On statistical convergence of double sequences of fuzzy valued functions

Abstract

We define the notions of pointwise and uniform statistical convergence of double sequences of fuzzy valued functions and obtain relationships between these two kinds of convergence. We further introduce the notion of equi-statistical convergence of double sequences of fuzzy valued functions and show that uniformly statistically convergent double sequence is equi-statistically convergent while the converse is not true in general. We present several interesting results related to these kinds of convergence and their representations of sequences of α-level cuts. We provide some interesting illustrative examples in support of our results.

Keywords

Fuzzy valued function double sequence statistical convergence equi-statistical convergence

1 Introduction and preliminaries

In 1965, Zadeh [35] for the first time introduced the term “Fuzzy sets” for the class of objects with a continuum of grades of membership or in other words degrees of membership. He used this notion for the addition of inclusion, intersection, union, convexity, complement, relation, etc., in the classical notion sets and established the notion Fuzzy Sets. From then onward, fuzzy logic and fuzzy sets are successfully applied by the researchers of the various fields such as Decision theory, Control Engineering and also explored in Artificial Intelligence. The array of its application is continuously expanding since then. In the last 50 years, the sequences of fuzzy numbers and its convergence properties have been nicely demonstrated by many researchers.

Since 1951 when Fast [12] and Steinhaus [33] defined statistical convergence for sequences of real numbers, several generalizations and applications of this notion have been investigated. However, the idea of statistical convergence was first appeared, under the name of almost convergence, in the first edition of the celebrated monograph of Zygmund [38]. Schoenberg [32] and Šalát [29] gave some basic properties of statistical convergence. The sequence x = (x_n) is statistically convergent to ℓ if $lim_{n} n^{- 1} | {k \leq n : | x_{k} - ℓ | \geq ɛ} | = 0 for all ɛ > 0,$ where the vertical bars indicate the number of elements in the enclosed set. The notion of statistically Cauchy sequence was first defined by Fridy [13] and he showed that it is equivalent to statistical convergence. Statistical convergence for single sequences extended to double sequences by Mursaleen and Edely [23] with the help of two dimensional analogue of natural density of subsets of $ℕ \times ℕ$ while the notion of convergence for double sequences of real numbers has been defined by Pringsheim’s [28]. Balcerzak et al. [8] discussed various kinds of statistical convergence and ideal convergence for sequences of functions with values in $ℝ$ or in a metric space. The notions of pointwise as well as uniform statistical convergence of double sequences of real-valued functions were introduced by Gökhan et al. [14]. Nuray and Şavas [27] studied statistical convergence for single sequence of fuzzy numbers and an appealing related characterization theorem was proved by Şavas [30]. Aytar and Pehlvian [7] showed that the statistical convergence of a sequence of fuzzy numbers with respect to the supremum metric is equivalent to the uniform statistical convergence of the sequences of functions which are defined via the endpoints of α-cuts of the same fuzzy numbers sequence. The notions of pointwise statistical convergence and statistically Cauchy of sequences of fuzzy mappings were introduced by Altin et al. [2] where some properties related to solidity, monotonicity and symmetricity are obtained. Quite recently, Gong et al. [16] studied pointwise and uniform statistical convergence and equi-statistical convergence for single sequences of fuzzy valued functions and obtained relationships between these convergence. For some other recent works related to these concepts, we refer the interested reader to [3 , 24, 25].

In spite of above, fuzzy sets also play an important role in the development of the theory of hemirings, in particular, the notion of falling fuzzy h-interior ideals of a hemiring has been defined and studied by Zhan et al. [37] (see also [36]).

Let E be a nonempty set. According to Zadeh, a fuzzy subset of E is a nonempty subset ${(t, \bar{x} (t)) : t \in E}$ of E × J (= [0, 1]) for some function $\bar{x} : E \to J (= [0, 1]) .$ A function $\bar{x} : ℝ \to J (= [0, 1])$ is called a fuzzy number if it satisfies the following properties:

$\bar{x}$ is convex i.e. $\bar{x} (t) \geq \bar{x} (s) \land \bar{x} (r) = min {\bar{x} (s), \bar{x} (r)},$ where s < t < r .

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

$\bar{x}$ is upper semi-continuous i.e. for each $ɛ > 0, {\bar{x}}^{- 1} ([0, a + ɛ))$ for all a ∈ [0, 1] is open in the usual topology of $ℝ .$

$[\bar{x}]^{0} = cl ({t \in ℝ : \bar{x} (t) > 0})$ is compact, where cl is the closure operator.

We denote the set of all fuzzy numbers on $ℝ$ by $F (ℝ) .$ The set $ℝ$ of real numbers can be embedded in $F (ℝ)$ if we define $\bar{r} \in F (ℝ)$ by $\bar{r} (t) = {\begin{matrix} 1, & if t = r : \\ 0, & if t \neq r \end{matrix}$

For 0 < α ≤ 1, α-cut of $\bar{x}$ is defined by $[\bar{x}]_{α} = {t \in ℝ : \bar{x} (t) \geq α} = [{\bar{x}}_{α}^{-}, {\bar{x}}_{α}^{+}]$ is a closed and bounded interval of $ℝ .$ As in [26], the Hausdorff distance between two fuzzy numbers $\bar{x}$ and $\bar{y}$ is given by $\begin{matrix} D (\bar{x}, \bar{y}) & = & sup_{α \in [0, 1]} d ([\bar{x}]_{α}, [\bar{y}]_{α}) \\ = & sup_{α \in [0, 1]} max {| {\bar{x}}_{α}^{-} - {\bar{y}}_{α}^{-} |, | {\bar{x}}_{α}^{+} - {\bar{y}}_{α}^{+} |}, \end{matrix}$ where $D : F (ℝ) \times F (ℝ) \to [0, + \infty)$ and d is the Hausdorff metric. Also, if $f, g : [a, b] \to F (ℝ)$ are fuzzy number valued functions, then the distance between f and g is given by $D^{*} (f, g) = sup_{x \in [a, b]} D (f, g)$ . For any $\bar{x}, \bar{y}, \bar{z}, \bar{u} \in F (ℝ)$ we know that

$(F (ℝ), D)$ is a complete metric space.

$D (γ \bar{x}, γ \bar{y}) = | γ | D (\bar{x}, \bar{y}); γ \in ℝ .$

$D (\bar{x} + \bar{u}, \bar{y} + \bar{u}) = D (\bar{x}, \bar{y}) .$

$D (\bar{x} + \bar{z}, \bar{y} + \bar{u}) \leq D (\bar{x}, \bar{y}) + D (\bar{x}, \bar{u}) .$

Lemma 1.1. [26] Let $\bar{x} \in F (ℝ)$ and $[\bar{x}]_{α} = [{\bar{x}}_{α}^{-}, {\bar{x}}_{α}^{+}] .$ Then the following conditions are satisfied:

${\bar{x}}_{α}^{-}$ is a left continuous monotone-nondecreasing function on (0, 1] .

${\bar{x}}_{α}^{+}$ is a right continuous monotone-nonincreasing function on (0, 1] .

${\bar{x}}_{α}^{-}$ and ${\bar{x}}_{α}^{+}$ are right continuous at α = 0 .

${\bar{x}}_{1}^{-} \leq {\bar{x}}_{1}^{+} .$

Let us recall that the convergence of double sequence means convergence in the Pringsheim’s sense (abbreviated as “P-convergent”) (see [28]). A double sequence x = (x_k,l) has a Pringsheim limit L, in symbols, we shall write P - lim x = L, provided that given an ɛ > 0 there exists an $N \in ℕ$ such that $| x_{k, l} - L | < ε$ whenever k, l > N .

For $K \subset ℕ \times ℕ$ and $i, j \in ℕ,$ δ_ij (K) is called the (ij) ^th partial double density of K, if $δ_{ij} (K) = \frac{| K \cap {(1, 1), (2, 2), . . .} |}{ij} .$

If $δ_{2} (K) = lim_{m, n \to \infty} \frac{1}{mn} | {k \leq m, l \leq n : (k, l) \in K} |$ in Pringsheim’s sense, i.e $(δ_{2} (K) = P - lim_{i, j \to + \infty} δ_{ij} (K))$ exists, it is called the double natural density of K . Also, the set $T = {K \subset ℕ \times ℕ, δ_{2} (K) = 0}$ is called the zero double density set.

The statistical analogue for double sequences has been defined and studied by Mursaleen and Edely [23] and Tripathy [34] independently. The idea was extended to double sequences $({\bar{x}}_{mn})$ of fuzzy numbers by Şavas and Mursaleen [31] as follows. We say that $({\bar{x}}_{mn})$ is statistically convergent to a fuzzy numbers ${\bar{x}}_{0}$ if for any ɛ > 0, we have $\begin{matrix} δ_{2} ({(m, n) \in ℕ \times ℕ : D ({\bar{x}}_{mn}, {\bar{x}}_{0}) \geq ɛ}) = 0, \end{matrix}$ i.e., the set ${(m, n) \in ℕ \times ℕ : D ({\bar{x}}_{mn}, {\bar{x}}_{0}) \geq ɛ} \in T$ . We shall write $S_{2} - lim {\bar{x}}_{mn} = {\bar{x}}_{0}$ or ${\bar{x}}_{mn} \overset{S_{2}}{\to} {\bar{x}}_{0} .$

We remark that $S_{2} - lim {\bar{x}}_{mn} = {\bar{x}}_{0}$ if and only if for given ɛ > 0, ∃ a subset $M \subset ℕ \times ℕ$ satisfying δ₂ (M) =0 such that $D ({\bar{x}}_{mn}, {\bar{x}}_{0}) < ɛ$ for all $(m, n) \in ℕ \times ℕ \ M .$

2 Pointwise and uniform convergence of double sequences of fuzzy valued functions

In this section, we define the notions of pointwise and uniform convergence of double sequences of fuzzy valued functions, and prove some classical results in this settings. Throughout this manuscript, we shall suppose that a fuzzy valued function $\bar{f} : [a, b] \to F (ℝ)$ and double sequence of fuzzy valued functions ${\bar{f}}_{mn} : [a, b] \to F (ℝ)$ for all $m, n \in ℕ .$ For convenience, we shall use the notations FVF and DSFVF instead of writing fuzzy valued function and double sequence of fuzzy valued functions, respectively.

Definition 2.1. A DSFVF $({\bar{f}}_{mn})$ is said to pointwise convergent to $\bar{f}$ if for each x ∈ [a, b] and for each ɛ > 0 there exists a positive integer N (x, ɛ) such that $D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ$ for all m, n > N. In symbols, we shall write $lim_{m, n \to \infty} {\bar{f}}_{mn} (x) = \bar{f} (x)$ or ${\bar{f}}_{m n} (x) \overset{p}{\to} \bar{f} (x)$ . The FVF $\bar{f}$ is called the Pringsheim limit function of the $({\bar{f}}_{mn})$ and we say that $({\bar{f}}_{mn})$ converges (pointwise) to $\bar{f}$ on [a, b] .

Lemma 2.2. Let $({\bar{f}}_{mn})$ be a DSFVF and $\bar{f}$ be a FVF defined on [a, b] . Then, for each x ∈ [a, b], $P - lim_{m, n \to \infty} {\bar{f}}_{mn} (x) = \bar{f} (x)$ (2.1) exists iff for all strictly increasing functions $m, n : ℕ \to ℕ$ the ordinary limit $lim_{r \to \infty} {\bar{f}}_{m (r) n (r)} (x) = \bar{f} (x) exists .$ (2.2)

Proof. It is obvious that (2.1) implies (2.2). Conversely suppose that the limit in (2.1) does not exist. Then for every FVF $\bar{f}$ there exists ɛ > 0 such that for each N, there exist integers m, n for which $D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq ɛ \forall x \in [a, b] .$ (2.3) For N = 1, we consider the corresponding values of m and n for which (2.3) is true by m (1) and n (1) , respectively. Put N₁ = max {m (1) , n (1)} +1 and suppose that a pair of values of m and n satisfying (2.3) by m (2) and n (2) , respectively. Continuing this process, we obtain an infinite sequence of values {m (r) , n (r)} for which (2.3) is true. Therefore limit of (2.2) does not exist.

Definition 2.3. A DSFVF $({\bar{f}}_{mn})$ is said to converge uniformly to $\bar{f}$ on [a, b] if, for every ɛ > 0, ∃ an integer N (= N (ɛ)) such that $D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ$ for all x ∈ [a, b] and all m, n > N. We denote this symbolically by ${\bar{f}}_{m n} \overset{u}{⇉} \bar{f}$ .

Theorem 2.4. Let $\bar{f}$ be FVF and $({\bar{f}}_{mn})$ be a DSFVF on [a, b] . Then ${\bar{f}}_{m n} \overset{u}{⇉} \bar{f}$ on [a, b] if and only if $d_{m n} \overset{p}{\to} 0,$ where $d_{mn} = sup_{x \in [a, b]} D ({\bar{f}}_{mn} (x), \bar{f} (x)) .$

Proof. Let ${\bar{f}}_{m n} \overset{u}{⇉} \bar{f}$ on [a, b] . Then, for every ɛ > 0, ∃ an integer N (= N (ɛ)) such that $D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ$ for all x ∈ [a, b] and all m, n > N which yields $sup_{x \in [a, b]} D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ$ for all m, n > N, (ɛ is arbitrary). Hence $d_{m n} \overset{p}{\to} 0$ .

Next, we suppose that $d_{m n} \overset{p}{\to} 0,$ i.e., $sup_{x \in [a, b]} D ({\bar{f}}_{mn} (x), \bar{f} (x)) \overset{p}{\to} 0 .$ Now, for every ɛ > 0, one obtains $sup_{x \in [a, b]} D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ .$ It follows that $D ({\bar{f}}_{mn} (x), \bar{f} (x)) \leq sup_{x \in [a, b]} D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ$ for all x ∈ [a, b]. Thus ${\bar{f}}_{m n} \overset{u}{⇉} \bar{f}$ on [a, b] .

Remark 2.5. Clearly, uniform convergence implies pointwise convergence with the same limit on [a, b] but not conversely. For the validity of this assertion, one construct the example below.

Example 2.6. For any x ∈ [0, 1], we define ${\bar{f}}_{mn} (x) = {\begin{matrix} \bar{(\frac{mnx}{1 + m^{2} n^{2} x^{2}})}, & for x \in [0, 1]; \\ \bar{0} & otherwise . \end{matrix}$

Then $({\bar{f}}_{mn})$ converges pointwise to $\bar{f} = \bar{0} .$ But for all $m, n \in ℕ$ $\begin{matrix} sup_{x \in [0, 1]} D ({\bar{f}}_{mn} (x), \bar{0}) & = & sup_{x \in [0, 1]} \frac{mnx}{1 + m^{2} n^{2} x^{2}} \\ = & sup_{x \in [0, 1]} \frac{1}{\frac{1}{mnx} + mnx} = \frac{1}{2} \neq 0 . \end{matrix}$

Hence $({\bar{f}}_{mn})$ is not uniformly convergent to $\bar{0}$ on [0, 1] .

Definition 2.7. A DSFVF $({\bar{f}}_{mn})$ is said to uniformly Cauchy on [a, b] if, for every ɛ > 0, ∃ an integer N (= N (ɛ)) such that $D ({\bar{f}}_{mn} (x), {\bar{f}}_{jk} (x)) < ɛ$ for all x ∈ [a, b] and all m, n, j, k > N .

Theorem 2.8. Let $({\bar{f}}_{mn})$ be a DSFVF on [a, b] . Then $({\bar{f}}_{mn})$ is uniformly convergent if and only if it is uniformly Cauchy.

Proof. Let $({\bar{f}}_{mn})$ is uniformly convergent to $\bar{f}$ on [a, b] . Then, for each ɛ > 0, ∃ a positive integer N (= N (ɛ)) such that $D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ / 2$ for all x ∈ [a, b] and all m, n > N . One writes $\begin{matrix} D ({\bar{f}}_{mn} (x), {\bar{f}}_{jk} (x)) \leq D ({\bar{f}}_{mn} (x), \bar{f} (x)) \\ + D ({\bar{f}}_{jk} (x), \bar{f} (x)) < \frac{ɛ}{2} + \frac{ɛ}{2} = ɛ, \end{matrix}$ for all x ∈ [a, b] and all m, n, j, k > N. Hence $({\bar{f}}_{mn})$ is uniformly Cauchy on [a, b] .

Next, let $({\bar{f}}_{mn})$ is uniformly Cauchy on [a, b]. Then, for each ɛ > 0, ∃ $N (= N (ɛ)) \in ℕ$ such that $D ({\bar{f}}_{mn} (x), {\bar{f}}_{jk} (x)) < ɛ$ (2.4) holds for all x ∈ [a, b] and for all m, n, j, k > N. Since $({\bar{f}}_{mn} (x))$ is a Cauchy sequence for every x ∈ [a, b] and by the completeness of $F (R)$ we have $P - lim_{m, n \to \infty} {\bar{f}}_{mn} (x) = \bar{f} (x) .$ In (2.4), fixed m, n > N and taking limit j, k→ ∞, we have $D ({\bar{f}}_{mn} (x), \bar{f} (x)) < \frac{ɛ}{2} < ɛ$ for all x ∈ [a, b] and all m, n > N . Hence $({\bar{f}}_{mn})$ is uniformly convergent to $\bar{f}$ on [a, b] .

Theorem 2.9. Let $({\bar{f}}_{mn})$ is uniformly convergent to $\bar{f} .$ Suppose that x ∈ [a, b] and $lim_{t \to x} {\bar{f}}_{mn} (t) = {\bar{a}}_{mn} \forall t \in [a, b] (m, n \in ℕ) .$ (2.5)

Then, a double sequence of fuzzy numbers $({\bar{a}}_{mn})$ is convergent and $lim_{t \to x} \bar{f} (t) = {\bar{a}}_{mn} for all t \in [a, b] .$ (2.6)

Proof. Suppose that $({\bar{f}}_{mn})$ is uniformly convergent to $\bar{f}$ on [a, b] . Therefore $({\bar{f}}_{mn})$ is uniformly Cauchy on [a, b] . Then, for each ɛ > 0, ∃ $N (= N (ɛ)) \in ℕ$ such that $D ({\bar{f}}_{mn} (t), {\bar{f}}_{jk} (t)) < ɛ$ (2.7) for all t ∈ [a, b] and for all m, n, j, k > N. Fixing m, n, j, k > N in (2.7) and taking limit t → x, then from (2.5) we have $D ({\bar{a}}_{mn}, {\bar{a}}_{jk}) < ɛ .$ Therefore double sequence of fuzzy numbers $({\bar{a}}_{mn})$ is a Cauchy sequence and so convergent, say, $lim_{m, n \to \infty} a_{mn} = \bar{a} .$ Then, for every ɛ > 0, ∃ a positive integer N₀ (= N₀ (ɛ)) such that $D ({\bar{f}}_{mn} (t), {\bar{f}}_{jk} (t)) < \frac{ɛ}{3} \forall t \in [a, b] and \forall m, n > N_{0}$ and $D ({\bar{a}}_{mn}, \bar{a}) < \frac{ɛ}{3}$ for all m, n > N₀. For fixed m, n > N₀ and $lim_{t \to x} {\bar{f}}_{mn} (t) = {\bar{a}}_{mn}$ ∀ t ∈ [a, b], ∃ a punctured neighborhood $\tilde{A} (x)$ of x such that $D ({\bar{f}}_{mn} (t), {\bar{a}}_{mn}) < \frac{ɛ}{3} \forall t \in \tilde{A} (x) \cap [a, b] .$

Then, for every m, n > N₀ and for each $t \in \tilde{A} (x) \cap [a, b],$ one writes $\begin{matrix} D (\bar{f} (t), \bar{a}) & \leq & D (\bar{f} (t), {\bar{f}}_{mn}) + D ({\bar{f}}_{mn} (t), {\bar{a}}_{mn}) \\ + D ({\bar{a}}_{mn}, \bar{a}) \\ < & ɛ / 3 + ɛ / 3 + ɛ / 3 = ɛ . \end{matrix}$

Hence $lim_{t \to x} \bar{f} (t) = \bar{a} = lim_{m, n \to \infty} {\bar{a}}_{mn} for all t \in [a, b] .$

Theorem 2.10. If $({\bar{f}}_{mn})$ is uniformly convergent to $\bar{f}$ on [a, b] , then $\bar{f}$ is continuous on [a, b] .

Proof. Let x ∈ [a, b] be arbitrary. Since each ${\bar{f}}_{mn}$ is continuous on [a, b] , then $lim_{t \to x} {\bar{f}}_{mn} (t) = {\bar{f}}_{mn} (x) \forall t \in [a, b] .$

Therefore by Theorem 2.9, $({\bar{f}}_{mn})$ is convergent and for all t ∈ [a, b] we have $\begin{matrix} lim_{t \to x} \bar{f} (t) & = & lim_{m, n \to \infty} {\bar{f}}_{mn} (x) \\ = & \bar{f} (x) \forall m, n \in ℕ . \end{matrix}$

Therefore $\bar{f}$ is continuous at x ∈ [a, b] and hence $\bar{f}$ is continuous on [a, b] (since x was arbitrary).

Theorem 2.11. (Dini’s type theorem) Let D be a compact subset of $ℝ$ and let $({\bar{f}}_{mn})$ be a double sequence of fuzzy valued continuous functions on D and pointwise converges to a fuzzy valued continuous function $\bar{f}$ on D . Suppose that $({\bar{f}}_{mn})$ be monotonic on D, i.e., ${\bar{f}}_{m + 1 n + 1} (x) \leq {\bar{f}}_{mn} (x)$ or ${\bar{f}}_{m + 1 n + 1} (x) \geq {\bar{f}}_{mn} (x)$ for every x ∈ D and for all m, n = 1, 2, 3, . . .. Then $({\bar{f}}_{mn})$ is uniformly convergent to $\bar{f}$ on D .

Proof. Let x ∈ D and $({\bar{f}}_{mn})$ be monotonic decreasing on D and let ${\bar{g}}_{mn} (x) = {\bar{f}}_{mn} (x) - \bar{f} (x)$ (on the other hand, let $({\bar{f}}_{mn})$ be monotonic increasing on D and let ${\bar{g}}_{mn} (x) = \bar{f} (x) - {\bar{f}}_{mn} (x))$ . Then, ${\bar{g}}_{mn} (x)$ is continuous on D and $lim_{m, n \to \infty} {\bar{g}}_{mn} (x) = \bar{0}$ for each x ∈ D, and $({\bar{g}}_{mn})$ is monotonic decreasing on D . To show that $({\bar{g}}_{mn})$ is uniformly convergent to $\bar{0}$ on D . Let ɛ > 0 . Since $lim_{m, n \to \infty} {\bar{g}}_{mn} (x) = \bar{0}$ for each x ∈ D, there exist $m_{x},, n_{x} \in ℕ$ such that $\bar{0} \leq {\bar{g}}_{m_{x} n_{x}} (t) < \frac{ɛ}{2} for every t \in D .$ (2.8)

Since ${\bar{g}}_{m_{x} n_{x}} (t)$ is continuous at x ∈ D, by relation (2.8), ∃ an open neighborhood $\bar{U} (x)$ of x such that $\bar{0} \leq {\bar{g}}_{m_{x} n_{x}} (t) < ɛ for every t \in \bar{U} (x) \cap D \equiv \bar{A} (x) .$

Since $({\bar{g}}_{mn} (t))$ is monotonic decreasing for every $t \in \bar{A} (x),$ we have ${\bar{g}}_{mn} (t) < ɛ for all m > m_{x}, n > n_{x} .$

Since D is a compact subset of $ℝ,$ there exists finite set {x₁, x₂, . . . x_j} such that $D \subset \bar{A} (x_{1}) \cup \bar{A} (x_{2}) \cup . . . \cup \bar{A} (x_{j}) .$ We define N₀ = max {m_{x
₁}, m_{x
₂}, . . . , m_{x
_j}} and N₁ = max {n_{x
₁}, n_{x
₂}, . . . n_{x
_j}} . Then, we have ${\bar{g}}_{mn} (t) < ɛ \forall m > N_{0}, n > N_{1} and \forall t \in D .$

Hence $({\bar{g}}_{mn})$ is uniformly convergent to $\bar{0}$ on D .

3 Statistical convergence of double sequences of fuzzy valued functions

We define the notions of pointwise and uniform statistical convergence as well as the notion of equi-statistical convergence of double sequences of fuzzy valued functions and discuss some fundamental results in this new settings.

Definition 3.1. A DSFVF $({\bar{f}}_{mn})$ is said to pointwise statistically convergent to $\bar{f}$ in Pringsheim’s sense, if for every x ∈ [a, b] and for every ɛ > 0 ∃ M_x ∈ T such that for all $(m, n) \in ℕ \times ℕ \ M_{x}$ we have $D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ$ . Here, one writes $p S_{2} - \lim {\bar{f}}_{m n} (x) = \bar{f} (x)$ or ${\bar{f}}_{m n} \overset{p S_{2}}{\to} \bar{f}$ on [a, b] .

Theorem 3.2. Let $({\bar{f}}_{mn})$ and $({\bar{g}}_{mn})$ be two DSFVF defined on [a, b] . If both ${\bar{f}}_{m n} (x) \overset{p S_{2}}{\to} \bar{f} (x)$ and ${\bar{g}}_{m n} (x) \overset{p S_{2}}{\to} \bar{g} (x)$ on [a, b]. Then

$({\bar{f}}_{m n} (x) + {\bar{g}}_{m n} (x)) \overset{p S_{2}}{\to} \bar{f} (x) + \bar{g} (x)$ on [a, b] ,

$c {\bar{f}}_{m n} (x) \overset{p S_{2}}{\to} c \bar{f} (x)$ on [a, b] , where $c \in ℝ .$

Proof. (a) Since ${\bar{f}}_{m n} (x) \overset{p S_{2}}{\to} \bar{f} (x)$ and ${\bar{g}}_{mn} (x) \overset{p S_{2}}{\to} \bar{g} (x)$ on [a, b] . Therefore, for every x ∈ [a, b] and for every ɛ > 0, there exists M_x ∈ T such that for all $(m, n) \in ℕ \times ℕ \ M_{x}$ , we have $D ({\bar{f}}_{mn} (x), \bar{f} (x)) < \frac{ɛ}{2} and D ({\bar{g}}_{mn} (x), \bar{g} (x)) < \frac{ɛ}{2} .$

It follows that $\begin{matrix} D ({\bar{f}}_{mn} (x) + {\bar{g}}_{mn} (x), \bar{f} (x) + \bar{g} (x)) \\ \leq D ({\bar{f}}_{mn} (x), \bar{f} (x)) + D ({\bar{g}}_{mn} (x), \bar{g} (x)) < ɛ . \end{matrix}$

Hence $({\bar{f}}_{mn} (x) + {\bar{g}}_{mn} (x)) \overset{p S_{2}}{\to} \bar{f} (x) + \bar{g} (x)$ on x ∈ [a, b] .

(b) Suppose ${\bar{f}}_{mn} (x) \overset{p S_{2}}{\to} \bar{f} (x)$ on [a, b]. Therefore, for every x ∈ [a, b] and for every ɛ > 0, there exists M_x ∈ T such that for all $(m, n) \in ℕ \times ℕ \ M_{x}$ , we have $D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ$ . The choice of c = 0 is obvious so we are choosing $(0 \neg =) c \in ℝ$ . Therefore, for each x ∈ [a, b], we obtain $D (c {\bar{f}}_{mn} (x), c \bar{f} (x)) = | c | D ({\bar{f}}_{mn} (x), \bar{f} (x)) < | c | ɛ .$

Hence $c {\bar{f}}_{mn} (x) \overset{p S_{2}}{\to} c \bar{f} (x)$ on [a, b] , where $c \in ℝ .$

Corollary 3.3. Let $({\bar{f}}_{mn})$ be a DSFVF on [a, b] such that ${\bar{f}}_{mn} (x) \overset{p S_{2}}{\to} \bar{f} (x)$ on [a, b] and [c, d] ⊂ [a, b]. Then ${\bar{f}}_{mn} (x) \overset{p S_{2}}{\to} \bar{f} (x)$ on [c, d] .

Theorem 3.4. A DSFVF $({\bar{f}}_{mn})$ is pointwise statistically convergent to $\bar{f}$ on [a, b] if and only if ∃ a subset $K_{x} = {(m, n)} \subseteq ℕ \times ℕ$ for each x ∈ [a, b] such that δ₂ (K_x) =1 and $lim_{\begin{matrix} m, n \to \infty \\ ; (m, n) \in K_{x} \end{matrix}} {\bar{f}}_{mn} (x) = \bar{f} (x)$ for each x ∈ [a, b] .

Proof. Suppose that $({\bar{f}}_{mn})$ is pointwise statistically convergent to $\bar{f}$ on [a, b] . Then, ${pS}_{2} - lim_{m, n} {\bar{f}}_{mn} (x) = \bar{f} (x)$ on [a, b] . We are writing $\begin{matrix} K_{x, j} & = & {(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq \frac{1}{j}} \\ L_{x, j} & = & {(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) < \frac{1}{j}} \end{matrix}$ for j = 1, 2, 3 . . . and for each x ∈ [a, b] . Then we have

δ₂ (K_x,j) =0,

L_x,1 ⊃ L_2,x ⊃ . . . ⊃ L_x,j ⊃ L_x,j+1 ⊃ . . .

δ₂ (L_x,j) =1, for j = 1, 2, 3 . . . and for each x ∈ [a, b] .

We have to show that for (m, n) ∈ L_x,j, $({\bar{f}}_{mn})$ is convergent (pointwise) to $\bar{f}$ on [a, b]. On the contrary, suppose that $({\bar{f}}_{mn})$ is not convergent (pointwise) to $\bar{f}$ on [a, b]. It follows that for every ɛ > 0 we have $D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq ɛ$ for infinitely many terms and for some x ∈ [a, b] . Let $L_{x, ɛ} = {(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ}$ and $ɛ > \frac{1}{j}$ for j = 1, 2, 3 . . . and for each x ∈ [a, b] . Then δ₂ (L_x,ɛ) =0 and by the relation (ii) we have L_x,j ⊂ L_x,ɛ . Therefore δ₂ (L_x,j) =0 which contradicts (iii). Hence $({\bar{f}}_{mn})$ is convergent to $\bar{f}$ on [a, b] .

Next, we suppose that ∃ a subset $K_{x} = {(m, n)} \subset ℕ \times ℕ$ for each x ∈ [a, b] such that δ₂ (K_x) =1 and $lim_{\begin{matrix} m, n \to \infty \\ ; (m, n) \in K_{x} \end{matrix}} {\bar{f}}_{mn} (x) = \bar{f} (x)$ , i.e., for every x ∈ [a, b] and for every ɛ > 0 ∃ an integer N (x, ɛ) such that $D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ \forall m, n > N .$

Therefore, for each x ∈ [a, b] and for each ɛ > 0, $\begin{matrix} K_{x, ɛ} & = & {(m, n) : D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq ɛ} \\ \subseteq & ℕ \times ℕ \ {(m_{N + 1}, n_{N + 1}), (m_{N + 2}, n_{N + 2}), . . .} . \end{matrix}$

Thus, we have δ₂ (K_x,ɛ) ≤1 - 1 =0. This yields that ${\bar{f}}_{mn} \overset{p S_{2}}{\to} \bar{f}$ on [a, b].

Corollary 3.5. If a DSFVF $({\bar{f}}_{mn})$ is pointwise statistically convergent to $\bar{f}$ on [a, b] , then there exists a DSFVF $({\bar{g}}_{mn})$ such that $lim_{m, n \to \infty} {\bar{g}}_{mn} (x) = \bar{f} (x)$ on [a, b] and $δ_{2} ({(m, n) : {\bar{f}}_{mn} (x) = {\bar{g}}_{mn} (x)}) = 1$ for each x ∈ [a, b] .

Definition 3.6. A DSFVF $({\bar{f}}_{mn})$ is said to uniformly statistically convergent to $\bar{f}$ on [a, b] if for every ɛ > 0, ∃ M ∈ T so that for all $(m, n) \in ℕ \times ℕ \ M$ we have $D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ$ which holds for all x ∈ [a, b]. One writes ${uS}_{2} - lim {\bar{f}}_{mn} (x) = \bar{f} (x)$ on [a, b] or ${\bar{f}}_{mn} \overset{u S_{2}}{⇉} \bar{f}$ on [a, b] .

Remark 3.7. From the above definition, we have ${\bar{f}}_{mn} \overset{u S_{2}}{⇉} \bar{f}$ on [a, b] if, for every ɛ > 0, $δ_{2} ({(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq ɛ}) = 0$ which holds for all x ∈ [a, b].

Theorem 3.8. Suppose that ${\bar{f}}_{mn} \overset{u S_{2}}{⇉} \bar{f}$ on [a, b] , then ${\bar{f}}_{mn} \overset{p S_{2}}{\to} \bar{f}$ on [a, b] .

Proof. The proof of this theorem directly follows from the definitions.

Theorem 3.9. Let $\bar{f}$ be a FVF and $({\bar{f}}_{mn})$ be a DSFVF on [a, b]. Then ${\bar{f}}_{mn} \overset{u S_{2}}{⇉} \bar{f}$ on [a, b]if and only if $c_{m n} \overset{p S_{2}}{\to} 0$ , where $c_{mn} = sup_{x \in [a, b]} D ({\bar{f}}_{mn} (x), \bar{f} (x)) .$

Proof. Assume that $c_{m n} \overset{p S_{2}}{\to} 0$ , i.e., $sup_{x \in [a, b]} D ({\bar{f}}_{mn} (x), \bar{f} (x)) \overset{p S_{2}}{\to} 0 .$ Then, for each x ∈ [a, b] and for every ɛ > 0, we have $D ({\bar{f}}_{mn} (x), \bar{f} (x)) \leq sup_{x \in [a, b]} D ({\bar{f}}_{mn} (x), \bar{f} (x)),$ so that $\begin{matrix} {(m, n) & \in & ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) \\ \geq & ɛ \forall x \in [a, b]} \\ \subseteq & {(m, n) \in ℕ \times ℕ : \\ sup_{x \in [a, b]} D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq ɛ} . \end{matrix}$

Hence, ${\bar{f}}_{mn} \overset{u S_{2}}{⇉} \bar{f}$ .

Next, suppose that ${\bar{f}}_{mn} \overset{u S_{2}}{⇉} \bar{f}$ on [a, b] . For every ɛ > 0, we are writing $\begin{matrix} H_{1} & = & {(m, n) \in ℕ \times ℕ \\ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq ɛ \forall x \in [a, b]} \\ H_{2} & = & {(m, n) \in ℕ \times ℕ \\ : sup_{x \in [a, b]} D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq ɛ} . \end{matrix}$

If $(m, n) \in ℕ \times ℕ \ H_{1},$ then $D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ$ ∀ x ∈ [a, b] which implies $sup_{x \in [a, b]} D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ$ as ɛ is arbitrary. This gives that $(m, n) \in ℕ \times ℕ \ H_{2}$ . Therefore, we have $ℕ \times ℕ \ H_{1} \subseteq ℕ \times ℕ \ H_{2} \Rightarrow H_{2} \subseteq H_{1}$ which gives δ₂ (H₂) ≤ δ₂ (H₁) =0 . Hence $c_{m n} \overset{p S_{2}}{\to} 0$ .

Corollary 3.10. If $({\bar{f}}_{mn})$ is a DSFVF and $\bar{f}$ is a FVF on [a, b], then

${\bar{f}}_{m n} \overset{u}{⇉} \bar{f}$ ${\bar{f}}_{m n} \overset{p}{⇉} \bar{f}$ $\Rightarrow {\bar{f}}_{mn} \overset{p S_{2}}{⇉} \bar{f}$ on [a, b] ,

${\bar{f}}_{m n} \overset{u}{⇉} \bar{f}$ $\Rightarrow {\bar{f}}_{mn} \overset{u S_{2}}{⇉} \bar{f}$ $\Rightarrow {\bar{f}}_{mn} \overset{p S_{2}}{⇉} \bar{f}$ on [a, b] .

Definition 3.11. A DSFVF $({\bar{f}}_{mn})$ is said to be statistically Cauchy if for each x ∈ [a, b] and for each ɛ > 0 ∃ positive integers M = M (ɛ, x) and N = N (ɛ, x) such that $δ_{2} ({(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), {\bar{f}}_{MN} (x)) \geq ɛ}) = 0 .$

Theorem 3.12. Let $({\bar{f}}_{mn})$ be a DSFVF defined on [a, b] . Then $({\bar{f}}_{mn})$ is (pointwise) statistically convergent on [a, b] if and only if $({\bar{f}}_{mn})$ is statistically Cauchy on [a, b] .

Proof. Suppose that $({\bar{f}}_{mn})$ is (pointwise) statistically convergent to $\bar{f}$ on [a, b] . Then, ∀ x ∈ [a, b] and ∀ ɛ > 0 ∃ M_x ∈ T such that $\forall (m, n) \in ℕ \times ℕ \ M_{x}$ , we have $D ({\bar{f}}_{mn} (x), \bar{f} (x)) < \frac{ɛ}{2}$ . If we choose $M, N \in ℕ$ such that $D ({\bar{f}}_{MN} (x), \bar{f} (x)) < \frac{ɛ}{2}$ ∀ x ∈ [a, b]. Therefore, $\begin{matrix} D ({\bar{f}}_{mn} (x), {\bar{f}}_{MN} (x)) \\ \leq D ({\bar{f}}_{mn} (x), \bar{f} (x)) + D ({\bar{f}}_{MN} (x), \bar{f} (x)) < ɛ . \end{matrix}$

Thus $({\bar{f}}_{mn})$ is statistically Cauchy on [a, b] .

Next, suppose that $({\bar{f}}_{mn})$ is statistically Cauchy on [a, b] . Then, we choose $M, N \in ℕ$ such that the band $I = [{\bar{f}}_{MN} (x) - 1, {\bar{f}}_{MN} (x) + 1]$ contains ${\bar{f}}_{mn} (x)$ for all $(m, n) \in ℕ \times ℕ$ and for each x ∈ [a, b] . Now choose K, L such that $I_{0} = [{\bar{f}}_{KL} (x) - \frac{1}{2}, {\bar{f}}_{KL} (x) + \frac{1}{2}]$ contains ${\bar{f}}_{mn} (x)$ for all $(m, n) \in ℕ \times ℕ$ and for each x ∈ [a, b] . We assert that I₁ = I ∩ I₀ contains ${\bar{f}}_{mn} (x)$ for all $(m, n) \in ℕ \times ℕ$ and for each x ∈ [a, b] , so A₁ = A₂ ∪ A₃, where $\begin{matrix} A_{1} & = & {(m, n) \in ℕ \times ℕ : {\bar{f}}_{mn} (x) \notin I \cap I_{0}} \\ A_{2} & = & {(m, n) \in ℕ \times ℕ : {\bar{f}}_{mn} (x) \notin I} \\ A_{3} & = & {(m, n) \in ℕ \times ℕ : {\bar{f}}_{mn} (x) \notin I_{0}} . \end{matrix}$

Therefore δ₂ (A₁) ≤ δ₂ (A₂) + δ₂ (A₃) =0. Thus I₁ is a closed band of height less than equal to 1 that contains ${\bar{f}}_{mn} (x)$ for all $(m, n) \in ℕ \times ℕ$ and for each x ∈ [a, b] . We proceed by choosing M (2) and N (2) so that $I_{2} = [{\bar{f}}_{M (2) N (2)} (x) - \frac{1}{4}, {\bar{f}}_{M (2) N (2)} (x) + \frac{1}{4}]$ contains ${\bar{f}}_{mn} (x)$ for all $(m, n) \in ℕ \times ℕ$ and for each x ∈ [a, b] . By the preceding argument I₃ = I₁ ∩ I₂ contains ${\bar{f}}_{mn} (x)$ and the height of I₃ is less than equal to $\frac{1}{2} .$ Continuing inductively we construct a sequence (I_i) of closed band such that I_i ⊇ I_i+1 for each i and I_i has height not greater than 2^1-i, and ${\bar{f}}_{mn} (x) \in I_{i}$ . Thus, ∃ a FVF $\bar{f} (x)$ such that $\bar{f} (x) = ⋂_{i = 1}^{\infty} I_{i} .$ We now have to show that ${\bar{f}}_{mn} (x)$ is (pointwise) statistically convergent to $\bar{f} (x)$ on [a, b]. Let ɛ > 0 be given. Then, ∃ j such that ɛ > 2^1-j . It follows from the above argument that ${\bar{f}}_{mn} (x) \in I_{j}$ for all $(m, n) \in ℕ \times ℕ$ and for each x ∈ [a, b] . Then $\begin{matrix} δ_{2} ({(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq ɛ}) \\ \leq δ_{2} ({(m, n) \in ℕ \times ℕ : {\bar{f}}_{mn} (x) \notin I_{j}}) = 0, \end{matrix}$ for each x ∈ [a, b]. Hence completes the proof.

Theorem 3.13. A DSFVF $({\bar{f}}_{mn})$ is pointwise statistically convergent to a FVF $\bar{f}$ if and only if $[{\bar{f}}_{mn}]_{α}$ is uniformly statistically convergent to $[\bar{f}]_{α}$ with respect to α .

Proof. Suppose that $({\bar{f}}_{mn})$ is pointwise statistically convergent to $\bar{f}$ on [a, b]. For each x ∈ [a, b] and for every ɛ > 0 ∃ M_x ∈ T so that $\forall (m, n) \in ℕ \times ℕ \ M_{x}$ , we have $D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ$ whichimplies $sup_{α \in [0, 1]} max {| ({\bar{f}}_{mn} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} |, | ({\bar{f}}_{mn} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} |} < ɛ$ for all α ∈ [0, 1]. But $[{\bar{f}}_{mn} (x)]_{α} = [({\bar{f}}_{mn} (x))_{α}^{-}, ({\bar{f}}_{mn} (x))_{α}^{+}]$ and $[\bar{f} (x)]_{α} = [(\bar{f} (x))_{α}^{-}, (\bar{f} (x))_{α}^{+}],$ therefore we have $| ({\bar{f}}_{mn} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | < ɛ$ and $| ({\bar{f}}_{mn} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} | < ɛ$ for each $(m, n) \in ℕ \times ℕ \ M_{x}$ and for all α ∈ [0, 1]. Hence $[{\bar{f}}_{mn} (x)]_{α}$ is uniformly statistically convergent to $[\bar{f} (x)]_{α}$ with respect to α.

Next, we suppose that for all α ∈ [0, 1] , $[{\bar{f}}_{mn}]_{α}$ is uniformly statistically convergent to $[\bar{f}]_{α}$ with respect to α . Then, for every ɛ > 0 and for any x ∈ [a, b], we are writing $\begin{matrix} A_{1} & = & {(m, n) \in ℕ \times ℕ : \\ | ({\bar{f}}_{mn} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | \geq ɛ} \\ A_{2} & = & {(m, n) \in ℕ \times ℕ : \\ | ({\bar{f}}_{mn} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} | \geq ɛ} \\ A_{3} & = & {(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq ɛ} . \end{matrix}$

Now for any $(m, n) \in A_{1}^{c} \cap A_{2}^{c},$ for any x ∈ [a, b] and α ∈ [0, 1] , we have $| ({\bar{f}}_{mn} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | < ɛ$ and $| ({\bar{f}}_{mn} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} | < ɛ .$ Therefore $D ({\bar{f}}_{mn} (x), \bar{f} (x)) = sup_{α \in [0, 1]} max {| ({\bar{f}}_{mn} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} |, | ({\bar{f}}_{mn} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} |} < ɛ$ which yields $(m, n) \in A_{3}^{c}$ . Therefore, δ₂ (A₃) ≤ δ₂ (A₁) + δ₂ (A₂) =0 . Hence $({\bar{f}}_{mn})$ is pointwise statistically convergent to $\bar{f}$ on [a, b] .

Theorem 3.14. Suppose that ${\bar{f}}_{mn} \overset{p S_{2}}{\to} \bar{f}$ on [a, b] , where $({\bar{f}}_{mn})$ is equi-continuous on [a, b]. Then $\bar{f}$ is continuous and ${\bar{f}}_{mn} \overset{u S_{2}}{⇉} \bar{f}$ on [a, b] .

Proof. First we have to prove that $\bar{f}$ is continuous on [a, b] . Let ɛ > 0 and for any x₀ in [a, b] . Since $({\bar{f}}_{mn})$ are equi-continuous on [a, b] , there exists δ > 0 such that $D ({\bar{f}}_{mn} (x), {\bar{f}}_{mn} (x_{0})) < \frac{ɛ}{3}$ where x₀ - δ < x < x₀ + δ for all $(m, n) \in ℕ \times ℕ .$ Also, since ${\bar{f}}_{mn} \overset{p S_{2}}{\to} \bar{f}$ on x ∈ [a, b], so $\begin{matrix} {(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x_{0}), \bar{f} (x_{0})) \geq \frac{ɛ}{3}} \\ ⋃ {(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq \frac{ɛ}{3}} \in T . \end{matrix}$

Thus, for each x ∈ [a, b] and for each ɛ > 0 ∃ M_x ∈ T so that $(m, n) \in ℕ \times ℕ \ M_{x}$ , we have $D ({\bar{f}}_{mn} (x_{0}), \bar{f} (x_{0})) < ɛ / 3$ and $D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ / 3 .$ Then $\begin{matrix} D (\bar{f} (x), \bar{f} (x_{0})) \leq D (\bar{f} (x), {\bar{f}}_{mn} (x_{0})) \\ + D ({\bar{f}}_{mn} (x_{0}), {\bar{f}}_{mn} (x)) + D ({\bar{f}}_{mn} (x), \bar{f} (x)) . \end{matrix}$

Using above three strict inequalities, we obtain $D (\bar{f} (x), \bar{f} (x_{0})) < ɛ$ (since x₀ was arbitrary). This proves that $\bar{f}$ is continuous on [a, b] .

It is left to prove that ${\bar{f}}_{mn} \overset{u S_{2}}{⇉} \bar{f}$ on [a, b] . Since $\bar{f}$ is continuous on [a, b] , so it is uniformly continuous on [a, b] and $({\bar{f}}_{mn})$ are equi-continuous on [a, b]. Then for any x, y ∈ [a, b] and for every ɛ > 0 ∃ δ > 0 such that $D ({\bar{f}}_{mn} (x), {\bar{f}}_{mn} (y)) < ɛ / 3$ and $D (\bar{f} (x), \bar{f} (y)) < ɛ / 3$ whenever y - δ < x < y + δ . Since [a, b] is compact, we can choose a finite cover (x₁ - δ, x₁ + δ) , (x₂ - δ, x₂ + δ) , (x₃ - δ, x₃ + δ) , . . . , (x_i - δ, x_i + δ) from the covers of [a, b] . Since ${\bar{f}}_{mn} \overset{p S_{2}}{\to} \bar{f}$ on x ∈ [a, b] , there exists a set M_{x
_k} ∈ T such that $D ({\bar{f}}_{mn} (x_{k}), \bar{f} (x_{k})) < ɛ / 3$ for all (m, n) ∉ M_{x
_k} and for k ∈ {1, 2, . . . i} . Then, for any (m, n) ∉ M_{x
_k} and for some k ∈ {1, 2, . . . , i}, one obtains $\begin{matrix} D ({\bar{f}}_{mn} (x), \bar{f} (x)) \leq D ({\bar{f}}_{mn} (x), {\bar{f}}_{mn} (x_{k})) \\ + D ({\bar{f}}_{mn} (x_{k}), \bar{f} (x_{k})) + D (\bar{f} (x_{k}), \bar{f} (x)) < ɛ, \end{matrix}$ for x ∈ (x_k - δ, x_k + δ). This shows that ${\bar{f}}_{mn} \overset{u S_{2}}{⇉} \bar{f}$ on [a, b] .

Definition 3.15. A DSFVF $({\bar{f}}_{mn})$ is called equi-statistically convergent to a FVF $\bar{f}$ if for given ɛ > 0, $G_{ij, ɛ} = δ_{ij} ({(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq ɛ})$ with respect to x ∈ [a, b] is uniformly convergent to zero function. In symbols, we shall write ${\bar{f}}_{m n} \overset{e S_{2}}{↠} \bar{f} .$

The proof of the following two theorems are straightforward from definitions.

Theorem 3.16. Suppose that ${\bar{f}}_{m n} \overset{e S_{2}}{↠} \bar{f}$ if and only if ∀ɛ, β > 0, $\exists k, l \in ℕ$ , ∀i ≥ k, j ≥ l, ∀x ∈ [a, b], $δ_{ij} ({(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq ɛ}) < β .$

Theorem 3.17. A DSFVF ${\bar{f}}_{m n} \overset{e S_{2}}{↠} \bar{f}$ , then ${\bar{f}}_{m n} \overset{p S_{2}}{\to} \bar{f}$ on [a, b] .

Theorem 3.18. Let $({\bar{f}}_{mn})$ be a DSFVF and let $\bar{f}$ be a FVF on [a, b] . If ${\bar{f}}_{m n} \overset{u S_{2}}{⇉} \bar{f},$ then ${\bar{f}}_{m n} \overset{e S_{2}}{↠} \bar{f}$ . Then, for every ɛ > 0 ∃ M ∈ T so that for all $(m, n) \in ℕ \times ℕ \ M$ , we have $D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ$ for all x ∈ [a, b] . Since M ∈ T, there exists $k, l \in ℕ$ such that δ_ij (M) < ɛ for all i ≥ k, j ≥ k . Let x ∈ [a, b] and $(m, n) \in ℕ \times ℕ .$ Therefore, we obtain $D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq ɛ \Rightarrow (m, n) \in M .$

Thus $\begin{matrix} δ_{ij} ({(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq ɛ}) \\ \leq δ_{ij} (M) < ɛ \forall x \in [a, b] . \end{matrix}$

This gives that ${\bar{f}}_{m n} \overset{e S_{2}}{↠} \bar{f} .$

Corollary 3.19. Let $({\bar{f}}_{mn})$ be a DSFVF and $\bar{f}$ be a FVF on [a, b] . Then $[a, b] . $ T h e n $ {\bar{f}}_{m n} \overset{u S_{2}}{⇉} \bar{f} \Rightarrow {\bar{f}}_{m n} \overset{e S_{2}}{↠} \bar{f} \Rightarrow {\bar{f}}_{m n} \overset{p S_{2}}{\to} \bar{f} .$ The converse implications does not hold in general.

Proof. To prove above result, we shall consider the examples as given below.

Example 3.20. Consider the DSFVF $({\bar{f}}_{mn})$ and is defined by ${\bar{f}}_{mn} (x) = \bar{(e^{- mnx})}$ for x ∈ [0, 1]. Then, we have $D ({\bar{f}}_{mn} (x), \bar{0}) = e^{- mnx}$ for each $(m, n) \in ℕ \times ℕ$ and x ∈ [0, 1] . Therefore, $({\bar{f}}_{mn})$ converges pointwise to $\bar{f} = \bar{0}$ and so ${\bar{f}}_{m n} \overset{p S_{2}}{\to} \bar{0} .$ But for each $(k, l) \in ℕ \times ℕ,$ we consider (k, l) ∈ [mn, 2mn - 1] . Therefore, for all $x \in [0, \frac{1}{2 mn - 1}]$ , we have $\begin{matrix} D ({\bar{f}}_{kl} (x), \bar{0}) & = & e^{- mnx} \\ \geq & e^{- (2 mn - 1) x} \\ \geq & \frac{1}{e} (as x \in [0, \frac{1}{2 mn - 1}]) \geq \frac{1}{3} . \end{matrix}$

Thus, for all $x \in [0, \frac{1}{2 mn - 1}]$ , one obtains $\begin{matrix} A_{mn} (x) & = & δ_{2} ({(k, l) \in [mn, 2 mn - 1] \\ : D ({\bar{f}}_{kl} (x), \bar{0}) \geq \frac{1}{3}}) ⟶ 1 (\neq 0) . \end{matrix}$

Hence $({\bar{f}}_{mn})$ is not equi-statistically convergent to $\bar{0}$ on [0, 1] .

Example 3.21. Let us define a DSFVF $({\bar{f}}_{mn})$ by ${\bar{f}}_{mn} (x) = {\begin{matrix} \bar{(\frac{m^{2} n^{2} x}{1 + m^{3} n^{3} x^{2}})}, & for x \in [\frac{1}{mn + 2}, \frac{1}{mn + 1}]; \\ \bar{0} & otherwise . \end{matrix}$

Then, for every x ∈ [0, 1], we have $δ_{2} ({(m, n) \in ℕ \times ℕ : {\bar{f}}_{mn} (x) \neq \bar{0}}) = 0 .$

Therefore, for every ɛ > 0, $\begin{matrix} \frac{1}{pq} | {m \leq p, n \leq q : D ({\bar{f}}_{mn} (x), \bar{0}) \geq ɛ} | \\ \leq \frac{1}{pq} | {(m, n) \in ℕ \times ℕ : {\bar{f}}_{mn} (x) \neq \bar{0}} \\ \cap {(1, 1), (2, 2), . . ., (m, n)} | \leq \frac{1}{pq} ⟶ 0 . \end{matrix}$

This yields that $({\bar{f}}_{mn})$ is pointwise statistically convergent to $\bar{f} = \bar{0} .$ But $\begin{matrix} sup_{x \in [0, 1]} D ({\bar{f}}_{mn} (x), \bar{0}) & = & sup_{x \in [0, 1]} \frac{m^{2} n^{2} x}{1 + m^{3} n^{3} x^{2}} \\ = & sup_{x \in [0, 1]} \frac{1}{\frac{1}{m^{2} n^{2} x} + mnx} \neq 0, \end{matrix}$ for all $m, n \in ℕ$ and hence $({\bar{f}}_{mn})$ is not uniformly statistically convergent to $\bar{0}$ on [0, 1] .

Theorem 3.22. A DSFVF $({\bar{f}}_{mn})$ is uniformly statistically convergent to a FVF $\bar{f}$ on [a, b] if and only if $[{\bar{f}}_{mn} (x)]_{α}$ is uniformly statistically convergent to $[\bar{f} (x)]_{α}$ with respect to α and x .

Proof. Let ɛ > 0 be given. Suppose that ${\bar{f}}_{m n} \overset{u S_{2}}{⇉} \bar{f}$ on [a, b] . Then, there exists M ∈ T such that for all $(m, n) \in ℕ \times ℕ \ M$ , we have $D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ$ holds for all x ∈ [a, b] which gives $sup_{α \in [0, 1]} max {| ({\bar{f}}_{mn} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} |, | ({\bar{f}}_{mn} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} |} < ɛ$ for all x ∈ [a, b]. Therefore, we obtain $| ({\bar{f}}_{mn} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | \leq D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ$ and $| ({\bar{f}}_{mn} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} | \leq D ({\bar{f}}_{mn} (x), \bar{f} (x)) < ɛ .$ for all α ∈ [0, 1] and for all x ∈ [a, b]. Since $[{\bar{f}}_{mn} (x))]_{α} = [({\bar{f}}_{mn} (x))_{α}^{-}, ({\bar{f}}_{mn} (x))_{α}^{+}]$ and $[\bar{f} (x))]_{α} = [(\bar{f} (x))_{α}^{-}, (\bar{f} (x))_{α}^{+}] .$ Then, for every ɛ > 0 and for all x ∈ [a, b] , we have $\begin{matrix} {(m, n) \in ℕ \times ℕ : | ({\bar{f}}_{mn} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | \geq ɛ} \\ \subseteq {(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq ɛ} \end{matrix}$ and $\begin{matrix} {(m, n) \in ℕ \times ℕ : | ({\bar{f}}_{mn} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} | \geq ɛ} \\ \subseteq {(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq ɛ} . \end{matrix}$

This implies that $[{\bar{f}}_{mn} (x)]_{α}$ is uniformly statistically convergent to $[\bar{f} (x)]_{α}$ with respect to α and x .

Next, suppose that $[{\bar{f}}_{mn} (x)]_{α}$ is uniformly statistically convergent to $[\bar{f} (x)]_{α}$ with respect to α and x . Then, for any ɛ > 0 and all x ∈ [a, b] , we are writing $\begin{matrix} P & = & {(m, n) \in ℕ \times ℕ : \\ | ({\bar{f}}_{mn} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | \geq ɛ} \\ Q & = & {(m, n) \in ℕ \times ℕ : \\ | ({\bar{f}}_{mn} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} | \geq ɛ} \\ R & = & {(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq ɛ} . \end{matrix}$

Clearly, P, Q ∈ T. For every ɛ > 0 ∃ P ∈ T such that $(m, n) \in ℕ \times ℕ \ P$ , we have $\begin{matrix} | ({\bar{f}}_{mn} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | < ɛ \\ \forall α \in [0, 1] and \forall x \in [a, b] \end{matrix}$ and, similarly, for each ɛ > 0 ∃ Q ∈ T so that $(m, n) \in ℕ \times ℕ \ Q$ , $\begin{matrix} | ({\bar{f}}_{mn} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} | < ɛ \\ \forall α \in [0, 1] and \forall x \in [a, b] . \end{matrix}$

Therefore $\begin{matrix} D ({\bar{f}}_{mn} (x), \bar{f} (x)) = sup_{α \in [0, 1]} max {| ({\bar{f}}_{mn} (x))_{α}^{-} \\ - (\bar{f} (x))_{α}^{-} |, | ({\bar{f}}_{mn} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} |} < ɛ, \end{matrix}$ for any x ∈ [a, b]. This implies that $(m, n) \in ℕ \times ℕ \ R \Rightarrow (ℕ \times ℕ \ P) \cap (ℕ \times ℕ \ Q) \subseteq (ℕ \times ℕ \ R) \Rightarrow R \subseteq P \cup Q$ which yields δ₂ (R) ≤ δ₂ (P) + δ₂ (Q) =0 . This completes the proof.

Theorem 3.23. Let $({\bar{f}}_{mn})$ be DSFVF and $\bar{f}$ be a FVF. If $({\bar{f}}_{mn})$ is equi-statistically convergent to $\bar{f}$ on [a, b] and each ${\bar{f}}_{mn} (m, n \in ℕ)$ are continuous at x₀ then $\bar{f}$ is continuous at x₀, where x₀ is fixed in [a, b].

Proof. Let ɛ > 0 be given. Since $({\bar{f}}_{mn})$ is equi-statistically convergent to $\bar{f}$ on [a, b], we can find numbers $i, j \in ℕ$ such that $\begin{matrix} δ_{ij} ({(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq \frac{ɛ}{3}}) < \frac{1}{2}, \end{matrix}$ for all x ∈ [a, b]. Consider the set $\begin{matrix} P (x) = {(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) < \frac{ɛ}{3}}, \end{matrix}$ for x ∈ [a, b]. Therefore, we have $δ_{ij} (P (x)) > \frac{1}{2}$ for all x ∈ [a, b]. Also since each ${\bar{f}}_{mn} (m, n \in ℕ)$ are continuous at x₀, then for some δ > 0, $\begin{matrix} x_{0} - δ < x < x_{0} + δ \\ \Rightarrow D ({\bar{f}}_{pq} (x), {\bar{f}}_{pq} (x_{0})) < \frac{ɛ}{3}, \end{matrix}$ for all p ∈ {1, 2, . . . , m}; q ∈ {1, 2, 3, . . . , n}. For fix x ∈ (x₀ - δ, x₀ + δ) , since $δ_{ij} (P (x)) > \frac{1}{2},$ and $δ_{ij} (P (x_{0})) > \frac{1}{2},$ we can find (k, l) ∈ P (x) ∩ P (x₀) . Therefore, we have $\begin{matrix} D (\bar{f} (x), \bar{f} (x_{0})) \\ \leq D (\bar{f} (x), {\bar{f}}_{kl} (x)) + D ({\bar{f}}_{kl} (x), {\bar{f}}_{kl} (x_{0})) \\ + D ({\bar{f}}_{kl} (x_{0}), \bar{f} (x_{0})) \\ < ɛ / 3 + ɛ / 3 + ɛ / 3 = ɛ . \end{matrix}$

Thus $x_{0} - δ < x < x_{0} + δ \Rightarrow D (\bar{f} (x), \bar{f} (x_{0})) < ɛ,$ i.e., $\bar{f}$ is continuous at x₀ .

Theorem 3.24. Let $({\bar{f}}_{mn})$ be a DSFVF and $\bar{f}$ be a FVF. If $({\bar{f}}_{mn})$ is equi-statistically convergent to $\bar{f}$ on [a, b] . Then the following hold:

${eS}_{2} - lim_{x \to x_{0}} {\bar{f}}_{mn} (x) = lim_{x \to x_{0}} \bar{f} (x) .$

${uS}_{2} - lim_{x \to x_{0}} {\bar{f}}_{mn} (x) = lim_{x \to x_{0}} \bar{f} (x) .$

If $({\bar{f}}_{mn})$ are continuous at some point x₀ ∈ [a, b] , then $\bar{f}$ is continuous x₀ .

Proof. (a) Let ɛ > 0 be given. Since $({\bar{f}}_{mn})$ is equi-statistically convergent to $\bar{f}$ on [a, b] , there exist $i, j \in ℕ$ such that $\begin{matrix} δ_{ij} ({(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq \frac{ɛ}{3}}) < \frac{1}{2}, \end{matrix}$ for all x ∈ [a, b]. Consider the set $\begin{matrix} P (x) = {(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) < \frac{ɛ}{3}}, \end{matrix}$ for x ∈ [a, b]. Therefore, we obtain $δ_{ij} (P (x)) > \frac{1}{2}$ for all x ∈ [a, b]. Let $lim_{x \to x_{0}} {\bar{f}}_{mn} (x) = {\bar{v}}_{mn} .$ Then, for some δ₁ > 0, we have $x_{0} - δ < x < x_{0} + δ \Rightarrow D ({\bar{f}}_{mn} (x), {\bar{v}}_{mn}) < \frac{ɛ}{3} .$

Denote $lim_{x \to x_{0}} \bar{f} (x) = \bar{v} .$ Then, for some δ₂ > 0, we have $x_{0} - δ < x < x_{0} + δ \Rightarrow D (\bar{f} (x), \bar{v}) < \frac{ɛ}{3} .$

We take δ = max {δ₁, δ₂} and (m, n) ∈ P (x) . Then, for δ > 0, we have $\begin{matrix} D ({\bar{v}}_{mn}, \bar{v}) & \leq & D ({\bar{v}}_{mn}, {\bar{f}}_{mn} (x)) + D ({\bar{f}}_{mn} (x), \bar{f} (x)) \\ + D (\bar{f} (x), \bar{v}) \\ < & ɛ / 3 + ɛ / 3 + ɛ / 3 = ɛ \end{matrix}$ which holds for all x satisfying x ∈ (x₀ - δ, x₀ + δ). Thus $x_{0} - δ < x < x_{0} + δ \Rightarrow D ({\bar{v}}_{mn}, \bar{v}) < ɛ .$

Since ɛ > 0 is arbitrary, therefore we have ${eS}_{2} - lim_{m, n \to \infty} {\bar{v}}_{mn} = \bar{v} .$

(b) It follows from the fact that equi-statistical convergence implies uniform statistical convergence.

Remark 3.25. The above Theorem 3.24 does not hold for pointwise convergence. To prove this assertion, we shall consider the following example.

Example 3.26. Let us consider DSFVF $({\bar{f}}_{mn})$ which is defined by ${\bar{f}}_{mn} (x) = \bar{(x^{mn})}, x \in [0, 1] .$

We see that $[{\bar{f}}_{mn} (x)]_{α} = x^{mn}$ for each α ∈ [0, 1]. Therefore $({\bar{f}}_{mn})$ is pointwise convergent and hence pointwise statistically convergent, i.e., ${\bar{f}}_{mn} (x) \overset{p S_{2}}{\to} \bar{f} (x)$ as m, n → ∞ , where $\bar{f} (x)$ is given by $\begin{matrix} \bar{f} (x) = {\begin{matrix} \bar{0}, & for x \in [0, 1); \\ \bar{1} & for x = 1 . \end{matrix} \end{matrix}$

Clearly $({\bar{f}}_{mn} (x))$ are continuous in [0, 1] , but the limit function $\bar{f} (x)$ is not.

Theorem 3.27. A DSFVF $({\bar{f}}_{mn})$ is equi-statistically convergent to a FVF $\bar{f}$ if and only if $[{\bar{f}}_{mn} (x)]_{α} \overset{e S_{2}}{⇉} [\bar{f} (x)]_{α}$ for x ∈ [a, b], 0 ≤ α ≤ 1 .

Proof. Suppose that $({\bar{f}}_{mn})$ is equi-statistically convergent to $\bar{f} .$ Then, for any ɛ > 0 and β > 0 there exist $k, l \in ℕ$ for all i ≥ k, j ≥ l and for any x ∈ [a, b] such that $\begin{matrix} δ_{ij} ({(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq ɛ}) < β \end{matrix}$ which yields $\begin{matrix} δ_{ij} ({(m, n) \in ℕ \times ℕ : sup_{α \in [0, 1]} max {| ({\bar{f}}_{mn} (x))_{α}^{-} \\ - (\bar{f} (x))_{α}^{-} |, | ({\bar{f}}_{mn} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} |} \geq ɛ}) < β \end{matrix}$ for any α ∈ [0, 1]. Therefore, we obtain $\begin{matrix} δ_{ij} ({(m, n) \in ℕ \times ℕ : | ({\bar{f}}_{mn} (x))_{α}^{-} \\ - (\bar{f} (x))_{α}^{-} | \geq ɛ}) < β \end{matrix}$ and $\begin{matrix} δ_{ij} ({(m, n) \in ℕ \times ℕ : | ({\bar{f}}_{mn} (x))_{α}^{+} \\ - (\bar{f} (x))_{α}^{+} | \geq ɛ}) < β, \end{matrix}$ for any α ∈ [0, 1]. Hence $[{\bar{f}}_{mn} (x)]_{α} \overset{e S_{2}}{⇉} [\bar{f} (x)]_{α}$ for any x ∈ [a, b] and 0 ≤ α ≤ 1 .

Next, we assume that $[{\bar{f}}_{mn} (x)]_{α} \overset{e S_{2}}{⇉} [\bar{f} (x)]_{α}$ for any x ∈ [a, b] and any 0 ≤ α ≤ 1 . Let ɛ > 0 and β > 0 be given. Then, there exist $k_{1}, l_{1} \in ℕ$ such that $\begin{matrix} δ_{ij} ({(m, n) \in ℕ \times ℕ : | ({\bar{f}}_{mn} (x))_{α}^{-} \\ - (\bar{f} (x))_{α}^{-} | \geq ɛ}) < β, \end{matrix}$ for all i ≥ k₁, j ≥ l₁ and any x ∈ [a, b]. Similarly, there exist $k_{2}, l_{2} \in ℕ$ such that $\begin{matrix} δ_{ij} ({(m, n) \in ℕ \times ℕ : | ({\bar{f}}_{mn} (x))_{α}^{+} \\ - (\bar{f} (x))_{α}^{+} | \geq ɛ}) < β, \end{matrix}$ for all i ≥ k₂, j ≥ l₂ and any x ∈ [a, b]. Take k = max {k₁, k₂} and l = max {l₁, l₂} . Then $\begin{matrix} δ_{ij} ({(m, n) \in ℕ \times ℕ : | ({\bar{f}}_{mn} (x))_{α}^{-} \\ - (\bar{f} (x))_{α}^{-} | \geq ɛ}) < β \end{matrix}$ and $\begin{matrix} δ_{ij} ({(m, n) \in ℕ \times ℕ : | ({\bar{f}}_{mn} (x))_{α}^{+} \\ - (\bar{f} (x))_{α}^{+} | \geq ɛ}) < β, \end{matrix}$ hold for all i ≥ k, j ≥ l and any x ∈ [a, b]. Therefore, we obtain $\begin{matrix} δ_{ij} ({(m, n) \in ℕ \times ℕ : sup_{α \in [0, 1]} max {| ({\bar{f}}_{mn} (x))_{α}^{-} \\ - (\bar{f} (x))_{α}^{-} |, | ({\bar{f}}_{mn} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} |} \geq ɛ}) < β \end{matrix}$ and hence $δ_{ij} ({(m, n) \in ℕ \times ℕ : D ({\bar{f}}_{mn} (x), \bar{f} (x)) \geq ɛ}) < β,$ for all i ≥ k, j ≥ l and any x ∈ [a, b]. This completes the proof.

4 Application

The significance of the concept of summability has been strikingly demonstrated in various contexts, for example, in analytic continuation, quantum mechanics, probability theory, Fourier analysis, approximation theory, and fixed point theory. Also the theory of statistical summability on one hand and approximation theory on the other hand are successfully linked to obtain necessary and sufficient conditions for the uniform convergence of L_n (f) $(n \in ℕ)$ to a function f by using the test function f_i defined by f_i (x) = xⁱ (i = 0, 1, 2), where $(L_{n})_{n \in ℕ}$ is a sequence of positive linear operators on the space of real valued continuous functions on a compact subset X of real numbers and such approximation theorem was first introduced by Korovkin [18] admittedly the classical Korovkin theorem is one of the most interesting result in approximation theory.

We write C [a, b] and C_F [a, b] for the set of all real-valued and fuzzy-valued continuous functions on [a, b], respectively. Let T : C_F [a, b] → C_F [a, b] be an operator. Then, the operator T is said to be fuzzy liner, if for every $μ, ν \in ℝ$ , f, g ∈ C_F [a, b] and x ∈ [a, b], $T (μ ⊙ f \oplus ν ⊙ g; x) = μ ⊙ T (f; x) \oplus ν ⊙ T (g; x)$ holds. Also, T is called fuzzy positive linear operator if it is fuzzy linear and the condition T (f ; x) ⪯ T (g ; x) is satisfied for any f, g ∈ C_F [a, b] and for all x ∈ [a, b] with f (x) ⪯ g (x). The fuzzy analogue of classical Korovkin approximation theorem was introduced by Anastassiou [4] as follows:

Theorem 4.1. Let (T_n) be a sequence of fuzzy positive linear operators from C_F [a, b] into itself. Assume that there exists a corresponding sequence $({\tilde{T}}_{n})$ of positive linear operators from C [a, b] into itself with the property ${(T_{n} (f; x))}_{α}^{\pm} = T_{n} (f_{α}^{\pm}, x)$ for all x ∈ [a ; b], α ∈ [0, 1] and f ∈ C_F [a, b]. Assume further that $lim_{n} ∥ {\tilde{T}}_{n} (f_{i}; x) - f_{i} (x) ∥ = 0 for each i = 0, 1, 2,$ where f₀ (x) =1, f₁ (x) = x and f₂ (x) = x². Then, for all f ∈ C_F [a, b], we have $lim_{n} D^{*} (T_{n} (f; x), f (x)) = 0 .$

The Korovkin’s type approximation theorem for fuzzy positive linear operators through statistical convergence was presented by Anastassiou and Duman [5]. Note that, in contrast to the case for single sequences, a convergent double sequence need not be bounded, so the studies on statistical convergence of double sequences has a rapid growth and an emerging area in mathematical research. We conclude that our results are more general than the one proved earlier for single sequences by Gong et al. [16]. As an application, researchers who linked two theories such as the theory of approximation and the theory of statistical summability may prove fuzzy analogue of Korovkin’s type approximation theorem for several test functions by using our convergence methods.

Footnotes

Acknowledgments

The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.

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