Statistical weighted matrix summability of fuzzy mappings and associated approximation results

Abstract

This paper is devoted to studying weighted A-statistical convergence and statistical weighted A-summability of fuzzy sequences and their representations of sequences of λ-levels, which are intervals. We obtain necessary and sufficient conditions for the matrix A to be weighted fuzzy regular and derive some inclusion relations concerning these newly proposed methods. Furthermore we prove a fuzzy Korovkin type approximation theorem using statistically weighted A-summability and estimate the rates of weighted A-statistical convergence by means of the fuzzy modulus of continuity. Finally, based on a fuzzy analogue of Meyer-König and Zeller operators, we present an illustrative example to show that our proposed methods are stronger than the existing literature related to fuzzy Korovkin type approximation theorem.

Keywords

Weighted fuzzy regular matrix Korovkin type approximation theorems rate of convergence

1. Introduction

The process of summation of infinite series is one of the most effective research methods of mathematical theory and plays an important role in science and engineering. At the end of the nineteenth century, many researchers focused on various alternative methods associated with the theory of infinite series. These methods were termed as Summability Methods. In order to obtain significant results based on summability methods, the theory of sequence spaces can be considered as a powerful tool.

With the rapid development of sequence spaces many researchers have focused on the idea of statistical convergence, which was initially introduced by Fast [1] and Steinhaus [2] independently. It is well known that this type convergence is used to relax to classical convergence condition and to achieve validity of convergence condition only for a majority of elements.

The concept of fuzzy statistical convergence has been studied by several authors. In 1995, Nuray [24] introduced the notion of lacunary statistical convergence of sequences of fuzzy numbers. Later, Savaş [25] gave the notion of strongly λ-summable and λ-statistical convergence of sequences of fuzzy numbers. In the year 2006, Aytar et al. [26] extended the concepts of statistical limit superior and limit inferior to statistically bounded fuzzy mappings. Furthermore, the concept of statistical summability (C, 1) for fuzzy real numbers has been recently studied by Altın et al. [27]. In recent years, summability methods have been investigated the fuzzy mappings point of view and linked with Abel summability by [28], Hölder summability by [29], power series methods of summability by [30] and many others.

The main focus of this study is to extend the concepts of A-statistical convergence and statistical A-summability of real sequences to the fuzzy set theory. Since the fuzzy results are obtained via level sets of fuzzy numbers, the proposed results in the interval context are presented here for the first time. Further, our present investigation deals essentially with various summability techniques involving fuzzy sequences and shows how these methods lead to a number of fuzzy approximation results.

2. Preliminaries

We shall now consider briefly some of the recent progress in understanding the development of statistical convergence and its weighted versions for real sequences.

Let K be a subset of the set $ℕ$ of natural numbers. Then, the asymptotic density δ (K) of K is defined as $δ (K) = lim_{n \to \infty} \frac{1}{n} | {k ≦ n : k \in K} |,$ provided that the limit exists. Here, and in what follows, | · | denotes the cardinality of the enclosed set. A sequence x = (x _k) is called statistically convergent (st-convergent) to the number L, denoted by st - lim x = L, if, for each ɛ > 0, the set $K (ɛ) = {k \leq n : | x_{k} - L | ≧ ɛ}$ has asymptotic density zero (see [2]), i.e., $\begin{matrix} lim_{n \to \infty} \frac{1}{n} | {k \leq n : | x_{k} - L | ≧ ɛ} | = 0 . \end{matrix}$ For more details of statistical convergence and its extensions, we refer to [3 –9].

A further generalization of statistical convergence is the weighted statistical convergence, which has been investigated by many mathematicians (see [10]). More recently, this notion was modified by Srivastava et al. [11] and was further extended by Kadak [12].

Let p = (p _k) be a sequence of nonnegative numbers such that p ₀ > 0 and $P_{n} = \sum_{k = 0}^{n} p_{k} \to \infty$ as n→ ∞. We say that a sequence x = (x _k) is weighted statistically convergent to L if, ∀ε∀ > 0, $lim_{n \to \infty} \frac{1}{P_{n}} | {k \leq P_{n} : p_{k} | x_{k} - L | \geq ε∀} | = 0 .$

In the year 2013, Belen and Mohiuddine [13] introduced a new method for weighted statistical convergence in terms of the de la Vallee-Poussin mean and called λ-statistical convergence (see also [14]). Recently, this notion has been modified by adding the condition lim inf p _k > 0 (see [15]). Aktuğlu [16] gave the concept of αβ-statistical convergence, as a new generalization of statistical convergence, via two sequences (α (n)) and (β (n)) of positive numbers which satisfy the following conditions: [(i)] α and β are both non-decreasing sequences, [(ii)] β (n) ≥ α (n), [(iii)] β (n) - α (n)⟶ ∞ as n⟶ ∞. Assume that Λ denotes the set of pairs (α, β) satisfying the conditions (i) to (iii). For (α, β) ∈ Λ and $K \subseteq ℕ$ , a sequence x = (x _k) is said to be αβ-statistically convergent to the number L if, for each ɛ > 0, $lim_{n \to \infty} \frac{1}{β (n) - α (n) + 1} | {k \in P_{n}^{α, β} : | x_{k} - L | ≧ ε∀} | = 0,$ where $P_{n}^{α, β} = [α (n), β (n)]$ .

Based on a non-negative regular matrix A = (a _nk), the concept of statistical convergence was extended by Kolk [17].

A sequence x = (x _k) is called A-statistically convergent to number L if, for every ε∀ > 0, $lim_{n} \sum_{k : | x_{k} - L | \geq ε∀} a_{nk} = 0 .$ For related concepts, one may refer to [18 –21] and the references therein.

In the year 2016, Mohiuddine [22] introduced the concepts weighted A-statistical convergence and statistically weighted A-summability, and also proved a Korovkin type approximation theorem via weighted A-statistical convergence.

2.1 Fuzzy summability

A fuzzy number is a function on the real axis (see [23]), i.e., a mapping $u : ℝ \to [0, 1]$ which is normal, fuzzy convex, upper semi-continuous and the closure of the set ${x \in ℝ : u (x) > 0}$ is compact in the usual topology of $ℝ$ (see [23]). We denote the set of all fuzzy numbers by $ℝ_{F}$ . Let λ-level set [u] _λ of $u \in ℝ_{F}$ is defined by $[u]_{0} = \bar{{t \in ℝ : u (t) > λ}} and [u]_{λ} = {t \in ℝ : u (t) \geq λ} .$

The set [u] _λ is closed, bounded and non-empty interval for each λ ∈ [0, 1] which is defined by [u] _λ : = [u ^- (λ), u ⁺ (λ)]. It is clear that any $r \in ℝ$ can be considered as a fuzzy number $\tilde{r}$ defined by $\tilde{r} (x) = 1 (x = r) and \tilde{r} (x) = 0 (x \neq r) .$ For instance, $\tilde{0} \in ℝ_{F}$ is defined as $\tilde{0} (x) = 1$ if x = 0, and $\tilde{0} (x) = 0$ otherwise.

Let $u, v, w \in ℝ_{F}$ and $k \in ℝ$ . Then the operations addition, scalar multiplication and product defined on $ℝ_{F}$ by $\begin{matrix} u \oplus v = w & \Leftrightarrow & [w]_{λ} = [u]_{λ} + [v]_{λ} \forall λ \in [0, 1] \\ \Leftrightarrow & w^{\pm} (λ) = u^{\pm} (λ) + v^{\pm} (λ) \end{matrix}$ and $[k ⊙ u]_{λ} = k [u]_{λ}, u ⊙ v = w \Leftrightarrow [w]_{λ} = [u]_{λ} [v]_{λ}$ where $\begin{matrix} w^{-} (λ) & = & min {u^{-} v^{-}, u^{-} v^{+}, u^{+} v^{-}, u^{+} v^{+}}, \\ w^{+} (λ) & = & max {u^{-} v^{-}, u^{-} v^{+}, u^{+} v^{-}, u^{+} v^{+}} . \end{matrix}$

For $u, v \in ℝ_{F}$ and all λ ∈ [0, 1], the relation ⪯ given in $ℝ_{F} \times ℝ_{F}$ by $u ⪯ v \Leftrightarrow u^{-} (λ) \leq v^{-} (λ) and u^{+} (λ) \leq v^{+} (λ),$ is a partial order relation. A fuzzy number u is called non-negative if and only if u (x) =0 for all x < 0. The set of all non-negative fuzzy numbers will be denoted by $ℝ_{F}^{+}$ . Now, we can define the metric D on $ℝ_{F}$ with respect to the Hausdorff metric d as $\begin{matrix} D (u, v) & = & sup_{λ \in [0, 1]} d ([u]_{λ}, [v]_{λ}) \\ = & sup_{λ \in [0, 1]} max {| u^{-} (λ) - v^{-} (λ) |, \\ | u^{+} (λ) - v^{+} (λ) |} . \end{matrix}$

In this case, $(ℝ_{F}, D)$ is complete metric space (see [31]).

Definition 1. [32] Let (u _k) be a sequence of fuzzy numbers. Then, the expression is called a series of fuzzy numbers and also is said to be convergent to a fuzzy number $u \in ℝ_{F}$ with [u] _λ : = [u ^- (λ), u ⁺ (λ)], if the sequence of partial sums ${S_{n}}_{n \in ℕ}$ , converges to u with respect to metric D on $ℝ_{F}$ . In this case, we say that the series converges to u, and we write $\oplus \sum_{k = 0}^{\infty} u_{k} = \lim_{n \to \infty} S_{n} = u .$

In addition, since ${⌈ \oplus \sum_{k = 0}^{n} u_{k}^{-} ⌉}_{λ} = + \sum_{k = 0}^{n} {[u_{k}]}_{λ} = [\sum_{k = 0}^{n} u_{k}^{-} (λ), \sum_{k = 0}^{n} u_{k}^{+} (λ)],$ the series is converges if and only if $\sum_{k = 0}^{n} u_{k}^{\pm} (λ) \to u^{\pm} (λ) as n \to \infty, uniformly in λ \in [0, 1] .$

3 Weighted A-summability of sequences of fuzzy numbers

We begin by recalling some basic definitions and preliminaries on fuzzy matrix transformations.

Let w (F) be the set of all fuzzy mappings. The certain sets ℓ_∞ (F) and c (F) of fuzzy mappings are defined by

$\begin{matrix} ℓ_{\infty} (F) & : = & {(u_{k}) \in w (F) : sup_{k \in ℕ} D (u_{k}, \bar{0}) < \infty}, \\ c (F) & : = & {(u_{k}) \in w (F) : \exists l \in ℝ_{F} ∋ lim_{k \to \infty} D (u_{k}, l) = 0} . \end{matrix}$

An infinite matrix A = (a _ij) of fuzzy numbers is a double sequence of fuzzy numbers defined by a function $A : ℕ \times ℕ \to ℝ_{F}$ . Let ν ₁ (F) and ν ₂ (F) be two subsets of w (F) and let A = (a _nk) be an infinite fuzzy matrix. Then, A defines a matrix mapping from ν ₁ (F) into ν ₂ (F), if for each u = (u _k) ∈ ν ₁ (F), the sequence Au = {(Au) _n} (also called the A-transform of u) exists and is in ν ₂ (F); where

${[{(A ⊙ u)}_{n}]}_{λ} : = {[\oplus \sum_{k = 0}^{\infty} a_{n k} ⊙ u_{k}]}_{λ} .$ (3.1)

Moreover, if $a_{nk}, u_{k} \in ℝ_{F}^{+}$ for all $k, n \in ℕ$ , then the A-transform of u is defined by

$\begin{matrix} [(A ⊙ u)_{n}]_{λ} = [\sum_{k = 0}^{\infty} a_{nk}^{-} (λ) u_{k}^{-} (λ), \sum_{k = 0}^{\infty} a_{nk}^{+} (λ) u_{k}^{+} (λ)] . \end{matrix}$

The class of all matrices A : ν ₁ (F) → ν ₂ (F) will be denoted by (ν ₁ (F) : ν ₂ (F)). Thus, A ∈ (ν ₁ (F) : ν ₂ (F)) if and only if the series on the right side of (3.1) converges for each $n \in ℕ$ and every u ∈ ν ₁ (F). We say that a sequence u = (u _k) of fuzzy numbers is A-summable to the fuzzy number ℓ if Au converges to ℓ.

Theorem 1. [33] Let $A = (a_{nk}) \in ℝ_{F}^{+}$ for all $n, k \in ℕ$ . Then A is fuzzy regular matrix if and only if $\begin{matrix} sup_{n \in ℕ} \sum_{k} D (a_{nk}, \bar{0}) < \infty, \end{matrix}$ (3.2) $\begin{matrix} lim_{n \to \infty} a_{nk} = \bar{0}, (k \in ℕ) \end{matrix}$ (3.3) and $\lim_{n \to \infty} \oplus \sum_{k} a_{n k} = \bar{1} .5.3 p c$ (3.4)

Definition 2. Let u = (u _k) be a sequence of fuzzy numbers and also let $p = (p_{k})_{k \in ℕ}$ be a sequence of positive real numbers such that lim inf p _k > 0, p ₀ > 0, $\begin{matrix} P_{n} = \sum_{k \in I_{n}^{α β}} p_{k} \to \infty as n \to \infty \end{matrix}$ and $σ_{n}^{(α, β)} (t) = \frac{1}{P_{n}} ⊙ (\oplus \sum_{k \in I_{n}^{α β}, t \in H} p_{k} ⊙ u_{k} (t)) (t \in H \subseteq ℝ)$ where $I_{n}^{α β} = [α (n), β (n)]$ is a closed interval for (α, β) ∈ Λ. The weighted interval means $[σ_{n}^{(α, β)} (t)]_{λ}$ here are of the form $\begin{matrix} [σ_{n}^{(α, β)} (t)]_{λ} = [\frac{1}{P_{n}} \sum_{k = α (n)}^{β (n)} p_{k} u_{k}^{-} (λ), \frac{1}{P_{n}} \sum_{k = α (n)}^{β (n)} p_{k} u_{k}^{+} (λ)] . \end{matrix}$

Definition 3. A sequence u = (u _n) of fuzzy numbers is said to be weighted αβ-summable, on a set $H \subseteq ℝ$ , if there is a fuzzy number $ℓ \in ℝ_{F}$ such that $\begin{matrix} lim_{n \to \infty} D (σ_{n} (t), ℓ (t)) = 0 (t \in H \subseteq ℝ) . \end{matrix}$

Note that the above limit is given level-wise by $\begin{matrix} lim_{n \to \infty} sup_{λ \in [0, 1]} max {| \frac{1}{P_{n}} \sum_{k = α (n)}^{β (n)} p_{k} u_{k}^{-} (λ) - ℓ^{-} (λ) | \\ , | \frac{1}{P_{n}} \sum_{k = α (n)}^{β (n)} p_{k} u_{k}^{+} (λ) - ℓ^{+} (λ) |} = 0 \end{matrix}$ uniformly in λ ∈ [0, 1].

By ω ^αβ (F), we denote the set of all weighted summable sequences of fuzzy numbers. In symbol, we may write ω ^αβ (F)- lim u _k =ℓ. For α (n) =1, β (n) = n and p _k = 1, then ω ^αβ (F)-summability reduces to Cesàro summability for fuzzy sequences (see [34]).

Definition 4. Let A = (a _nk) be an infinite matrix of fuzzy numbers and (α, β) ∈ Λ. A fuzzy sequence u = (u _n) is said to be weighted A-summable to ℓ, on the set H, if the A-transform of u is weighted αβ-summable to ℓ. We say that the sequence u = (u _n) is pointwise weighted A-summable to $ℓ \in ℝ_{F}$ , denoted by $ω_{A}^{α β} (F) - lim u = ℓ$ , that is

$\lim_{m \to \infty} D \frac{1}{P_{m}} ⊙ \oplus \sum_{n \in I_{m}^{α β}} p_{n} \oplus \sum_{k = 1}^{\infty} a_{n k} (t) ⊙ u_{k} (t), ℓ (t) = 0$ (3.5) for every t ∈ H.

For simplicity in notation, we shall use the conventions that

$A_{m}^{α β} (u (t)) = \frac{1}{P_{m}} ⊙ \oplus \sum_{n \in I_{m}^{α β}} p_{n} \oplus \sum_{k = 1}^{\infty} a_{n k} (t) ⊙ u_{k} (t) .$

Moreover, if $a_{nk}, u_{k} \in ℝ_{F}^{+}$ for all $k, n \in ℕ$ , then (3.6) is defined level-wise by $\begin{matrix} [A_{m}^{α β} (u (t))]_{λ} \\ = & [(A_{m}^{α β})^{-} (λ), (A_{m}^{α β})^{+} (λ)] \\ = & [\frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} \sum_{k = 1}^{\infty} p_{n} a_{nk}^{-} u_{k}^{-}, \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} \sum_{k = 1}^{\infty} p_{n} a_{nk}^{+} u_{k}^{+}] . \end{matrix}$ (3.6)

Definition 5. Let A = (a _nk) be an infinite matrix of fuzzy numbers and (α, β) ∈ Λ. The matrix A is said to be weighted regular if Au ∈ ω ^αβ (F) for all u = (u _k) ∈ c (F) with ω ^αβ (F)- lim Au = lim u and denoted by A ∈ (c (F) : ω ^αβ (F)). This means that, $A_{m}^{α β} (u)$ exists for every $m \in ℕ$ and u ∈ c (F) and $A_{m}^{α β} (u) \to ℓ$ as m → ∞ whenever u _k → ℓ as k → ∞.

As a new characterization of weighted non-negative regular fuzzy matrix, we give the following theorem.

Theorem 2. Let A = (a _nk) be an infinite fuzzy matrix and (α, β) ∈ Λ. Then, A ∈ (c (F) : ω ^αβ (F)) if and only if $\begin{matrix} sup_{m} \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} p_{n} \sum_{k = 1}^{\infty} D (a_{nk}, \bar{0}) < \infty, \end{matrix}$ (3.7) $\begin{matrix} lim_{m \to \infty} \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} p_{n} D (a_{nk}, \bar{0}) = 0, \end{matrix}$ (3.8) $\lim_{m \to \infty} \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} p_{n} D (\oplus \sum_{k = 1}^{\infty} a_{n k}, \bar{0}) = 1.$ (3.9)

Proof. Asume that (3.7), (3.8) and ((3.9)) hold, and u _k→ ℓ as k→ ∞. Since $u_{k}^{\pm} (λ)$ are uniformly bounded in λ ∈ [0, 1], then $| u_{k}^{\pm} (λ) | \leq M^{\pm} (λ)$ holds for all $k \in ℕ$ . Further, since Au exists the series $\frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} \sum_{k = 1}^{\infty} p_{n} a_{nk}^{+} (λ) \to ℓ^{+} (λ)$ (3.10) uniformly in λ′s as m→ ∞. Hence A _n belongs to the β-dual of c (F) for each $n \in ℕ$ (see [33]). It is conclude that (3.7) holds if and only if $sup_{m} \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} p_{n} \sum_{k = 1}^{\infty} sup_{λ \in [0, 1]} | a_{nk}^{+} (λ) | < \infty .$ Again, (3.8) holds if and only if $\begin{matrix} lim_{m \to \infty} \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} p_{n} sup_{λ \in [0, 1]} | a_{nk}^{+} (λ) | = 0, for each k \in ℕ, \end{matrix}$ and (3.9) holds if and only if $\begin{matrix} lim_{m} \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} p_{n} sup_{λ \in [0, 1]} | \sum_{k = 1}^{\infty} a_{nk}^{+} (λ) | = 1 . \end{matrix}$

Thus, the matrices $A_{λ}^{+} = a_{nk}^{+} (λ)$ are regular for all λ ∈ [0, 1]. Since u _k→ ℓ as k→ ∞ then $u_{k}^{+} (λ) \to ℓ^{+} (λ)$ , uniformly in λ′s. Now, we will show that Au ∈ ω ^αβ (F). Taking into account the hypothesis one can observe that $\begin{matrix} | \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} \sum_{k} p_{n} a_{nk}^{+} (λ) u_{k}^{+} (λ) - ℓ^{+} (λ) | \\ = | \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} \sum_{k} p_{n} a_{nk}^{+} (λ) [u_{k}^{+} (λ) - ℓ^{+} (λ)] \\ + ℓ^{+} (λ) (\frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} \sum_{k} p_{n} a_{nk}^{+} (λ) - 1) | \\ \leq \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} \sum_{k} p_{n} a_{nk}^{+} (λ) | u_{k}^{+} (λ) - ℓ^{+} (λ) | \\ + \frac{1}{P_{m}} | ℓ^{+} (λ) | \sum_{n \in I_{m}^{α β}} p_{n} | \sum_{k} a_{nk}^{+} (λ) - 1 | \\ \leq \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} p_{n} \sum_{k = 1}^{s} sup_{λ \in [0, 1]} | a_{nk}^{+} (λ) | | u_{k}^{+} (λ) - ℓ^{+} (λ) | \\ + \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} p_{n} \sum_{k = s + 1}^{\infty} sup_{λ \in [0, 1]} | a_{nk}^{+} | | u_{k}^{+} - ℓ^{+} | \\ + \frac{1}{P_{m}} sup_{λ \in [0, 1]} | ℓ^{+} (λ) | \sum_{n \in I_{m}^{α β}} p_{n} sup_{λ \in [0, 1]} | \sum_{k} a_{nk}^{+} (λ) - 1 | \\ \leq \frac{s M^{\pm} (λ)}{P_{m}} \sum_{k = 1}^{s} \sum_{n \in I_{m}^{α β}} p_{n} sup_{λ \in [0, 1]} | a_{nk}^{+} (λ) | \\ + \frac{ε∀}{P_{m}} \sum_{n \in I_{m}^{α β}} p_{n} sup_{λ \in [0, 1]} \sum_{k = s + 1}^{\infty} | a_{nk}^{+} (λ) | \to 0 \end{matrix}$ as m→ ∞, uniformly in λ ∈ [0, 1]. Therefore, Au ∈ ω ^αβ (F).

Conversely, let A = (a _nk) ∈ (c (F) : ω ^αβ (F)) and u = (u _k) ∈ c (F). Then, since Au exists and the inclusion (c (F) : ω ^αβ (F)) ⊂ (c (F) : ℓ _∞ (F)) holds, the necessity of (3.7) is trivial. Now, consider the sequence $u^{(n)} = {u_{k}^{(n)}} \in c_{0} (F)$ defined by $\begin{matrix} u_{k}^{(n)} : = {\begin{matrix} \bar{1} & , & (n = k), \\ \bar{0} & , & (n \neq k), \end{matrix} (3.10) \end{matrix}$ for all $n \in ℕ$ and $v = (v_{k}) = (\bar{1})$ in the set c (F). Then, since Au ⁽ⁿ⁾ and Av ∈ ω ^αβ (F), the necessities of (3.8) and (3.9) are obvious.

4 Statistical weighted A-summability of fuzzy mappings

In this section, we give the definitions of weighted A-statistical convergence and statistically weighted A-summability of fuzzy mappings.

Let A = (a _nk) be a nonnegative weighted regular fuzzy matrix, (α, β) ∈ Λ and $K \subset ℕ$ . Then, the fuzzy weighted A-density of the set K is defined by $δ_{A}^{F} (K) = \lim_{m \to \infty} \frac{1}{P_{m}} ⊙ \oplus \sum n \in I_{m}^{α β} \oplus \sum_{k \in K} p_{n} ⊙ a_{n k} (t) n = [\lim_{m \to \infty} \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} \sum_{k \in K} p_{n} a_{n k}^{-} (λ), \lim_{m \to \infty} \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} \sum_{k \in K} p_{n} a_{n k}^{+} (λ)]$ (4.1) provided that the limits (4.1) exist uniformly in λ ∈ [0, 1].

Definition 6. Let A = (a _nk) be a non-negative weighted regular fuzzy matrix and (α, β) ∈ Λ. A sequence u = (u _k) of fuzzy numbers is said to be weighted A-statistically convergent, on a set H, to the fuzzy number ℓ, if, for every ε∀ > 0 $lim_{m \to \infty} \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} p_{n} \sum_{k : D (u_{k} (t), ℓ (t)) \geq ε∀, t \in H} a_{nk}^{\pm} (λ) = 0$ uniformly in λ ∈ [0, 1]. We denote it by $S_{A}^{α β} (F) - lim_{m} u = ℓ .$

Definition 7. Let A = (a _nk) be a non-negative weighted regular fuzzy matrix and (α, β) ∈ Λ. We say that a sequence u = (u _k) is statistically weighted A-summable to the fuzzy number ℓ, if δ (E _ε∀) =0, where $\begin{matrix} E_{ε∀} = {m \in ℕ : sup_{λ \in [0, 1]} max {| (A_{m}^{α β})^{-} (λ) - ℓ^{-} (λ) | \\ , | (A_{m}^{α β})^{+} (λ) - ℓ^{+} (λ) |} \geq ε∀} \end{matrix}$ where $(A_{m}^{α β})^{\pm} (λ) = \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} \sum_{k = 1}^{\infty} p_{n} a_{nk}^{\pm} (λ) u_{k}^{\pm} (λ) .$ Equivalently, $\begin{matrix} lim_{m \to \infty} \frac{1}{m} | E_{ε∀} | = 0 . \end{matrix}$

In this case, we write it as $ω_{A}^{α β} (F) (stat) - lim u_{k} = ℓ .$ Clearly, u = (u _k) is statistically weighted A-summable to ℓ if and only if st- $lim_{m} (A_{m}^{α β})^{\pm} (λ) = ℓ^{\pm} (λ)$ , uniformly in λ ∈ [0, 1].

Now, we prove the following theorem determining the relation between weighted A-statistical convergence and statistically weighted A-summability of fuzzy mappings.

Theorem 3. Let A = (a _nk) be a non-negative weighted regular fuzzy matrix and (α, β) ∈ Λ. If u = (u _k) is weighted A-statistically convergent to the fuzzy number ℓ, then it is statistically weighted A-summable to the same fuzzy limit ℓ but not conversely.

Proof. Let us set $K (ε∀) : = {k : D (u_{k} (t), ℓ (t)) \geq ε∀, t \in H}$ and $K^{C} (ε∀) : = {k : D (u_{k} (t), ℓ (t)) < ε∀, t \in H} .$

It follows from the weighted regularity of the matrix A that $\begin{matrix} | \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} \sum_{k = 1}^{\infty} p_{n} a_{nk}^{\pm} (λ) u_{k}^{\pm} (λ) - ℓ^{\pm} (λ) | \\ \leq & | \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} \sum_{k = 1}^{\infty} p_{n} a_{nk}^{\pm} (λ) [u_{k}^{\pm} (λ) - ℓ^{\pm} (λ)] | \\ + & | ℓ^{\pm} (λ) | | \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} \sum_{k = 1}^{\infty} p_{n} a_{nk}^{\pm} (λ) - 1 | \\ \leq & \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} p_{n} \sum_{k \in K (ε∀)} a_{nk}^{\pm} (λ) | u_{k}^{\pm} (λ) - ℓ^{\pm} (λ) | \\ + & \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} p_{n} \sum_{k \in K^{C} (ε∀)} a_{nk}^{\pm} (λ) | u_{k}^{\pm} (λ) - ℓ^{\pm} (λ) | \\ + & | ℓ^{\pm} (λ) | \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} p_{n} sup_{λ \in [0, 1]} | \sum_{k = 1}^{\infty} a_{nk}^{\pm} (λ) - 1 | . \end{matrix}$

Thus we find that $\begin{matrix} | \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} \sum_{k = 1}^{\infty} p_{n} a_{nk}^{\pm} (λ) u_{k}^{\pm} (λ) - ℓ^{\pm} (λ) | \\ \leq & M^{\pm} (λ) \frac{| K (ε∀) |}{P_{m}} \sum_{n \in I_{m}^{α β}} p_{n} \sum_{k \in K (ε∀)} sup_{λ \in [0, 1]} a_{nk}^{\pm} (λ) \\ + & ε∀ \frac{| K^{C} (ε∀) |}{P_{m}} \sum_{n \in I_{m}^{α β}} p_{n} \sum_{k \in K^{C} (ε∀)} sup_{λ \in [0, 1]} a_{nk}^{\pm} (λ) . \end{matrix}$

Since $S_{A}^{α β} (F) - lim u = ℓ$ , then $lim_{m} | \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} \sum_{k = 1}^{\infty} p_{n} a_{nk}^{\pm} (λ) u_{k}^{\pm} (λ) - ℓ^{\pm} (λ) | = 0,$ uniformly in λ, which implies that u is weighted A-summable to ℓ and hence it is statistically weighted A-summable to the same fuzzy limit ℓ.□

Now, we give the following example to show that the converse of this theorem is not true.

Example 1. Define the matrix A = (a _nk) of fuzzy numbers by $a_{nk} : = {\begin{matrix} v_{k} & , (k = n), \\ \bar{0} & , (k \neq n), \end{matrix}$ with $v_{k} (t) : = {\begin{matrix} k (k + 1) (t - 1) - 1 & , & (1 + \frac{1}{k (k + 1)} \leq t \leq 1 + \frac{2}{k (k + 1)}) \\ 3 - k (k + 1) (t - 1) & , & (1 + \frac{2}{k (k + 1)} \leq t \leq 1 + \frac{3}{k (k + 1)}) \\ 0 & , & (otherwise) \end{matrix}$ for all $k \in ℕ$ . It is clear that $a_{nk}^{-} (λ) = 1 + \frac{λ + 1}{k (k + 1)}$ and $a_{nk}^{+} (λ) = 1 + \frac{3 - λ}{k (k + 1)}$ .

Let α (m) =1, β (m) = m and p _n = 1 for all n ∈ [1, m], i. e. P _m = m. Then, since $D (a_{nk}, \bar{0}) = {\begin{matrix} 1 + \frac{3}{k (k + 1)} & , & (k = n) \\ 0 & , & (k \neq n) \end{matrix},$ under above conditions, we have $sup_{m} \frac{1}{m} \sum_{n = 1}^{m} \sum_{k = 1}^{\infty} D (a_{nk}, \bar{0}) < \infty,$ which implies that (3.7) holds. Besides, for each $k \in ℕ$ , $\begin{matrix} lim_{m \to \infty} \frac{1}{m} \sum_{n = 1}^{m} D (a_{nk}, \bar{0}) = 0, \end{matrix}$ which shows that ((3.8)) holds. By using similar arguments we also obtain that $\begin{matrix} lim_{m} \frac{1}{m} \sum_{n = 1}^{m} sup_{λ \in [0, 1]} | \sum_{k = 1}^{\infty} a_{nk}^{+} (λ) - 1 | = 0, \end{matrix}$ which implies ((3.9)) holds. Hence, A = (a _nk) is weighted regular fuzzy matrix since it satisfies all conditions of Theorem 3.

Now, define the sequences $u_{k} : [0, 1] \to ℝ_{F}$ , whose λ-cuts $u_{k}^{\pm} (λ) : [0, 1] \to ℝ$ are given by

$\begin{matrix} u_{k}^{\pm} (λ) = {\begin{matrix} 1, & (k = N^{3} - N^{2}, N^{3} - N^{2} + 1, \dots, N^{3} - 1) \\ - N^{2}, & (k = N^{3}; N = 2, 3, 4, \dots) \\ 0 & (otherwise) . \end{matrix} \end{matrix}$ Then

$\begin{matrix} | \frac{1}{m} \sum_{n = 1}^{m} \sum_{k = 1}^{n} a_{nk}^{+} (λ) u_{k}^{+} (λ) - 1 | \\ = & {\begin{matrix} \frac{s (3 - λ)}{m (m + 1)} & (m = N^{3} - N^{2} + s; s = 0, 1, 2, \dots, N^{2} - 1) \\ 0 & (otherwise) . \end{matrix} \end{matrix}$

Thus, $lim_{m} | \frac{1}{m} \sum_{n = 1}^{m} \sum_{k = 1}^{n} a_{nk}^{+} (λ) u_{k}^{+} (λ) - 1 | = 0, uniformly in λ,$ which yields (u _k) is weighted A-summable to $ℓ = \bar{1}$ and hence it is statistically weighted A-summable to the same fuzzy limit. On the other hand, since $\begin{matrix} lim_{m \to \infty} \frac{1}{m} \sum_{n = 1}^{m} \sum_{k \in K^{'} (ε∀)} a_{nk}^{+} (λ) \neq 0, \end{matrix}$ where $K^{'} (ε∀) : = {k : D (u_{k}, \bar{1}) \geq ε∀}$ , then, the sequence (u _k) is not weighted A-statistically convergent to the fuzzy number $ℓ = \bar{1}$ .

This completes the proof of Theorem 3.

5 Application to Korovkin type approximation theorem

It is known that the theory of approximation by positive linear operators is an important research field in mathematics. One of the most interesting results on uniform approximation by polynomials were introduced by Korovkin [35]. Based on a family of continuous functions (also called test functions) Korovkin established the necessary and sufficient conditions for the uniform convergence of a sequence of positive linear operators acting on C [a, b]. In the last few years many research papers investigated on the approximations of membership functions of fuzzy numbers, with the aim to obtain strong application background in various areas of fuzzy analysis. For more details on fuzzy Korovkin type approximation results, we refer to [36 –38] and the references therein.

In this section we shall obtain a fuzzy Korovkin type approximation theorem via statistically weighted A-summability. We also present an illustrative example using a fuzzy analogue of Meyer-König and Zeller operators to show that our proposed approximation methods are more powerfull than classical and statistical versions of fuzzy Korovkin type results.

Let $f : [a, b] \to ℝ_{F}$ be a fuzzy number valued function. We say that f is fuzzy continuous at some point x ₀ of its domain if and only if whenever u _n → u ₀, then D (f (u _n), f (u ₀)) →0, as n→ ∞. A fuzzy number valued function is said to be fuzzy continuous at every point x ∈ [a, b], if it is fuzzy continuous on [a, b]. The set of all fuzzy-valued continuous functions and real valued continuous functions on [a, b] will be denoted by $C_{F} [a, b]$ and C [a, b], respectively. As usual, C [a, b] is equipped with the supremum norm $f_{C [a, b]} = sup_{x \in [a, b]} | f (x) | .$ It is pointed out here that $C_{F} [a, b]$ is only a cone, not a vector space.

Given two fuzzy number valued functions $F, H : [a, b] \to ℝ_{F}$ such that $F_{λ} (t) = [f_{λ}^{-} (t), f_{λ}^{+} (t)]$ and $H_{λ} (t) = [h_{λ}^{-} (t), h_{λ}^{+} (t)]$ . Then, the fuzzy distance $D^{*} : ℝ_{F} \times ℝ_{F} \to [0, + \infty]$ is defined by (see [38]) $\begin{matrix} D^{*} (F, H) = sup_{t \in [a, b]} sup_{λ} max {| f_{λ}^{-} (t) - h_{λ}^{-} (t) |, \\ | f_{λ}^{+} (t) - h_{λ}^{+} (t) |} . \end{matrix}$

Now, let $L : C_{F} [a, b] \to C_{F} [a, b]$ is an operator. Then, L is called a fuzzy linear operator, if $\begin{matrix} L (c_{1} ⊙ f (t) \oplus c_{2} ⊙ g (t); x) = c_{1} ⊙ \\ L (f (t); x) \oplus c_{2} ⊙ L (g (t); x) \end{matrix}$ holds for every $c_{1}, c_{2} \in ℝ$ , $f, g \in C_{F} [a, b]$ and x, t ∈ [a, b]. We say that L is a fuzzy positive linear operator if it is fuzzy linear and, the condition L (f (t); x) ≥ L (g (t); x) holds for any $f, g \in C_{F} [a, b]$ and all x ∈ [a, b] with f (t) ≥ g (t).

Throughout the paper, we consider the test functions e _i given by $e_{i} (x) = (\frac{x}{1 - x})^{i}$ , i = 0, 1, 2 for x ∈ I = [0, B] where B ≤ 1/2.

Theorem 4. [38] Let {T _k} _k≥1 be a sequence of fuzzy positive linear operators acting from $C_{F} [a, b]$ into itself. Assume that there exists a corresponding sequence ${{\tilde{T}}_{k}}_{k \geq 1}$ of positive linear operators from C [a, b] into itself with the level-wise property $\begin{matrix} {[T_{k} (f; x)]}_{λ}^{\pm} = {\tilde{T}}_{k} (f_{λ}^{\pm}; x) (5.1) \end{matrix}$ for all λ ∈ [0, 1], ∀x, t ∈ [a, b], $k \in ℕ$ and $f \in C_{F} [a, b]$ . Assume further that $\begin{matrix} lim_{k \to \infty} ∥ {\tilde{T}}_{k} ({\tilde{e}}_{i}; x) - {\tilde{e}}_{i} ∥_{C [a, b]} = 0, \end{matrix}$ where ${\tilde{e}}_{i} (x) = x^{i}$ , i = 0, 1, 2. Then, for all $f \in C_{F} [a, b]$ , we have $lim_{k \to \infty} D^{*} (T_{k} (f (t); x), f (x)) = 0 .$

Now, we first give the following analogue of fuzzy Korovkin theorem by using statistically weighted A-summability of fuzzy mappings.

Theorem 5. Let A = (a _nk) be a non-negative weighted regular matrix of fuzzy numbers, (α, β) ∈ Λ and let {T _m} be a sequence of fuzzy positive linear operators acting from $C_{F} (I)$ into itself. Suppose that there exists a corresponding sequence ${{\tilde{T}}_{m}}_{m \in ℕ}$ of positive linear operators from C (I) into itself with the property (1). Assume further that $\begin{matrix} ω_{A}^{α β} (F) (stat) - lim_{m} ∥ {\tilde{T}}_{m} (e_{i}; x) - e_{i} ∥_{C (I)} = 0, \end{matrix}$ (5.2) for i = 0, 1, 2. Then, for all $f \in C_{F} (I)$ , we get $\begin{matrix} ω_{A}^{α β} (F) (stat) - lim_{m} D^{*} (T_{m} (f (t); x), f (x)) = 0 . (5.3) \end{matrix}$ (5.3)

Proof. Suppose that the conditions (5.2) hold for each of the functions $1, \frac{x}{1 - x}, (\frac{x}{1 - x})^{2}$ belongs to C (I). Let $f \in C_{F} (I)$ and x ∈ I be fixed. Since $f_{λ}^{\pm} \in C (I)$ by the hypothesis, for a given ε∀ > 0, there exists a number δ = δ (ε∀) >0 such that $\begin{matrix} | f_{λ}^{\pm} (t) - f_{λ}^{\pm} (x) | < ε∀ \end{matrix}$ 5.4 whenever $| \frac{t}{1 - t} - \frac{x}{1 - x} | < δ$ . Then we immediately get, for all t ∈ I satisfying $| \frac{t}{1 - t} - \frac{x}{1 - x} | \geq δ$ that $\begin{matrix} | f_{λ}^{\pm} (t) - f_{λ}^{\pm} (x) | < ɛ + \frac{2 M_{λ}^{\pm}}{δ^{2}} φ (t, x) \end{matrix}$ where $φ (t, x) = (\frac{t}{1 - t} - \frac{x}{1 - x})^{2}$ and $M_{λ}^{\pm} : = ∥ f_{λ}^{\pm} ∥_{C (I)}$ . It follows from the fuzzy linearity and positivity of the operators ${\tilde{T}}_{m}$ , for each $m \in ℕ$ , that: $\begin{matrix} | {\tilde{T}}_{m} (f_{λ}^{\pm} (t); x) - f_{λ}^{\pm} (x) | \\ = & | {\tilde{T}}_{m} (f_{λ}^{\pm} (t) - f_{λ}^{\pm} (x); x) + f_{λ}^{\pm} (x) ({\tilde{T}}_{m} (e_{0}; x) - e_{0}) | \\ \leq & {\tilde{T}}_{m} (| f_{λ}^{\pm} (t) - f_{λ}^{\pm} (x) |; x) + M_{λ}^{\pm} | {\tilde{T}}_{m} (e_{0}; x) - e_{0} | \\ \leq & | {\tilde{T}}_{m} (ɛ + \frac{2 M_{λ}^{\pm}}{δ^{2}} φ (t, x); x) | + M_{λ}^{\pm} | {\tilde{T}}_{m} (e_{0}; x) - e_{0} | \\ \leq & ɛ + (ɛ + M_{λ}^{\pm} + \frac{2 M_{λ}^{\pm}}{δ^{2}}) | {\tilde{T}}_{m} (e_{0}; x) - e_{0} (x) | \\ + \frac{4 M_{λ}^{\pm}}{δ^{2}} | {\tilde{T}}_{m} (e_{1}; x) - e_{1} (x) | + \frac{2 M_{λ}^{\pm}}{δ^{2}} | {\tilde{T}}_{m} (e_{2}; x) - e_{2} (x) | . \end{matrix}$

Then taking supremum over x ∈ I in the last inequality that $\begin{matrix} ∥ {\tilde{T}}_{m} (f_{λ}^{\pm}; x) - f_{λ}^{\pm} ∥_{C (I)} \leq ɛ + N_{λ}^{\pm} (ɛ) \\ \sum_{i = 0}^{2} ∥ {\tilde{T}}_{m} (e_{i}; x) - e_{i} ∥_{C (I)} \end{matrix}$ where $N_{λ}^{\pm} (ɛ) : = {ɛ + M_{λ}^{\pm} + \frac{4 M_{λ}^{\pm}}{δ^{2}}}$ . Now replacing ${\tilde{T}}_{m} (\cdot; x)$ by $\begin{matrix} A_{m}^{α β} (\cdot; x) = \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} p_{n} \sum_{k = 1}^{\infty} a_{nk}^{\pm} (λ) {\tilde{T}}_{m} (\cdot; x), \\ λ \in [0, 1] . \end{matrix}$

We also write $\begin{matrix} ∥ A_{m}^{α β} (f_{λ}^{\pm}) - f_{λ}^{\pm} ∥_{C (I)} \leq ɛ + N_{λ}^{\pm} (ɛ) \\ \sum_{i = 0}^{2} ∥ A_{m}^{α β} (e_{i}) - e_{i} ∥_{C (I)} \end{matrix}$ (5.5)

Besides, it follows from (5.1) that $\begin{matrix} D^{*} (T_{m} (f (t); x), f (x)) \\ = & sup_{x \in I} D (T_{m} (f (t)), f (x)) \\ = & sup_{λ \in [0, 1]} max {∥ {\tilde{T}}_{m} (f_{λ}^{-}; x) - f_{λ}^{-} ∥_{C (I)}, \\ ∥ {\tilde{T}}_{m} (f_{λ}^{+}; x) - f_{λ}^{+} ∥_{C (I)}} . \end{matrix}$

Combining the last equality with (5.5), we have $\begin{matrix} D^{*} (T_{m} (f (t); x), f (x)) \leq ɛ + N (ɛ) \\ \sum_{i = 0}^{2} ∥ A_{m}^{α β} (e_{i}; x) - e_{i} ∥_{C (I)} \end{matrix}$ where $N (ɛ) : = D (N (ɛ), \bar{0}) = sup_{λ \in [0, 1]} max {| N_{λ}^{-} (ɛ) |, | N_{λ}^{+} (ɛ) |} .$

For a given ɛ′ > 0, choose a number ɛ > 0 such that ɛ < ɛ′. Then, setting

$\begin{matrix} D^{*} (T_{m} (f (t); x), f (x)) \geq ɛ^{'}}, \\ x) - e_{i} ∥_{C (I)} \geq \frac{ɛ^{'} - ɛ}{3 N (ɛ)}}, i = 0, 1, 2 . \end{matrix}$

Then $H \subset H_{0} \cup H_{1} \cup H_{2}$ which guarantees that, for each $m \in ℕ$ , $\begin{matrix} \in I_{m}^{α β} \end{matrix} \sum_{k \in H} p_{n} a_{nk}^{\pm} (λ) \leq \frac{1}{P_{m}} \sum_{n \in I_{m}^{α β}} p_{n} \sum_{i = 0}^{2} \sum_{k \in H_{i}} a_{nk}^{\pm} (λ) .$ Therefore, we immediately conclude that $ω_{A}^{α β} (F) (stat) - lim_{m \to \infty} D^{*} (T_{m} (f (t); x), f (x)) = 0,$ whence the result.□ Now we present an example using a fuzzy analogue of Meyer-König and Zeller (MKZ) operators (cf. [39]). Firstly, we will construct fuzzy MKZ-operator on [0, B′] where B′ ∈ (0, 1). Let us take the following sequence of fuzzy Meyer-König and Zeller operators: $\begin{matrix} M_{n}^{F} (f; x) = \oplus \sum_{k = 0}^{\infty} f (\frac{k}{k + n + 1}) ⊙ b_{k}^{n} (x) \end{matrix}$ where $b_{k}^{n} (x) = n + k k x^{k} (1 - x)^{n + 1}$ , $f \in C_{F} [0, B^{'}]$ , x ∈ [0, B′] and $n \in ℕ$ . This operator is called fuzzy MKZ operator and we write $\begin{matrix} {[M_{n}^{F} (f; x)]}_{λ}^{\pm} = {\tilde{M}}_{n} (f_{λ}^{\pm}; x) = \\ \sum_{k = 0}^{\infty} b_{k}^{n} (x) f_{λ}^{\pm} (\frac{k}{k + n + 1}) \end{matrix}$ where $f_{λ}^{\pm} \in C [0, B^{'}]$ . Observe easily that (see [40]) $\begin{matrix} x) & = & 1; \\ x) & = & \frac{x}{1 - x}; \\ x) & = & \frac{n + 2}{n + 1} (\frac{x}{1 - x})^{2} + \frac{1}{n + 1} (\frac{x}{1 - x}) . \end{matrix}$ Considering the sequence {T _n} of positive fuzzy linear operators given by $\begin{matrix} T_{n} (f; x) = u_{n} ⊙ M_{n}^{F} (f; x) \end{matrix}$ where u = {u _n} is defined as in Example 1. Since $ω_{A}^{α β} (F) (stat) - lim u_{n} = \bar{1}$ , we observe that $\begin{matrix} x) - e_{0} ∥_{C [0, B^{'}]} = 0, \\ x) - e_{1} ∥_{C [0, B^{'}]} = 0, \\ x) - e_{2} ∥_{C [0, B^{'}]} = 0 . \end{matrix}$ So, by Theorem 1, we obtain, for all $f \in C_{F} [0, B^{'}]$ , $ω_{A}^{α β} (F) (stat) - lim_{m} D^{*} (T_{m} (f (t); x), f (x)) = 0 .$

6 Rates of weighted A-statistical convergence of fuzzy mappings

In this section, we estimate the rate of weighted A-statistical convergence of fuzzy positive linear operator by means of the fuzzy modulus of continuity.

Definition 8. Let A = (a _nk) be a weighted non-negative regular fuzzy matrix, (α, β) ∈ Λ and let $(s_{m})_{m \in ℕ}$ be a positive non-increasing sequence of real numbers. A sequence u = (u _k) of fuzzy numbers is weighted A-statistically convergent to a fuzzy number ℓ with the rate of o (s _m), if, for every ε∀ > 0, $\begin{matrix} \infty \end{matrix} \frac{1}{s_{m} P_{m}} \sum_{n \in I_{m}^{α β}} p_{n} \sum_{k : D (u_{k} (t), ℓ (t)) \geq ε∀, t \in H} a_{nk}^{\pm} (λ) = 0$ uniformly in λ ∈ [0, 1]. In this case, we denote it by $D (u_{k}, ℓ) = S_{A}^{α β} (F) - lim o (s_{m})$ . Lemma 1. Let A = (a _nk) be a weighted non-negative regular matrix of fuzzy numbers, (α, β) ∈ Λ and let (a _m), (b _m) be two positive non-increasing sequences of real numbers. Let u = (u _k) and v = (v _k) be two sequences of fuzzy numbers such that $D (u_{k}, ℓ_{1}) = S_{A}^{α β} (F) - lim o (a_{m}) and D (u_{k}, ℓ_{2}) = S_{A}^{α β} (F) - o (b_{m}) .$ Then

$D (u_{k}, ℓ_{1}) + D (v_{k}, ℓ_{2}) = S_{A}^{α β} (F) - o (c_{m})$

$D (u_{k}, ℓ_{1}) D (v_{k}, ℓ_{2}) = S_{A}^{α β} (F) - o (a_{m} b_{m})$

$D (u_{k}, ℓ_{1}) ξ = S_{A}^{α β} (F) - o (a_{m})$ , for any scalar ξ,

where c _m = max {a _m, b _m}. Proof. Let

D (u_{k}, ℓ_{1}) = S_{A}^{α β} (F) - lim o (a_{m})

and

D (u_{k}, ℓ_{2}) = S_{A}^{α β} (F) - o (b_{m}) .

For a given ε∀ > 0, let us set

\begin{matrix} S : & = & {k : D (u_{k}, ℓ_{1}) + D (v_{k}, ℓ_{2}) \geq ε∀}, \\ S_{0} : & = & {k : D (u_{k}, ℓ_{1}) \geq \frac{ε∀}{2}}, \\ S_{1} : & = & {k : D (v_{k}, ℓ_{2}) \geq \frac{ε∀}{2}} . \end{matrix}

Then observe that

S \subset S_{0} \cup S_{1}

which gives for

m \in ℕ

\begin{matrix} \frac{1}{P_{m}} {\sum_{n \in I_{m}^{α β}} \sum_{k \in S} p_{n} a_{nk}^{\pm} (λ)} \leq \frac{1}{P_{m}} \\ \sum_{i = 0}^{1} {\sum_{n \in I_{m}^{α β}} \sum_{k \in S_{i}} p_{n} a_{nk}^{\pm} (λ)} . \end{matrix}

Thus

\begin{matrix} \frac{1}{c_{m} P_{m}} {\sum_{n \in I_{m}^{α β}} \sum_{k \in S} p_{n} a_{nk}^{\pm} (λ)} \leq \frac{1}{a_{m} P_{m}} \\ {\sum_{n \in I_{m}^{α β}} \sum_{k \in S_{0}} p_{n} a_{nk}^{\pm} (λ)} \\ + \frac{1}{b_{m} P_{m}} {\sum_{n \in I_{m}^{α β}} \sum_{k \in S_{1}} p_{n} a_{nk}^{\pm} (λ)} . \end{matrix}

Now letting m→ ∞ in the last inequality and using hypothesis, we have

\begin{matrix} lim_{m \to \infty} \frac{1}{c_{m} P_{m}} {\sum_{n \in I_{m}^{α β}} \sum_{k \in S} p_{n} a_{nk}^{\pm} (λ)} = 0, \\ uniformly in λ . \end{matrix}

This step completes the proof of the case (1). Since the proofs of other cases are similar, we omitted. Now, we recall that the following basic definition and notation on fuzzy modulus of continuity to get the rates of

S_{A}^{α β} (F)

-convergence by using Definition 8. The (first) fuzzy modulus of continuity for a function

f \in C_{F} (I)

is defined as follows:

ω_{1}^{F} (f, δ) : = sup_{| x - y | \leq δ} {D (f (x), f (y)) : x, y \in I},

for any δ > 0. We obtain, for all $f_{λ}^{\pm} \in C (I)$ , that $\begin{matrix} δ) (\frac{| x - y |}{δ} + 1) . (6.1) \end{matrix}$ Theorem 6. Let A = (a _nk) be a weighted non-negative regular matrix, (α, β) ∈ Λ and let {T _m} be a sequence of fuzzy positive linear operators from $C_{F} (I)$ into itself. Assume that there exists a corresponding sequence ${{\tilde{T}}_{m}}$ from C [a, b] into itself with the property (1). Assume further that (a _m) and (b _m) are positive non-increasing sequences of real numbers and the followings hold:

${\tilde{T}}_{m} (e_{0}; x) - e_{0} ∥_{C (I)} = S_{A}^{α β} (F) - o (a_{m})$ on I,

$ω_{1}^{F} (f, ξ_{m}) = S_{A}^{α β} (F) - o (b_{m})$ on I, where $ξ_{m} : = ∥ {\tilde{T}}_{m} (φ; x) ∥_{C (I)}^{1 / 2}$ with $φ (t) = (\frac{t}{1 - t} - \frac{x}{1 - x})^{2} .$

Then, for all

f \in C_{F} (I)

, we have

\begin{matrix} D^{*} (T_{m} (f (t); x), f (x)) = S_{A}^{α β} (F) - o (c_{m}), as m \to \infty, \end{matrix}

where c _m = max {a _m, b _m}.∥Proof. Let

f \in C_{F} (I)

and x, t ∈ I. Since

{\tilde{T}}_{k}

is positive linear and monotone, we obtain, for any δ > 0, that

\begin{matrix} T_{m} (f_{λ}^{\pm}; x) - f_{λ}^{\pm} | \\ δ) {\tilde{T}}_{m} (\frac{| \frac{t}{1 - t} - \frac{x}{1 - x} |}{δ} + 1; x) + N | {\tilde{T}}_{m} (e_{0}) - e_{0} | \\ T_{m} (\frac{(\frac{t}{1 - t} - \frac{x}{1 - x})^{2}}{δ^{2}} + 1; x) + N | {\tilde{T}}_{m} (e_{0}) - e_{0} | \\ δ) {\frac{1}{δ^{2}} {\tilde{T}}_{m} (φ^{2} (t)) + {\tilde{T}}_{m} (e_{0})} + N | {\tilde{T}}_{m} (e_{0}) - e_{0} | \end{matrix}

where

N = D^{*} (f, \bar{0})

. Now, putting

δ = ξ_{m} = ∥ {\tilde{T}}_{m} (φ; x) ∥_{C (I)}^{1 / 2}

in the last inequality, one obtains

\begin{matrix} D^{*} (T_{m} (f (t); x), f (x)) \\ \leq & ω_{1}^{F} (f, ξ_{m}) ∥ {\tilde{T}}_{m} (e_{0}; x) - e_{0} ∥_{C (I)} + 2 ω_{1}^{F} (f, ξ_{m}) \\ + & N ∥ {\tilde{T}}_{m} (e_{0}; x) - e_{0} ∥_{C (I)} . \end{matrix}

∥For a given ɛ > 0, consider the following sets:

\begin{matrix} D^{*} (T_{m} (f (t); x), f (x)) \geq ɛ}, \\ ξ_{m}) ∥ {\tilde{T}}_{m} (e_{0}; x) - e_{0} ∥_{C (I)} \geq \frac{ɛ}{3}}, \\ x) - e_{0} ∥_{C (I)} \geq \frac{ɛ}{3 N}}, \\ ξ_{m}) \geq \frac{ɛ}{6}} . \end{matrix}

Using the hypothesis (i)-(ii) and Lemma (3), we have

\begin{matrix} D^{*} (T_{m} (f (t); x), f (x)) = S_{A}^{α β} (F) - o (c_{m}), as m \to \infty . \end{matrix}

□

7 Conclusion

In this work, we have extended the concepts of A-statistical convergence and statistical A-summability of real sequences to the fuzzy set theory. These results in the interval context are presented here for the first time as the fuzzy results are obtained via level sets of fuzzy numbers. In our results we have used various summability techniques involving fuzzy sequences and showed how these methods lead to a number of fuzzy approximation results. We have obtained the necessary and sufficient conditions for the matrix A to be weighted fuzzy regular and derived some inclusion relations concerning these newly proposed methods. Furthermore we proved a fuzzy Korovkin type approximation theorem using statistically weighted A-summability and estimated the rates of weighted A-statistical convergence by means of the fuzzy modulus of continuity supported by an illustrative example to show that our proposed methods are stronger than the existing ones.

References

Fast , Sur la convergence statistique, Colloq Math 2 (1951), 241–244.

Steinhaus , Sur la convergence ordinaire et la convergence asymptotique, Colloq Math 2 (1951), 73–74.

Salat , On statistically convergent sequences of real numbers, Math Slovaca 30(2) (1980), 139–150.

J.A.

Fridy and

Orhan , Lacunary statistical convergence, Pacific J Math 160 (1993), 43–51.

Mursaleen ,

Karakaya ,

Ertürk and

Gürsoy , Weighted statistical convergence and its application to Korovkin type approximation theorem, Appl Math Comput 218 (2012), 9132–9137.

Connor , On strong matrix summability with respect to a modulus and statistical convergence, Canad Math Bull 32 (1989), 194–198.

Kadak , Weighted statistical convergence based on generalized difference operator involving (p, q)-gamma function and its applications to approximation theorems, J Math Anal Appl 448 (2017), 1633–1650.

Kadak , On relative weighted summability in modular function spaces and associated approximation theorems, Positivity 21(4) (2017), 1593–1614.

Kadak , Generalized weighted invariant mean based on fractional difference operator with applications to approximation theorems for functions of two variables, Results in Mathematics 72(3) (2017), 1181–1202.

10.

Karakaya and

T.A.

Chishti , Weighted statistical convergence, Iran J Sci Technol, Trans A, Sci 33 (2009), 219–223.

11.

H.M.

Srivastava ,

Mursaleen and

Khan , Generalized equi-statistical convergence of positive linear operators and associated approximation theorems, Math Comput Model 55 (2012), 2041–2051.

12.

Kadak ,

N.L.

Braha and

H.M.

Srivastava , Statistical weighted B-summability and its applications to approximation theorems, Appl Math Comput 302 (2017), 80–96.

13.

Belen and

S.A.

Mohiuddine , Generalized weighted statistical convergence and application, Appl Math Comput 219 (2013), 9821–9826.

14.

Mursaleen , θ-Statistical convergence, Math Slovaca 50 (2000), 111–115.

15.

Ghosal , Generalized weighted random convergence in probability, Appl Math Comput 249 (2014), 502–509.

16.

Aktuglu , Korovkin type approximation theorems proved via αβ-statistical convergence, J Comput Appl Math 259 (2014), 174–181.

17.

Kolk , Matrix summability of statistically convergent sequences, Analysis 13 (1993), 77–83.

18.

O.H.H.

Edely and

Mursaleen , On statistical asummabil-ity, Math Comput Model 49 (2009), 672–680.

19.

Duman and

Orhan , Rates of A-statistical convergence of operators in the space of locally integrable functions, Appl Math Lett 21(5) (2008), 431–435.

20.

M.K.

Khan and

Orhan , Matrix characterization of Asta-tistical convergence, J Math Anal Appl 335(1) (2007), 406–417.

21.

Demirci , Strong A-summability and A-statistical convergence, Indian J Pure Appl Math 27(6) (1996), 589–593.

22.

S.A.

Mohiuddine , Statistical weighted A-summability with application to Korovkin's type approximation theorem, J Inequal Appl 2016 (2016). Article ID 101.

23.

L.A.

Zadeh , Fuzzy sets, Inform and Control 8 (1965), 338–353.

24.

Nuray , Lacunary statistical convergence of sequences of fuzzy numbers, Fuzzy Sets Syst 99(3) (1998), 353–356.

25.

Savaş , On strongly θ-summable sequences of fuzzy numbers, Inform Sci 125 (2000), 181–186.

26.

Aytar ,

M.A.

Mammadov and

Pehlivan , Statistical limit inferior and limit superior for sequences of fuzzy numbers, Fuzzy Sets Syst 157(1) (2006), 976–985.

27.

Altin ,

Mursaleen and

Altinok , Statistical summa-bility (C, 1) for sequences of fuzzy real numbers and a Tauberian theorem, J Intell Fuzzy Syst 21(6) (2010), 379–384.

28.

Ö.

Talo , Abel summability of sequences of fuzzy numbers, Soft Computing 20(3) (2016), 1041–1046.

29.

Canak , Holder summability method of fuzzy numbers and atauberian theorem, Iran J Fuzzy Syst 11(4) (2014), 87–93.

30.

S.A.

Sezer and

Canak , Power series methods of summa-bility for series of fuzzy numbers and related Tauberian theorems, Soft Computing 21(4s) (2017), 1057–1064.

31.

M.L.

Puri and

D.A.

Ralescu , Differentials for fuzzy functions, J Math Anal Appl 91 (1983), 552–558.

32.

Y.K.

Kim and

B.M.

Ghil , Integrals of fuzzy-number-valued functions, Fuzzy Sets Syst 86 (1997), 213–222.

33.

Ö.

Talo and

Başar , Determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transformations, Comput Math Appl 58 (2009), 717–733.

34.

Altinok ,

Altin and

Işik , Statistical convergence and strong p-Cesaro summability of order β in sequences of fuzzy numbers, Iran J Fuzzy Syst 9(2) (2012), 63–73.

35.

P.P.

Korovkin , Linear Operators and Approximation Theory, Hindustan Publishing Corporation, Delhi, 1960.

36.

Karaisa and

Kadak , On αβ-statistical convergence for sequences of fuzzy mappings and Korovkin type approximation theorem, Filomat 31(12) (2017), 3749–3760.

37.

G.A.

Anastassiou , Fuzzy Mathematics, Approximation Theory (Studies in Fuzziness and Soft Computing), Springer, 2010.

38.

G.A.

Anastassiou and

Duman , Statistical fuzzy approximation by fuzzy positive linear operators, Comput Math Appl 55(3) (2008), 573–580.

39.

Meyer-König and

Zeller , Bernsteinsche potenzreihen, Studia Math 19 (1960), 89–94.

40.

Altin ,

Doğru and

Taşdelen , The generalization of Meyer-Konig and Zeller operators by generating functions, J Math Anal Appl 312 (2005), 181–194.