On lacunary statistical convergence of double sequences and some properties in fuzzy normed spaces

Abstract

In this study, we have investigated the concepts of lacunary summable, lacunary statistical convergence and lacunary statistically Cauchy sequence for double sequences in fuzzy normed spaces. Also, we have investigated some properties and relationships between these concepts.

Keywords

Double sequences statistical convergence fuzzy numbers fuzzy normed spaces lacunary sequences

1 Introduction and Background

The concept of convergence of real sequences has been extended to statistical convergence independently by Fast [10] and Schoenberg [29]. This concept was extended to the double sequences by Mursaleen and Edely [17]. Çakan and Altay [5] presented multidimensional analogues of the results given by Fridy and Orhan [13].

The concept of ordinary convergence of a sequence of fuzzy numbers was firstly introduced by Matloka [14] and proved some basic theorems for sequences of fuzzy numbers. Nanda [18] studied the sequences of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers form a complete metric space. Ṣencimen and Pehlivan [30] introduced the notions of statistically convergent sequence and statistically Cauchy sequence in a fuzzy normed linear space. Türkmen and C˙ınar [32] studied lacunary statistical convergence in fuzzy normed linear spaces. Mohiuddine et al. [16] studied Statistical convergence of double sequences in fuzzy normed spaces. Recently, Savaş [27, 28] studied On I -lacunary statistical convergence of weight g of fuzzy numbers and On lacunary p-summable convergence of weight g for fuzzy numbers via ideal and Altınok [3] studied f-Statistical Convergence of order β for Sequences of Fuzzy Numbers.

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) taking values 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.

A fuzzy set u on $ℝ$ is called a fuzzy number if it has the following properties:

1. u is normal, that is, there exists an $x_{0} \in ℝ$ such that u (x₀) =1;

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

3. u is upper semicontinuous;

4. $supp u = cl {x \in ℝ : u (x) > 0},$ or denoted by [u] ₀, is compact.

Now, we recall the basic definitions and concepts [1 , 30–34].

Let $L (ℝ)$ be set of all fuzzy numbers. If $u \in L (ℝ)$ and u (t) = 0 for t < 0, then u is called a non-negative fuzzy number. We write $L^{*} (ℝ)$ by the set of all non-negative fuzzy numbers. We can say that $u \in L^{*} (ℝ)$ iff $u_{α}^{-} \geq 0$ for each α ∈ [0, 1]. Clearly we have $\tilde{0} \in L (ℝ) .$ For $u \in L (ℝ),$ the α level set of u is defined by ${[u]}_{α} = {\begin{matrix} {x \in ℝ : u (x) \geq α}, & if α \in (0, 1] \\ supp u, & if α = 0 . \end{matrix} ∥$

A partial order ⪯ on $L (ℝ)$ is defined by u ⪯ v if $u_{α}^{-} \leq v_{α}^{-}$ and $u_{α}^{+} \leq v_{α}^{+}$ for all α ∈ [0, 1].

Arithmetic operation for $t \in ℝ$ , ⊕, ⊖, ⊙ and ø on $L (ℝ) \times L (ℝ)$ are defined by $(u \oplus v) (t) = sup_{s \in ℝ} {u (s) \land v (t - s)}$ , $(u ⊖ v) (t) = sup_{s \in ℝ} {u (s) \land v (s - t)}$ , $(u ⊙ v) (t) = sup_{\begin{matrix} s \in ℝ, s \neq 0 \end{matrix}} {u (s) \land v (t / s)}$ and $(u ø v) (t) = sup_{s \in ℝ} {u (st) \land v (s)}$ .

For $k \in ℝ^{+}$ , ku is defined as ku (t) = u (t/k) and $0 u (t) = \tilde{0},$ $t \in ℝ .$

Some arithmetic operations for α-level sets are defined as follows: $u, v \in L (ℝ)$ and ${[u]}_{α} = [u_{α}^{-}, u_{α}^{+}]$ and ${[v]}_{α} = [v_{α}^{-}, v_{α}^{+}],$ α ∈ (0, 1]. Then, ${[u \oplus v]}_{α} = [u_{α}^{-} + v_{α}^{-}, u_{α}^{+} + v_{α}^{+}]$ , ${[u ⊖ v]}_{α} = [u_{α}^{-} - v_{α}^{+}, u_{α}^{+} - v_{α}^{-}]$ , ${[u ⊙ v]}_{α} = [u_{α}^{-} . v_{α}^{-}, u_{α}^{+} . v_{α}^{+}]$ and ${[\tilde{1} ø u]}_{α} = [\frac{1}{u_{α}^{+}}, \frac{1}{u_{α}^{-}}]$ , $u_{α}^{-} > 0 .$

For $u, v \in L (ℝ),$ the supremum metric on $L (ℝ)$ defined as $D (u, v) = sup_{0 \leq α \leq 1} max {| u_{α}^{-} - v_{α}^{-} |, | u_{α}^{+} - v_{α}^{+} |} .$

It is known that D is a metric on $L (ℝ)$ and $(L (ℝ), D)$ is a complete metric space.

A sequence x = (x_k) of fuzzy numbers is said to be convergent to the fuzzy number x₀, if for every ɛ > 0 there exists a positive integer k₀ such that D (x_k, x₀) < ɛ for k > k₀ and a sequence x = (x_k) of fuzzy numbers convergens to levelwise to x₀ iff $lim_{k \to \infty} {[x_{k}]}_{α} = {[x_{0}]}_{α}^{-}$ and $lim_{k \to \infty} {[x_{k}]}_{α} = {[x_{0}]}_{α}^{+}$ , where ${[x_{k}]}_{α} = [{(x_{k})}_{α}^{-}, {(x_{k})}_{α}^{+}]$ and ${[x_{0}]}_{α} = [{(x_{0})}_{α}^{-}, {(x_{0})}_{α}^{+}]$ , for every α ∈ (0, 1).

Let X be a vector space over $ℝ,$ $∥ . ∥ : X \to L^{*} (ℝ)$ and the mappings L; R (respectively, left norm and right norm): [0, 1] × [0, 1] → [0, 1] be symetric, nondecreasing in both arguments and satisfy L (0, 0) = 0 and R (1, 1) =1.

The quadruple (X, ||. ||, L, R) is called fuzzy normed linear space (briefly (X, ||. ||) FNS) and ||. || a fuzzy norm if the following axioms are satisfied

$∥ x ∥ = \tilde{0}$ iff x = 0,

||rx|| = |r| ⊙ ||x|| for x ∈ X, $r \in ℝ,$

For all x, y ∈ X (a) x + y (s + t) ≥ L (x (s),y (t)), whenever $s \leq {∥ x ∥}_{1}^{-}, t \leq {∥ y ∥}_{1}^{-}$ and $s + t \leq {∥ x + y ∥}_{1}^{-}$ , (b) x + y (s + t) ≤ R (x (s), y (t)), whenever $s \geq {∥ x ∥}_{1}^{-}, t \geq {∥ y ∥}_{1}^{-}$ and $s + t \geq {∥ x + y ∥}_{1}^{-}$ .

Let (X, ||. ||_C) be an ordinary normed linear space. Then, a fuzzy norm ||. || on X can be obtained by $| | x | | (t) = {\begin{matrix} 0 & \begin{matrix} if 0 \leq t \leq a | | x | |_{C} or t \geq b | | x | |_{C} \end{matrix} \\ \frac{t}{(1 - a) | | x | |_{C}} - \frac{a}{1 - a} & a | | x | |_{C} \leq t \leq | | x | |_{C} \\ \frac{- t}{(b - 1) | | x | |_{C}} + \frac{b}{b - 1} & | | x | |_{C} \leq t \leq b | | x | |_{C} \end{matrix} ∥$ where ||x||_C is the ordinary norm of x (≠ 0), 0 < a < 1 and 1 < b < ∞. For x = θ, define $| | x | | = \tilde{0} .$ Hence, (X, ||. ||) is a fuzzy normed linear space.

Let us consider the topological structure of an FNS (X). For any ɛ > 0, α ∈ [0, 1] and x ∈ X, the (ɛ, α)- neighborhood of x is the set $N_{x} (ɛ, α) = {y \in X : ∥ x - y ∥_{α}^{+} < ɛ}$ .

Let (X, ||. ||) be an FNS. A sequence ${(x_{n})}_{n = 1}^{\infty}$ in X is convergent to x ∈ X with respect to the fuzzy norm on X and we denote by $x_{n} \overset{FN}{\to} x,$ provided that $(D) - lim_{n \to \infty} | | x_{n} - x | | = \tilde{0};$ i.e., for every ɛ > 0 there is an $N (ɛ) \in ℕ$ such that $D (| | x_{n} - x | |, \tilde{0}) < ɛ$ for all n ≥ N (ɛ). This means that for every ɛ > 0 there is an $N (ɛ) \in ℕ$ such that for all n ≥ N (ɛ), $sup_{α \in [0, 1]} | | x_{n} - x | |_{α}^{+} = | | x_{n} - x | |_{0}^{+} < ɛ .$

Let (X, ||. ||) be an FNS. A sequence (x_k) in X is statistically convergent to L ∈ X with respect to the fuzzy norm on X and we denote by $x_{n} \overset{FS}{\to} x,$ provided that for each ɛ > 0, we have $δ ({k \in ℕ : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ}) = 0 .$ This implies that for each ɛ > 0, the set $K (ɛ) = {k \in ℕ : | | x_{k} - L | |_{0}^{+} \geq ɛ}$ has natural density zero; namely, for each ɛ > 0, $| | x_{k} - L | |_{0}^{+} < ɛ$ for almost all k.

By a lacunary sequence we mean an increasing integer sequence θ = {k_r} such that k₀ = 0 and 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].

Let (X, ||. ||) be an FNS and θ ={ k_r } be lacunary sequence. A sequence $x = (x_{k})_{k \in ℕ}$ in X is said to be lacunary summable with respect to fuzzy norm on X if there is an L ∈ X such that $lim_{r \to \infty} \frac{1}{h_{r}} (\sum_{k \in I_{r}} D (| | x_{k} - L | |, \tilde{0})) = 0 .$ In this case, we can write x_k → L ((N_θ) _FN) or $x_{k} \overset{{(N_{θ})}_{FN}}{⟶} L$ and ${(N_{θ})}_{FN} = {x = (x_{k}) : lim_{r \to \infty} \frac{1}{h_{r}} (\sum_{k \in I_{r}} D (| | x_{k} - L | |, \tilde{0})) = 0, for some L} .$

A sequence x = (x_k) in X is said to be lacunary statistically convergent or FS_θ-convergent to L ∈ X with respect to fuzzy norm on X if for each ɛ > 0 $lim_{r \to \infty} \frac{1}{h_{r}} | {k \in I_{r} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} | = 0$ where |A| denotes the number of elements of the set $A \subseteq ℕ$ . In this case, we write $x_{k} \overset{{FS}_{θ}}{⟶} L$ or x_k → L (FS_θ) or ${FS}_{θ} - lim_{k \to \infty} x_{k} = L$ . This implies that for each ɛ > 0, the set $K (ɛ) = {k \in I_{r} : | | x_{k} - L | |_{0}^{+} \geq ɛ}$ has natural density zero, namely, for each ɛ > 0, $| | x_{k} - L | |_{0}^{+} < ɛ$ , for almost all k.

A double sequence x = (x_jk) is said to be Pringsheim’s convergent (or P-convergent) if for given ɛ > 0 there exists an integer N such that |x_jk - l| < ɛ, whenever j, k > N. We shall write this as $lim_{j, k \to \infty} x_{jk} = l,$ where j and k tending to infinity independent of each other.

A double sequence x = (x_jk) is said to be bounded if there exists a positive real number M such that for all $k, j \in ℕ$ , |x_jk| < M, that is, $∥ x ∥_{\infty} = sup_{k, j} | x_{jk} | < \infty$ . We let the set of all bounded double sequences by L_∞.

Let $K \subset ℕ \times ℕ$ . Let K_mn be the number of (j, k) ∈ K such that j ≤ m, k ≤ n. If the sequence ${\frac{K_{mn}}{m . n}}$ has a limit in Pringsheim’s sense then we say that K has double natural density and is denoted by $\begin{matrix} δ_{2} (K) = lim_{m, n \to \infty} \frac{K_{mn}}{m . n} . \end{matrix}$

A double sequence x = (x_jk) is said to be statistically convergent to the numberl if for each ɛ > 0, the set { (j, k): j ≤ m and k ≤ n, |x_jk - l| ≥ɛ } has double natural density zero. In this case, we write ${st}_{2} - lim_{j, k} x_{jk} = l$ .

Let (X, ||. ||) be an FNS. Then a double sequence (x_jk) is said to be convergent to x ∈ X with respect to the fuzzy norm on X if for every ɛ > 0 there exist a number N = N (ɛ) such that $D (| | x_{jk} - x | |, \tilde{0}) < ɛ, for all j, k \geq N .$

In this case, we write $x_{jk} \overset{FN}{⟶} x$ . This means that, for every ɛ > 0 there exist a number N = N (ɛ) such that $sup_{α \in [0, 1]} | | x_{jk} - x | |_{α}^{+} = | | x_{jk} - x | |_{0}^{+} < ɛ,$ for all j, k ≥ N. In terms of neighnorhoods, we have $x_{jk} \overset{FN}{⟶} x$ provided that for any ɛ > 0, there exists a number N = N (ɛ) such that $x_{jk} \in N_{x} (ɛ, 0),$ whenever j, k ≥ N.

Let (X, ||. ||) be an FNS. A double sequence (x_jk) is said to be statistically convergent to x ∈ X with respect to the fuzzy norm on X if for every ɛ > 0, $δ_{2} ({(j, k) \in ℕ \times ℕ : D (| | x_{jk} - x | |, \tilde{0}) \geq ɛ}) = 0 .$

This implies that, for each ɛ > 0 the set $K (ɛ) = {(j, k) \in ℕ \times ℕ : | | x_{jk} - x | |_{0}^{+} \geq ɛ}$ has naturaly density zero; namely, for each ɛ > 0 $| | x_{jk} - x | |_{0}^{+} < ɛ$ for almost all j, k. In this case, we write ${st}_{2} (FN) - lim (x_{jk} - x) = \tilde{0}$ or $x_{jk} \overset{{st}_{2} (FN)}{⟶} x .$

Let (X, ||. ||) be an FNS. Then a double sequence (x_jk) is said to be statistically Cauchy with respect to the fuzzy norm on X if for every ɛ > 0, there exist N (ɛ) and M (ɛ) such that for all j, p ≥ N and k, q ≥ M, $δ_{2} ({(j, k) \in ℕ \times ℕ : j \leq n and k \leq m, | | x_{jk} - x_{pq} | |_{0}^{+} \geq ɛ}) = 0 .$

The double sequence [y] is a double subsequence of the double sequence [x] provided that there exist two increasing double index sequences {n_j} and {k_j} such that if z_j = x_{n_j,k_j}, then y is formed by $\begin{matrix} z_{1} & z_{2} & z_{5} & z_{10} \\ z_{4} & z_{3} & z_{6} & - \\ z_{9} & z_{8} & z_{7} & - \\ - & - & - & - \end{matrix}$

The double sequence θ₂ = {(k_r, j_u)} is called double lacunary sequence if there exist two increasing sequence of integers such that k₀ = 0, h_r = k_r - k_r-1 → ∞ and $j_{0} = 0, {\bar{h}}_{u} = j_{u} - j_{u - 1} \to \infty,$ as r, u → ∞.

We use following notations in the sequel: $k_{ru} = k_{r} j_{u}, h_{ru} = h_{r} {\bar{h}}_{u},$ $I_{ru} = {(k, j) : k_{r - 1} < k \leq k_{r} and j_{u - 1} < j \leq j_{u}}, q_{r} = \frac{k_{r}}{k_{r - 1}} and {\bar{q}}_{u} = \frac{j_{u}}{j_{u - 1}} .$

Let θ₂ be a double lacunary sequence. The double sequence x = (x_jk) is $S_{θ_{2}}^{''}$ -convergent to L provided that for every ɛ > 0, $P - lim_{r, u \to \infty} \frac{1}{h_{ru}} | {(k, j) \in I_{ru} : | x_{kj} - L | \geq ɛ} | = 0 .$ In this case, write $S_{θ_{2}}^{''} - lim x = L$ or $x_{k, l} ⟶ L (S_{θ_{2}}^{''})$ .

2 Main Results

In this section, we introduce the concepts of lacunary summable, lacunary statistically convergence and lacunary statistically Cauchy sequence for double sequences in fuzzy normed spaces. Also, we investigate some properties and relationships between these concepts.

Throughout the paper, we consider (X, ∥·∥) be an FNS and θ₂ = {(k_r, j_u)} be a double lacunary sequence.

2.1 Definition

A double sequence $x = (x_{mn})_{(m, n) \in ℕ \times ℕ}$ in X is said to be lacunary summable with respect to fuzzy norm on X if there is an L ∈ X such that $lim_{r, u \to \infty} \frac{1}{h_{ru}} (\sum_{(m, n) \in I_{ru}} D (| | x_{mn} - L | |, \tilde{0})) = 0 .$ In this case, we write x_mn → L ((N_{θ
₂}) _FN) or $x_{mn} \overset{{(N_{θ_{2}})}_{FN}}{⟶} L$ and (N_{θ
₂}) _FN = { $(x_{mn}) : lim_{r, u \to \infty} \frac{1}{h_{ru}} (\sum_{(m, n) \in I_{ru}} D (| | x_{mn} - L | |, \tilde{0})) =$ 0, for some L}

2.2 Definition

A double sequence x = (x_mn) in X is said to be lacunary statistically convergent or FS_{θ
₂}-convergent to L ∈ X with respect to fuzzy norm on X if for each ɛ > 0 $lim_{r, u \to \infty} \frac{1}{h_{ru}} | {(m, n) \in I_{ru} : D (| | x_{mn} - L | |, \tilde{0}) \geq ɛ} | = 0 .$ (1) In this case, we write $x_{mn} \overset{{FS}_{θ_{2}}}{⟶} L$ or x_mn → L (FS_{θ
₂}) or ${FS}_{θ_{2}} - lim_{m, n \to \infty} x_{mn} = L$ . This implies that, for each ɛ > 0, the set $K (ɛ) = {(m, n) \in I_{ru} : {∥ x_{mn} - L ∥}_{0}^{+} \geq ɛ}$ has natural density zero, namely, for each ɛ > 0, ${∥ x_{mn} - L ∥}_{0}^{+} < ɛ$ , for almost all (m, n). In terms of neighborhoods, we have $x_{mn} \overset{{FS}_{θ_{2}}}{⟶} L$ if for every ɛ > 0, $δ_{2} ({(m, n) \in I_{ru} : x_{mn} \notin N_{L} (ɛ, 0)}) = 0,$ that is, for each ɛ > 0, $(x_{mn}) \in N_{L} (ɛ, 0)$ for almost all (m, n).

A useful interpretation of the above definition is the following; $x_{mn} \overset{{FS}_{θ_{2}}}{⟶} L \Leftrightarrow {FS}_{θ_{2}} - lim | | x_{mn} - L | |_{0}^{+} = 0 .$

Note that ${FS}_{θ_{2}} - lim | | x_{mn} - L | |_{0}^{+} = 0$ implies that ${FS}_{θ_{2}} - lim | | x_{mn} - L | |_{α}^{-} = {FS}_{θ_{2}} - lim | | x_{mn} - L | |_{α}^{+} = 0,$ for each α ∈ [0, 1], since $0 \leq | | x_{mn} - L | |_{α}^{-} \leq | | x_{mn} - L | |_{α}^{+} \leq | | x_{mn} - L | |_{0}^{+}$ holds for every $m, n \in ℕ$ and for each α ∈ [0, 1].

The set of all lacunary statistically convergent double sequence with respect to fuzzy norm on X will be denoted by FS_{θ
₂} = { x: for some L, FS_{θ
₂} - lim x = L }.

2.3 Theorem

We have the following:

(i) x_mn → L ((N_{θ
₂}) _FN) ⇒ x_mn → L (FS_{θ
₂}).

(ii) (N_{θ
₂}) _FN is a proper subset of FS_{θ
₂}.

Proof. (i) If x_mn → L ((N_{θ
₂}) _FN), then for given ɛ > 0 $\begin{matrix} \sum_{(m, n) \in I_{ru}} D (∥ x_{mn} - L ∥, \tilde{0}) \\ \geq & \underset{D (∥ x_{mn} - L ∥, \tilde{0}) > ɛ}{\sum_{(m, n) \in I_{ru}}} D (∥ x_{mn} - L ∥, \tilde{0}) \\ \geq & ɛ . | {(m, n) \in I_{ru} : D (∥ x_{mn} - L ∥, \tilde{0}) \geq ɛ} | . \end{matrix}$ Therefore, we have $lim_{r, u \to \infty} \frac{1}{h_{ru}} | {(m, n) \in I_{ru} : D (| | x_{mn} - L | |, \tilde{0}) \geq ɛ} | = 0 .$ This implies that x_mn → L (FS_{θ
₂}).

(ii) In order to indicate that the inclusion (N_{θ
₂}) _FN ⊆ FS_{θ
₂} in (i) is proper, let a double lacunary sequence θ₂ be given and define a sequence x = (x_mn) as follows: $x_{mn} = {\begin{matrix} (m, n), \begin{matrix} if k_{r - 1} < m < k_{r - 1} + [\sqrt{h_{r}}], \\ j_{u - 1} < n < j_{u - 1} + [\sqrt{{\bar{h}}_{u}}], \\ (r, u = 1, 2, . . .) \end{matrix} \\ (0, 0), otherwise . \end{matrix} ∥$

Note that, x is not bounded. We have, for every ɛ > 0 and for each x ∈ X, $\begin{matrix} \frac{1}{h_{r} {\bar{h}}_{u}} | {(m, n) \in I_{ru} : D (| | x_{mn} - 0 | |, \tilde{0}) \geq ɛ} | \\ = \frac{[\sqrt{h_{r}}] [\sqrt{{\bar{h}}_{u}}]}{h_{r} {\bar{h}}_{u}} \to 0, as r, u \to \infty . \end{matrix}$ That is, x_mn → 0 (FS_{θ
₂}). On the other hand $\begin{matrix} \frac{1}{h_{r} {\bar{h}}_{u}} \sum_{(m, n) \in I_{ru}} D (| | x_{mn} - 0 | |, \tilde{0}) \\ = & \frac{1}{h_{r} {\bar{h}}_{u}} \sum_{(m, n) \in I_{ru}} | | x_{mn} | |_{0}^{+} \\ = & \frac{1}{h_{r} {\bar{h}}_{u}} \cdot \frac{[\sqrt{h_{r}}] . ([\sqrt{h_{r}}] + 1) [\sqrt{{\bar{h}}_{u}}] . ([\sqrt{{\bar{h}}_{u}}] + 1)}{4} \\ \to & \frac{1}{4} \neq 0 . \end{matrix}$ Hence, x_mn ↛ 0 ((N_{θ
₂}) _FN).

2.4 Theorem

Let θ₂ be a double lacunary sequence. Then, x = (x_mn) ∈ l_∞ and x_mn → L (FS_{θ
₂}) ⇒ x_mn → L ((N_{θ
₂}) _FN).

Proof. Suppose that x ∈ l_∞ and x_mn → L (FS_{θ
₂}). Then, we say that $D (| | x_{mn} - L | |, \tilde{0}) < M$ , for all (m, n). Given ɛ > 0, we get $\begin{matrix} \frac{1}{h_{ru}} underset (m, n) \in I_{ru} \sum D (| | x_{mn} - L | |, \tilde{0}) \\ = & \frac{1}{h_{ru}} \underset{D (| | x_{mn} - L | |, \tilde{0}) \geq ɛ}{\sum_{(m, n) \in I_{ru}}} D (| | x_{mn} - L | |, \tilde{0}) \\ + & \frac{1}{h_{ru}} \underset{D (| | x_{mn} - L | |, \tilde{0}) < ɛ}{\sum_{(m, n) \in I_{ru}}} D (| | x_{mn} - L | |, \tilde{0}) \\ \leq & \frac{M}{h_{ru}} \cdot | {(m, n) \in I_{ru} : D (| | x_{mn} - L | |, \tilde{0}) \geq ɛ} | + ɛ . \end{matrix}$ Hence, x_mn → L ((N_{θ
₂}) _FN).

2.5 Theorem

Let θ₂ be a double lacunary sequence. Then, FS_{θ
₂} ∩ l_∞ = (N_{θ
₂}) _FN ∩ l_∞.

Proof. This follows from consequences Theorem 2.3 and Theorem 2.4.

2.6 Theorem

Let θ₂ be a double lacunary sequence. If $lim inf_{r} q_{r} > 1$ and $lim inf_{u} {\bar{q}}_{u} > 1$ , then ${FS}_{2} - lim_{m, n \to \infty} x_{mn} = L$ implies ${FS}_{θ_{2}} - lim_{m, n \to \infty} x_{mn} = L$ .

Proof. Suppose that $lim inf_{r} q_{r} > 1$ and $lim inf_{u} {\bar{q}}_{u} > 1$ . Then, there exist δ > 0, τ > 0 such that q_r > 1 + δ for sufficiently large r and ${\bar{q}}_{u} > 1 + τ$ for sufficiently large u which implies that $\frac{h_{r}}{k_{r}} \geq \frac{δ}{1 + δ}$ and $\frac{{\bar{h}}_{u}}{j_{u}} \geq \frac{τ}{1 + τ}$ . If x_mn → L (FS₂), then for every ɛ > 0 and sufficiently large r and u, we have $\begin{matrix} \frac{1}{k_{ru}} | {m \leq k_{r} and n \leq j_{u} : D (| | x_{mn} - L | |, \tilde{0}) \geq ɛ} | \\ \geq \frac{1}{k_{ru}} | {(m, n) \in I_{ru} : D (| | x_{mn} - L | |, \tilde{0}) \geq ɛ} | \\ \geq (\frac{δ}{1 + δ}) (\frac{τ}{1 + τ}) \cdot \frac{1}{h_{ru}} | {(m, n) \in I_{ru} : D (| | x_{mn} - L | |, \tilde{0}) \geq ɛ} | . \end{matrix}$ So this result indicates that the double sequence x = (x_mn) is lacunary statistically convergence with respect to fuzzy norm.

2.7 Theorem

Let θ₂ be a double lacunary sequence. If $lim sup_{r} q_{r} < \infty$ and $lim inf_{u} {\bar{q}}_{u} < \infty$ , then ${FS}_{θ_{2}} - lim_{m, n \to \infty} x_{mn} = L$ implies ${FS}_{2} - lim_{m, n \to \infty} x_{mn} = L$ .

Proof. If $lim sup_{r} q_{r} < \infty$ and $lim inf_{u} {\bar{q}}_{u} < \infty$ , then there is an M, N > 0 such that q_r < M and ${\bar{q}}_{u} < N$ , for all r, u. Suppose that x_mn → L (FS_{θ
₂}) and let ${FN}_{ru} = | {(m, n) \in I_{ru} : D (| | x_{mn} - L | |, \tilde{0}) \geq ɛ} | .$ By (1), given ɛ > 0, there is an $r_{0}, u_{0} \in ℕ$ such that $\frac{{FN}_{ru}}{h_{ru}} < ɛ$ for all r > r₀, u > u₀.

Now, let H: = max{ FN_ru: 1 ≤ r ≤ r₀, 1 ≤ u ≤ u₀ } and let t and v be any integers satisfying k_r-1 < t ≤ k_r and j_u-1 < v ≤ j_u. Then, we write $\begin{matrix} \frac{1}{tv} | {k \leq t, j \leq v : D (| | x_{kj} - L | |, \tilde{0}) \geq ɛ} | \\ \leq \frac{1}{k_{r - 1} j_{u - 1}} | {k \leq k_{r}, j \leq j_{u} : D (| | x_{kj} - L | |, \tilde{0}) \geq ɛ} | \\ = \frac{1}{k_{r - 1} j_{u - 1}} {{FN}_{11} + {FN}_{12} + {FN}_{21} + \\ {FN}_{22} + . . . + {FN}_{r_{0} u_{0}} + . . . + {FN}_{ru}} \\ \leq \frac{H}{k_{r - 1} j_{u - 1}} r_{0} u_{0} + \\ \frac{1}{k_{r - 1} j_{u - 1}} {h_{r_{0}} {\bar{h}}_{u_{0} + 1} \frac{{FN}_{r_{0}, u_{0} + 1}}{h_{r_{0}} {\bar{h}}_{u_{0} + 1}} \\ + h_{r_{0} + 1} {\bar{h}}_{u_{0}} \frac{{FN}_{r_{0} + 1, u_{0}}}{h_{r_{0} + 1} {\bar{h}}_{u_{0}}} + . . . + h_{r} {\bar{h}}_{u} \frac{{FN}_{ru}}{h_{r} {\bar{h}}_{u}}} \\ \leq \frac{r_{0} u_{0} H}{k_{r - 1} j_{u - 1}} + \\ \frac{1}{k_{r - 1} j_{u - 1}} (sup_{underset u > u_{0} r > r_{0}} \frac{{FN}_{ru}}{h_{r} {\bar{h}}_{u}}) {h_{r_{0}} {\bar{h}}_{u_{0} + 1} + \\ h_{r_{0} + 1} {\bar{h}}_{u_{0}} + . . . + h_{r} {\bar{h}}_{u}} \\ \leq \frac{r_{0} u_{0} H}{k_{r - 1} j_{u - 1}} + ɛ \frac{(k_{r} - k_{r_{0}}) (j_{u} - j_{u_{0}})}{k_{r - 1} j_{u - 1}} \end{matrix}$ $\begin{matrix} \leq & \frac{r_{0} u_{0} H}{k_{r - 1} j_{u - 1}} + ɛ q_{r} {\bar{q}}_{u} \\ \leq & \frac{r_{0} u_{0} H}{k_{r - 1} j_{u - 1}} + ɛ MN . \end{matrix}$

Hence, we have FS₂ - lim x = L.

2.8 Theorem

Let θ₂ be a double lacunary sequence. If $1 < lim_{r} inf q_{r} \leq lim_{r} sup q_{r} < \infty$ and $1 < lim_{u} inf$ ${\bar{q}}_{u} \leq lim_{u} sup {\bar{q}}_{u} < \infty$ , then FS₂ = FS_{θ
₂}.

Proof. This follows from Theorem 2.6 and Theorem 2.7.

2.9 Theorem

Let (X, || · ||) be an FNS and θ₂ be a double lacunary sequence. (x_mn) and (y_mn) be double sequences in X such that x_mn → L₁ (FS_{θ
₂}) and y_mn → L₂ (FS_{θ
₂}), where L₁, L₂ ∈ X. Then, we have the following;

(x_mn + y_mn) → L₁ + L₂ (FS_{θ
₂}),

$β x_{mn} \to β L_{1} ({FS}_{θ_{2}}), (β \in ℝ),$

FS_{θ
₂} - lim ||x_mn|| = ||L₁||.

Proof. (i) Assume that x_mn → L₁ (FS_{θ
₂}) and y_mn → L₂ (FS_{θ
₂}). Since $| | . | |_{0}^{+}$ is a norm in the usual sense, we get ${∥ (x_{mn} + y_{mn}) - (L_{1} + L_{2}) ∥}_{0}^{+} \leq {∥ x_{mn} - L_{1} ∥}_{0}^{+} + {∥ y_{mn} - L_{2} ∥}_{0}^{+}$ (2) for all (m, n) ∈ I_ru. Now, let us write $\begin{matrix} K (ɛ) = {(m, n) \in I_{ru} : | | (x_{mn} + y_{mn}) - (L_{1} + L_{2}) | |_{0}^{+} \geq ɛ}, \\ K_{1} (ɛ) = {(m, n) \in I_{ru} : | | x_{mn} - L_{1} | |_{0}^{+} \geq \frac{ɛ}{2}} and \\ K_{2} (ɛ) = {(m, n) \in I_{ru} : | | y_{mn} - L_{2} | |_{0}^{+} \geq \frac{ɛ}{2}} . \end{matrix}$ From (2) that K (ɛ) ⊆ K₁ (ɛ) ∪ K₂ (ɛ). Now, by assumption, we have δ₂ (K₁ (ɛ)) = δ₂ (K₂ (ɛ)) = 0. This yields δ₂ (K (ɛ)) = 0 which completes the proof.

(ii) It is obvious.

(iii) Since $| | . | |_{α}^{-}$ and $| | . | |_{α}^{+}$ are norms in the usual sense, we have the inequalities $0 \leq | | | x_{mn} | |_{α}^{-} - | | L_{1} | |_{α}^{-} | \leq | | x_{mn} - L_{1} | |_{α}^{-}$ and $0 \leq | | | x_{mn} | |_{α}^{+} - | | L_{1} | |_{α}^{+} | \leq | | x_{mn} - L_{1} | |_{α}^{+}$ for all α ∈ [0, 1] and (m, n) ∈ I_ru. Hence, $0 \leq max {| | | x_{mn} | |_{α}^{-} - | | L_{1} | |_{α}^{-} |, | | | x_{mn} | |_{α}^{+} - | | L_{1} | |_{α}^{+} |} \leq | | x_{mn} - L_{1} | |_{α}^{+}$ for all α ∈ [0, 1] and (m, n) ∈ I_ru. Taking supremum over α ∈ [0, 1], we get $0 \leq D (| | x_{mn} | |, | | L_{1} | |) \leq | | x_{mn} - L_{1} | |_{0}^{+} .$ Hence, we have FS_{θ
₂} - lim ||x_mn|| = ||L₁||.

2.10 Definition

Let θ₂ be a double lacunary sequence. The double sequence x = (x_mn) is said to be an FS_{θ
₂}-Cauchy double sequence if there exists a double subsequence $\bar{x} = {x_{{\bar{m}}_{r} {\bar{n}}_{u}}}$ of x such that $({\bar{m}}_{r}, {\bar{n}}_{u}) \in I_{ru}$ for each (r, u), ${FS}_{θ_{2}} - lim_{r, u} x_{{\bar{m}}_{r} {\bar{n}}_{u}} = L$ and for every ɛ > 0 $lim_{r, u} \frac{1}{h_{ru}} | {(m, n) \in I_{ru} : D (| | x_{mn} - x_{{\bar{m}}_{r} {\bar{n}}_{u}} | |, \tilde{0}) \geq ɛ} | = 0 .$

2.11 Theorem

Let (X, ||. ||) be an FNS and θ₂ ={ (k_r, j_u) } be a double lacunary sequence. The double sequence x = (x_mn) is lacunary statistically convergent with respect to fuzzy norm on X if and only if x = (x_mn) is lacunary statistical Cauchy sequence.

Proof. Let (x_mn) → L (FS_{θ
₂}) and $K^{t, v} = {(m, n) \in ℕ \times ℕ : D (| | x_{mn} - L | |, \tilde{0}) < \frac{1}{tv}}$ for each (t, v) ∈ ℕ × ℕ. We obtain the following $K^{t + 1, v + 1} \subseteq K^{t, v} {and \frac{| K^{t, v} \cap I_{ru} |}{h_{ru}} \to 1, {as r, u \to \infty .$ This implies that there exist k₁ and l₁ such that r ≥ k₁ and u ≥ l₁ and $\frac{| K^{1, 1} \cap I_{ru} |}{h_{ru}} > 0$ , that is, K^1,1 ∩ I_ru ≠ ∅. We next choose k₂ > k₁ and l₂ > l₁ such that r > k₂ and u > l₂ implies that K^2,2 ∩ I_ru ≠ ∅. Thus, for each pair (r, u) such that k₁ ≤ r < k₂ and l₁ ≤ u < l₂. We select $({\bar{m}}_{r}, {\bar{n}}_{u}) \in I_{ru}$ such that $({\bar{m}}_{r}, {\bar{n}}_{u}) \in K^{r, u} \cap I_{ru}$ that is $D (| | x_{{\bar{m}}_{r} {\bar{n}}_{u}} - L | |, \tilde{0}) < 1 .$ In general, we choose k_t+1 > k_t and l_v+1 > l_v such that r > k_t+1 and u > l_v+1. This implies K^t+1,v+1 ∩ I_ru ≠ ∅. Thus, for all (r, u) such that for k_t ≤ r < k_t+1 and l_v ≤ u < l_v+1 choose $(\bar{m_{r}}, \bar{n_{u}}) \in I_{ru}$ , that is, $D (| | x_{mn} - L | |, \tilde{0}) < \frac{1}{tv} .$ Thus, $({\bar{m}}_{r}, {\bar{n}}_{u}) \in I_{ru}$ for each pair (r, u) and $D (| | x_{{\bar{m}}_{r} {\bar{n}}_{u}} - L | |, \tilde{0}) < \frac{1}{tv}$ implies ${FS}_{θ_{2}} - lim_{r, u} x_{{\bar{m}}_{r} {\bar{n}}_{u}} = L .$ Also, for each ɛ > 0 $\begin{matrix} \frac{1}{h_{ru}} | {(m, n) \in I_{ru} : D (| | x_{mn} - x_{{\bar{m}}_{r} {\bar{n}}_{u}} | |, \tilde{0}) \geq ɛ} | \\ \leq \frac{1}{h_{ru}} | {(m, n) \in I_{ru} : D (| | x_{mn} - L | |, \tilde{0}) \geq \frac{ɛ}{2}} | \\ + \frac{1}{h_{ru}} | {(m, n) \in I_{ru} : D (| | x_{{\bar{m}}_{r} {\bar{n}}_{u}} - L | |, \tilde{0}) \geq \frac{ɛ}{2}} | \end{matrix}$

Since x_mn → L (FS_{θ
₂}) and ${FS}_{θ_{2}} - lim_{r, u} x_{{\bar{m}}_{r} {\bar{n}}_{u}} = L,$ it follows that x is an FS_{θ
₂}-Cauchy double sequence.

Now suppose that x is an FS_{θ
₂}-Cauchy double sequence. Then $\begin{matrix} | {(m, n) \in I_{ru} : D (| | x_{mn} - L | |, \tilde{0}) \geq ɛ} | \\ \leq | {(m, n) \in I_{ru} : D (| | x_{mn} - x_{{\bar{m}}_{r} {\bar{n}}_{u}} | |, \tilde{0}) \geq \frac{ɛ}{2}} | \\ + | {(m, n) \in I_{ru} : D (| | x_{{\bar{m}}_{r} {\bar{n}}_{u}} - L | |, \tilde{0}) \geq \frac{ɛ}{2}} | \end{matrix}$ Therefore, x_mn → L (FS_{θ
₂}). Thus the theorem is proven.

3 Conclusion

In this study, the definitions of lacunary summable, lacunary statistical convergence, and lacunary statistical Cauchy sequence for double sequences were given in fuzzy normed spaces.

We have also shown that the relationship between lacunary statistical convergence and lacunary statistical Cauchy sequence is similar in fuzzy normed spaces as in classical spaces.

In further studies, the ideal convergence of double sequences can be defined and examined in fuzzy normed spaces.

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