On generalized difference sequence spaces of fuzzy numbers

The idea of difference sequence space was introduced by Kizmaz (1981) and this concept was generalized by Tripathy and Esi (2006). In this article we introduced the paranormed sequence spaces c(f,Λ,Δm,p), F c0 (f,Λ,Δm,p) and F ∞  (f,Λ,Δm,p) of fuzzy numbers associated with the multiplier sequence Λ= (λk) defined by a modulus function f. We study some of their properties like solidity, symmetricity, completeness etc. and prove some inclusion results.


Introduction
The concept of fuzzy set was introduced by L A. Zadeh in the year 1965.Based on this, sequences of fuzzy numbers have been introduced by different authors and many important properties have been investigated.Applying the notion of fuzzy real numbers, different classes of fuzzy real-valued sequences have been introduced and investigated by Tripathy andBaruah (2009, 2010a andb), Tripathy and Borgohain (2008), Tripathy and Dutta (2010a), Tripathy and Sarma (2011) and many researchers on sequence spaces.
A fuzzy real number X is a fuzzy subset of the real line R, i.e., a mapping X: R → I= [0,1] associating each real number t, with its grade of membership X(t).The α-level set of a fuzzy real number X is denoted by [X] α , 0 < α ≤ 1, where [X] α = {t∈R : X(t) ≥ α}.The 0-level set is the closure of the strong 0-cut i.e. [X] 0 = cl({t∈R : X(t) > 0}).
A fuzzy real number X is said to be upper-semi continuous if for each ε>0, X -1 ([0, a +ε)) is open in the usual topology of R.
If there exists t∈R such that X(t) = 1, then the fuzzy real number X is called normal.
A fuzzy real number X is said to be convex, if X(t) ≥ X(s)∧X(r) = min{X(s), X(r)}, where s ≤ t ≤ r.
The class of all upper semi-continuous, normal and convex fuzzy real numbers is de-noted by R(I).
The absolute value of |X|∈R(I) is defined by (one may refer to Kaleva and Seikkala (1984)).
The additive identity and multiplicative identity of R(I) are denoted by 0 and 1 respectively.
Let D be the set of all closed and bounded intervals.Define d : R(I)×R(I) →R by d(X,Y)=
The notion of modulus function was introduced by Nakano (1953).It was further investigated with applications to sequence spaces by Tripathy and Chandra (2011) and many others.
Definition 2.1.A modulus function f is a mapping (iii) f is increasing; (iv) f is continuous from right at 0. Hence f is continuous everywhere in [0, ∞) by (ii) and (iv).
The idea of difference sequences for real numbers was introduced by Kizmaz (1981) and it was further generalized by Tripathy and Esi (2006) as follows: Let w F be the class of all sequences of fuzzy numbers.Throughout the paper w F , c F , F c 0 and denote the classes of all, convergent, null and bounded sequences of fuzzy real numbers respectively.Definition 2.2.A sequence (X n ) of fuzzy real numbers is said to converge to the fuzzy real number X 0 if for every ε > 0 there exists a positive Definition 2.5.A fuzzy real valued sequence space E F is said to be monotone if E F contains the canonical pre-images of all its step spaces.
Remark 2.1.If a class of sequences of fuzzy numbers is solid, then it is monotone.
Definition 2.6.A fuzzy real valued sequence For (a k ) and (b k ) two sequences of complex terms and p = (p k )∈ ∞  , we have the following known inequality: where Recently the paranormed sequence spaces c(f,Λ,Δ m ,p), c 0 (f,Λ,Δ m ,p) and ∞  (f,Λ,Δ m ,p) are introduced by Tripathy and Chandra (2011).We now give the fuzzy analogue of these classes of sequences as follows.
Definition 2.8.Let f be a modulus function, then for a given multiplier sequence Λ=(λ k ), we introduce the following fuzzy real valued sequence spaces.
When f(x) = x, for all x∈[0, ∞), λ k = 1 and p k = 1 for all k∈N, we represent these classes of sequences by c F (Δ m ), Similarly taking different combinations of restrictions, we will get different paranormed sequences of fuzzy numbers from these classes of sequences.

Theorem
Now for a, b∈R, we have 1) and ( 2).This shows that (aX k + bY k )∈ F c 0 (f,Λ,Δ m ,p) and hence the class of sequences F c 0 (f,Λ,Δ m ,p) is closed under the addition and multiplication.
where M= max (1, sup k p k ) and X=(X k ).
Next we show the continuity of the product.Let α be fixed and g(X)→0.Then it is obvious that g(αX)→0.Let α →0 and X be fixed.Since f is continuous, we have , as α → 0, for all k∈N.
Therefore g is a paranorm.Proof: Let (X k ) be a given sequence and (α k ) be a sequence of scalars such that |α k | ≤1, for all k∈ N. Then we have, for all k∈N.The solidness of F c 0 (f,Λ,p) and F ∞  (f,Λ,p) follows from the above inequality.
The monotonicity of these two classes of sequences follows by Remark 2.1.
The first part of the proof follows from the following examples.
Example 3.1: for all k∈N; p k = 1 for all k odd and p k = 2 for all k even.Consider the sequence (X k ) defined by X k = A for all k∈N, where Then clearly (X k )∈c F (f,Λ,Δ 2 ,p).For E, a class of sequences, consider its J-step space E J defined as follows.
When (X k )∈E J , then its canonical pre-image (Y k )∈E J is given by Thus the class of sequences c F (f,Λ,p) is not monotone.Hence is not solid.Hence the class of sequences c F (f,Λ,p) is not monotone in general by Remark 2.1.
Example 3.2: Let f(x) = x, for all x∈ [0, ∞); m = 3, λ k = 2+k -1 , p k = 2 for all k odd and p k =3 for all k even.Consider the sequence (X k ) defined by Similar examples can be constructed to show that the classes of sequences

Conclusion
In this article we have introduced and studied different properties of the classes of sequences c F (f,Λ,Δ m ,p), F c 0 (f,Λ,Δ m ,p) and F ∞  (f,Λ,Δ m ,p) of fuzzy numbers and have investigated their different properties.The idea applied can be used for introducing many other classes of sequences and study their similar properties.
for all x∈[0, ∞), the above classes of sequences are denoted by c F (Λ,Δ m ,p), F c 0 (Λ,Δ m ,p) and F ∞  (Λ,Δ m ,p) respectively.When λ k = 1 for all k∈N, these classes of sequences are denoted by c 3.1.The classes of sequences c F (f,Λ,Δ m ,p), F c 0 (f,Λ,Δ m ,p) and F ∞  (f,Λ,Δ m ,p) are closed under addition and multiplication.Proof: We prove the theorem for the class of sequences F c 0 (f,Λ,Δ m ,p).The other classes can be proved similarly.Suppose

F
∞ (f, Λ, Δ m ,p) are are paranormed spaces, paranormed by g, define by p) and the inclusions are proper.Proof: Easy, so omitted.Theorem 3.4.The classes of sequences c F (f,Λ,Δ m ,p), F c 0 (f,Λ,Δ m ,p) and F ∞  (f,Λ,Δ m ,p) are neither solid nor monotone in general, but the class of sequences, F c 0 (f,Λ,p) and F ∞  (f,Λ,p) are solid and as such are monotone.
Λ,Δ 3, p) is not solid.Thus the class of sequences F ∞  (f,Λ,p) is not monotone in general by Remark 2.1.Similarly one can construct examples to show that the class of sequences F c 0 (f,Λ,Δ m ,p) is neither solid nor monotone in general.Acta Scientiarum.Technology Maringá, v. 35, n. 1, p. 117-121, Jan.-Mar., 2013 Theorem 3.5.The classes of sequences c F (f,Λ,Δ m ,p), F c 0 (f,Λ,Δ m ,p) and F ∞  (f,Λ,Δ m ,p) are not convergence free in general.Proof: The result follows from the following example.Example 3.3.Let f(x) = x, for all x∈ [0, ∞), m = 2, λ k = 2, for all k∈N, p k = 2 for all k odd and p k = 3 for all k even.Consider the sequence (X k ) defined as in Example 3.1.Clearly X k ∈c F (f,Λ,Δ 2 ,p).Consider the sequence (Y k ) defined by Y k = A for all k odd and Y k = A k for all k even, where

F c 0
(f,Λ,Δ m ,p) and F ∞  (f, Λ,Δ m , p) are not convergence free.Theorem 3.6.The classes of sequences c F (f,Λ,Δ m ,p), F c 0 (f,Λ,Δ m ,p) and F ∞  (f,Λ,Δ m ,p) are not symmetric in general.Proof.The result follows from the following example.Example 3.4.Let f(x) =x, for all x∈[0, ∞); m = 2, λ k =3, p k = 2 for all k odd and p k = 3 for all k even.Consider the sequence (X k ) = (A, B, A, B,…), where the fuzzy number A is defined as in Example 3.1 and the fuzzy number B is defined by k ) of (X k ) defined by (Y k ) = (A, B, B, A, A, B, B, A, A, ...).Then (Y k )∉c F (f,Λ,Δ 2 ,p).Hence the class of sequences c F (f,Λ,Δ m ,p) is not symmetric.Similar examples can be constructed to show that the classes of sequences F c 0 (f,Λ,Δ m ,p) and F ∞  (f,Λ,Δ m ,p) are not symmetric.
, Tripathy and