x) → ∞, as x → ∞ (one may refer to Korasnoselkii and Rutitsky [14]).
An Orlicz function M is said to satisfy Δ 2 -condition for all values of x, if there exists a constant K > 0, such that M(Lx) ≤ KLM(x), for all x > 0 and for L > 1.
Throughout the paper w F , F , F ∞ represent the classes of all, absolutely summable, and bounded sequences of fuzzy real numbers, respectively.

Definitions and Background
Let C(R n ) = {A ⊂ R n : A is compact and convex}.Then the space C(R n ) has linear structure induced by the operations A fuzzy real number on R n is a function X : R n → I(= [0, 1]) associating each real number t ∈ R n with its grade of membership X(t).
A fuzzy real number X is called convex if X(t) ≥ X(s) ∧ X(r) = min(X(s), X(r)), where s < t < r.
If there exists t 0 ∈ R n such that X(t 0 ) = 1, then the fuzzy real number X is called normal.
A fuzzy real number X is said to be upper semicontinuous if for each ε > 0, X −1 ([0, a + ε)), for all a ∈ I is open in the usual topology of R n .

The linear structure of C(R n ) induces the addition X + Y and scalar multiplication
Define, for each 1 ≤ q < ∞, and ) is a complete, separable, and locally compact metric space.
A sequence X = (X k ) of fuzzy real numbers is said to converge to the fuzzy number X 0 , if for every ε > 0, there exists Let X = (X n ) be a sequence, then S(X) denotes the set of all permutations of the elements of ( A sequence space E is said to be monotone if E contains the canonical preimages of all its step spaces.

Lemma 2. A class of sequences E is solid which implies that E is monotone.
Lindenstrauss and Tzafriri [15] used the notion of Orlicz function and introduced the sequence space: The space M with the norm, becomes a Banach space, which is called an Orlicz sequence space.The space M is closely related to the space p , which is an Orlicz sequence space with M(x) = x p , for 1 ≤ p < ∞.
In this paper we introduce the following difference sequence spaces: for some ρ > 0

Theorem 3. The classes of sequences
Proof.We establish the result for the class of sequences The proof for the other cases will follow similarly.It can easily be verified that F ∞ (M, Δ n m ) is a metric space by the metric η defined above.Next we show that it is a complete metric space.
Let (X (i) ) be a Cauchy sequence in F ∞ (M, Δ n m ) such that Let ε > 0 be given.For a fixed x 0 > 0, choose r > 0 such that M(rx 0 /2) ≥ 1.Then there exits a positive integer By the definition of η, we have, which implies Hence (X (i) r ), for r = 1, 2, 3, . . ., mn are Cauchy sequence in R n (I) and hence are convergent in R n (I), since R n (I) is a complete metric space. Let Since M is continuous, we get, which implies (Δ n m X (i) k ) is a Cauchy sequence in R n (I) and so is convergent in R n (I), since R n (I) is complete metric space.

Advances in Fuzzy Systems
Proceeding in this way inductively, we get Next taking j → ∞, keeping i fixed, and by the continuity of M, we have the following from (20): Now on taking the infimum of such ρ's, we get inf Hence from ( 17) on taking limit as j → ∞, we get Since M is continuous and nondecreasing, so we get ) is a complete metric space.The other cases can be established similarly.This completes the proof of the theorem.

Result 1. The classes of sequences
, are neither solid nor monotone in general.
Proof.The result follows from the following example.

Example 4. Consider the sequence space
Consider the sequence (X k ) defined by Then, we have 2 ).Consider the sequence (α k ) of scalars defined by For k = i 2 , we have For k / = i 2 , we have which implies Similarly the other cases can be established.The classes of sequences are not monotone followed by Lemma 2. and c F 0 (M, n m ) are not symmetric in general.

Result 2. The classes of sequences
Proof.The result follows from the following example.

Advances in Fuzzy Systems
Proof.Let (X k ) ∈ F ∞ (M, Δ n−1 m ).Then we have, Now we have Proceeding in this way, we have Z(M, Δ i m ) ⊂ Z(M, Δ n m ), for 0 ≤ i < n, for Z = F ∞ , c F , and c F 0 .This completes the proof.

Proposition 6 .Example 7 .
for each fixed ρ > 0. (40) Hence, (Y k ) / ∈ F ∞ (M, Δ 2 ).Thus the classes of sequences F ∞ (M, Δ n m ), c F (M, Δ n m ), and c F 0 (M, Δ n m ) are not symmetric in general.Note 1.For m = n = 0, the class of sequences F ∞ (M) and c F (M) are symmetric.For m ≤ 1 and n ≤ 1, the class of sequences c F 0(M, Δ n m ) is symmetric.The classes of sequences F ∞ (M, Δ n m ), c F (M, Δ n m ), c F 0 (M, Δ n m ) are not convergence-free in general.Proof.The result follows from the following example.Let m = 4 and n = 1.Let M(x) = x 3 , for all

Proposition 9 .
One has Z(M, Δ i m ) ⊂ Z(M, Δ n m ), for 0 ≤ i < n, for Z = F∞ , c F , and c F 0 .
are not convergence-free in general.