New type of generalized difference sequence of fuzyy numbers involving lacunary sequences

Abstract

In this paper, we introduce a new type of almost statistical convergence of generalized difference sequences of fuzzy numbers involving lacunary sequences. We give the relations between the lacunary strongly almost Cesàro type convergence and lacunary almost statistical convergence. Furthermore, we study some of their properties like completeness, solidity, symmetricity etc. We also give some inclusion relations related to these classes.

Keywords

Almost statistical convergence difference sequence fuzzy numbers solidness symmetricity convergence free AMS Mathematical Subject Classification: 40A05;40A25;40A30;40C05;03E72

1 Introduction

The concept of statistical convergence was initially introduced by Fast [13] and Schoenberg [28], independently. It has a wide range of applications in field of sequences spaces like Fourier analysis, Ergodic theory,Number theory and Summability theory ([6 , 34]). Moreover, statistical convergence is closely related to the concept of convergence inprobability.

The existing literature on almost statistical convergence and strongly almost convergence has been restricted to real or complex sequences, but Altınok et al. [2] extended the idea to apply to sequences of fuzzy numbers and also Altın et al. [1], Altınok et al.[3], Çolak et al. [8], Et et al. [12], Gökhan et al. [17],Nuray [25], Savas [27], Talo and Başar [29], Tripathy et al. [33] studied the sequences of fuzzy numbers.

The main purpose of the present paper is to introduce and examine the spaces ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p)$ and ${\hat{S}}^{F} (Δ_{m}^{n}, θ),$ where $Δ_{m}^{n}$ is a difference operator, θ = (k _r) is a lacunary sequence and p is any sequence of strictly positive real numbers so as to fill up the existinggaps in the literature. It is particularly interesting to use a difference operator and a lacunary sequence for introducing a sequence space of fuzzy numbers. In the second section of this study, we give a brief overview about statistical convergence, fuzzy numbers and using the generalized difference operator $Δ_{m}^{n}$ and the lacunary sequence θ = (k _r) we define the concepts of lacunary almost $Δ_{m}^{n} -$ statistical convergence and lacunary strongly almost $Δ_{m}^{n} -$ convergence of sequences of fuzzy numbers. In section 3 we establish some inclusion relations between ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p)$ and ${\hat{S}}^{F} (Δ_{m}^{n}, θ)$ , between ${\hat{S}}^{F} (Δ_{m}^{n}, θ)$ and ${\hat{S}}^{F} (Δ_{m}^{n})$ .

2 Definitions and preliminaries

The definitions of statistical convergence and strong p-Cesàro convergence of a sequence of real numbers were introduced in the literature independently of one another and followed different lines of development since their first appearance. It turns out, however, that the two definitions can be simply related to one another in general and are equivalent for bounded sequences. The idea of statistical convergence depends on the density of subsets of the set $ℕ$ of natural numbers. The density of a subset E of $ℕ$ is defined by $δ (E) = \lim_{n \to \infty} \frac{1}{n} \sum_{k = 1}^{n} χ E^{(k)} provided the limit exists,$

where χ _E is the characteristic function of E. It is clear that any finite subset of $ℕ$ has zero natural density and δ (E ^c) = 1 - δ (E).

A sequence (x _k) of complex numbers is said to be statistically convergent to ℓ if for every ɛ > 0, In this case we write $stat - lim_{k \to \infty} x_{k} = ℓ$ or $S - lim_{k \to \infty} x_{k} = ℓ .$

Fuzzy sets are considered with respect to a nonempty base set X of elements of interest. The essential idea is that each element x ∈ X is assigned a membership grade u (x) a value in [0, 1], with u (x) =0 corresponding to nonmembership, 0 < u (x) <1 to partial membership, and u (x) =1 to full membership. According to Zadeh [35] a fuzzy subset of X is a nonempty subset {(x, u (x)) : x ∈ X} of X × [0, 1] for some function u : X ⟶ [0, 1]. The function u itself is often used for the fuzzy set.

Let $C (ℝ^{n})$ denote the family of all nonempty, compact, convex subsets of $ℝ^{n} .$ 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 $A, B \in C (ℝ^{n})$ and $λ \in ℝ .$ If $α, β \in ℝ$ and $A, B \in C (ℝ^{n})$ , then $α (A + B) = α A + α B, (α β) A = α (β A), 1 A = A$

and if α, β ≥ 0, then (α + β) A = αA + βA . The distance between A and B is defined by the Hausdorff metric $δ_{\infty} (A, B) = max {sup_{a \in A} inf_{b \in B} ∥ a - b ∥, sup_{b \in B} inf_{a \in A} ∥ a - b ∥},$

where ∥ .∥ denotes the usual Euclidean norm in $ℝ^{n} .$ It is well known that $(C (ℝ^{n}), δ_{\infty})$ is a complete metric space.

Denote $L (ℝ^{n}) = {u : ℝ^{n} ⟶ [0, 1] : u satisfies (i) - (iv) below},$

where
u is normal, that is, there exists an $x_{0} \in ℝ^{n}$ such that u (x ₀) =1 ;

u is fuzzy convex, that is, for $x, y \in ℝ^{n}$ and 0≤ λ ≤ 1, u (λx + (1 - λ) y) ≥ min [u (x) , u (y)] ;

u is upper semicontinuous;

the closure of ${x \in ℝ^{n} : u (x) > 0},$ denoted by [u] ⁰, is compact.

If $u \in L (ℝ^{n})$ , then u is called a fuzzy number, and $L (ℝ^{n})$ is said to be a fuzzy number space.

For 0 < α ≤ 1, the α-level set [u] ^α of $u \in L (ℝ^{n})$ is defined by

$[u]^{α} = {x \in ℝ^{n} : u (x) \geq α} .$

Then from (i) - (iv), it follows that the α-level sets [u] ^α are in the space $C (ℝ^{n}) .$

For the addition and scalar multiplication in $L (ℝ^{n}),$ we have

$[u + v]^{α} = [u]^{α} + [v]^{α}, [ku]^{α} = k [u]^{α}$

where $u, v \in L (ℝ^{n}),$ $k \in ℝ .$

The aritmetic operations for α-level sets are defined as follows:

For $u, v \in L (ℝ^{n}),$ now and what follows we will denote α-level sets by ${[u]}^{a} = [a_{1}^{α}, b_{1}^{α}],$ ${[v]}^{a} = [a_{2}^{α}, b_{2}^{α}],$ α ∈ [0, 1] . Then we have $\begin{matrix} {[u + v]}^{α} & = [a_{1}^{α} + a_{2}^{α}, b_{1}^{α} + b_{2}^{α}] \\ {[u - v]}^{α} & = [a_{1}^{α} - b_{2}^{α}, b_{1}^{α} - a_{2}^{α}] \\ {[u . v]}^{α} & = [min_{i, j \in [1, 2]} a_{i}^{α} b_{j}^{α}, max_{i, j \in [1, 2]} a_{i}^{α} b_{j}^{α}] . \end{matrix}$

Define, for each 1 ≤ q < ∞ , $d_{q} (u, v) = {(\overset{1}{\int_{0}} {[δ_{\infty} ([u]^{α}, [v]^{α})]}^{q} d α)}^{1 / q}$ and $d_{\infty} (u, v) = sup_{0 \leq α \leq 1} δ_{\infty} ([u]^{α}, [v]^{α}),$ where δ _∞ is the Hausdorff metric. Clearly $d_{\infty} (u, v) = lim_{q \to \infty} d_{q} (u, v)$ with d _q ≤ d _s if q ≤ s ([9] and kloeden, [19]).

For simplicity in notation, throughout the paper d will denote the notation d _q with 1 ≤ q ≤ ∞ .

By a lacunary sequence θ = (k _r) ; r = 0, 1, 2, …, where k ₀ = 0, we mean an increasing sequence of non-negative integers with h _r = (k _r - k _r-1)→ ∞ as r → ∞ . The intervals determined by θ will be denoted by I _r = (k _r-1, k _r] and $q_{r} = \frac{k_{r}}{k_{r - 1}}$ . Lacunary sequences have been discussed in ([5 , 25]).

A sequence X = (X _k) of fuzzy numbers is a function X from the set $ℕ$ of all natural numbers into $L (ℝ^{n}) .$ Thus, a sequence (X _k) of fuzzy numbers is a correspondence from the set of natural numbers to a set of fuzzy numbers, i.e., to each natural number k there corresponds a fuzzy number X (k). It is more common to write X _k rather than X (k) and to denote the sequence by (X _k) rather than X. The fuzzy number X _k is called the k-th term of the sequence.

Let X = (X _k) be a sequence of fuzzy numbers. The sequence X = (X _k) of fuzzy numbers is said to be bounded if the set ${X_{k} : k \in ℕ}$ of fuzzy numbers is bounded and convergent to the fuzzy number X ₀, written as $lim_{k} X_{k}$ = X ₀, if for every ɛ > 0 there exists a positive integer k ₀ such that d (X _k, X ₀) < ɛ for k > k ₀ . Let $ℓ_{\infty}^{F}$ and $c^{F}$ denote the set of all bounded sequences and all convergent sequences of fuzzy numbers, respectively [22].

The famous space $\hat{c}$ of all almost convergent sequences was introduced by Lorentz [20] and a sequence x = (x _k) is said to be strongly almost con-vergent to a number ℓ (see Maddox [21]) if

$\lim_{n} \frac{1}{n} \sum_{k = 1}^{n} | x_{k + i} - ℓ | = 0 uniformly in i.$

The difference sequence spaces ℓ_∞ (Δ), c (Δ) and c ₀ (Δ), consisting of all real valued sequences x = (x _k) such that Δx = (x _k - x _k+1) in the sequence spaces ℓ_∞, c and c ₀, were defined by Kızmaz [18]. The idea of difference sequences was generalized by Et and Çolak [11] and colak, Başar and Altay [4] and altay2 Tripathy et al. ([31], [32]) and many others.

Let $w^{F}$ be the set of all sequences of fuzzy numbers. The operator $Δ_{m}^{n} : w^{F} \to w^{F}$ is defined by (Δ ⁰ X) _k = X _k, ${(Δ_{m}^{1} X)}_{k} = {(Δ_{m} X)}_{k} = X_{k} - X_{k + m},$ and ${(Δ_{m}^{n} X)}_{k} = \sum_{υ = 0}^{n} (- 1)^{υ} n υ x_{k + m υ}$ for all $k \in ℕ .$ Throughout the paper m, n will denote any positive integers and for convenience we will write $Δ_{m}^{n} X_{k}$ instead of ${(Δ_{m}^{n} X)}_{k} .$

Definition 2.1. [31]Let X = (X _k) be a sequence of fuzzy numbers. Then the sequence X = (X _k) is said to be $Δ_{m}^{n} -$ bounded if the set ${Δ_{m}^{n} X_{k} : k \in ℕ}$ of fuzzy numbers is bounded, and $Δ_{m}^{n} -$ convergent to the fuzzy number X ₀, written as $lim_{k} Δ_{m}^{n} X_{k} = X_{0}$ , if for every ɛ > 0 there exists a positive integer k ₀ such that $d (Δ_{m}^{n} X_{k}, X_{0}) < ɛ$ for all k > k ₀ . By $ℓ_{\infty}^{F} (Δ_{m}^{n})$ and $c^{F} (Δ_{m}^{n})$ we denote the sets of all $Δ_{m}^{n} -$ bounded sequences and all $Δ_{m}^{n} -$ convergent sequences of fuzzy numbers, respectively.

Definition 2.2. Let θ = (k _r) be a lacunary sequence and X = (X _k) be a sequence of fuzzy numbers. Then the sequence X = (X _k) of fuzzy numbers is said to be lacunary almost $Δ_{m}^{n} -$ statistically convergent to the fuzzy number X ₀, if for every ɛ > 0, $\begin{matrix} lim_{r \to \infty} h_{r}^{- 1} | {k \in I_{r} : d (Δ_{m}^{n} X_{k + i}, X_{0}) \geq ɛ} | = 0, \\ uniformly in i \in ℕ . \end{matrix}$

The set of all lacunary almost $Δ_{m}^{n} -$ statistically convergent sequences of fuzzy numbers is denoted by ${\hat{S}}^{F} (Δ_{m}^{n}, θ) .$ In this case we write $X_{k} \to X_{0} ({\hat{S}}^{F}$ $(Δ_{m}^{n}, θ)) .$ In the special case θ = (2^r) , we shall write ${\hat{S}}^{F} (Δ_{m}^{n})$ instead of ${\hat{S}}^{F} (Δ_{m}^{n}, θ) .$

Definition 2.3. Let θ = (k _r) be a lacunary sequence, X = (X _k) be a sequence of fuzzy numbers and p = (p _k) be any sequence of strictly positive real numbers. We define the following sets ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p) = {\begin{matrix} [l] l X = (X_{k}) \in w^{F} : lim_{r \to \infty} \frac{1}{h_{r}} \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}} = 0, \\ uniformly in i \in ℕ \end{matrix}},$ $\begin{matrix} {\hat{w}}_{0}^{F} (Δ_{m}^{n}, θ, p) & = {\begin{matrix} [l] l X = (X_{k}) \in w^{F} : lim_{r \to \infty} \frac{1}{h_{r}} \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}, \bar{0})]}^{p_{k}} = 0, \\ uniformly in i \in ℕ \end{matrix}}, \\ {\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p) & = {\begin{matrix} [l] l X = (X_{k}) \in w^{F} : sup_{r, i} \frac{1}{h_{r}} \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}, \bar{0})]}^{p_{k}} < \infty, \\ uniformly in i \in ℕ \end{matrix}}, \end{matrix}$

where

$\bar{0} (t) = {\begin{matrix} [c] cc 1, & t = (0, 0, 0, \dots, 0) \\ 0, & otherwise . \end{matrix}$

If $X \in {\hat{w}}^{F} (Δ_{m}^{n}, θ, p),$ we say that X is lacunary strongly almost $Δ_{m}^{n} -$ convergent to the fuzzy number X ₀ and is written as $X_{k} \to X_{0} ({\hat{w}}^{F} (Δ_{m}^{n}, θ, p)) .$

We get the following sequence spaces from the above sequence spaces giving particular values to m, n, θand p .
${\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p) = {\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, p),$ ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p) = {\hat{w}}^{F} (Δ_{m}^{n}, p)$ and ${\hat{w}}_{0}^{F} (Δ_{m}^{n}, θ, p) = {\hat{w}}_{0}^{F} (Δ_{m}^{n}, p)$ when θ = (2^r) ,

If p _k = 1 for all $k \in ℕ$ then ${\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p) = {\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ),$ ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p) = {\hat{w}}^{F} (Δ_{m}^{n}, θ),$ and ${\hat{w}}_{0}^{F} (Δ_{m}^{n}, θ, p) = {\hat{w}}_{0}^{F} (Δ_{m}^{n}, θ),$

If m = 1 then ${\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p) = {\hat{w}}_{\infty}^{F} (Δ^{n}, θ, p),$ ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p) = {\hat{w}}^{F} (Δ^{n}, θ, p)$ and ${\hat{w}}_{0}^{F} (Δ_{m}^{n}, θ, p) = {\hat{w}}_{0}^{F} (Δ^{n}, θ, p) .$

A sequence space $E^{F}$ is said to be normal (or solid) if $(X_{k}) \in E^{F}$ and (Y _k) is such that $d (Y_{k}, \bar{0}) \leq d (X_{k}, \bar{0})$ implies $(Y_{k}) \in E^{F} .$

A sequence space $E^{F}$ is said to be monotone if $E^{F}$ contains the canonical pre-image of all its step spaces.

Remark. If a sequence space $E^{F}$ is solid, then $E^{F}$ is monotone.

A sequence space $E^{F}$ is said to be symmetric if $(X_{π (n)}) \in E^{F},$ whenever $(X_{k}) \in E^{F},$ where π is a permutation of $ℕ$ .

A sequence space $E^{F}$ is said to be convergence free if $(Y_{k}) \in E^{F},$ whenever $(X_{k}) \in E^{F}$ and $X_{k} = \bar{0}$ implies $Y_{k} = \bar{0} .$

The sequence spaces ${\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p),$ ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p),$ ${\hat{w}}_{0}^{F} (Δ_{m}^{n}, θ, p)$ and ${\hat{S}}^{F} (Δ_{m}^{n}, θ)$ contain some unbounded sequences of fuzzy numbers which are divergent, too. To show that let m = 1, θ = (2^r) and p _k = 1 for all $k \in ℕ .$ Then the sequence $X = (X_{k}) = ({\bar{k}}^{n})$ belongs to ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p)$ , but the sequence X is divergent and is not bounded.

For the classical sets, (x _k) converges to ℓ implies $(Δ_{m}^{n} x_{k})$ converges to 0, but this case does not hold for the sequences of fuzzy numbers. We give the following example for this.

Example 1. Let θ = (2^r) , p _k = 1 for all $k \in ℕ$ and n = m = 1 . Consider the sequence (X _k) as follows: $X_{k} (t) = {\begin{matrix} [c] cc \frac{k}{k + 1} t + \frac{1 - k}{1 + k}, & if t \in [\frac{k - 1}{k}, 2] \\ - \frac{k}{k + 1} t + \frac{3 k + 1}{1 + k}, & if t \in [2, \frac{3 k + 1}{k}] \\ 0, & otherwise \end{matrix}$

Then the sequence X = (X _k) is convergent to the fuzzy number L, where $L (t) = {\begin{matrix} [l] ll t - 1, & if t \in [1, 2] \\ - t + 3, & if t \in [2, 3] \\ 0, & otherwise \end{matrix} .$

We find the α-level sets of X _k and ΔX _k as follows respectively: ${[X_{k}]}^{α} = [\frac{k - 1}{k} + \frac{k + 1}{k} α, \frac{3 k + 1}{k} - \frac{k + 1}{k} α]$ and $\begin{matrix} {[Δ X_{k}]}^{α} = [\frac{- 2 k^{2} - 4 k - 1}{k^{2} + k} + (\frac{k + 1}{k} + \frac{k + 2}{k + 1}) α, \\ \frac{2 k^{2} + 4 k + 1}{k^{2} + k} - (\frac{k + 1}{k} + \frac{k + 2}{k + 1}) α] . \end{matrix}$

Then we have ΔX _k → L ₁, where ${[L_{1}]}^{α} = [- 2 + 2 α, 2 - 2 α] \neq \bar{0}$ .(See fig. 1)

3 Main results

In this section we give some inclusion relations between ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p)$ and ${\hat{S}}^{F} (Δ_{m}^{n}, θ)$ , between ${\hat{S}}^{F} (Δ_{m}^{n})$ and ${\hat{S}}^{F} (Δ_{m}^{n}, θ) .$

Theorem 3.1. Let the sequence (p_k) be bounded. Then ${\hat{w}}_{0}^{F} (Δ_{m}^{n}, θ, p) \subset {\hat{w}}^{F} (Δ_{m}^{n}, θ, p) \subset {\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p)$ and the inclusions are strict.

Proof. The inclusion ${\hat{w}}_{0}^{F} (Δ_{m}^{n}, θ, p) \subset {\hat{w}}^{F} (Δ_{m}^{n}, θ, p)$ is obvious. So, we will only show that ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p) \subset {\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p) .$ Let $X \in {\hat{w}}^{F} (Δ_{m}^{n}, θ, p) .$ Then we have $\begin{matrix} \frac{1}{h_{r}} \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}, \bar{0})]}^{p_{k}} \\ \leq \frac{D}{h_{r}} \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}} \\ + \frac{D}{h_{r}} \sum_{k \in I_{r}} {[d (X_{0}, \bar{0})]}^{p_{k}} \\ \leq \frac{D}{h_{r}} \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}} \\ + D max {1, sup {[d (X_{0}, \bar{0})]}^{H}}, \end{matrix}$ where $sup_{k} p_{k} = H,$ and D = max(1, 2^H-1). Thus we get $X \in {\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p) .$

To show that the inclusion is strict, consider the following example:

Let θ = (2^r) , p _k = 1 for all $k \in ℕ$ and n = m = 1 . Consider the sequence (X _k) of fuzzy numbers asfollows: $\begin{matrix} X_{k} (t) = {\begin{matrix} \frac{2}{k} t + 1, & - \frac{k}{2} \leq t \leq 0 \\ - \frac{2}{k} t + 1, & 0 \leq t \leq \frac{k}{2} \\ 0, & otherwise \end{matrix}}, \\ \begin{matrix} if k = 10^{j}, (j = 1, 2, 3, \dots) \\ \bar{0}, otherwise . \end{matrix} \end{matrix}$

Then, for α ∈ (0, 1] , we have α-level sets of X _k and ΔX _k as follows: $\begin{matrix} {[X_{k}]}^{α} = \\ {\begin{matrix} [c] cc [\frac{k}{2} (α - 1), \frac{k}{2} (1 - α)], & if k = 10^{j}, (j = 1, 2, 3, \dots) \\ [0, 0], & otherwise \end{matrix} \end{matrix}$ and $\begin{matrix} {[Δ X_{k}]}^{α} = \\ {\begin{matrix} [c] ll [\frac{k}{2} (α - 1), \frac{k}{2} (1 - α)], & if k = 10^{j}, (j = 1, 2, 3, \dots) \\ [\frac{(k + 1)}{2} (α - 1), \frac{(k + 1)}{2} (1 - α)], & if k + 1 = 10^{j}, \\ [0, 0], & otherwise . \end{matrix} \end{matrix}$

Now it is easy to see that $- \bar{2} < {[σ_{n}]}^{α} < \bar{2}$ for α ∈ (0, 1] and all $n \in ℕ$ , where ${[σ_{n}]}^{α} = \frac{1}{n} \sum_{k = 1}^{n} {[Δ X_{k}]}^{α} .$ Thus, the sequence (σ _n) of fuzzy numbers is bounded but is not convergent.(See Fig. 2)

Theorem 3.2. The spaces ${\hat{w}}_{0}^{F} (Δ_{m}^{n}, θ, p),$ ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p),$ ${\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p)$ and ${\hat{S}}^{F} (Δ_{m}^{n}, θ)$ are closed under the operations of addition and scalarmultiplication.

Proof. It is easy so it is omitted.

Theorem 3.3. The spaces ${\hat{w}}_{0}^{F} (Δ_{m}^{n}, θ, p),$ ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p),$ ${\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p)$ are complete metric space with the metric $\begin{matrix} δ_{Δ} (X, Y) = \sum_{k = 1}^{n} d (X_{k}, Y_{k}) + sup_{r, i} \\ {(h_{r}^{- 1} \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}, Δ_{m}^{n} Y_{k + i})]}^{p_{k}})}^{\frac{1}{K}} \end{matrix}$ where $K = max (1, H = sup_{k} p_{k}) .$

Proof. We will prove only for space ${\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p) .$ The others can be shown by using similar techniques. Let (X ^s) be a Cauchy sequence in space ${\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p),$ where $X^{s} = {(X_{j}^{s})}_{(j)} = (X_{1}^{s}, X_{2}^{s}, \dots) \in {\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p)$ for each $s \in ℕ .$ Then $\begin{matrix} δ_{Δ} (X^{s}, X^{t}) = \sum_{k = 1}^{n} d (X_{k}^{s}, X_{k}^{t}) + sup_{r, i} \\ {(h_{r}^{- 1} \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}^{s}, Δ_{m}^{n} X_{k + i}^{t})]}^{p_{k}})}^{\frac{1}{K}} \\ \to 0, as s, t \to \infty . \end{matrix}$

Thus $\sum_{k = 1}^{n} d (X_{k}^{s}, X_{k}^{t}) \to 0$ and $h_{r}^{- 1} \sum_{k \in I_{r}} [d (Δ_{m}^{n} X_{k + i}^{s}, Δ_{m}^{n} X_{k + i}^{t})]^{p_{k}} \to 0$ as s, t → ∞ , for each fixed $i \in ℕ .$ Now from $\begin{matrix} d (X_{j + n}^{s}, X_{j + n}^{t}) \leq d (Δ^{n} X_{j}^{s}, Δ^{n} X_{j}^{t}) + n 0 \\ d (X_{j}^{s}, X_{j}^{t}) + \dots + n n - 1 d (X_{j + n - 1}^{s}, X_{j + n - 1}^{t}) \end{matrix}$ we have $d (X_{j}^{s}, X_{j}^{t}) \to 0,$ as s, t → ∞ , for each fixed $j \in ℕ .$ Hence ${(X_{j}^{s})}_{s} = (X_{j}^{1}, X_{j}^{2}, \dots)$ is a Cauchy sequence in $L (ℝ^{n}) .$ Since $L (ℝ^{n})$ is complete, it is convergent $lim_{s \to \infty} X_{j}^{s} = X_{j}$ say, for each $j \in ℕ .$ Since (X ^s) is a Cauchy sequence, for each ɛ > 0, there exists n ₀ = n ₀ (ɛ) such that $δ_{Δ} (X^{s}, X^{t}) < ɛ for all s, t \geq n_{0} .$

Hence for each $i \in ℕ$ we get $\begin{matrix} \sum_{k = 1}^{n} d (X_{k}^{s}, X_{k}^{t}) < ɛ and h_{r}^{- 1} \\ \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}^{s}, Δ_{m}^{n} X_{k + i}^{t})]}^{p_{k}} < ɛ^{K}, (s, t \geq n_{0}) . \end{matrix}$

So we have $lim_{t \to \infty} \sum_{k = 1}^{n} d (X_{k}^{s}, X_{k}^{t}) = \sum_{k = 1}^{n} d (X_{k}^{s}, X_{k}) < ɛ$ and $\begin{matrix} lim_{t \to \infty} h_{r}^{- 1} \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}^{s}, Δ_{m}^{n} X_{k + i}^{t})]}^{p_{k}} = \\ h_{r}^{- 1} \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}^{s}, Δ_{m}^{n} X_{k + i})]}^{p_{k}} < ɛ^{K} \end{matrix}$ for all $n, i \in ℕ$ and s ≥ n ₀ . This implies that δ _Δ (X ^s, X) <2ɛ, for all s ≥ n ₀, that is X ^s → X as s → ∞ , where X = (X _j) . Finally, we obtain $X \in {\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p)$ from following inequality $\begin{matrix} h_{r}^{- 1} \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}, \bar{0})]}^{p_{k}} \begin{matrix} [c] c \leq \end{matrix} h_{r}^{- 1} \\ {\sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}^{n_{0}}, \bar{0}) + d (Δ_{m}^{n} X_{k + i}^{n_{0}}, Δ_{m}^{n} X_{k + i})]}^{p_{k}}} \\ \begin{matrix} [c] c \leq \end{matrix} D . h_{r}^{- 1} \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}^{n_{0}}, \bar{0})]}^{p_{k}} \\ + D . h_{r}^{- 1} \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}^{n_{0}}, Δ_{m}^{n} X_{k + i})]}^{p_{k}} \end{matrix}$ where $sup_{k} p_{k} = H,$ and D = max(1, 2^H-1) . This shows that ${\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p)$ is a complete metric space.

Theorem 3.4. Let 0 < p_k ≤ q_k and $(\frac{q_{k}}{p_{k}})$ be bounded. Then ${\hat{w}}^{F} (Δ_{m}^{n}, θ, q) \subseteq {\hat{w}}^{F} (Δ_{m}^{n}, θ, p)$ .

Proof. It is easy so it is omitted.

Theorem 3.5. Let θ = (k_r) be a lacunary sequence with 1< $lim inf_{r} q_{r} \leq lim sup_{r} q_{r} < \infty,$ then we have ${\hat{w}}^{F} (Δ_{m}^{n}, θ) = {\hat{w}}^{F} (Δ_{m}^{n}, θ, p) .$

Proof. Suppose $lim inf_{r} q_{r} > 1,$ then there exists δ > 0 such that $q_{r} = \frac{k_{r}}{k_{r - 1}} \geq 1 + δ$ for all r ≥ 1 . Then for $X \in {\hat{w}}^{F} (Δ_{m}^{n}, θ)$ we write $\begin{matrix} A_{r} & = \frac{1}{h_{r}} \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}} \\ = \frac{1}{h_{r}} \sum_{k = 1}^{k_{r}} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}} \\ - \frac{1}{h_{r}} \sum_{k = 1}^{k_{r - 1}} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}} \end{matrix}$ $\begin{matrix} = \frac{k_{r}}{h_{r}} (k_{r}^{- 1} \sum_{k = 1}^{k_{r}} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}}) \\ - \frac{k_{r - 1}}{h_{r}} (k_{r - 1}^{- 1} \sum_{k = 1}^{k_{r - 1}} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}}) . \end{matrix}$

Since h _r = k _r - k _r-1, we have $\frac{k_{r}}{h_{r}} \leq \frac{1 + δ}{δ}$ and $\frac{k_{r - 1}}{h_{r}} \leq \frac{1}{δ}$ . The terms $k_{r}^{- 1} \sum_{k = 1}^{k_{r}} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}}$ and $k_{r - 1}^{- 1} \sum_{k = 1}^{k_{r - 1}} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}}$ both converge to zero uniformly in i, and hence it follows that A _r → 0 as r → ∞ , uniformly in i . Hence ${\hat{w}}^{F} (Δ_{m}^{n}, θ) \subset {\hat{w}}^{F} (Δ_{m}^{n}, θ, p) .$

Now suppose that $lim sup_{r} q_{r} < \infty .$ Then, there exists β > 0 such that q _r < β for all r ≥ 1 . Let $X \in {\hat{w}}^{F} (Δ_{m}^{n}, θ, p)$ and ɛ > 0 . There exists R > 0 such that for every j ≥ R $A_{j} = \frac{1}{h_{r}} \sum_{k \in I_{j}} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}} < ɛ .$

We can also find K > 0 such that A _j ≤ K for all j = 1, 2, …. Now let ℓ be any integer with k _r-1 < ℓ ≤ k _r, where r > R . Then $\begin{matrix} \frac{1}{ℓ} \sum_{k = 1}^{n} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}} \\ \leq \frac{1}{k_{r - 1}} \sum_{k = 1}^{k_{r}} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}} \\ = \frac{1}{k_{r - 1}} {\sum_{k \in I_{1}} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}} \\ + \sum_{k \in I_{2}} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}} \\ + \dots + \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}}} \\ = \frac{k_{1}}{k_{r - 1}} A_{1} + \frac{k_{2} - k_{1}}{k_{r - 1}} A_{2} + \dots + \frac{k_{R} - k_{R - 1}}{k_{r - 1}} A_{R} \\ + \dots + \frac{k_{r} - k_{r - 1}}{k_{r - 1}} A_{r} = (sup_{j \geq 1} A_{j}) \\ \frac{k_{R}}{k_{r - 1}} + (sup_{j \geq R} A_{j}) \frac{k_{r} - k_{R}}{k_{r - 1}} < K \frac{k_{R}}{k_{r - 1}} + ɛ β . \end{matrix}$

Since k _r-1→ ∞ as r → ∞ , it follows that $\frac{1}{ℓ} \sum_{k = 1}^{n} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}} \to 0$ and consequently ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p) \subset {\hat{w}}^{F} (Δ_{m}^{n}, θ) .$

Theorem 3.6. If $lim_{k \to \infty} p_{k} > 0$ and X is lacunary strongly almost $Δ_{m}^{n} -$ convergent to the fuzzy numberX₀, that is $X_{k} \to X_{0} ({\hat{w}}^{F} (Δ_{m}^{n}, θ, p)),$ then X₀ is unique.

Proof. Let $lim_{k \to \infty} p_{k} = s > 0$ and suppose that $X_{k} \to X_{0} ({\hat{w}}^{F} (Δ_{m}^{n}, θ, p)), X_{k} \to X_{1} ({\hat{w}}^{F} (Δ_{m}^{n}, θ, p))$ such that X ₀ ≠ X ₁. Then $lim_{r \to \infty} h_{r}^{- 1} \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}} = 0,$ and $lim_{r \to \infty} h_{r}^{- 1} \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}, X_{1})]}^{p_{k}} = 0, uniformly in i .$

Then we have $\begin{matrix} h_{r}^{- 1} \sum_{k \in I_{r}} {[d (X_{0}, X_{1})]}^{p_{k}} \begin{matrix} [c] c \leq \end{matrix} \frac{D}{h_{r}} \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}, X_{0})]}^{p_{k}} \\ + \frac{D}{h_{r}} \sum_{k \in I_{r}} {[d (Δ_{m}^{n} X_{k + i}, X_{1})]}^{p_{k}} \begin{matrix} [c] c \to \end{matrix} 0, \\ uniformly in i (n \to \infty), \end{matrix}$ hence $lim_{r} h_{r}^{- 1} \sum_{k \in I_{r}} {[d (X_{0}, X_{1})]}^{p_{k}} = 0 .$

Also, since clearly $lim_{k} {[d (X_{0}, X_{1})]}^{p_{k}} = {[d (X_{0}, X_{1})]}^{s}$ we have, from the last two equality [d (X ₀, X ₁)] ^s = 0 . Thus the limit is unique.

Theorem 3.7. Let θ = (k_r) be a lacunary sequence and X = (X_k) be a sequence of fuzzy numbers, then we have ${\hat{w}}^{F} (Δ_{m}^{n}, θ) = ℓ_{\infty}^{F} (Δ_{m}^{n}) .$

Proof. Let $X \in {\hat{w}}^{F} (Δ_{m}^{n}, θ)$ . Then there exists a constant K ₁ > 0 such that $\begin{matrix} \frac{1}{h_{1}} [d (Δ_{m}^{n} X_{1 + i}, \bar{0})] \leq \frac{1}{h_{r}} \sum_{k \in I_{r}} d (Δ_{m}^{n} X_{k + i}, \bar{0}) \\ \leq K_{1} for each r, i \end{matrix}$ and so we have $X \in ℓ_{\infty}^{F} (Δ_{m}^{n}) .$ Conversely, let $X \in ℓ_{\infty}^{F} (Δ_{m}^{n}) .$ Then there exists a constant K ₂ > 0 such that $d (Δ_{m}^{n} X_{j}, \bar{0}) \leq K_{2}$ for all j, and so $\begin{matrix} \frac{1}{h_{r}} \sum_{k \in I_{r}} d (Δ_{m}^{n} X_{k + i}, \bar{0}) \leq \frac{K_{2}}{h_{r}} \\ \sum_{k \in I_{r}} 1 \leq K_{2} for each k and i . \end{matrix}$

Thus $X \in {\hat{w}}^{F} (Δ_{m}^{n}, θ) .$

The proofs of the following theorems are easy, therefore we give them without proof.

Theorem 3.8. Let θ = (k_r) be a lacunary sequence with $1 < lim inf_{r} q_{r} \leq lim sup_{r} q_{r} < \infty,$ then we have ${\hat{S}}^{F} (Δ_{m}^{n}) = {\hat{S}}^{F} (Δ_{m}^{n}, θ) .$

Theorem 3.9. Let X = (X_k) be a sequence of fuzzy numbers and $0 < h = inf_{k} p_{k} \leq p_{k} \leq sup_{k} p_{k} = H,$ then ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p) \subset {\hat{S}}^{F} (Δ_{m}^{n}, θ)$ .

Theorem 3.10. Let X = (X_k) be a $Δ_{m}^{n} -$ bounded sequence of fuzzy numbers, then ${\hat{S}}^{F} (Δ_{m}^{n}, θ) \subset {\hat{w}}^{F} (Δ_{m}^{n}, θ, p)$ .

Theorem 3.11. The sequence spaces ${\hat{w}}_{0}^{F} (θ)$ and ${\hat{w}}_{\infty}^{F} (θ)$ are solid and hence monotone, but the sequence spaces ${\hat{S}}^{F} (Δ_{m}^{n}, θ)$ , ${\hat{w}}_{0}^{F} (Δ_{m}^{n}, θ, p)$ , ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p)$ and ${\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p)$ are not solid.

Proof. Let $(X_{k}) \in {\hat{w}}^{F} (θ)$ and (Y _k) be such that $d (Y_{k + i}, \bar{0}) \leq d (X_{k + i}, \bar{0})$ for each $k, i \in ℕ$ . Then we get $\frac{1}{h_{r}} \sum_{k \in I_{r}} d (Y_{k + i}, \bar{0}) \leq \frac{1}{h_{r}} \sum_{k \in I_{r}} d (X_{k + i}, \bar{0}) .$

Hence ${\hat{w}}^{F} (θ)$ is solid and hence monotone. The space ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p)$ is not solid. This follows from the following example:

Let p _n = 1 for all $n \in ℕ$ , θ = (2^r) and m = 1 . Let us consider the sequences $X = (X_{n}) = (\bar{n}) = (\bar{1}, \bar{2}, \bar{3}, \dots) \in$ ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p)$ and

$α_{n} = {\begin{matrix} [c] cc 0, & if n is odd \\ 1, & if n is even \end{matrix}$

then $(α_{n} X_{n}) = (\bar{0}, \bar{2}, \bar{0}, \bar{4}, \bar{0}, \bar{6}, \dots) \notin$ ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p) .$ Hence ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p)$ is not solid.

Theorem 3.12. Let μ denotes any of the sequence spaces ${\hat{S}}^{F} (Δ_{m}^{n}, θ)$ , ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p)$ , ${\hat{w}}_{o}^{F} (Δ_{m}^{n}, θ, p)$ and ${\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p) .$ Then, the following statements hold:
μ is not symmetric,

μ is not convergence free.

Proof. Since the proof can be obtained for the spaces ${\hat{S}}^{F} (Δ_{m}^{n}, θ),$ ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p),$ ${\hat{w}}_{o}^{F} (Δ_{m}^{n}, θ, p)$ in the similar way, we consider only the space ${\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p) .$

a)Let p _k = 1 for all $k \in ℕ,$ θ = (2^r) and m = 2 . Consider the sequence $(X_{k}) = (\bar{1}, \bar{2}, \bar{3}, \dots) \in$ ${\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p)$ . Define $(Y_{k}) = (\bar{1}, \bar{3}, \bar{2}, \bar{5}, \bar{4}, \bar{6}, \dots)$ by a rearrangement of (X _k) . Then $(Y_{k}) \notin {\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ) .$

b) Let p _k = 1 for all $k \in ℕ$ and m = 3 . Suppose that θ = (2^r) . Define the sequence {X _n (t)} by $X_{n} (t) = {\begin{matrix} [c] cc \frac{nt + n + 1}{n + 1}, & - 1 - \frac{1}{n} \leq t \leq 0, \\ \frac{n + 1 - nt}{n + 1}, & 0 \leq t \leq 1 + \frac{1}{n}, \\ 0, & otherwise . \end{matrix}$

Then, we have $\begin{matrix} Δ_{3} X_{n} (t) = \\ {\begin{matrix} [c] cc \frac{(t + 2) n^{2} + 3 nt + 8 n + 3}{2 n^{2} + 8 n + 3}, & - 2 - \frac{1}{n} - \frac{1}{n + 3} \leq t \leq 0, \\ \frac{(- t + 2) n^{2} - 3 tn + 8 n + 3}{2 n^{2} + 8 n + 3}, & 0 \leq t \leq 2 + \frac{1}{n} + \frac{1}{n + 3}, \\ 0, & otherwise . \end{matrix} \end{matrix}$ Therefore, $lim_{r \to \infty} Δ_{3} X_{n} (t) = X (t),$ where X (t) is defined by $X (t) = {\begin{matrix} [c] cc \frac{(t + 2)}{2}, & - 2 \leq t \leq 0, \\ \frac{(2 - t)}{2}, & 0 \leq t \leq 2, \\ 0, & otherwise . \end{matrix}$

Hence, ${X_{n} (t)} \in {\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ) .$ Now, consider {Y _n (t)} defined by $Y_{n} (t) = {\begin{matrix} [c] cc \begin{matrix} [c] c \frac{(t + n)}{n}, \end{matrix} & - n \leq t \leq 0, {% \end{matrix} \begin{matrix} [c] c \frac{(n - t)}{n}, \end{matrix} & 0 \leq t \leq n, 0, & otherwise .$ ∥At this stage, $Δ_{3} Y_{n} (t) = {\begin{matrix} [c] cc \frac{t + 2 n + 3}{2 n + 3}, & - 2 n - 3 \leq t \leq 0, \\ \frac{2 n + 3 - t}{2 n + 3}, & 0 \leq t \leq 2 n + 3, \\ 0, & otherwise . \end{matrix}$ ∥It is clear that ${Y_{n} (t)} \notin {\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p) .$ This shows that the space ${\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p)$ is not convergence free.

4 Conclusion

The sequences of fuzzy numbers were introduced and studied by Matloka [22] and the first difference sequences of fuzzy numbers was studied by Savaş [27], Talo and Başar [30]. Now in this paper we study the m ^th difference sequences of fuzzy numbers for some sequence classes. The results which we obtained in this study are much more general than those obtained by others. To do this we introduce some of fairly wide classes of sequences of fuzzy numbers using the generalized difference operator $Δ_{m}^{n}$ and a lacunary sequence θ = (k _r) . Furthermore using these concepts we establish some inclusion relations between ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p)$ and ${\hat{S}}^{F} (Δ_{m}^{n}, θ)$ , between ${\hat{S}}^{F} (Δ_{m}^{n}, θ)$ and ${\hat{S}}^{F} (Δ_{m}^{n})$ and show that the sequence spaces ${\hat{w}}^{F} (Δ_{m}^{n}, θ, p)$ , ${\hat{w}}_{o}^{F} (Δ_{m}^{n}, θ, p)$ and ${\hat{w}}_{\infty}^{F} (Δ_{m}^{n}, θ, p)$ are complete metric spaces with suitable metric.

References

1.
Altin Y Et M Başarir M 2007 On some generalized difference sequences of fuzzy numbers Kuwait J Sci Engrg 34 1A 1 14

2.
Altinok H Altin Y Et M 2004 Lacunary almost statistical convergence of fuzzy numbers Thai J Math 2 2 265 274

3.
Altinok H Çolak R Et M 2009 λ-difference sequence spaces of fuzzy numbers Fuzzy Sets and Systems 160 21 3128 3139

4.
Başar F Altay B 2003 On the space of sequences of p-bounded variation and related matrix mappings Ukrainian Math J 55 1 136 147

5.
Bilgin T 2004 Lacunary strongly Δ-convergent sequences of fuzzy numbers Inform Sci 160 1-4 201 206

6.
Braha NL 2011 A new class of sequences related to the ℓ_p spaces defined by sequences of Orlicz functions J Inequal Appl 10 Art. ID 539745

7.
Connor JS 1988 The statistical and strong p-Cesáro convergence of sequences Analysis 8 1-2 47 63

8.
Çolak R Altinok H Et M 2009 Generalized difference sequences of fuzzy numbers Chaos Solitons Fractals 40 3 1106 1117

9.
Diamond P Kloeden P 1990 Metric spaces of fuzzy sets Fuzzy Sets and Systems 35 2 241 249

10.
Et M 2003 Strongly almost summable difference sequences of order μ defined by a modulus Studia Sci Math Hungar 40 4 463 476

11.
Et M Çolak R 1995 On some generalized difference sequence spaces Soochow J Math 21 4 377 386

12.
Et M Altinok H Çolak R 2006 On λ-statistical convergence of difference sequences of fuzzy numbers Inform Sci 176 15 2268 2278

13.
Fast H 1951 Sur la convergence statistique Colloquium Math 2 241 244

14.
Freedman AR Sember JJ Raphael M 1978 Some Cesáro type summability Proc London Math Soc 37 3 508 520

15.
Fridy JA 1985 On statistical convergence Analysis 5 4 301 313

16.
Fridy JA Orhan C 1993 Lacunary statistical convergence Pacific J Math 160 1 43 51

17.
Gökhan A Et M Mursaleen M 2009 Almost lacunary statistical and strongly almost lacunary convergence of sequences of fuzzy numbers Math Comput Modelling 49 3-4 548 555

18.
Kizmaz H 1981 On certain sequence spaces Canad Math Bull 24 2 169 176

19.
Kelava O Seikkala S 1984 On fuzzy metric spaces Fuzzy Sets and Systems 12 3 215 229

20.
Lorentz GG 1948 A contribution to the theory of divergent sequences Acta Math 80 167 190

21.
Maddox IJ 1978 A new type of convergence Math Proc Cambridge Philos Soc 83 1 61 64

22.
Matloka M 1986 Sequences of fuzzy numbers BUSEFAL 28 28 37

23.
Mursaleen M Mohiuddine SA 2009 Statistical convergence of double sequences in intuitionistic fuzzy normed spaces Chaos Solitons Fractals 41 5 2414 2421

24.
Mursaleen M Osama H Edely H 2009 On the invariant mean and statistical convergence Appl Math Lett 22 11 1700 1704

25.
Nuray F 1998 Lacunary statistical convergence of sequences of fuzzy numbers Fuzzy Sets and Systems 99 3 353 356

26.
Šalát T 1980 On statistically convergent sequences of real numbers Math Slovaca 30 2 139 150

27.
Savas E 2000 A note on sequence of fuzzy numbers Inform Sci 124 1-4 297 300

28.
Schoenberg IJ 1959 The integrability of certain functions and related summability methods Amer Math Monthly 66 361 375

29.
Talo Ö Başar F 2009 Determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transformations Comput Math Appl 58 4 717 733

30.
Talo Ö Başar F 2008 On the space bv_p(F) of sequences ofp-bounded variation of fuzzy numbers Acta Math Sin (Engl Ser) 24 7 1205 1212

31.
Tripathy BC Baruah A Et M Güngör M 2012 On almost statistical convergence of new type of generalized difference sequence of fuzzy numbers Iranian Journal of Science &Technology, IJST A2 147 155

32.
Tripathy BC Esi A Tripathy B 2005 On a new type of generalized difference Cesáro sequence spaces Soochow J Math 31 3 333 340

33.
Tripathy BC Dutta AJ 2010 Bounded variation double sequence space of fuzzy numbers Comput Math Appl 59 2 1031 1037

34.
Tripathy BC 1997 Matrix transformations between some class of sequences J Math Anal Appl 206 2 448 450

35.
Zadeh LA 1965 Fuzzy sets Information and Control 8 338 353