On Some New Generalized Difference Sequence Space of Fuzzy Numbers and Statistical Convergence

A sequence space is defined to be a linear space of real or complex sequences. Throughout the paper N, R and C denotes the set of nonnegative integers, the set of real numbers and the set of complex numbers respectively. Let ω denote the space of all sequences (real or complex) and let l∞ and c be Banach spaces of bounded and convergent sequences =0 = { } n n x x ∞ with supremum norm = | | sup n n x x   . Let T denote the shift operator on ω, that is, =1 = { } n n Tx x ∞ , 2 =2 = { } n n T x x ∞ and so on. A Banach limit L is defined on l∞ as a non-negative linear functional such that L is invariant i.e., L(Sx) = L(x) and L(e) = 1,e = (1,1,1,...) [1].


Introduction
A sequence space is defined to be a linear space of real or complex sequences. Throughout the paper ℕ, ℝ and ℂ denotes the set of nonnegative integers, the set of real numbers and the set of complex numbers respectively. Let ω denote the space of all sequences (real or complex) and let l ∞ and c be Banach spaces of bounded and convergent sequences Lorentz called a sequence {x n } almost convergent if all Banach limits of x, L(x), are same and this unique Banach limit is called F -limit of x. In his paper, Lorentz proved the following criterian for almost convergent sequences.
A sequence x = {x n } ∈ l ∞ is almost convergent with F -limit of L(x) if and only if We denote the set of almost convergent sequences by f. This was further studied by Ganie [2], ℕanda [3] and many others.
A complex number sequence x is said to be statistically convergent to the number L if for every ε > 0, where the vertical bars indicate the number of elements in the enclosed set. In this case we write S -limit x = L or x k → L(S). We shall also use S to denote the set of all statistically convergent sequences. The idea of statistical convergence was introduced by Fast [4] and was further studied by several authors [5][6][7][8][9].
By a lacunary sequence we mean an increasing integer sequence θ = {k r } such that k 0 = 0 and h r = k r -k r-1 → ∞ as r → ∞. Throughout this paper, the intervals determined by θ will be denoted by I r = (k r-1 , k r ] and the ratio k r / k r-1 will be abbreviated by q r .
Let θ be a lacunary sequence; the number sequence x is S θ -convergent to L provided that for every ε > 0, In this case we write S θ -limit x = L or x k → L(S θ ), and we define The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh [10] 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. Matloka [11] introduced bounded and convergent sequences of fuzzy numbers and studied their some properties and has shown that every convergent sequence of fuzzy numbers is bounded. Later on sequences of fuzzy numbers have been discussed by many others [4,[7][8][9][12][13][14][15][16][17][18].
Let D denote the set of all closed and bounded intervals X = [a 1 , a 2 ] on the real line ℝ. For X, Y ∈ D we define . It is known that (D, d) is a complete metric space.
Let I = [0,1]. A fuzzy real number X is a fuzzy set on ℝ and is a mapping X : ℝ → Ι associating each real number t with its grade membership X (t). A fuzzy real number X is called convex if A fuzzy real number X is called upper semi-continuous if for each ε > 0, X -1 ([0, a + ε)) for all a ∈ I and given ε > 0, X -1 ([0, a + ε)) is open in the usual topology of ℝ. The set of all upper semi-continuous, normal, convex fuzzy numbers is denoted by R(I). The α -level set of a fuzzy real number X for 0 < α ≤ 1 denoted by X α is defined by X α = {t ∈ R : X (t) ≥ α}. The 0 -level set is the closure of strong 0 -cut.
The absolute value of | | X of X ∈ ℝ(I) is defined by [19].
Then d defines a metric on ℝ(I) [19]. The additive identity and multiplicative identity in ℝ(I) are denoted by 0 and 1 respectively.
A 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 < α ≤ 1, the αlevel set X α = {t ∈ ℝ n : X(t) ≥ α} is non-void compact convex subset of ℝ n , with support X 0 = {t ∈ ℝ n : X(t) > 0} We denote by L (ℝ n ) the set of all Fuzzy number. The linear structure of L (ℝ n ) induces the addition X + Y and the scalar multiplication λX, λ ∈ ℝ, interms of α level sets, for each 0 ≤ α ≤ 1. ℕow, for each 1 ≤ q < ∞, we define and for q ≤ r, we have d q ≤ d r . Throughout the text, we will denote d q by d where 1 ≤ q < ∞.

Results
In this section, we shall introduce the notion of Fuzzy numbers by using generalized difference operator n m ∆ and the lacunary sequence k = (k r ) and study their properties.   Theorem 4.2: Let θ = (k r ) be a lacunary sequence; p = (p k ) be a sequence of strictly positive real numbers with 0 < h = inf p k ≤ p k ≤ supp k = H < ∞, then