Characterization of matrix classes involving some sets of summable double sequences of fuzzy numbers

Abstract

In this paper, we first study the notion of almost convergence of double sequences of fuzzy numbers using the idea of Pringsheim’s convergence and compare with the set of bounded double sequences of fuzzy numbers. The primary aim is to characterize matrix classes involving some sets of double sequences of fuzzy numbers and also the set of almost convergent double sequence of fuzzy numbers. The approach adopted in the investigation of main results should be helpful to develop theories to deal with fuzzification of summability theory and its applications.

Keywords

Double sequences of fuzzy numbers Pringsheim’s convergence and almost convergence algebraic duals four-dimensional infinite matrix of fuzzy numbers matrix transformations

1 Introduction and definitions

The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh [31] as a generalization of the classical notion of set and subsequently several authors have discussed various aspects of the theory and applications of fuzzy sets such as fuzzy topological spaces, similarity relations and fuzzy orderings, fuzzy measures of fuzzy events, fuzzy mathematical programming, etc. Working as a powerful mathematical tool for approximate reasoning, fuzzy set theory plays a significant role in decision making in complex phenomena, which are difficult to be described by traditional mathematics. Matloka [25] introduced bounded and convergent sequences of fuzzy numbers and studied their properties. Later on different types of spaces of sequences of fuzzy numbers have been studied for various purposes by Diamond and Kloeden [41], Kizmaz [24], Nanda [43], Dutta and Gogoi [22] and many others. For studies regarding fuzzy topological spaces and their algebraic properties, fuzzy decision making methods we refer to Qiu and Zhang [3], Qiu, Zhang and Lu [4], Qiu, Lu, Zhang and Lan [5], Ma, Liu and Zhan [45], Zhan and Zhu [27], Zhan, Ali and Mehmood [28], Zhan, Liu and Herawan [29], Zhan, Yu and Fotea [30], etc.

In recent years, the concept of summability methods of sequences of fuzzy numbers has obtained a lot of interests from various researchers. In [15], Nuray and Savaş have extended the concept of statistical convergence defined independently by Fast [23] and Schoenberg [26], to sequences of fuzzy numbers and showed that a sequence of fuzzy numbers is statistically convergent if and only if it is statistically Cauchy. Some interesting results related to statistical convergence of sequences of fuzzy numbers and Tauberian conditions can be found in Nuray [14], Nuray and Savaş [15], Savaş [7], Altinok [18], Altinok, Çolak and Altin [19], Altinok, Et and Çolak [20], Altinok, Altin and Isik [21], Et, Altinok and Altin [33], Mursaleen, Srivastava and Sharma [36], and Yavuz [9, 10], etc.

The initial works on double sequences of real or complex terms are found in Bromwich [44]. Hardy [17] introduced the notion of regular convergence for double sequences of real or complex terms. Later on Basarir and Solancan [32], Başar [11], Mòricz [12], Mursaleen and Mohiuddine [35] and several other authors investigated the works on double sequence. In [6] Savaş extended this notion for convergent double sequences of fuzzy numbers and studied some of their properties. He further showed that the set of all convergent double sequences of fuzzy numbers is complete. The fuzzy analogue of statistically convergent and statistically Cauchy double sequences of fuzzy numbers were introduced and studied by Savaş and Mursaleen [8]. For some interesting works on statistical convergence, summability methods and Tauberian theorems for double sequences of fuzzy numbers, we refer to Savaş [6], Savaş and Mursaleen [8], Mursaleen and Mohiuddine [34], Önder, Çanak and Totur [46], Talo and Bayazit [40].

Definition 1.1. (Goetschel and Voxman [42]) A fuzzy number is a fuzzy set on the real axis, i.e., a mapping $u : ℝ \to [0, 1]$ , which satisfies the following four conditions:

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

u is fuzzy convex, i.e., u [λx+ (1 - λ) y] ≥ min { u (x), u (y) } for all $x, y \in ℝ and for all λ \in [0, 1] .$

u is upper semi-continuous.

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

We denote the set of all fuzzy numbers on $ℝ by E^{1}$ and call it the space of fuzzy numbers. λ-level set [u] _λ of u ∈ E¹ is defined by ${[u]}_{λ} = {\begin{matrix} {t \in ℝ : u (t) \geq 0}, & (0 < λ \leq 1), \\ \bar{{t \in ℝ : u (t) > 0,},} & (λ = 0) . \end{matrix}$

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

Definition 1.2. (Talo and Başar [39]) Let u, v, w ∈ E¹ and $k \in ℝ .$ Then the operations addition, scalar multiplication and product defined on E¹ by $\begin{matrix} u + v = w \\ \Leftrightarrow {[w]}_{λ} = {[u]}_{λ} + {[v]}_{λ} \end{matrix}$ for all λ ∈ [0, 1], $\begin{matrix} \Leftrightarrow w^{-} (λ) = u^{-} (λ) + v^{-} (λ), \\ w^{+} (λ) = u^{+} (λ) + v^{+} (λ) \end{matrix}$ for all λ ∈ [0, 1], $[ku]_{λ} = k [u]_{λ} for all λ \in [0, 1]$ and $uv = w \Leftrightarrow {[w]}_{λ} = {[u]}_{λ} {[v]}_{λ}$ for all λ ∈ [0, 1], where it is immediate that $\begin{matrix} w^{-} (λ) & = & min {u^{-} (λ) v^{-} (λ), u^{-} (λ) v^{+} (λ), \\ u^{+} (λ) v^{-} (λ), u^{+} (λ) v^{+} (λ)} \end{matrix}$ and $\begin{matrix} w^{+} (λ) & = & max {u^{-} (λ) v^{-} (λ), u^{-} (λ) v^{+} (λ), \\ u^{+} (λ) v^{-} (λ), u^{+} (λ) v^{+} (λ)} \end{matrix}$ for all λ ∈ [0, 1].

Definition 1.3. Let $W$ be the set of all closed bounded intervals A of real numbers such that A = [A₁, A₂]. Define the relation $d on W$ as follows: $d (A, B) = max {| A_{1} - B_{1} |, | A_{2} - B_{2} |} .$

where B = [B₁, B₂] ∈ 𝒲. Then $(W, d)$ is a complete metric space (see Diamond and Kloeden [41], Nanda [43]). Then Bede [2] defined the metric D on E¹ by means of Hausdorff metric d as $\begin{matrix} D (u, v) & = & sup_{λ \in [0, 1]} d ({[u]}_{λ}, {[v]}_{λ}) \\ = & sup_{λ \in [0, 1]} max {| u^{-} (λ) - v^{-} (λ) |, \\ | u^{+} (λ) - v^{+} (λ) |} . \end{matrix}$

One can see that $\begin{matrix} D (u, \bar{0}) & = & sup_{λ \in [0, 1]} \max {| u^{-} (λ) |, | u^{-} (λ) |} \\ = & max {| u^{-} (0) |, | u^{-} (0) |} . \end{matrix}$

The partial ordering relation on E¹ is defined as follows: $\begin{matrix} u ≼ v \Leftrightarrow {[u]}_{λ} ≼ {[v]}_{λ} \\ \Leftrightarrow u^{-} (λ) \leq v^{-} (λ) and u^{+} (λ) \leq v^{+} (λ) \end{matrix}$ for all λ ∈ [0, 1].

Definition 1.4. (Talo and Başar [39]) u ∈ E¹ is a non-negative fuzzy number if and only if u (x) = 0 for all x < 0. It is immediate that $u ≽ \bar{0}$ if u is a non-negative fuzzy number.

Lemma 1.1. (Bede [2]) Let u, v, w ∈ E¹ and $k \in ℝ .$ Then

(E¹, D) is a complete metric space.

D (ku, kv) = |k|D (u, v).

D (u + v, w + v) = D (u, w).

D (u + v, w + z) ≤ D (u, w) + D (v, z).

$| D (u, \bar{0}) - D (v, \bar{0}) | \leq D (u, v) \leq D (u, \bar{0}) + D (v, \bar{0})$ .

Lemma 1.2. (Talo and Başar [39]) The following statements hold:

$D (uv, \bar{0}) \leq D (u, \bar{0}) D (v, \bar{0}) for all u, v \in E^{1} .$

If u_k→ u, as k → ∞ then $D (u_{k}, \bar{0}) \to D (u, \bar{0})$ as k → ∞.

Lemma 1.3. (Talo and Başar [39],) If $u_{k} : ℝ \to [0, 1]$ are fuzzy sets for $k \in ℕ$ such that λ-level sets [u_k] _λ are nonempty, bounded and closed for every λ ∈ (0, 1], then the λ-level sets of sum are nonempty, bounded amd closed, and ${[\sum_{k = 0}^{n} u_{k}]}_{λ} = \sum_{k = 0}^{n} {[u_{k}]}_{λ} .$

Definition 1.5. (Savaş [6]) A double sequence x = (x_mn) of fuzzy numbers is a function $x : ℕ \times ℕ \to E^{1}$ . The fuzzy number x_mn denotes the value of the function at a point $(m, n) \in ℕ \times ℕ$ and is called the (m, n)-term of the double sequence.

By $2 W^{F}$ we denote the set of all double sequences of fuzzy numbers.

Recently Talo and Başar [39] introduced the notion of α-, β- and γ-duals of sets of single sequences of fuzzy numbers and computed the α-, β- and γ-duals of some classical sets of single sequences of fuzzy numbers.

Definition 1.6. (Savaş [6]) A double sequence of fuzzy numbers x = (x_mn) is said to be bounded if there exists a positive number M such that $D (x_{mn}, \bar{0}) < M$ for all $m, n \in ℕ$ , i.e, if $sup_{m, n \in ℕ} D (x_{mn}, \bar{0}) < \infty, where ℕ = {0, 1, 2, \dots} .$

We denote the set of all bounded double sequences of fuzzy numbers by $M_{u}^{F}$ . Also by $M_{U}^{F}$ , we denote the set of all bounded double sequences of nonnegtive fuzzy numbers, i.e., if $x = (x_{mn}) \in M_{U}^{F}$ then x = (x_mn) is bounded and $x_{mn} ≽ \bar{0}$ for all $m, n \in ℕ$ .

In [1], Pringsheim gave a notion of convergence of a double sequence known as Pringsheim convergence. In [6], Savaş extended this notion of convergence in the Pringsheim’s sense to double sequence of fuzzy numbers.

Definition 1.7. (Savaş [6]) Consider the double sequence of fuzzy numbers $(x_{m n}) \in 2 W^{F}$ . If for every ɛ > 0 there exists $n_{0} = n_{0} (ɛ) \in ℕ$ and l ∈ E¹ such that D (x_mn, l) < ɛ for all m, n > n₀, then we say that the double sequence is said to be convergent in the Pringsheim’s sense to the limit l and write $P - lim_{m, n \to \infty} x_{mn} = l .$

The number l is called the Pringsheim limit of (x_mn).

Precisely, a double sequence of fuzzy numbers (x_mn) converges to a fuzzy number l ∈ E¹ if x_mn tends to l as both m and n tend to ∞ independent of one another.

We denote the set of all P-convergent double sequences of fuzzy numbers by $C_{P}^{F}$ . Also by $C_{P 0}^{F},$ we denote all P-convergent to $\bar{0}$ double sequence of fuzzy numbers. In this case for every ɛ > 0 there exists $n_{0} = n_{0} (ɛ) \in ℕ$ such that $D (x_{mn}, \bar{0}) < ɛ$ for all m, n > n₀ and we write, $P - lim_{m, n \to \infty} x_{mn} = \bar{0} .$

Lorentz [16] introduced the concept of almost convergence for single sequences and for double sequences, it was introduced by Mòricz and Rhoades [13].

Definition 1.8. A double sequence of fuzzy numbers $(x_{m n}) \in 2 W^{F}$ is said to be almost convergence to a generalized limit l ∈ E¹ if for every ɛ > 0 there exists $n_{0} = n_{0} (ɛ) \in ℕ$ and l ∈ E¹ such that for allq, r > n₀, $D (\frac{1}{qr} \sum_{k = m}^{q + m - 1} \sum_{l = n}^{r + n - 1} x_{kl}, l) < ɛ,$ uniformly in m, n and we write $P - lim_{q, r \to \infty} \frac{1}{qr} \sum_{k = m}^{q + m - 1} \sum_{l = n}^{r + n - 1} x_{kl} = l,$ uniformly in m, n.

In this case, l is called the f₂ - limit of x.

We denote the set of all almost convergent double sequences of fuzzy numbers by $C_{f}^{F} .$

In case of double sequences of fuzzy numbers, convergence in the Pringsheim’s sense does not imply the boundedness, i.e, there are sequences in the space $C_{P}^{F}$ but not in $M_{u}^{F}$ .

Example 1.1. We define the sequence x = (x_mn) by $x_{mn} = {\begin{matrix} u_{n}, & m = 0, n \in N \\ \bar{0}, & otherwise \end{matrix} .$ where, $u_{n} = \sum_{k = 0}^{n} v_{k}$ and $v_{k} (t) = {\begin{matrix} 3 + \sqrt{k + 1} t, \frac{- 3}{\sqrt{k + 1}} \leq t \leq \frac{- 2}{\sqrt{k + 1}}, \\ 2 + \sqrt{k + 1} t, \frac{- 2}{\sqrt{k + 1}} \leq t \leq \frac{- 1}{\sqrt{k + 1}}, \\ 0 otherwise, \end{matrix}$ for all $k \in ℕ .$ It is clear that (x_mn) is P-convergent to $\bar{0} .$ Also we have that the endpoints of the λ-level set of the double sequence x = x_mn of fuzzy numbers are $x_{mn}^{-} (λ) = {\begin{matrix} u_{n}^{-} (λ) & if m = 0, n \in N \\ 0 & otherwise, \end{matrix}$ and $x_{mn}^{+} (λ) = {\begin{matrix} u_{n}^{+} (λ) & if m = 0, n \in N \\ 0 & otherwise . \end{matrix}$

From this, we can see that $\begin{matrix} sup_{m, n \in ℕ} D (x_{mn}, \bar{0}) \\ = sup_{m, n \in ℕ} sup_{λ \in [0, 1]} max {| x_{mn}^{-} (λ) - 0 |, | x_{mn}^{+} (λ) - 0 |} \\ = sup_{m, n \in ℕ} sup_{λ \in [0, 1]} max {| x_{mn}^{-} (λ) |, | x_{mn}^{+} (λ) |} \\ = sup_{m, n \in ℕ} max {| x_{mn}^{-} (0) |, | x_{mn}^{+} (0) |} \\ = sup_{n \in ℕ} max {| u_{n}^{-} (0) |, | u_{n}^{+} (0) |} . \end{matrix}$

Now, since the sequences $(u_{n}^{-} (0)) = (\sum_{k = 0}^{n} \frac{- 3}{\sqrt{k + 1}})$ and $(u_{n}^{+} (0)) = (\sum_{k = 0}^{n} \frac{- 1}{\sqrt{k + 1}})$ are divergent as n → ∞, the double sequence x = (x_mn) of fuzzy numbers is not bounded. Thus $x = (x_{mn}) \in C_{P}^{F} ∖ M_{u}^{F} .$

Following Talo and Başar [39], we give the following definitions, which we will use in the later part of the paper.

Definition 1.9. Let $(x_{k l}) \in 2 W^{F}$ Then the expression $\sum_{k, l} x_{kl}$ is called a series corresponding to the double sequence (x_kl) of fuzzy number. Denote $s_{mn} = \sum_{k = 0, l = 0}^{m, n} x_{kl} for all m, n \in ℕ .$

If the sequence (s_mn) converges to a fuzzy number u, then we say that the series $\sum_{k, l} x_{kl}$ converges to x and write $\sum_{k, l} x_{kl} = x .$

Definition 1.10. Let ₂μ^F be a space of double sequences of fuzzy numbers converging with respect to Pringsheim’s limit. The sum of a double series $\sum_{k, l} x_{kl}$ with respect to this rule is defined by $P - lim_{m, n \to \infty} \sum_{k, l} x_{kl} .$ which implies as m, n→ ∞ that $\sum_{k = 0, l = 0}^{m, n} x_{kl}^{-} (λ) \to x^{-} (λ)$ and $\sum_{k = 0, l = 0}^{m, n} x_{kl}^{+} (λ) \to x^{+} (λ),$ uniformly in λ ∈ [0, 1]. Conversely, if the fuzzy numbers $x_{kl} = {(x_{kl}^{-} (λ), x_{kl}^{+} (λ)) : λ \in [0, 1]},$ $\sum_{k, l} x_{kl}^{-} (λ) = x^{-} (λ),$ and $\sum_{k, l} x_{kl}^{+} (λ) = x^{+} (λ)$ converge uniformly in λ ∈ [0, 1], then x = {(x^- (λ), x⁺ (λ) : λ ∈ [0, 1]} defines a fuzzy number such that $x = \sum_{k = 0, l = 0}^{\infty, \infty} x_{kl} .$

Otherwise, we say the series of fuzzy numbers diverges. Additionally, the series $\sum_{k, l} x_{kl}$ of fuzzy numbers is bounded if the sequence (s_mn) is bounded.

Definition 1.11. Throughout the paper, the summations without limits run from 0 to ∞, for example $\sum_{k, l} x_{kl}$ means that $\sum_{k = 0}^{\infty} \sum_{l = 0}^{\infty} x_{kl} .$

We denote by ₂cs^F and ₂bs^F the set of all convergent and bounded series of fuzzy numbers respectively.

Now we define α -, β- and γ-duals of a set ₂E^F ∈ ₂W^F which are respectively denoted by {₂E^F } ^α, { ₂E^F } ^β and {₂E^F } ^γ as follows: $\begin{matrix} {2 E^{F}}^{α} & = & {(a_{m n}) \in 2 W^{F} : (a_{m n} b_{m n}) \in 2 ℓ_{1}^{F}, \\ forall (b_{m n}) \in 2 E^{F}}, \\ {2 E^{F}}^{β} & = & {(a_{m n}) \in 2 W^{F} : (a_{m n} b_{m n}) \in 2 c s^{F}, \\ forall (b_{m n}) \in 2 E^{F}}, \\ {2 E^{F}}^{γ} & = & {(a_{m n}) \in 2 W^{F} : (a_{m n} b_{m n}) \in 2 b s^{F}, \\ forall (b_{m n}) \in 2 E^{F}} . \end{matrix}$

2 Matrix transformations between some sets of double sequences of fuzzy numbers

An infinite matrix is one of the most general linear operators between two sequence spaces. The study of theory of matrix transformations has always been of great interest to mathematicians in the study of sequence spaces, which is motivated by special results in summability theory. Yeşilkayagil and Başar [37] and Zeltser, Mursaleen and Mohiuddine [38] gave some matrix transformations between some sets of double sequences of real numbers.

Let A = (a_mnkl) be any four dimensional matrix of fuzzy numbers. Then a double sequence of fuzzy numbers x = (x_kl) is said to be P - summable with respect to A if and only if ${(Ax)}_{mn} = \sum_{k, l} a_{mnkl} x_{kl}$ (1) exists for each $m, n \in ℕ .$

Let $μ_{1}^{F}, μ_{2}^{F} \subset 2 W^{F}$ and A = (a_mnkl) be any four dimensional infinite matrix of fuzzy numbers. Then A is said to map $μ_{1}^{F}$ to $μ_{2}^{F}$ , denoted by $A \in (μ_{1}^{F} : μ_{2}^{F})$ if and only if the series on the right hand side of (1) converges in the Pringsheim’s sense. Also by $A \in (μ_{1}^{F} : μ_{2}^{F}; P)$ we denote that A preserves the P - limit, that is A - limit of x is equal to the limit of x for all $x = (x_{kl}) \in μ_{1}^{F} .$

Lemma 2.1. Every almost convergent fuzzy double sequence x = (x_kl) is bounded, i.e., $C_{f}^{F} \subset M_{u}^{F}$ but the converse may not be true.

Proof. Let us consider, $x = (x_{kl}) \in C_{f}^{F}$ and $q_{0}, r_{0} \in ℕ$ with $\frac{1}{qr} D (\sum_{k = s}^{s + q - 1} \sum_{l = t}^{t + r - 1} x_{kl}, \bar{0}) \leq 1$ (2) where $q \geq q_{0}, r \geq r_{0}, s, t \in ℕ$ be given. Which implies $\frac{1}{qr} sup_{λ \in [0, 1]} | \sum_{k = s}^{s + q - 1} \sum_{l = t}^{t + r - 1} x_{kl}^{-} (λ) | \leq 1,$ (3) and $\frac{1}{qr} sup_{λ \in [0, 1]} | \sum_{k = s}^{s + q - 1} \sum_{l = t}^{t + r - 1} x_{kl}^{+} (λ) | \leq 1 .$ (4)

Then, for $s, t \in ℕ$ and for all λ ∈ [0, 1], we have $D (x_{st}, \bar{0}) = sup_{λ \in [0, 1]} max {| x_{st}^{-} (λ) |, | x_{st}^{+} (λ) |} .$

Now, for $s, t \in ℕ$ and for all λ ∈ [0, 1], we have, $\begin{matrix} | x_{st}^{+} (λ) | & = & | \sum_{k = s}^{s + q_{0} - 1} \sum_{l = t}^{t + r_{0} - 1} x_{kl}^{+} (λ) \\ - \sum_{k = s + 1}^{s + q_{0} - 1} \sum_{l = t + 1}^{t + r_{0} - 1} x_{kl}^{+} (λ) | \\ = & | \sum_{k = s}^{s + q_{0} - 1} \sum_{l = t}^{t + r_{0} - 1} (x_{kl}^{+} (λ) - L^{+} (λ)) \\ - \sum_{k = s + 1}^{s + q_{0} - 1} \sum_{l = t + 1}^{t + r_{0} - 1} (x_{kl}^{+} (λ) - L^{+} (λ)) \\ + L^{+} (λ) | \\ \leq & q_{0} r_{0} + (q_{0} + 1) (r_{0} + 1) \\ + | L^{+} (λ) | [from (4)] . \end{matrix}$

Similarly, $| x_{st}^{-} (λ) | \leq q_{0} r_{0} + (q_{0} + 1) (r_{0} + 1) + | L^{-} (λ) | .$

Thus, we have $x = (x_{kl}) \in M_{u}^{F} .$

For the converse part, let us define x = (x_kl) by $x_{kl} : = (= 3 pt \begin{matrix} \bar{1} & \bar{0} & \bar{0} & \bar{1} & \bar{1} & \bar{1} & \bar{1} & . & . & . & . \\ \bar{1} & \bar{0} & \bar{0} & \bar{1} & \bar{1} & \bar{1} & \bar{1} & . & . & . & . \\ \bar{1} & \bar{0} & \bar{0} & \bar{1} & \bar{1} & \bar{1} & \bar{1} & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . \\ \begin{matrix} . \\ . \end{matrix} & \begin{matrix} . \\ . \end{matrix} & \begin{matrix} . \\ . \end{matrix} & \begin{matrix} . \\ . \end{matrix} & \begin{matrix} . \\ . \end{matrix} & \begin{matrix} . \\ . \end{matrix} & \begin{matrix} . \\ . \end{matrix} & \begin{matrix} . \\ . \end{matrix} & \begin{matrix} . \\ . \end{matrix} & \begin{matrix} . \\ . \end{matrix} & \begin{matrix} . \\ . \end{matrix} \end{matrix})$ i.e., in each row, there is one $\bar{1}$ , then two $\bar{0}$ s, then four $\bar{1}$ s, then eight $\bar{0}$ s, then sixteen $\bar{1}$ s, etc. Then, we can see that $x \notin C_{f}^{F}$ but $x \in M_{u}^{F} .$ This step terminates the proof. ■

The main aim of this section is to give some results characterizing matrix transformations involving some classes of double sequences of fuzzy numbers.

Theorem 2.1. Let A = (a_mnkl) be any four dimensional matrix of fuzzy numbers. Then $A = (a_{mnkl}) \in (M_{u}^{F} : M_{u}^{F})$ if and only if $sup_{m, n} \sum_{k, l} D (a_{mnkl}, \bar{0}) < \infty .$ (5)

Proof. Suppose that $A = (a_{mnkl}) \in (M_{u}^{F} : M_{u}^{F})$ and $x = (x_{kl}) \in M_{u}^{F} .$ Since Ax exists, the series $\sum_{k, l} a_{mnkl} x_{kl}$ converges for each fixed $m, n \in ℕ$ . Hence $Ax \in {M_{u}^{F}}^{β}$ for all $n \in ℕ .$

Let us define the sequence $x = (x_{kl}) \in M_{u}^{F}$ by x_kl : = χ_[-1,1] for all $k, l \in ℕ .$

Then, $Ax \in M_{u}^{F}$ which yields for all $m, n \in ℕ$ and for all λ ∈ [0, 1] that is $\begin{matrix} \sum_{k, l} {(a_{mnkl} x_{kl})}^{+} (λ) \\ = \sum_{k, l} \max {a_{mnkl}^{-} (λ) x_{kl}^{-} (λ), a_{mnkl}^{-} (λ) x_{kl}^{+} (λ), \\ a_{mnkl}^{+} (λ) x_{kl}^{-} (λ), a_{mnkl}^{+} (λ) x_{kl}^{+} (λ)} \\ = \sum_{k, l} \max {- a_{mnkl}^{-} (λ), + a_{mnkl}^{-} (λ), \\ - a_{mnkl}^{+} (λ) + a_{mnkl}^{+} (λ)} \\ = \sum_{k, l} max {| a_{mnkl}^{-} (λ) |, | a_{mnkl}^{+} (λ) |} . \end{matrix}$

In the special case λ = 0, the sequence $\begin{matrix} {\sum_{k, l} D (a_{mnkl}, \bar{0})} \\ = \sum_{k, l} max {| a_{mnkl}^{-} (0) |, | a_{mnkl}^{+} (0) |}, \end{matrix}$ for all $m, n \in ℕ$ is bounded which means (5) holds.

Conversely, suppose that (5) holds and $x = (x_{kl}) \in M_{u}^{F} .$ Then, since $A_{mn} \in {M_{u}^{F}}^{β}$ for each $m, n \in ℕ, Ax$ exists. Therefore, by using Lemma 1.2, together with the condition (5), we get $\begin{matrix} D ({(Ax)}_{mn}, \bar{0}) \\ = D (\sum_{k, l} a_{mnkl} x_{kl}, \bar{0}) \\ \leq \sum_{k, l} D (a_{mnkl} x_{kl}, \bar{0}) \\ \leq \sum_{k, l} D (a_{mnkl}, \bar{0}) D (x_{kl}, \bar{0}) \\ \leq sup_{k, l} D (x_{kl}, \bar{0}) \sum_{k, l} D (a_{mnkl}, \bar{0}) < \infty \end{matrix}$ i.e., $Ax \in M_{u}^{F}$ .

This step concludes the proof. ■

Theorem 2.2. Let A = (a_mnkl) be any four dimensional matrix of fuzzy numbers. Then $A \in (M_{u}^{F} : C_{P 0}^{F})$ if and only if $P - lim_{m, n \to \infty} a_{mnkl} = \bar{0} for each k, l \in ℕ,$ (6)for every $l \in ℕ,$ there exists K $\in ℕ$ such that, $a_{mnkl} = \bar{0} for all m, n, k > K,$ (7)for every k $\in ℕ,$ there exists L $\in ℕ$ such that, $a_{mnkl} = \bar{0} for all m, n, l > L .$ (8)

Proof. Suppose that the conditions (6–8) hold, and let $x = (x_{kl}) \in M_{u}^{F} .$ Let $n_{0} = max {K, L} \in ℕ .$ Hence using Lemma (1.2) and the fact that for every ɛ > 0 there exists t₀ = t₀ (ɛ) such that, for all m, n > t₀, we have $\begin{matrix} D (\sum_{k, l} a_{mnkl} x_{kl}, \bar{0}) \\ \leq \sum_{k, l} D (a_{mnkl} x_{kl}, \bar{0}) \\ \leq \sum_{k, l}^{D} (a_{mnkl}, \bar{0}) D (x_{kl}, \bar{0}) \\ \leq sup_{k, l} D (x_{kl}, \bar{0}) \sum_{k, l}^{D} (a_{mnkl}, \bar{0}) \\ = sup_{k, l} D (x_{kl}, \bar{0}) [\sum_{k = 0}^{t_{0}} \sum_{l = 0}^{t_{0}} D (a_{mnkl}, \bar{0}) \\ + \sum_{k = 0}^{t_{0}} \sum_{l = t_{0} + 1}^{\infty} D (a_{mnkl}, \bar{0}) \\ + \sum_{k = t_{0} + 1}^{\infty} \sum_{l = 0}^{t_{0}} D (a_{mnkl}, \bar{0}) \\ + \sum_{k = t_{0} + 1}^{\infty} \sum_{l = t_{0} + 1}^{\infty} D (a_{mnkl}, \bar{0})] \\ < ɛ sup_{k, l} D (x_{kl}, \bar{0}) {(t_{0} + 1)}^{2}, \end{matrix}$ for all m, n > t₀, that is $Ax \in C_{P 0}^{F} .$

Conversely, suppose that $A = (a_{mnkl}) \in (M_{u}^{F} : C_{P 0}^{F})$ and $x = (x_{kl}) \in M_{u}^{F},$ Then Ax exists and belongs to $C_{P 0}^{F} .$

Let us define the double sequences of fuzzy numbers $\begin{matrix} e^{kl} & = & (e_{mn}^{kl}) : = {\begin{matrix} \bar{1}, & (k, l) = (m, n) \\ \bar{0}, & otherwise, \end{matrix} \\ e^{l} & : = & \sum_{k} e^{kl}, \\ e_{k} & : = & \sum_{l} e^{kl}, \\ e & : = & \sum_{k, l} e^{kl} \end{matrix}$ for all $k, l, m, n \in ℕ .$

If we take x = e^kl for each k, l ∈ N, x = e^l, x = e_k and x = e, respectively, then we have $\begin{matrix} P - lim_{m, n \to \infty} {({Ae}^{kl})}_{mn} \\ = P - lim_{m, n \to \infty} a_{mnkl} = \bar{0}, \\ for each k, l \in ℕ, \\ P - lim_{m, n \to \infty} {({Ae}^{l})}_{mn} \\ = P - lim_{m, n \to \infty} \sum_{k} a_{mnkl} = \bar{0}, \\ for each l \in ℕ, \end{matrix}$ (9) $\begin{matrix} P - lim_{m, n \to \infty} ({Ae}_{k})_{mn} \\ = P - lim_{m, n \to \infty} \sum_{l} a_{mnkl} = \bar{0}, \\ for each k \in ℕ, \\ P - lim_{m, n \to \infty} {(Ae)}_{mn} \\ = P - lim_{m, n \to \infty} \sum_{k, l} a_{mnkl} = \bar{0} . \end{matrix}$ (10)

If the condition (9) is valid, then there exists $K \in ℕ$ such that $a_{mnkl} = \bar{0},$ for all m, n, k > K, where K may depend on l. Otherwise there exist three non-decreasing unbounded sequences of positive integers (m_j), (n_j) and (k_j) for $j \in ℕ$ such that $a_{m_{j} n_{j} k_{j} l} \neq \bar{0} for each l \in ℕ .$ If we take x = e^l and $D (a_{m_{j} n_{j} k_{j} l}, \bar{0}) > D (\sum_{i = 0}^{j - 1} a_{m_{j} n_{j} k_{i} l}, \bar{0}) + K,$ then we have $\begin{matrix} D (\sum_{i = 0}^{j} a_{m_{j} n_{j} k_{i} l}, \bar{0}) \\ \geq D (a_{m_{j} n_{j} k_{j} l}, \bar{0}) - D (\sum_{i = 0}^{j - 1} a_{m_{j} n_{j} k_{i} l}, \bar{0}) \\ > K, \end{matrix}$ i.e., $D (\sum_{i = 0}^{j} a_{m_{j} n_{j} k_{i} l}, \bar{0}) \to \infty as j \to \infty .$

Again using the relation $\begin{matrix} {({Ae}^{l})}_{m_{j} n_{j}} & = & \sum_{k} a_{m_{j} n_{j} kl} \\ = & lim_{r \to \infty} \sum_{k = 0}^{r} a_{m_{j} n_{j} kl}, \end{matrix}$ we get that $P - lim_{m, n \to \infty} {(Ax)}_{mn} ↛ \bar{0},$ a contradiction. Thus, we can say that condition (7) is necessary. We can obtain the necessity of condition (8) by using similar arguments. Also, we see that, by taking x = e, the condition (10) is the consequence of (7) and (8), which further gives condition (6). This step concludes the proof. ■

In the next two characterization results, we consider four dimensional matrices of nonnegative fuzzy numbers and the set of bounded double sequences of nonnegative fuzzy numbers. We are still unaware if the same characterization is applicable for any four dimensional matrices of fuzzy numbers and the set of bounded double sequences of fuzzy numbers.

Theorem 2.3. Let A = (a_mnkl) be any four dimensional matrix of fuzzy numbers with $a_{mnkl} ≽ \bar{0}$ . Then $A \in (M_{U}^{F} : C_{P}^{F})$ if and only if the conditions (7) and (8) hold, and

there exists ${\bar{α}}_{kl} \in E^{1}$ such that, $P - lim_{m, n \to \infty} a_{mnkl} = {\bar{a}}_{kl},$ (11)

Proof. Suppose $A = (a_{mnkl}) \in (M_{U}^{F} : C_{P}^{F})$ and $x = (x_{kl}) \in M_{U}^{F}$ then Ax exists and belongs to $C_{P}^{F}$ .

Since $a_{mnkl} ≽ \bar{0}$ for all $m, n, k, l \in ℕ$ , $x_{kl} ≽ \bar{0}$ and $x_{kl}^{-} (λ) \leq x_{kl}^{+} (λ)$ for λ ∈ [0, 1], we have $\begin{matrix} {(a_{mnkl} x_{kl})}^{-} (λ) \\ = min {a_{mnkl}^{-} (λ) x_{kl}^{-} (λ), a_{mnkl}^{-} (λ) x_{kl}^{+} (λ), \\ a_{mnkl}^{+} (λ) x_{kl}^{-} (λ), a_{mnkl}^{+} (λ) x_{kl}^{+} (λ)} \\ = a_{mnkl}^{-} (λ) x_{kl}^{-} (λ) \end{matrix}$ and $\begin{matrix} {(a_{mnkl} x_{kl})}^{+} (λ) \\ = max {a_{mnkl}^{-} (λ) x_{kl}^{-} (λ), a_{mnkl}^{-} (λ) x_{kl}^{+} (λ), \\ a_{mnkl}^{+} (λ) x_{kl}^{-} (λ), a_{mnkl}^{+} (λ) x_{kl}^{+} (λ)} \\ = a_{mnkl}^{+} (λ) x_{kl}^{+} (λ) . \end{matrix}$

By definition we have, $\begin{matrix} D ({(Ax)}_{m + r, n + s}, {(Ax)}_{mn}) \\ = D (\sum_{k, l} a_{m + r, n + s, kl} x_{kl}, \sum_{k, l} a_{mnkl} x_{kl}) \\ = sup_{λ \in [0, 1]} \max {| \sum_{k, l} {(a_{m + r, n + s, kl} x_{kl})}^{-} (λ) \\ - \sum_{k, l} {(a_{mnkl} x_{kl})}^{-} (λ) |, \\ | \sum_{k, l} {(a_{m + r, n + s, kl} x_{kl})}^{+} (λ) - \sum_{k, l} {(a_{mnkl} x_{kl})}^{+} (λ) |} \end{matrix}$

Now, $\begin{matrix} | \sum_{k, l} {(a_{m + r, n + s, kl} x_{kl})}^{+} (λ) - \sum_{k, l} {(a_{mnkl} x_{kl})}^{+} (λ) | \\ = | \sum_{k, l} ({(a_{m + r, n + s, kl} x_{kl})}^{+} (λ) - {(a_{mnkl} x_{kl})}^{+} (λ)) | \\ = | \sum_{k, l} (a_{m + r, n + s, kl}^{+} (λ) x_{kl}^{+} (λ) - a_{mnkl}^{+} (λ) x_{kl}^{+} (λ)) | \\ = | \sum_{k, l} (a_{m + r, n + s, kl}^{+} (λ) - a_{mnkl}^{+} (λ)) x_{kl}^{+} (λ) | \\ = | {(Bx)}_{mn}^{+} (λ) |, \end{matrix}$ for all $m, n \in ℕ$ , where the matrix $B^{+} (λ) = ((b_{mnkl}^{rs})^{+} (λ))$ is defined by ${(b_{mnkl}^{rs})}^{+} (λ) = a_{m + r, n + s, kl}^{+} (λ) - a_{mnkl}^{+} (λ)$ for all $m, n, r, s, k, l \in ℕ$ .

Similarly, $\begin{matrix} | \sum_{k, l} {(a_{m + r, n + s, kl} x_{kl})}^{-} (λ) - \sum_{k, l} {(a_{mnkl} x_{kl})}^{-} (λ) | \\ = | {(Bx)}_{mn}^{-} (λ) |, \end{matrix}$ for all $m, n \in ℕ$ , where the matrix $B^{-} (λ) = ({(b_{mnkl}^{rs})}^{-} (λ))$ is defined by ${(b_{mnkl}^{rs})}^{-} (λ) = a_{m + r, n + s, kl}^{-} (λ) - a_{mnkl}^{-} (λ)$ for all $m, n, r, s, k, l \in ℕ$ . From the above we get, $P - lim_{m, n \to \infty} {(Bx)}_{mn} = \bar{0} .$ (12)

Thus $B \in (M_{U}^{F} : C_{P 0}^{F})$ . So, from condition (6) of Theorem 2.2, we have that $P - lim_{m, n \to \infty} a_{mnkl}$ exists for each $k, l \in ℕ$ , say ${\bar{a}}_{kl}$ .

Therefore, condition (11) is necessary.

Following the similar arguments used in the proof of Theorem 2.2, we can get the necessity of the conditions (7) and (8).

Conversely, let us suppose that the conditions (7), (8) and (11) hold and let $x = (x_{kl}) \in M_{U}^{F}$ .

Let $t_{0} = \max {L, K} \in ℕ$ . Hence for every $ɛ > 0, \exists t_{0} = t_{0} (ɛ) \in ℕ$ such that, for all λ ∈ [0, 1], $\begin{matrix} | \sum_{k, l = 0} {(a_{m n k l} x_{k l})}^{+} (λ) - \sum_{k, l = 0}^{t_{0}} {({\bar{a}}_{k l} x_{k l})}^{+} (λ) | \\ = | \sum_{k, l = 0}^{t_{0}} ({(a_{m n k l} x_{k l})}^{+} (λ) - {({\bar{a}}_{k l} x_{k l})}^{+} (λ)) | \\ = | \sum_{k, l = 0}^{t_{0}} (a_{m n k l} + (λ) x_{k l} + (λ) - {\bar{a}}_{k l} + (λ)) (x_{k l} + (λ)) | \\ = | \sum_{k, l = 0}^{t_{0}} (a_{m n k l} + (λ) - {\bar{a}}_{k l} + (λ)) (x_{k l} + (λ)) | \\ \leq ∥ x_{k l} + (λ) ∥_{\infty} \sum_{k, l = 0}^{t_{0}} | a_{m n k l} + (λ) - {\bar{a}}_{k l} + (λ) | \\ < ∥ x_{k l} + (λ) ∥_{\infty} {(t_{0} + 1)}^{2}, \end{matrix}$ for all m, n > t₀.

Thus $\begin{matrix} | \sum_{k, l = 0} {(a_{mnkl} x_{kl})}^{+} (λ) - \sum_{k, l = 0}^{t_{0}} {({\bar{a}}_{kl} x_{kl})}^{+} (λ) | \\ \to as m, n \to \infty . \end{matrix}$

Similarly, $\begin{matrix} | \sum_{k, l = 0} {(a_{mnkl} x_{kl})}^{-} (λ) - \sum_{k, l = 0}^{t_{0}} {({\bar{a}}_{kl} x_{kl})}^{-} (λ) | \\ \to 0 as m, n \to \infty . \end{matrix}$

From the above two relations, we get $P - lim_{m, n \to \infty} {(Ax)}_{mn} = \sum_{k, l = 0}^{t_{0}} {\bar{a}}_{kl} x_{kl} .$

So, we can establish that $A \in (M_{U}^{F} : C_{p}^{F}),$ which completes the proof. ■

Theorem 2.4. Let A = (a_mnkl) be any four dimensional matrix of fuzzy numbers with $a_{mnkl} ≽ \bar{0}$ . Then $A = (a_{mnkl}) \in (M_{U}^{F} : C_{f}^{F})$ if and only if the condition (5) holds and there exists ${\bar{a}}_{kl} \in E^{1}$ $f_{2} - lim_{m, n \to \infty} a_{mnkl} = {\bar{a}}_{kl}, for all k, l \in ℕ,$ (13)for every $m, n, l \in ℕ$ , there exists $K \in ℕ$ such that

$\begin{matrix} \frac{1}{(q + 1) (r + 1)} \sum_{i = 0}^{q} \sum_{j = 0}^{r} a_{m + i, n + j, kl} \\ = \bar{0}, for all q, r, k > K, \end{matrix}$ (14) for every $m, n, k \in ℕ$ , there exists $L \in ℕ$ such that

$\begin{matrix} \frac{1}{(q + 1) (r + 1)} \sum_{i = 0}^{q} \sum_{j = 0}^{r} a_{m + i, n + j, kl} \\ = \bar{0}, for all q, r, l > L . \end{matrix}$ (15)

Proof. Suppose that $A = (a_{mnkl}) \in (M_{U}^{F} : C_{f}^{F})$ and let $x = (x_{kl}) \in M_{U}^{F}$ . Then Ax exists and belongs to $C_{f}^{F} .$

Also, since, by Lemma 2.1 the inclusion $C_{f}^{F} \subset M_{u}^{F}$ holds, so the inclusion $(M_{U}^{F} : C_{f}^{F}) \subset (M_{u}^{F} : M_{u}^{F})$ holds. Therefore, from Theorem 2.1, the necessity of (5) is verified.

Now, let us define the sequence $e^{kl} = (e_{mn}^{kl})$ by $e^{kl} = (e_{mn}^{kl}) : = {\begin{matrix} \bar{1}, & (k, l) = (m, n), \\ \bar{0}, & otherwise . \end{matrix}$

We define the sequences e^l, e_k and e same as in Theorem 2.2. Now, since, $Ax \in C_{f}^{F} for all x \in M_{U}^{F},$ if we take x = e^kl for each $k, l \in ℕ$ , we obtain the necessity of the condition (13). If we take the sequences x = e^l, x = e_k and x = e, then we derive the conditions $\begin{matrix} f_{2} - lim_{m, n \to \infty} \sum_{l} a_{mnkl}, \\ exists for each k \in ℕ, \end{matrix}$ (16) $\begin{matrix} f_{2} - lim_{m, n \to \infty} \sum_{l} a_{mnkl}, \\ exists for each l \in ℕ, \end{matrix}$ (17) and

$\begin{matrix} f_{2} - lim_{m, n \to \infty} \sum_{l} a_{mnkl}, \\ exists . \end{matrix}$ (18)

Let us define the matrix $B = (B_{mn}) : = (b_{qrkl}^{mn})$ for all $m, n, q, r, k, l \in ℕ$ , by $b_{qrkl}^{mn} : = \frac{1}{(q + 1) (r + 1)} \sum_{i = 0}^{q} \sum_{j = 0}^{r} a_{m + i, n + j, kl}$

Then for $x = (x_{kl}) \in M_{U}^{F}$ we get, for all λ ∈ [0, 1], $\begin{matrix} {(B_{mn} x)}_{qr}^{+} (λ) = \sum_{k, l} {(b_{qrkl}^{mn} x_{kl})}^{+} (λ) \\ = \sum_{k, l} \frac{1}{(q + 1) (r + 1)} \sum_{i = 0}^{q} \sum_{j = 0}^{r} a_{m + i, n + j, kl}^{+} (λ) x_{kl}^{+} (λ) \\ = \frac{1}{(q + 1) (r + 1)} \sum_{i = 0}^{q} \sum_{j = 0}^{r} \sum_{k, l} a_{m + i, n + j, kl}^{+} (λ) x_{kl}^{+} (λ) \\ = \frac{1}{(q + 1) (r + 1)} \sum_{i = 0}^{q} \sum_{j = 0}^{r} \sum_{k, l} {(a_{m + i, n + j, kl} x_{kl})}^{+} (λ) \\ = \frac{1}{(q + 1) (r + 1)} \sum_{i = 0}^{q} \sum_{j = 0}^{r} \sum_{k, l} {(Ax)}_{m + i, n + j, kl}^{+} (λ) \end{matrix}$ where ${(B_{mn})}^{+} (λ) = ({(b_{qrkl}^{mn})}^{+} (λ))$ is defined by $\begin{matrix} {(b_{qrkl}^{mn})}^{+} (λ) \\ = \frac{1}{(q + 1) (r + 1)} \sum_{i = 0}^{q} \sum_{j = 0}^{r} a_{m + i, n + j, kl}^{+} (λ) . \end{matrix}$ for all $m, n, q, r, k, l \in ℕ$ and for all λ ∈ [0, 1].

Similarly we have, $\begin{matrix} {(B_{mn} x)}_{qr}^{-} (λ) \\ = \frac{1}{(q + 1) (r + 1)} \sum_{i = 0}^{q} \sum_{j = 0}^{r} \sum_{k, l} {(Ax)}_{m + i, n + j, kl}^{-} (λ) \end{matrix}$ where ${(B_{mn})}^{-} (λ) = ({(b_{qrkl}^{mn})}^{-} (λ))$ is defined by $\begin{matrix} {(b_{qrkl}^{mn})}^{-} (λ) \\ = \frac{1}{(q + 1) (r + 1)} \sum_{i = 0}^{q} \sum_{j = 0}^{r} a_{m + i, n + j, kl}^{-} (λ) . \end{matrix}$ for all $m, n, q, r, k, l \in ℕ$ and for all λ ∈ [0, 1].

Thus, from the above two relations, we get ${(B_{mn} x)}_{qr}^{=} \frac{1}{(q + 1) (r + 1)} \sum_{i = 0}^{q} \sum_{j = 0}^{r} \sum_{k, l} {(Ax)}_{m + i, n + j, kl}$ for all $m, n, q, r, k, l \in ℕ$ .

Then, since $Ax \in C_{f}^{F} for all x \in M_{U}^{F}$ , $f_{2} - lim_{A} x$ exists, that is, $P - lim_{q, r \to \infty} \frac{1}{(q + 1) (r + 1)} \sum_{i = 0}^{q} \sum_{j = 0}^{r} {(Ax)}_{m + i, n + j}$ exists, uniformly in m and n. Thus we can conclude that B_mnx exists for all $x \in M_{U}^{F}$ , that is, $B_{mn} \in (M_{U}^{F} : C_{P}^{F})$ . Thus, from Theorem 2.3, $there exists {\bar{a}}_{kl} \in E^{1} such that$ $P - lim_{q, r \to \infty} b_{qrkl}^{mn} = {\bar{a}}_{kl}, for all m, n, k, l \in ℕ,$ (19) for every $m, n, l \in ℕ$ , there exists $K \in ℕ,$ such that, $b_{qrkl}^{mn} = \bar{0}, for all q, r, k > K,$ (20) for every $m, n, k \in ℕ$ , there exists $L \in ℕ,$ such that, $b_{qrkl}^{mn} = \bar{0}, for all q, r, l > L .$ (21)

We see that the relation (19) is equivalent to the condition (13). Also, the conditions (13) and (20) include the condition (16). In a similar manner, the conditions (13) and (21) include the condition (17). Finally, the conditions (13), (20) and (21) include the condition (18).

Using similar arguments used in the proof of Theorem 2.3, we can prove the sufficiency part. ■

Footnotes

Acknowledgments

The authors would like to express their gratitude to the University Grants Commission, New Delhi, India for offering fellowship to the co-author (J. Gogoi) via award letter no F./2015-16/NFO-2015-17-OBC-ASS-36722/(SA-III/Website).

The authors are very much grateful to the anonymous referees for their constructive comments and suggestions.

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