Applications of Tauberian theorem in Orlicz spaces of double difference sequences of fuzzy numbers

Abstract

In this paper we introduce and study some difference double sequence spaces of fuzzy numbers by using Hausdorff metric associated with sequence of Orlicz functions. We make an effort to study some algebraic, topological properties and inclusion relations between these sequence spaces. We show that the new formed sequence spaces are complete with respect to the metric $D^{M, Δ^{m}, p, u} (x, y)$ . Finally, we establish some relation between Δ^r-statistically convergent and Δ^r-statistically (C, 1, 1) summable double sequences of fuzzy numbers under the slowly oscillating and statistically slowly oscillating conditions in certain sense, respectively.

Keywords

Orlicz function paranorm double sequence of fuzzy numbers statistical convergence slowly oscillating sequences

1 Introduction and Preliminaries

After the pioneering work on fuzzy sets and fuzzy set operations which were discovered and introduced by Zadeh [34] almost fifty years ago, several authors discussed various aspects of the theory and applications of fuzzy sets such as fuzzy topological spaces and fuzzy mathematical programming. Nanda [17] has studied the spaces of bounded and convergent sequences of fuzzy numbers and proved every Cauchy sequence of fuzzy numbers is convergent. In recent years, there has been an interest on the summability methods of sequences of fuzzy numbers. Subrahmanyam [26] has defined the Cesàro summability of sequences of fuzzy numbers and proved fuzzy analogues of some Tauberian theorems for sequences of fuzzy numbers. In [14], Mòricz obtained necessary and sufficient Tauberian conditions under which ordinary convergence follows from Cesàro convergence of single sequences of real numbers. Finally, Talo and Çakan [27] proved fuzzy analogues of the results in [14]. For other interesting results related to Tauberian type theorems for Cesàro summability method see [3–5 , 28]. Moreover, Tripathy and Dutta [29] have defined the space of (C, 1, 1) summable double sequences of fuzzy numbers and obtained some results regarding it.

But in many applications it appears to have several limitations so in view of the recent applications of fuzzy numbers in the theory of different convergence of sequences, it seems very natural to extend the interesting concept of statistical convergence further by using Orlicz functions. Our main goal is to study the double difference sequence spaces of order r of fuzzy numbers defined by an Orlicz function so as to fill up the existing gaps in the literature. We give a brief over view about Δ^r-statistically convergence, fuzzy numbers and sequences of fuzzy numbers. In addition, we study the linearity, paranorm, Δ^r-statistical (C, 1, 1) summability of these sequence spaces which were not studied earlier. It is very interesting to define these sequence spaces via Hausdroff metric and an Orlicz function. We have also define the slow oscillation of a double sequence of fuzzy numbers in different senses and prove that the slow oscillation in some sense is a Tauberian condition for (C, 1, 1) summability method. The spaces which we define in the present paper are much more general than those already have in the literature.

In these days many mathematicians focusing on the study of quotient spaces of fuzzy numbers. Recently, Qiu et al. [30] studied the algebraic properties of quotient space of fuzzy number up to the refined equivalence defined by Hong, presented a way to construct its so called Mareš cores for any fuzzy numbers, and show some characterizations of the equivalence classes of fuzzy numbers. Based on these results, they defined a new convergence structure under which the quotient space has many nice properties. They have also given some applications to fuzzy derivatives of fuzzy functions for their main results. In [31], Qiu discussed the properties of the symmetric fuzzy numbers, showed an equivalent characterization of convex fuzzy sets, and presented a way to construct a symmetric convex fuzzy set. They also reply one of Mareš’s open question in [31]. In [32], Qiu et al. introduced a metric on the quotient space of fuzzy numbers and then deal with the fuzzy mapping of real variable whose values are equivalence classes of fuzzy numbers. They studied differentiability and integrability of such functions and have given an existence and uniqueness theorem for a solution to a fuzzy differential equation.

In our study we deal with some applications of Tauberian theorem in Orlicz double difference sequence spaces of fuzzy numbers. We prove that these sequence spaces are complete metric spaces. As in [32], Qiu et al. also introduced a metric on quotient space of fuzzy numbers and proved some properties. In our paper we study Δ^r-statistical convergence and prove some results based on this convergence. In [31], Qiu also studied a new convergence on quotient spaces and have given several applications to fuzzy derivatives of fuzzy numbers. Similarly, we can make the use of [30 –32] and construct the quotient space of double difference sequences and study different properties.

Let $ℕ$ , $ℝ$ and $ℂ$ be the set of all positive integers, real numbers and complex numbers, respectively. The initial work on double sequences was founded by Bromwich [2], Hardy [8] introduced the notion of regular convergence for double sequences. A double sequence x = (x_kl) is said to be convergent in the Pringsheim’s sense (or P-convergent) if for given ∊ ≥ 0 there exists $N \in ℕ$ such that |x_kl - L| < ∊ whenever k, l > N. We shall write this as $lim_{k, l \to \infty} x_{kl} = L,$ where k and l tending to infinity independent of each other. Throughout this paper limit of a double sequence means limit in the Pringsheim’s sense. A double sequence x is bounded if $∥ x ∥ = sup_{k, l \geq 0} | x_{kl} | < \infty$ . We denote the space of double sequences which are bounded convergent and the space of bounded double sequences by $c_{2}^{\infty}$ and $l_{2}^{\infty},$ respectively.

The notion of difference sequence spaces was introduced by Kızmaz [9] who studied the difference sequence spaces l_∞ (Δ), c (Δ) and c₀ (Δ). The notion was further generalized by Et and Çolak [7] by introducing the spaces l_∞ (Δ^r), c (Δ^r) and c₀ (Δ^r). Let w (F) be the set of all sequences of fuzzy numbers and let r be a non-negative integer. The operator Δ^r : w (F) → w (F) is defined for Z = c, c₀ and l_∞, we have the following sequence spaces $Z (Δ^{r}) = {x = (x_{k}) \in w : (Δ^{r} x_{k}) \in Z},$ where Δ^rx = (Δ^rx_k) = (Δ^r-1x_k - Δ^r-1x_k+1) and Δ⁰x_k = x_k for all $k \in ℕ$ , which is equivalent to the following binomial representation $Δ^{r} x_{k} = \sum_{v = 0}^{r} (- 1)^{v} (\begin{matrix} r \\ v \end{matrix}) x_{k + v} .$

Similarly, we can define difference operators on double sequence spaces as: $\begin{matrix} Δ^{r} x_{k, l} & = & Δ^{r - 1} x_{k, l} - Δ^{r - 1} x_{k, l + 1} - Δ^{r - 1} x_{k + 1, l} \\ + Δ^{r - 1} x_{k + 1, l + 1} . \end{matrix}$

For more details about sequence spaces (see [10 , 20–23]) and references therein.

The study of Orlicz sequence spaces was initiated with a certain specific purpose in Banach space theory. Indeed, Lindberg [12] got interested in Orlicz spaces in connection with finding Banach spaces with symmetric Schauder bases having complementary subspaces isomorphic to c₀ or l_p (1 ≤ p < ∞).

An Orlicz function M : [0, ∞) → [0, ∞) is a continuous, non-decreasing and convex such that M (0) =0, M (x) >0 for x > 0 and M (x)⟶ ∞ as x⟶ ∞. Lindenstrauss and Tzafriri [13] used the idea of Orlicz function to construct the following sequence space $\begin{matrix} ℓ_{M} & = & {x = (x_{k}) \in w : \sum_{k = 1}^{\infty} M (\frac{| x_{k} |}{ρ}) < \infty, \\ for some ρ > 0}, \end{matrix}$ where w denotes the classes of all sequences. The space ℓ_M with the norm $| | x | | = inf {ρ > 0 : \sum_{k = 1}^{\infty} M (\frac{| x_{k} |}{ρ}) \leq 1}$ becomes a Banach space which is called an Orlicz sequence space. A sequence $ℳ = (ℳ)$ of Orlicz functions is said to be Musielak-Orlicz function (see [15, 16]).

A fuzzy number is a fuzzy set on the real axis that is, a mapping $x : ℝ \to [0, 1]$ which satisfies the following four conditions:

x is normal, i.e., there exists an $t_{0} \in ℝ$ such that x (t₀) =1.

x is fuzzy convex, i.e., x [λt₀ + (1 - λ) t₁] ≥ min {x (t₀), x (t₁)} for all $t_{0}, t_{1} \in ℝ$ and for all λ ∈ [0, 1].

x is upper semi-continuous.

The set $[x]_{0} = \bar{{t \in ℝ : x (t) > 0}}$ is compact, (cf. Zadeh [34]), where $\bar{{t \in ℝ : x (t) > 0}}$ denotes the closure of the set ${t \in ℝ : x (t) > 0}$ in the usual topology of $ℝ$ .

We denote the set of all fuzzy numbers on $ℝ$ by E¹ and call it as the space of fuzzy numbers.

For x ∈ E¹, the λ -level set of x is defined by is defined by $[x]_{λ} = {\begin{matrix} {t \in ℝ : x (t) \geq λ}, 0 < λ \leq 1, \\ \bar{{t \in ℝ : x (t) > λ}}, λ = 0 . \end{matrix}$

The set [x] _λ is closed, bounded and non-empty interval for each λ ∈ [0, 1] which is defined by [x] _λ = [x^- (λ), x⁺ (λ)]. $ℝ$ can be embedded in E¹, since each $r \in ℝ$ can be regarded as a fuzzy number $\bar{r}$ defined by $\bar{r} (t) = {\begin{matrix} 1, t = r \\ 0, t \neq r . \end{matrix}$

Let W be the set of all closed bounded intervals. For the particular case when A = [a^-, a⁺], B = [b^-, b⁺] are two intervals, the Hausdorff distance on W by $d (A, B) = max {| a^{-} - b^{-} |, | a^{+} - b^{+} |} .$

Then it can easily be observed that d is a metric on W (cf. Diamond and Kloeden [6]) and (W, d) is a complete metric space, (cf. Nanda [17]). Now, it can be define a metric D on the space of fuzzy numbers with the help of the Hausdorff metric d.

Definition 1.1. [1] Let $D : E^{1} \times E^{1} \to ℝ^{+},$ $\begin{matrix} D (x, y) \\ = sup_{λ \in [0, 1]} max {| x^{-} (λ) - y^{-} (λ) |, | x^{+} (λ) - y^{+} (λ) |} \\ = sup_{λ \in [0, 1]} d ([x]_{λ}, [y]_{λ}) . \end{matrix}$

Then D is called the Hausdorff distance between fuzzy numbers x and y.

Proposition 1.2. [1] Let x, y, z, t ∈ E¹ and $k \in ℝ .$ Then the following statements hold true.

(E¹, D) is a complete metric space.

D (x + z, y + z) = D (x, y), i.e, D is translation invariant.

D (kx, ky) = |k|D (x, y).

D (x + y, z + t) ≤ D (x, z) + D (y, t).

$| D (x, \bar{0}) - D (y, \bar{0}) | \leq D (x, y) \leq D (x, \bar{0}) + D (y, \bar{0}) .$

In [24], Savaş introduced the following definitions which we need in the sequel:

Definition 1.3. A double sequence x = (x_kl) of fuzzy numbers is a function x from $ℕ \times ℕ$ into the set E₁. We denote the set of all double sequences of fuzzy numbers by w₂ (F).

Definition 1.4. A double sequence x = (x_kl) of fuzzy numbers is said to be convergent in Pringsheim’s sense (or P-convergent) to the fuzzy number μ₀ written as $lim_{k, l} x_{kl} = μ_{0}$ , if for every ∊ > 0 there exists a positive integer n₀ (∊) such that D (x_kl, μ) < ∊ whenever k, l ≥ n₀. The number μ₀ is called the Pringsheim limit of x. By c₂ (F), we denote the set of all double sequences of fuzzy numbers.

Let X be a linear space. A function p: X→ $ℝ$ is called paranorm, if

p (x) ≥0 for all x ∈ X,

p (- x) = p (x) for all x ∈ X,

p (x + y) ≤ p (x) + p (y) for all x, y ∈ X,

if (λ_n) is a sequence of scalars with λ_n→ λ as n → ∞ and (x_n) is a sequence of vectors with p (x_n - x)→0 as n → ∞, then p (λ_nx_n - λx)→0 as n → ∞.

A paranorm p for which p (x) =0 implies x = 0 is called total paranorm and the pair (X, p) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [33] Theorem 10.4.2, pp. 183).

Let $ℳ = (ℳ_{ℓ})$ be a Musielak-Orlicz function, p = (p_kl) be a bounded sequence of positive real numbers and u = (u_kl) be a sequence of positive real numbers. In the present paper we defined the following sequence spaces: $\begin{matrix} c^{F} (D, M, Δ^{r}, p, u) = {x = (x_{kl}) \in w^{2} (F) : \\ lim_{m, n} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, x_{0})}{ρ})]^{p_{kl}} \\ = 0, x_{0} \in E^{1} for some ρ > 0}, \\ c_{0}^{F} (D, M, Δ^{r}, p, u) = {x = (x_{kl}) \in w^{2} (F) : \\ lim_{m, n} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ})]^{p_{kl}} \\ = 0, \bar{0} \in E^{1} for some ρ > 0}, \\ l_{\infty}^{F} (D, M, Δ^{r}, p, u) = {x = (x_{kl}) \in w^{2} (F) : \\ sup_{m, n} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ})]^{p_{kl}} \\ < \infty, \bar{0} \in E^{1} for some ρ > 0} . \end{matrix}$

The following inequality will be used throughout the paper. If 0 ≤ p_kl ≤ sup p_kl = H, G = max(1, 2^H-1) then $| a_{kl} + b_{kl} |^{p_{kl}} \leq K (| a_{kl} |^{p_{kl}} + | b_{kl} |^{p_{kl}})$ (1.1) for all k, l and $a_{kl}, b_{kl} \in ℂ$ . Also |a|^{p
_kl} ≤ max(1, |a|^H) for all $a \in ℂ$ .

The main aim of this paper is to introduce and study some difference double sequence spaces of fuzzy numbers by using the sequence of Orlicz functions. We make an effort to study some topological properties and inclusion relations related to these sequence spaces. Further, we establish some relation between Δ^r-statistically convergent and Δ^r-statistically (C, 1, 1) summable double sequences of fuzzy numbers and prove some interestingrelations.

2 Main results

Theorem 2.1. Let $ℳ = (ℳ_{ℓ})$ be a sequence of Orlicz functions, p = (p_kl) be a bounded sequence of positive real numbers and u = (u_kl) be a sequence of positive real numbers. Then $c^{F} (D, M, Δ^{r}, p, u)$ , $c_{0}^{F} (D, M, Δ^{r}, p, u)$ and $l_{\infty}^{F} (D, M, Δ^{r}, p, u)$ are linear spaces over the complex field $ℂ$ .

Proof. We shall prove the assertion for $l_{\infty}^{F} (D, M, Δ^{r}, p, u)$ only and others can be proved similarly. Let $x = (x_{kl}), y = (y_{kl}) \in l_{\infty}^{F} (D, M, Δ^{r}, p, u)$ and $α, β \in ℂ$ . Then there exist positive numbers ρ₁ and ρ₂ such that $\begin{matrix} sup_{m, n} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ_{1}})]^{p_{kl}} \\ < \infty, for some ρ_{1} > 0 \end{matrix}$ and $\begin{matrix} sup_{m, n} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} y_{kl}, \bar{0})}{ρ_{2}})]^{p_{kl}} \\ < \infty, for some ρ_{2} > 0 . \end{matrix}$

Let ρ₃ = max(2|α|ρ₁, 2|β|ρ₂). Since $ℳ = (ℳ_{ℓ})$ is a non-decreasing and convex so by using inequality (1.1), we have $\begin{matrix} sup_{m, n} \frac{1}{(m + 1) (n + 1)} \\ \times \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{α D (u_{kl} Δ^{r} x_{kl}, \bar{0}) + β D (u_{kl} Δ^{r} y_{kl}, \bar{0})}{ρ_{3}})]^{p_{kl}} \\ \leq K sup_{m, n} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ_{1}})]^{p_{kl}} \\ + K sup_{m, n} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} y_{kl}, \bar{0})}{ρ_{2}})]^{p_{kl}} \\ < \infty . \end{matrix}$

Thus, $α x + β y \in l_{\infty}^{F} (D, M, Δ^{r}, p, u)$ . This proves that $l_{\infty}^{F} (D, M, Δ^{r}, p, u)$ is a linear space. Similarly, we can prove that $c_{0}^{F} (D, M, Δ^{r}, p, u)$ and $c^{F} (D, M, Δ^{r}, p, u)$ are also linear spaces. □

Theorem 2.2. Let $ℳ = (ℳ_{ℓ})$ be a sequence of Orlicz functions, p = (p_kl) be a bounded sequence of positive real numbers and u = (u_kl) be a sequence of positive real numbers. Then $l_{\infty}^{F} (D, M, Δ^{r}, p, u)$ is paranormed space with the paranorm, $\begin{matrix} g (x) & = & inf {(ρ)^{\frac{p_{kl}}{G}} : (sup_{m, n} \frac{1}{(m + 1) (n + 1)} \\ \times \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ})]^{p_{kl}})^{\frac{1}{G}} \leq 1, \\ for some ρ > 0}, \end{matrix}$ where 0 ≤ p_kl ≤ sup p_kl = H, and G = max(1, H).

Proof. (i) Clearly g (x) ≥0 for x = (x_kl) $\in l_{\infty}^{F} (D, M, Δ^{r}, p, u)$ . Since M_kl (0) =0, we get g (0) =0.

(ii) g (- x) = g (x).

(iii) Let x = (x_kl) and y = (y_kl) $\in l_{\infty}^{F} (D, M, Δ^{r}, p, u)$ , then there exist positive numbers ρ₁ and ρ₂ such that $\begin{matrix} sup_{m, n} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ_{1}})]^{p_{kl}} \leq 1 \end{matrix}$ and $\begin{matrix} sup_{m, n} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} y_{kl}, \bar{0})}{ρ_{2}})]^{p_{kl}} \leq 1 . \end{matrix}$

Let ρ = ρ₁ + ρ₂. Then by using Minkowski’s inequality, we have $\begin{matrix} sup_{m, n} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} (x_{kl} + y_{kl}), \bar{0})}{ρ})]^{p_{kl}} \\ \leq (\frac{ρ_{1}}{ρ_{1} + ρ_{2}}) sup_{m, n} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ_{1}})]^{p_{kl}} \\ + (\frac{ρ_{2}}{ρ_{1} + ρ_{2}}) sup_{m, n} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} y_{kl}, \bar{0})}{ρ_{2}})]^{p_{kl}} \\ \leq 1 \end{matrix}$ and thus, $\begin{matrix} g (x + y) \\ = inf {(ρ)^{\frac{p_{kl}}{G}} : (sup_{m, n} \frac{1}{(m + 1) (n + 1)} \\ \times \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} (x_{kl} + y_{kl}), \bar{0})}{ρ})]^{p_{kl}})^{\frac{1}{G}} \leq 1} \\ \leq inf {(ρ_{1})^{\frac{p_{kl}}{G}} : (sup_{m, n} \frac{1}{(m + 1) (n + 1)} \\ \times \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ_{1}})]^{p_{kl}})^{\frac{1}{G}} \leq 1} \\ + inf {(ρ_{2})^{\frac{p_{kl}}{G}} : (sup_{m, n} \frac{1}{(m + 1) (n + 1)} \\ \times \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} y_{kl}, \bar{0})}{ρ_{2}})]^{p_{kl}})^{\frac{1}{G}} \leq 1} . \end{matrix}$

Therefore, g (x + y) ≤ g (x) + g (y). Finally, we prove that the scalar multiplication is continuous. Let λ be any complex number. By definition,

$\begin{matrix} g (λ x) & = & inf {(ρ)^{\frac{p_{kl}}{G}} : (sup_{m, n} \frac{1}{(m + 1) (n + 1)} \\ \times \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} λ x_{kl}, \bar{0})}{ρ})]^{p_{kl}})^{\frac{1}{G}} \leq 1} \\ = & inf {(| λ | t)^{\frac{p_{kl}}{G}} : (sup_{m, n} \frac{1}{(m + 1) (n + 1)} \\ \times \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{t})]^{p_{kl}})^{\frac{1}{G}} \leq 1}, \end{matrix}$

where $t = \frac{ρ}{| λ |} > 0$ . Since |λ|^{p
_kl} ≤ max(1, |λ|^supp_kl), we have $\begin{matrix} g (λ x) \leq max (1, | λ |^{sup p_{kl}}) \\ inf {(t)^{\frac{p_{kl}}{G}} : (sup_{m, n} \frac{1}{(m + 1) (n + 1)} \\ \times \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{t})]^{p_{kl}})^{\frac{1}{G}} \leq 1} . \end{matrix}$

So, the fact that the scalar multiplication is continuous follows from the above inequality. This completes the proof of the theorem. □

Theorem 2.3. Let $ℳ = (ℳ_{ℓ})$ be a sequence of Orlicz functions, u = (u_kl) be a sequence of positive real numbers. If p = (p_kl) and q = (q_kl) are bounded sequences of positive real numbers with 0≤ p_kl ≤ q_kl < ∞ for each k, l, then $c_{0}^{F} (D, M, Δ^{r}, p, u) \subseteq c_{0}^{F} (D, M, Δ^{r}, q, u)$ .

Proof. Let $x \in c_{0}^{F} (D, M, Δ^{r}, p, u)$ . Then $\begin{matrix} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} \\ [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ})]^{p_{kl}} ⟶ 0 \end{matrix}$ as m, n → ∞. This implies that $\begin{matrix} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} \\ [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ})]^{p_{kl}} \leq 1, \end{matrix}$ for sufficiently large values of k, l. Since M_kl is increasing and p_kl ≤ q_kl we have $\begin{matrix} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ})]^{q_{kl}} \\ \leq \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ})]^{p_{kl}} \\ ⟶ 0 as m, n \to \infty . \end{matrix}$

Hence, $x \in c_{0}^{F} (D, M, Δ^{r}, q, u)$ . This completes the proof. □

Theorem 2.4. Let $ℳ = (ℳ_{ℓ})$ be a sequence of Orlicz functions and $ϱ = lim_{t \to \infty} \frac{M_{kl} (t)}{t} > 0 \forall k, l \in ℕ .$ Then $c_{0}^{F} (D, M, Δ^{r}, p, u) \subseteq c_{0}^{F} (D, Δ^{r}, p, u) .$

Proof. In order to prove that $c_{0}^{F} (D, M, Δ^{r}, p, u) \subseteq c_{0}^{F} (D, Δ^{r}, p, u)$ . Let ϱ > 0. By definition of ϱ, we have M_kl (t) ≥ ϱ (t), for all t ≥ 0. Since ϱ > 0, we have $t \leq \frac{1}{ϱ} M_{kl} (t)$ for all t ≥ 0.

Let $x = (x_{kl}) \in c_{0}^{F} (D, M, Δ^{r}, p, u)$ . Thus, we have $\begin{matrix} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ}]^{p_{kl}} \\ \leq \frac{1}{ϱ} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ})]^{p_{kl}} \\ ⟶ 0 as m, n \to \infty \end{matrix}$ which implies that $x = (x_{kl}) \in c_{0}^{F} (D, Δ^{r}, p, u)$ . This completes the proof. □

Theorem 2.5. Let $ℳ^{'} = (ℳ_{ℓ}^{'})$ and $ℳ^{″} = (ℳ_{ℓ}^{″})$ are sequences of Orlicz functions, then $\begin{matrix} c_{0}^{F} (D, M^{'}, Δ^{r}, p, u) \\ \cap c_{0}^{F} (D, M^{″}, Δ^{r}, p, u) \\ \subseteq c_{0}^{F} (D, (M^{'} + M^{″}), Δ^{r}, p, u) . \end{matrix}$

Proof. Let $x = (x_{kl}) \in c_{0}^{F} (D, M^{'}, Δ^{r}, p, u) \cap c_{0}^{F} (D, M^{″}, Δ^{r}, p, u)$ . Therefore, $\begin{matrix} \frac{1}{(m + 1) (n + 1)} \\ \sum_{k, l = 0}^{m, n} [M_{kl}^{'} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ})]^{p_{kl}} ⟶ 0 \end{matrix}$ as m, n→ ∞ and $\begin{matrix} \frac{1}{(m + 1) (n + 1)} \\ \sum_{k, l = 0}^{m, n} [M_{kl}^{″} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ})]^{p_{kl}} ⟶ 0 \end{matrix}$ as m, n → ∞. Then, we have $\begin{matrix} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [(M_{kl}^{'} + M_{kl}^{″}) (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ})]^{p_{kl}} \\ \leq K {\frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl}^{'} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ})]^{p_{kl}}} \\ + K {\frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl}^{″} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ})]^{p_{kl}}} \\ ⟶ 0 as m, n \to \infty . \end{matrix}$ Thus, $\begin{matrix} \frac{1}{(m + 1) (n + 1)} \\ \sum_{k, l = 0}^{m, n} [(M_{kl}^{'} + M_{kl}^{″}) (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ})]^{p_{kl}} \\ ⟶ 0 as m, n \to \infty . \end{matrix}$

Therefore, $x = (x_{kl}) \in c_{0}^{F} (D, (M^{'} + M^{″}), Δ^{r}, p, u)$ and this completes the proof. □

Theorem 2.6. Let $ℳ = (ℳ_{ℓ})$ and $ℳ^{'} = (ℳ_{ℓ}^{'})$ be two sequences of Orlicz functions, then $c_{0}^{F} (D, M^{'}, Δ^{r}, p, u) \subseteq c_{0}^{F} (D, M^{'} \circ M^{″}, Δ^{r}, p, u) .$

Proof. Let $x = (x_{kl}) \in c_{0}^{F} (D, M^{'}, Δ^{r}, p, u)$ . Then we have $\begin{matrix} lim_{m, n} \frac{1}{(m + 1) (n + 1)} \\ \sum_{k, l = 0}^{m, n} [M_{kl}^{'} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ})]^{p_{kl}} = 0 . \end{matrix}$

Let ∊ > 0 and choose δ > 0 with 0 < δ < 1 such that M_kl (t) < ∊, for 0 ≤ t ≤ δ.

Write $y_{kl} = [M_{kl}^{'} (\frac{D (u_{kl} Δ^{r} x_{kl}, \bar{0})}{ρ})]$ and consider $\begin{matrix} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl} (y_{kl})]^{p_{kl}} \\ = \frac{1}{(m + 1) (n + 1)} \sum_{1} [M_{kl} (y_{kl})]^{p_{kl}} \\ + \frac{1}{(m + 1) (n + 1)} \sum_{2} [M_{kl} (y_{kl})]^{p_{kl}}, \end{matrix}$ where the first summation is over y_kl ≤ δ and second summation is over y_kl > δ. Since M_kl is continuous, we have $\frac{1}{(m + 1) (n + 1)} \sum_{1} [M_{kl} (y_{kl})]^{p_{kl}} < ∊^{H}$ (2.1) and for y_kl > δ, we use the fact that $y_{kl} < \frac{y_{kl}}{δ} \leq 1 + \frac{y_{kl}}{δ} .$

By the definition, we have for y_kl > δ $M_{kl} (y_{kl}) < 2 M_{kl} (1) \frac{y_{kl}}{δ} .$ Hence,

$\begin{matrix} \frac{1}{(m + 1) (n + 1)} \sum_{1} [M_{kl} (y_{kl})]^{p_{kl}} \\ \leq max (1, (2 M_{kl} (1) δ^{- 1})^{H}) \frac{1}{(m + 1) (n + 1)} \\ \times \sum_{1} [y_{kl}]^{p_{kl}} . \end{matrix}$ (2.2)

From Equations (2.1) and (2.2), we have $c_{0}^{F} (D, M^{'}, Δ^{r}, p, u) \subseteq c_{0}^{F} (D, M^{'} \circ M^{″}, Δ^{r}, p, u) .$

This completes the proof. □

Theorem 2.7. Let $D^{ℳ, Δ,} γ (D, ℳ, Δ^{m}, p, u)$ by $\begin{matrix} D^{ℳ, Δ,} : γ (D, ℳ, Δ^{m}, p, u) \\ \times γ (D, ℳ, Δ^{m}, p, u) ℝ^{+} \\ (x, y) D^{ℳ, Δ,} (x, y) = \sup_{m, n} [\frac{1}{(m + 1) (n + 1)} \\ \times \sum_{k, l = 0}^{m, n} {[M_{k l} (\frac{D (u_{k l} Δ^{r} x_{k l}, u_{k l} Δ^{r} y_{k l})}{ρ})]}^{p_{k l}}], \end{matrix}$ where $x = (x_{kl}), y = (y_{kl}) \in γ (D, M, Δ^{m}, p, u)$ denotes any of the spaces $l_{\infty}^{F} (D, M, Δ^{m}, p, u)$ , $c^{F} (D, M, Δ^{m}, p, u)$ or $c_{0}^{F} (D, M, Δ^{m}, p, u)$ . Then $(γ (H, ℳ, Δ^{m}, p, u), D^{ℳ, Δ,})$ is complete metric space with the metric $D^{ℳ, Δ,} (x, y)$ .

Proof. Since the proof is similar for the spaces $l_{\infty}^{F} (D, M, Δ^{m}, p, u)$ and $c_{0}^{F} (D, M, Δ^{m}, p, u)$ , we prove the theorem only for the space $c^{F} (D, M, Δ^{m}, p, u)$ . Let x = (x_kl), y = (y_kl) and $z = (z_{kl}) \in c^{F} (D, M, Δ^{m}, p, u)$ . It is easy to see that $D^{ℳ, Δ,}$ is a metric on $c^{F} (D, M, Δ^{m}, p, u)$ . To prove completeness, let (xⁱ) be a Cauchy sequence in $c^{F} (D, M, Δ^{m}, p, u)$ , where $x^{i} = (x_{kl}^{i})$ for each $i \in ℕ .$ Then, for any ∊ > 0, there exists a positive integer $n_{0} \in ℕ$ such that $\begin{matrix} D^{ℳ, Δ,} (x^{i}, x^{j}) = \sup_{m, n} [\frac{1}{(m + 1) (n + 1)} \\ \times \sum_{k, l = 0}^{m, n} {[M_{k l} (\frac{D (u_{k l} Δ^{r} x_{k l}^{i}, u_{k l} Δ^{r} x_{k l}^{j})}{ρ})]}^{p_{k l}}] \\ < forall i, j \geq n_{0} \end{matrix}$

For every fixed $k, l \in ℕ$ and for all i, j ≥ n₀ $\begin{matrix} \frac{1}{(m + 1) (n + 1)} \\ \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}^{i}, u_{kl} Δ^{r} x_{kl}^{j})}{ρ})]^{p_{kl}} \\ < ∊ . \end{matrix}$

Hence, for every fixed $k, l \in ℕ$ , by using the completeness of (E¹, D) in Proposition 1.2, we say the sequence $(Δ^{r} x_{kl}^{(i)}) = {Δ^{r} x_{kl}^{(1)}, Δ^{r} x_{kl}^{(2)}, . . .}$ is a Cauchy sequence and is uniformly convergent. For simplicity, let $lim_{m, n \to \infty} Δ^{r} x_{kl}^{(i)} = x_{kl}$ for all k, l ≥ 1. Considering k, l = 1, 2, 3,…, m, n we can easily conclude that $lim_{m, n \to \infty} x_{kl}^{(i)} = x_{kl}$ exists for all k, l ≥ 1. We must show that $x = (x_{kl}) \in c^{F} (D, M, Δ^{m}, p, u)$ . Now we can find that $\begin{matrix} lim_{j \to \infty} \frac{1}{(m + 1) (n + 1)} \\ \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}^{i}, u_{kl} Δ^{r} x_{kl}^{j})}{ρ})]^{p_{kl}} \\ < ∊ for all i \geq n_{0} and for all k, l \in ℕ . \end{matrix}$

Thus, $\begin{matrix} lim_{j \to \infty} \frac{1}{(m + 1) (n + 1)} \\ \times \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}^{i}, u_{kl} Δ^{r} x_{kl})}{ρ})]^{p_{kl}} \\ < ∊ for all i \geq n_{0} and for all k, l \in ℕ \end{matrix}$

This implies that $D^{ℳ, Δ,} (x^{i}, x) < forall i \geq n_{0}$ . Therefore, $x = (x_{kl}) \in c^{F} (D, M, Δ^{m}, p, u)$ . This completes the proof. □

3 Statistical summability (C,1,1) and Statistical convergence for fuzzy real numbers

In [25] Savaş and Mursaleen introducted the concept of statistical convergence for double sequences of fuzzy numbers. In this section we define the concepts of (C, 1, 1) summable method for sequence of fuzzy numbers and obtained fuzzy analogues of some classical Tauberian Theorems and some relations between Δ^r-statistically convergence and Δ^r-statistically (C, 1, 1) summable.

Let u = (u_kl) be a sequence of positive real numbers and x = (x_kl) be a double sequence of fuzzy numbers. Then its (C, 1, 1) Δ^r-means are defined by $ϱ_{kl}^{Δ^{r}} = \frac{1}{(m + 1) (n + 1)} \sum_{k = 0}^{m} \sum_{l = 0}^{n} u_{kl} Δ^{r} x_{kl},$ for all non negative integers m and n.

Definition 3.1. Let u = (u_kl) be a sequence of positive real numbers. Then a double sequence x = (x_kl) of fuzzy numbers is said to be (C, 1, 1) Δ^r-summable to the fuzzy number μ₀, if $lim_{k, l \to \infty} ϱ_{kl}^{Δ^{r}} = μ_{0} .$

The notion of statistical convergence depends on the density of subsets of $ℕ$ . A subset of $ℕ$ is said to have density δ (E) if $δ (E) = lim_{n \to \infty} \frac{1}{n} \sum_{k = 1}^{n} χ_{E} (k) exists .$

Let $K \subseteq ℕ \times ℕ$ be two dimensional set of positive integers and let K_mn be the numbers of (k, l) in K such that k ≤ m and l ≤ n, (see [19]). Then the lower natural density of K is defined as $P - lim_{m, n} inf \frac{| K_{mn} |}{(m + 1) (n + 1)} = δ_{2} (K) .$

In the case when the sequence $(\frac{K_{mn}}{(m + 1) (n + 1)})$ has a limit in Pringsheim’s sense, we say that K has a double natural density and it is defined by $P - lim_{m, n} \frac{| K_{mn} |}{(m + 1) (n + 1)} = δ_{2} (K) .$

Definition 3.2. Let u = (u_kl) be a sequence of positive real numbers, then a double sequence x = (x_kl) of fuzzy numbers is said to be Δ^r-statistically convergent in Pringsheim’s sense to the fuzzy number μ₀ if for any ∊ > 0, $\begin{matrix} lim_{m, n \to \infty} \frac{1}{(m + 1) (n + 1)} | {k \leq m and l \leq n : \\ D (u_{kl} Δ^{r} x_{kl}, μ_{0}) \geq ∊} | = 0, \end{matrix}$ where the vertical bars indicate the number of elements in the closed set. In this case, we write $S_{2}^{F} (Δ^{r}, u) - lim_{k, l \to \infty} u_{kl} Δ^{r} x_{kl} = μ_{0}$ and the set of all Δ^r-statistically convergent sequences is denoted by $S_{2}^{F} (Δ^{r}, u)$ .

Let u = (u_kl) be a sequence of positive real numbers, then a double sequence (x_kl) of fuzzy numbers is called Δ^r-statistically (C, 1, 1) summable to μ₀ if $S_{2}^{F} (Δ^{r}, u) - lim_{k, l \to \infty} ϱ_{kl}^{Δ^{r}} = μ_{0} .$

Now we define the concept of slow oscillation for the double sequence of fuzzy numbers in certain sense as follows:

Let u = (u_kl) be a sequence of positive real numbers. A double sequence (x_kl) of fuzzy numbers is said to be Δ^r-slowly oscillating in sense (1, 0) if

$\begin{matrix} lim_{α ↓ 1} \underset{k, l \to \infty}{lim sup} max_{k + 1 \leq i \leq α_{k}} \\ D (u_{kl} Δ^{r} x_{il}, u_{kl} Δ^{r} x_{kl}) = 0 . \end{matrix}$ (3.1)

We say that a double sequence (x_kl) of fuzzy numbers is said to be Δ^r-slowly oscillating in sense (1, 0) if (3.1) satisfied with $\begin{array}{l} \max_{\begin{matrix} k + 1 \leq i \leq α_{k} \\ l + 1 \leq j \leq α_{l} \end{matrix}} D (u_{k l} Δ^{r} x_{i j}, u_{k l} Δ^{r} x_{k j}) instead of \\ \max_{k + 1 \leq i \leq α_{k}} D (u_{k l} Δ^{r} x_{i l}, u_{k l} Δ^{r} x_{k l}) . \end{array}$ (3.2)

Now, let u = (u_kl) be a sequence of positive real numbers. Then, we say that a double sequence (x_kl) of fuzzy numbers with difference operator of order r satisfies the two-sided Tauberian condition of Hardy in sense (1, 0) if there exist positive constants n₀ and L such that

$\begin{matrix} iD (u_{kl} Δ^{r} x_{il}, u_{kl} Δ^{r} x_{i - 1, l}) \leq L \\ whenever i, l > n_{0} . \end{matrix}$ (3.3)

It is clear that if (3.3) holds, then (x_kl) is Δ^r-slowly oscillating in both sense (1, 0) and the strong (1, 0). Similarly, we can define Δ^r-slowly oscillating of double sequence (x_kl) of fuzzy numbers in sense (0, 1) and the strong (0, 1). In addition to these, a double sequence (x_kl) of fuzzy number with difference operator of order r which satisfies the two-sided Tauberian condition of Hardy type in sense (0, 1) can be defined and it is Δ^r-slowly oscillating in both sense (0, 1) and strong sense (0, 1).

Theorem 3.3. Consider $ℳ = (ℳ_{ℓ})$ be a sequenceof Orlicz functions, p = (p_kl) be a bounded sequence of positive real numbers and u = (u_kl) be a sequence of positive real numbers and $sup_{kl} p_{kl} = H < \infty$ . Then $c^{F} (D, M, Δ^{r}, p, u) \subset S_{2}^{F} (Δ^{r}, u)$ .

Proof. Let $x \in c^{F} (D, M, Δ^{r}, p, u)$ . Take ∊ > 0 and $∊_{1} = \frac{∊}{ρ}$ , ∑₁ denote the sum over k ≤ m and l ≤ n with D (u_klΔ^rx_kl, μ₀) ≥ ∊ and ∑₂ denote the sum over k ≤ n and l ≤ n with D (u_klΔ^rx_kl, μ₀) < ∊. Then, we obtain $\begin{matrix} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, x_{0})}{ρ})]^{p_{kl}} \\ = \frac{1}{(m + 1) (n + 1)} (\sum_{1} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, x_{0})}{ρ})]^{p_{kl}} \\ + \sum_{2} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, x_{0})}{ρ})]^{p_{kl}}) \\ \geq \frac{1}{(m + 1) (n + 1)} \sum_{1} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, x_{0})}{ρ})]^{p_{kl}} \\ \geq \frac{1}{(m + 1) (n + 1)} \sum_{1} {[M_{kl} (∊_{1})]^{p_{kl}}} \\ \geq \frac{1}{(m + 1) (n + 1)} \sum_{1} min ({[M_{kl} (∊_{1})]^{h}}, {[M_{kl} (∊_{1})]^{H}}) \\ = \frac{1}{(m + 1) (n + 1)} | {k \leq m and l \leq n : D (u_{kl} Δ^{r} x_{kl}, μ_{0}) \\ \geq ∊} | min ({[M_{kl} (∊_{1})]^{h}}, {[M_{kl} (∊_{1})]^{H}}) . \end{matrix}$

Hence, $x \in S_{2}^{F} (Δ^{r}, u)$ . This completes the proof of the theorem. □

Theorem 3.4. Consider $ℳ = (ℳ_{ℓ})$ be bounded sequence of Orlicz functions, p = (p_kl) be a bounded sequence of positive real numbers and u = (u_kl) be a sequence of positive real numbers and $0 < inf_{kl} p_{kl} \leq p_{kl} \leq sup_{kl} p_{kl} = H < \infty$ . Then $S_{2}^{F} (Δ^{r}, u) \subset c^{F} (D, M, Δ^{r}, p, u) .$

Proof. Suppose that $x \in c^{F} (D, M, Δ^{r}, p, u)$ . Take ∊ > 0, ∑₁ denote the sum over k ≤ m and l ≤ n with D (u_klΔ^rx_kl, μ₀) ≥ ∊ and ∑₂ denote the sum over k ≤ n and l ≤ n with D (u_klΔ^rx_kl, μ₀) < ∊. Since $ℳ = (ℳ_{ℓ})$ be bounded there exists an integer K such that M_kl (x) < K for all x ≥ 0. Then, we get $\begin{matrix} \frac{1}{(m + 1) (n + 1)} \sum_{k, l = 0}^{m, n} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, x_{0})}{ρ})]^{p_{kl}} \\ \leq \frac{1}{(m + 1) (n + 1)} (\sum_{1} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, x_{0})}{ρ})]^{p_{kl}} \\ + \sum_{2} [M_{kl} (\frac{D (u_{kl} Δ^{r} x_{kl}, x_{0})}{ρ})]^{p_{kl}}) \\ \leq \frac{1}{(m + 1) (n + 1)} \sum_{1} max {(\frac{K}{ρ})^{h}, (\frac{K}{ρ})^{H}} \\ + \frac{1}{(m + 1) (n + 1)} \sum_{2} [M_{kl} (\frac{∊}{ρ})]^{p_{kl}} \\ \leq max {(T)^{h}, (T)^{H}} \frac{1}{(m + 1) (n + 1)} | {k \leq m \\ and l \leq n : \\ D (u_{kl} Δ^{r} x_{kl}, μ_{0}) \geq ∊} | \\ + max {[M_{kl} (\frac{∊}{ρ})^{h}], [M_{kl} (\frac{∊}{ρ})^{H}]}, \\ where T = \frac{K}{ρ} . \end{matrix}$

Hence, $x \in c^{F} (D, M, Δ^{r}, p, u) .$ This completes the proof of the theorem. □

Theorem 3.5. For u = (u_kl) be a sequence of positive real numbers and let the double sequence (x_kl) of fuzzy numbers be bounded and Δ^r-statistically convergent to a fuzzy number μ₀, then (x_kl) is Δ^r-statistically (C, 1, 1) summable to the same number.

Proof. Let u = (u_kl) be a sequence of positive real numbers and suppose that the double sequence (x_kl) of fuzzy numbers be bounded and Δ^r-statistically convergent to a fuzzy number μ₀. Then there exists a positive constant C such that $sup_{k, l} D (u_{kl} Δ^{r} x_{kl}, μ_{0}) < C$ for all $k, l \in ℕ .$ By the definition of Δ^r-statistical convergence and for a given ∊ > 0, we have $lim_{m, n \to \infty} \frac{1}{(m + 1) (n + 1)} | K_{mn} | = 0,$ where K_mn = {k ≤ m and l ≤ n : D (u_klΔ^rx_kl, μ₀) ≥ ∊}.

Now by arithmetic mean $(ϱ_{kl}^{Δ^{r}})$ of (x_kl) for a given ∊ > 0, there exists a n₀ = n₀ (∊) ≥0 such that for m, n > n₀ (∊), we can write $\begin{matrix} D (ϱ_{kl}^{Δ^{r}}, μ_{0}) \\ = D (\frac{1}{(m + 1) (n + 1)} \sum_{k = 0}^{m} \sum_{l = 0}^{n} u_{kl} Δ^{r} x_{kl}, μ_{0}) \\ = D (\frac{1}{(m + 1) (n + 1)} \sum_{k = 0}^{m} \sum_{l = 0}^{n} u_{kl} Δ^{r} x_{kl}, \\ \frac{1}{(m + 1) (n + 1)} \sum_{k = 0}^{m} \sum_{l = 0}^{n} μ_{0}) \\ \leq \frac{1}{(m + 1) (n + 1)} \sum_{k = 0}^{m} \sum_{l = 0}^{n} D (u_{kl} Δ^{r} x_{kl}, μ_{0}) \\ = \frac{1}{(m + 1) (n + 1)} {\sum_{(k, l) \in K_{mn}} + \sum_{(k, l) \notin K_{mn}}} \\ \times D (u_{kl} Δ^{r} x_{kl}, μ_{0}) \\ = \frac{1}{(m + 1) (n + 1)} \sum_{(k, l) \in K_{mn}} D (u_{kl} Δ^{r} x_{kl}, μ_{0}) \\ + \frac{1}{(m + 1) (n + 1)} \sum_{(k, l) \notin K_{mn}} D (u_{kl} Δ^{r} x_{kl}, μ_{0}) \\ \leq C \frac{1}{(m + 1) (n + 1)} | K_{mn} | \\ + ∊ \frac{1}{(m + 1) (n + 1)} | K_{mn}^{c} | \\ \leq C \frac{1}{(m + 1) (n + 1)} | K_{mn} | + ∊ \\ \leq 2 ∊ . \end{matrix}$

Therefore, we have $\begin{matrix} \frac{1}{(M + 1) (N + 1)} | {k \leq M and l \leq N : \\ D (ϱ_{kl}^{Δ^{r}}, μ_{0}) \geq 2 ∊} | \\ \leq \frac{n_{0}}{N + 1} + \frac{n_{0}}{M + 1}, M, N > n_{0} . \end{matrix}$

It follows that the fact that for any given ∊ > 0, $\begin{matrix} lim_{M, N \to \infty} \frac{1}{(M + 1) (N + 1)} | {k \leq M and l \leq N : \\ D (ϱ_{kl}^{Δ^{r}}, μ_{0}) \geq 2 ∊} | = 0 . \end{matrix}$

In other words, $(ϱ_{kl}^{Δ^{r}})$ is Δ^r-statistically convergent to μ₀. Hence, we get that (x_kl) is Δ^r-statistically (C, 1, 1) summable to μ₀. □

The converse of the above theorem does not hold in general.

Example 3.6. Let us consider the double sequence x = (x_kl) of fuzzy numbers defined by $Δ^{r} x_{kl} = {\begin{matrix} ξ_{0}, & if k, l are even \\ φ_{0}, & if k, l are odd \\ \bar{0}, & otherwise, \end{matrix}$ where $\begin{matrix} ξ_{0} & = & {\begin{matrix} 5 + 6 t, & if t \in [- \frac{5}{6}, 0] \\ 0, & otherwise \end{matrix} and \\ φ_{0} & = & {\begin{matrix} 5 - 6 t, & if t \in [0, \frac{5}{6}] \\ 0, & otherwise . \end{matrix} \end{matrix}$

We can check that the endpoints of λ-level set of (x_kl) by taking u = (u_kl) =1 ∀ k, l and we take two cases for r, that is, r = 1 and r ≥ 2.

For the case r = 1, we have $u_{kl} Δ^{r} x_{kl}^{-} (λ) = {\begin{matrix} \frac{- 4 + 2 λ}{6}, & if k, l are even \\ 0, & otherwise \end{matrix}$ and $u_{kl} Δ^{r} x_{kl}^{+} (λ) = {\begin{matrix} \frac{4 - 2 λ}{6}, & if k, l are odd \\ 0, & otherwise . \end{matrix}$

It follows from the definition of (C, 1, 1) Δ^r-mean that $\begin{matrix} ϱ_{kl}^{- Δ^{r}} (λ) \\ = {\begin{matrix} \frac{(- 4 + 2 λ)}{6} \frac{(k + 2) (l + 2)}{4 (k + 1) (l + 1)}, & if k, l are even \\ \frac{(- 4 + 2 λ)}{6} \frac{(k + 2)}{4 (k + 1)}, & if k is even, l is odd \\ \frac{(- 4 + 2 λ)}{24}, & if k, l are odd \\ \frac{(- 4 + 2 λ)}{6} \frac{(l + 2)}{4 (l + 1)}, & if k is odd, l is even \end{matrix} \end{matrix}$ and $\begin{matrix} ϱ_{kl}^{+ Δ^{r}} (λ) \\ = {\begin{matrix} \frac{(4 - 2 λ)}{6} \frac{kl}{4 (k + 1) (l + 1)}, & if k, l are even \\ \frac{(4 - 2 λ)}{6} \frac{k}{4 (k + 1)}, & if k is even, l is odd \\ \frac{(4 - 2 λ)}{24}, & if k, l are odd \\ \frac{(4 - 2 λ)}{6} \frac{l}{4 (l + 1)}, & if k is odd, l is even . \end{matrix} \end{matrix}$

Now, for the case r ≥ 2, we have $u_{kl} Δ^{r} x_{kl}^{-} (λ) = {\begin{matrix} - 2^{r} (1 - λ), & if k, l are even \\ 0, & otherwise \end{matrix}$ and $u_{kl} Δ^{r} x_{kl}^{+} (λ) = {\begin{matrix} 2^{r} (1 - λ), & if k, l are odd \\ 0, & otherwise . \end{matrix}$

Therefore, the double sequences $ϱ_{kl}^{- Δ^{r}} (λ)$ and $ϱ_{kl}^{+ Δ^{r}} (λ)$ converges to $\frac{(- 4 + 2 λ)}{24}$ and $\frac{(4 - 2 λ)}{24}$ as k, l → ∞, respectively. By taking μ₀ = (ξ₀ + φ₀)/4, we have $lim_{k, l \to \infty} D (ϱ_{kl}^{Δ^{r}}, μ_{0}) = 0$ . On the other hand, we can easily find that D (u_klΔ^rx_kl, μ₀) ≥ ∊ for every $k, l \in ℕ$ and 0 < ∊ < 1/6. In this case, we obtain $\begin{matrix} lim_{M, N \to \infty} \frac{1}{(M + 1) (N + 1)} | \\ {k \leq M and l \leq N : D (u_{kl} Δ^{r} x_{kl}, μ_{0}) \geq ∊} | \\ = lim_{M, N \to \infty} \frac{(M + 1) (N + 1)}{(M + 1) (N + 1)} = 1 \end{matrix}$ for every ∊ > 0. Hence, (x_kl) is not Δ^r-statisically convergent to μ₀.

Theorem 3.7. For u = (u_kl) be a sequence of strictly positive real numbers and assume that the double sequence (x_kl) of fuzzy numbers be Δ^r-statistically (C, 1, 1) summable to a fuzzy number μ₀. If (x_kl) satisfies the two-sided Tauberian conditions of Hardy type in senses (0, 1) and (1, 0), then (x_kl) is also Δ^r-statistically convergent to μ₀.

Proof. Let u = (u_kl) be a sequence of positive real numbers and suppose that the double sequence (x_kl) of fuzzy numbers which is Δ^r-statistically (C, 1, 1) summable to a fuzzy number μ₀ satisfies the two-sided Tauberian conditions of Hardy type in senses (0, 1) and (1, 0). Then there exist positive constants C₁, C₂ and n₀ = n₀ (∊) $kD (u_{kl} Δ^{r} x_{kl}, u_{kl} Δ^{r} x_{k - 1, l}) \leq C_{1}$ and $lD (u_{kl} Δ^{r} x_{kl}, u_{kl} Δ^{r} x_{k, l - 1}) \leq C_{2}$ whenever k, l > n₀. By using the condition in sense (1, 0) for every ∊ > 0 and k, l > n₀, we obtain $\begin{array}{l} \max_{\begin{matrix} k + 1 \leq i \leq α_{k} \\ l + 1 \leq j \leq α_{l} \end{matrix}} D (u_{k l} Δ^{r} x_{i j}, u_{k l} Δ^{r} x_{k j}) \\ = \max_{\begin{matrix} k + 1 \leq i \leq α_{k} \\ l + 1 \leq j \leq α_{l} \end{matrix}} \sum_{s = k + 1}^{i} D (u_{k l} Δ^{r} x_{s j}, u_{k l} Δ^{r} x_{s - 1, j}) \\ \leq \max_{\begin{matrix} k + 1 \leq i \leq α_{k} \\ l + 1 \leq j \leq α_{l} \end{matrix}} {(\sum_{s = k + 1}^{i} \frac{1}{s}) (\sup_{k + 1 \leq s \leq i} D (u_{k l} Δ^{r} x_{s j}, u_{k l} Δ^{r} x_{s - 1, j}))} \\ \leq \sum_{s = k + 1}^{α_{k}} \frac{C_{1}}{s} \leq C_{1} (\frac{α_{k} - k}{k}) \leq C_{1} (α - 1) \leq ϵ \end{array}$ whenever 1 < α ≤ 1 + ∊/C₁. Hence, the set $\begin{matrix} {n_{0} < k \leq M and n_{0} < l \leq N : \\ \max_{\begin{array}{l} k + 1 \leq α_{k} \\ l + 1 \leq j \leq α_{l} \end{array}} D (u_{k l} Δ^{r} x_{i j}, u_{k l} Δ^{r} x_{k j}) \geq \in \end{matrix}$ is empty. Therefore, if the two-sided Tauberian condition of Hardy type in sense (1, 0) holds, then (x_kl) is Δ^r-statistically slowly oscillating in the strong sense (0, 1). Similarly, we can find that (x_kl) which satisfies the two-sided Tauberian condition of Hardy type is Δ^r-statistically slowly oscillating in sense (0, 1). As a result, it follows from (Theorem 4.5 in [18]) that (x_kl) is Δ^r-statistically convergent to μ₀. □

Footnotes

Acknowledgments

The authors are very thankful to the referee for his carefully reading the manuscript and valuable suggestions which improved the presentation of the paper.

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