Some new classes of ideal convergent difference multiple sequences of fuzzy real numbers

Abstract

The idea of difference sequence spaces (for single sequences) was first introduced by Kizmaz in 1981 and the idea of triple sequences was first introduced by Sahiner et al. 2007. In this article, we introduce some new classes of ideal convergent difference multiple sequence spaces $_{3} (c^{I (F)}) (Δ, p),_{3} (c_{0}^{I (F)}) (Δ, p)$ and $_{3} (c_{0}^{I (F)}) (Δ, p),$ of fuzzy real numbers using a difference operator Δ, where p =〈 p_nlk 〉 is a triple sequence of bounded strictly positive numbers. We study some basic algebraic and topological properties of these spaces. We also investigate the relations related to these spaces. It is shown that the sequence spaces $_{3} (c^{I (F)}) (Δ, p),_{3} (c_{0}^{I (F)}) (Δ, p),_{3} (m^{I (F)}) (Δ, p)$ and $_{3} (m_{0}^{I (F)}) (Δ, p)$ are closed under addition and scalar multiplication also these spaces are sequence algebras. We have proved that the sequence space $_{3} (m_{0}^{I (F)}) (Δ, p)$ is solid as well as monotone. We have obtained the inclusion relation $_{3} (m_{0}^{I (F)}) (Δ, p) \subset_{3} (m^{I (F)}) (Δ, p) \subset_{3} (ℓ_{\infty}^{F}) (Δ, p)$ where the inclusions are strict. We have also proved that the sequence spaces $_{3} (m_{0}^{I (F)}) (Δ, p)$ and ₃ (m^I(F)) (Δ, p) are nowhere dense subsets of $_{3} (ℓ_{\infty}^{F}) (Δ, p) .$

Keywords

Ideal convergence fuzzy real valued triple sequence multiple sequences difference sequence difference operator complete solid monotone sequence algebra etc

1 Introduction

Fuzzy set theory, compared to other mathematical theories, is perhaps the most easily adaptable theory to practice. Instead of defining an entity in calculus by assuming that its role is exactly known, we can use fuzzy sets to define the same entity by allowing possible deviations and inexactness in its role. This representation suits well the uncertainties encountered in practical life, which make fuzzy sets a valuable mathematical tool. The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh [30] and subsequently several authors have discussed various aspects of the theory and applications of fuzzy sets. In fact the fuzzy set theory has become an area of active area of research in science and engineering for the last 40 years. Fuzzy set theory is a powerful hand set for modelling uncertainty and vagueness in various problems arising in the field of science and engineering. It extends the scope and results of classical mathematical analysis by applying fuzzy logic to conventional mathematical objects, such as functions, sequences and series etc. While studying fuzzy topological spaces, we face many situations where we need to deal with convergence of fuzzy numbers.

Standard ranking methods for generalized fuzzy numbers are found in [28] and [29]. Using the notion of fuzzy real numbers, different types of fuzzy real-valued sequence spaces have been introduced and studied by several mathematicians. The initial works on double sequences of real or complex terms are found in Bromwich [2]. Hardy [9] introduced the notion of regular convergence for double sequences of real or complex terms. Agnew [1] studied the summability theory of multiple sequences and obtained certain theorems which have already been proved for double sequences by the author himself. Móricz [15] extended statistical convergence from single to multiple real sequences and obtained some results for real double sequences. Matloka [14] introduced bounded and convergent sequences of fuzzy numbers, studied some of their properties and showed that every convergent sequence of fuzzy numbers is bounded. There are many applications of the sequences and difference sequences of numbers (real, complex and fuzzy numbers). The sequences of fuzzy numbers have unexpected and practical uses in many areas of science and engineering, including acoustics. They find application in measuring concert hall acoustics, radar echoes from planets, the travel times of deep-ocean sound waves for monitoring ocean temperature, and improving synthetic speech and the sounds associated with computer music. Furthermore, it is shown by Kawamura et al. [10] that the earthquake ground motions have very simple conditioned fuzzy set rules with non-fuzzy parameters of the first and second order differences Δx_k and Δ²x_k defined by membership functions. Therefore the difference sequences of fuzzy numbers are used, for example in the prediction of earthquakewaves.

2 Definitions and background

Throughout the article N, R and C denote the sets of natural, real and complex numbers respectively. Throughout w, c, c₀, ℓ _∞ denote the spaces of all, convergent, null and bounded sequences and ₂w, ₂ ℓ _∞, ₂c, ₂c₀ denote the spaces of all, bounded, convergent, null double sequences respectively.

Let C (Rⁿ) = {A ⊂ Rⁿ : A is compact and convex}. The space C (Rⁿ) has a linear structure induced by the operations A + B = {a + b : a ∈ A, b ∈ B} and λA = {λa : a ∈ A} for A, B ∈ C (Rⁿ) and λ ∈ R . The Hausdorff distance between A and B of C (Rⁿ) is defined as $δ_{\infty} (A, B) = max {sup_{a \in A} inf_{b \in B} ∥ a - b ∥, sup_{b \in B} inf_{a \in A} ∥ a - b ∥} .$

It is well known that (C (Rⁿ) , δ_∞) is a complete metric space.

A fuzzy real number on R is a mapping X : R → L (= [0, 1]) associating each real number t ∈ R with its grade of membership X (t). Every real number r can be expressed as a fuzzy real number $\bar{r}$ as follows: $\bar{r} (t) = {\begin{matrix} 1 & if t = r \\ 0 & otherwise \end{matrix}$

The α-level set of a fuzzy real number X, 0 < α ≤ 1, denoted by [X] ^α is defined as $[X]^{α} = {t \in R : X (t) \geq α} .$

A fuzzy real number X is called convex if X (t) ≥ X (s) ∧ X (r) = min(X (s) , X (r)) , where s < t < r . If there exists t₀ ∈ R such that X (t₀) =1, then the fuzzy real number X is called normal. A fuzzy real number X is said to be upper semi-continuous if for each ɛ > 0, ∖ X^-1 [0, a + ɛ)) , for all a ∈ L is open in the usual topology of R . The set of all upper semi continuous, normal, convex fuzzy number is denoted by R (L) . The additive identity and multiplicative identity in R (L) are denoted by $\bar{0}$ and $\bar{1}$ respectively.

The Arithmetic operations on R (L) are defined as follows:

If X, Y ∈ R (L), then $\begin{matrix} (X Θ Y) (t) & = & sup_{s \in R} {X (s) \land Y (s - t)}, t \in R, \\ (X \oplus Y) (t) & = & sup_{s \in R} {X (s) \land Y (t - s)}, t \in R, \\ (X \otimes Y) (t) & = & sup_{s \in R} {X (s) \land Y (t / s)}, t \in R, \\ (X / Y) (t) & = & sup_{s \in R} {X (st) \land Y (s)}, t \in R . \end{matrix}$

The absolute value |X |of X ∈ R (L) is defined as $| X | (t) = {\begin{matrix} max {X (t), X (- t)}, & if t \geq 0 \\ 0 & if t < 0 . \end{matrix}$

Let D be the set of all closed bounded intervals X = [X^L, X^R] on the real line R. Then X ≤ Y if and only if X^L ≤ Y^L and X^R ≤ Y^R . Also let d (X, Y) = max(|X^L - X^R|, |Y^L - Y^R|) . Then (D, d) is a complete metric space.

Let $\bar{d} : R (L) \times R (L) \to R$ be defined by $\bar{d} (X, Y) = sup_{0 \leq α \leq 1} d ([X]^{α}, [Y]^{α}), for X, Y \in R (L) .$

Then $\bar{d}$ defines a metric on R (L) .

In order to generalize the notion of convergence of real sequences, Kostyrko, Šalát and Wilczyński [12] introduced the idea of ideal convergence for single sequences in 2000-2001. Later on it was further developed by Šalát et al. [17], Tripathy et al. [20], Das et al. [3], Tripathy and Sen [26], Kumar et al. [14], Sen and Roy [19], V.S. Debnath et al. [4] and many others. The notion of Ideal convergence depends on the structure of the ideal I of the subset of the set of natural numbers. Let X be a non empty set. A non-void class I ⊆ 2^X (power set of X) is said to be an ideal if I is additive and hereditary, i.e. if I satisfies the following conditions:

(i) A, B ∈ I ⇒ A ∪ B ∈ I and (ii) A ∈ IandB ⊆ A ⇒ B ∈ I .

A non-empty family of sets F ⊆ 2^X is said to be a filter on X if

(i) ∅ ∉ F (ii) A, B ∈ F ⇒ A ∩ B ∈ F and (iii) A ∈ F and A ⊆ B ⇒ B ∈ F.

For any ideal I, there is a filter F (I) given by F (I) = {K ⊆ N : N ∖ K ∈ I} .

An ideal I ⊆ 2^X is said to be non-trivial if I≠ ∅ and X ∉ I. Clearly I ⊆ 2^X is a non-trivial ideal if and only if F = F (I) = {X - A : A ∈ I} is a filter on X.

A subset E of N × N × N is said to have density δ (E) if $δ (E) = lim_{p, q, r \to \infty} \sum_{n = 1}^{p} \sum_{l = 1}^{q} \sum_{k = 1}^{r} χ_{E} (n, l, k)$ exists. Throughout the article, the ideals of 2^N×N×N will be denoted by I₃ .

Example 1. Let I₃ (ρ) ⊂2^N×N×Ni.e. the class of all subsets of N × N × N of zero natural density. Then I₃ (ρ) is an ideal of 2^N×N×N.

Example 2. Let I₃ (P) be the class of all subsets of N × N × N such that D ∈ I₃ (P) implies that there exists n₀, l₀, k₀ ∈ N such that $\begin{matrix} D \subseteq N \times N \times N - {(n, l, k) \\ \in N \times N \times N : n \geq n_{0}, l \geq l_{0}, k \geq k_{0}} . \end{matrix}$

Then I₃ (P) is an ideal of 2^N×N×N .

A triple sequence can be defined as a function x : N × N × N → R (C) .

The different types of notions of triple sequences was introduced and investigated at the initial stage by Sahiner et al. [16], Dutta et al. [6], P. Kumar et al. [13] and others. Recently Savas and Esi [18] have introduced statistical convergence of triple sequences on probabilistic normed space. Later on, Esi [8] have introduced statistical convergence of triple sequences in topological groups. Some more works on triple sequences are found in [7, 8]. A fuzzy real-valued triple sequence X =〈 X_nlk 〉 is a triple infinite array of fuzzy real numbers X_nlk for all n, l, k ∈ N and is denoted by 〈X_nkl〉 where X_nlk ∈ R (L) .

A fuzzy real-valued triple sequence X =〈 X_nlk 〉 is said to be convergent in Pringsheims sense to the fuzzy real number X, if for every ∈>0, ∃, s n₀ = n₀ (ɛ) , l₀ = l₀ (ɛ) , k₀ = k₀ (ɛ) ∈ N such that $\bar{d} (X_{nlk}, X) < ɛ$ for all n ≥ n₀, l ≥ l₀, k ≥ k₀ .

A fuzzy real-valued triple sequence X =〈 X_nlk 〉 is said to be I₃-convergent to the fuzzy number X₀, if for all ɛ > 0, the set ${(n, l, k) \in N \times N \times N : \bar{d} (X_{nlk}, X_{0}) \geq ɛ} \in I_{3} .$ We write I₃ - lim X_nlk = X₀ .

A fuzzy real-valued triple sequence X =〈 X_nlk 〉 is said to be I₃-bounded if there exists a real number μ such that the set ${(n, l, k) \in N \times N \times N : \bar{d} (X_{nlk}, \bar{0}) > μ} \in I_{3} .$

Throughout $_{3} (w^{F}),_{3} (ℓ_{\infty}^{F}),_{3} (c^{F}),_{3} (c_{0}^{F})$ denote the spaces of all, bounded, convergent in Pringsheim’s sense, null in Pringsheim’s sense fuzzy real-valued triple sequences respectively.

A fuzzy real-valued triple sequence space E^F is said to be solid if 〈Y_nlk 〉 ∈ E^F whenever $\bar{d} (Y_{nlk}, \bar{0}) \leq \bar{d} (X_{nlk}, \bar{0})$ i.e. |Y_nlk| ≤ |X_nlk| for all n, l, k ∈ N and 〈X_nlk 〉 ∈ E^F .

A fuzzy real-valued triple sequence space E^F is said to be monotone if E^F contains the canonical pre-image of all its step spaces.

A fuzzy real-valued triple sequence E^F is said to be symmetric if S (X) ⊂ E^F, for all X ∈ E^F, where S (X) denotes the set of all permutations of the elements of X = 〈 X_nlk 〉 .

A fuzzy real-valued triple sequence space E^F is said to be sequence algebra if 〈X_nlk ⊗ Y_nlk 〉 ∈ E^F, whenever 〈X_nlk 〉 , 〈 Y_nlk 〉 ∈ E^F .

A fuzzy real-valued triple sequence space E^F is said to be convergence free if 〈Y_nlk 〉 ∈ E^F whenever 〈X_nlk 〉 ∈ E^F . and $X_{nlk} = \bar{0}$ implies $Y_{nlk} = \bar{0} .$

The notion of difference sequence spaces was introduced by Kizmaz [11] as follows: $\begin{matrix} Z (Δ) & = & {x = (x_{k}) \in w : (Δ x_{k}) \in Z}, for \\ Z & = & ℓ_{\infty}, c, c_{0}, where Δ x_{k} = x_{k} - x_{k + 1}, \\ for all k \in N . \end{matrix}$

Tripathy and Esi [23] introduced the notion of difference sequence space Δ_mx = (Δ_mx_k) = x_k - x_k+m, forall k ∈ N and m ∈ N be fixed. Later on it was studied by Tripathy and Sarma [25] and introduced difference double sequence spaces as follows: $\begin{matrix} Z (Δ) & = & {X = (X_{nk}) \in_{2} w : (Δ X_{nk}) \in Z}, for \\ Z & = & _{2} ℓ_{\infty},_{2} c,_{2} c_{0}, where \\ Δ X_{nk} & = & X_{n, k} - X_{n, k + 1} - X_{n + 1, k} + X_{n + 1, k + 1}, \\ for all n, k \in N \end{matrix}$

For triple sequences, we introduce the following:

$\begin{matrix} Δ X_{nlk} & = & X_{nlk} - X_{n, l + 1, k} - X_{n, l, k + 1} + X_{n, l + 1, k + 1} \\ - X_{n + 1, l, k} + X_{n + 1, l + 1, k} + X_{n + 1, l, k + 1} \\ - X_{n + 1, l + 1, k + 1} for all n, l, k \in N \end{matrix}$ (1)

Some works on difference sequence spaces can be found in [5 , 27].

Let p =〈 p_nkl 〉 be a triple sequence of bounded strictly positive numbers. We introduce the following I-convergent triple sequence spaces: $\begin{matrix} _{3} (c^{I (F)}) (Δ, p) \\ = {X = 〈 X_{nlk} 〉 \in_{3} (w^{F}) : I_{3} - lim [\bar{d} (Δ X_{nlk}, X_{0})]^{p_{nlk}} \\ = 0, for some X_{0} \in R (L)}, \end{matrix}$

$\begin{matrix} _{3} (c_{0}^{I (F)}) (Δ, p) \\ = {X = 〈 X_{nlk} 〉 \in_{3} (w^{F}) : I_{3} - lim [\bar{d} (Δ X_{nlk}, \bar{0})]^{p_{nlk}} = 0}, \end{matrix}$

Let $_{3} (ℓ_{\infty}^{F}) (Δ, p) = {X = 〈 X_{nlk} 〉 \in_{3} (w^{F}) : sup_{n, l, k} [\bar{d} (Δ X_{nlk}, \bar{0})]^{p_{nlk}} < \infty} .$

Then $_{3} (ℓ_{\infty}^{F}) (Δ, p)$ is a complete metric space with respect to the metric ρ defined by $ρ (X, Y) = sup_{n, l} [\bar{d} (X_{nl 1}, Y_{nl 1})]^{p_{nlk}} + sup_{n, k} [\bar{d} (X_{n 1 k}, Y_{n 1 k})]^{p_{nlk}} + sup_{l, k} [\bar{d} (X_{1 lk}, Y_{1 lk})]^{p_{nlk}} + sup_{n, l, k} [\bar{d} (Δ X_{nlk}, Δ Y_{nlk})]^{p_{nlk}}$ where M = max(1, H) , $H = \sup_{nlk} p_{nlk} < \infty .$

Also, we define the following sequence spaces: $_{3} (m^{I (F)}) (Δ, p) =_{3} (c^{I (F)}) (Δ, p) \cap_{3} (ℓ_{\infty}^{F}) (Δ, p)$ $_{3} (m_{0}^{I (F)}) (Δ, p) =_{3} (c_{0}^{I (F)}) (Δ, p) \cap_{3} (ℓ_{\infty}^{F}) (Δ, p) .$

For particular values of p and ideal I, these sequence spaces reduce to many well known sequence spaces.

For establishing the results of this chapter, we procure the following existing result.

Lemma.If a sequence spaceE^Fis solid, then it is monotone.

3 Main results

Theorem 1.T he classes of sequences $_{3} (c^{I (F)}) (Δ, p),_{3} (c_{0}^{I (F)}) (Δ, p),_{3} (m^{I (F)}) (Δ, p)$ and $_{3} (m_{0}^{I (F)}) (Δ, p)$ are closed under addition and scalar multiplication.

Proof. We prove the result for the space ₃ (m^I(F)) (Δ, p) .

Let 〈X_nlk 〉 , 〈 Y_nlk 〉 ₃ (m^I(F)) (Δ, p) and α, β be scalars. Then there exists X₀ and Y₀ ∈ R (L) such that $\begin{matrix} I_{3} - lim [\bar{d} (Δ X_{nlk}, X_{0})]^{p_{nlk}} = 0 and \\ I_{3} - lim [\bar{d} (Δ Y_{nlk}, Y_{0})]^{p_{nlk}} = 0 \end{matrix}$

Then ${[\bar{d} (Δ (α X_{nlk} + β Y_{nlk}), (α X_{0} + β Y_{0}))]}^{p_{nlk}} \leq D {[\bar{d} (Δ X_{nlk}, X_{0})]}^{p_{nlk}} + D {[\bar{d} (Δ Y_{nlk}, Y_{0})]}^{p_{nlk}}$ where D = max(1, 2^H-1) , $H = sup_{n, l, k} p_{nlk} < \infty .$

Then taking I-limit on both sides of the above inequality, we get $I_{3} - lim {[\bar{d} (Δ (α X_{nlk} + β Y_{nlk}), (α X_{0} + β Y_{0}))]}^{p_{nlk}} = 0 .$

Therefore (αX_nlk + βY_nlk) ∈ ₃ (m^I(F)) (Δ, p) .

Hence the class of sequences ₃ (m^I(F)) (Δ, p) is closed under the operations of addition and scalar multiplication.

Using similar technique the other cases can be established. ■

Theorem 2.Let the triple sequence 〈p_nkl〉 be bounded. Then $_{3} (m_{0}^{I (F)}) (Δ, p) \subset_{3} (m^{I (F)}) (Δ, p) \subset_{3} (ℓ_{\infty}^{F}) (Δ, p)$ and the inclusions are strict.

Proof. The inclusion $_{3} (m_{0}^{I (F)}) (Δ, p) \subset_{3} (m^{I (F)}) (Δ, p) \subset_{3} (ℓ_{\infty}^{F}) (Δ, p)$ is obvious.

In order to show that the inclusion $_{3} (m_{0}^{I (F)}) (Δ, p) \subset_{3} (ℓ_{\infty}^{F}) (Δ, p)$ is strict, we consider the following example.

Example 3. Let I₃ (P) be the class of all subsets of N × N × N such that A ∈ I₃ (P) implies that there exits n₀, l₀, k₀ ∈ N such that A ⊆ N × N × N - {(n, l, k) ∈ N × N × N : n ≥ n₀, l ≥ l₀, k ≥ k₀} .

Let $p_{nlk} = {\begin{matrix} \frac{1}{3}, & if 1 \leq n \leq n_{0}, 1 \leq l \leq l_{0}, 1 \leq k \leq k_{0} \\ 3, & otherwise \end{matrix}$

We consider the sequence 〈X_nlk〉 defined by: $X_{nlk} = \bar{1}, for 1 \leq n \leq n_{0},, 1 \leq l \leq l_{0}, 1 \leq k \leq k_{0} .$

For n > n₀, l > l₀, k > k₀ and (n + l + k) even, $X_{nlk} (t) = {\begin{matrix} \frac{nt - 3 n + 1}{n + 1}, & f o r 3 - n^{- 1} \leq t \leq 4 \\ 5 - t, & f o r 4 < t \leq 5 \\ 0, & otherwise \end{matrix}$

Otherwise $X_{nlk} (t) = {\begin{matrix} \frac{nt - 1}{3 n - 1}, & for n^{- 1} \leq t \leq 3 \\ 4 - t, & for 3 < t \leq 4 \\ 0, & otherwise \end{matrix}$

Then $〈 Δ X_{nlk} 〉 \in_{3} (ℓ_{\infty}^{F}) (Δ, p),$ but $〈 Δ X_{nlk} 〉 \notin_{3} (m_{0}^{I (F)}) (Δ, p) .$

Hence the inclusion $_{3} (m_{0}^{I (F)}) (Δ, p) \subset_{3} (ℓ_{\infty}^{F}) (Δ, p)$ is strict. ■

Following standard techniques, we can easily prove the following result.

Theorem 3. If $H = sup_{n, k} p_{nk} < \infty,$ then the classes of sequences ₃ (m^I(F)) (Δ, p) and $_{3} (m_{0}^{I (F)}) (Δ, p)$ are complete metric spaces with respect to the metric ρ defined by

$ρ (X, Y) = sup_{n, l} [\bar{d} (X_{nl 1}, Y_{nl 1})]^{p_{nlk}} + sup_{n, k} [\bar{d} (X_{n 1 k}, Y_{n 1 k})]^{p_{nlk}} + sup_{l, k} [\bar{d} (X_{1 lk}, Y_{1 lk})]^{p_{nlk}} + sup_{n, l, k} [\bar{d} (Δ X_{nlk}, Δ Y_{nlk})]^{p_{nlk}}$ where 〈X_nlk 〉 , 〈 Y_nlk 〉 ∈ ₃ (m^I(F)) (Δ, p) and $\begin{matrix} Δ X_{nlk} & = & X_{nlk} - X_{n, l + 1, k} - X_{n, l, k + 1} + X_{n, l + 1, k + 1} \\ - X_{n + 1, l, k} + X_{n + 1, l + 1, k} + X_{n + 1, l, k + 1} \\ - X_{n + 1, l + 1, k + 1} \end{matrix}$

Theorem 4. The classes of sequences $_{3} (m_{0}^{I (F)}) (Δ, p)$ is solid as well as monotone.

Proof. Let $〈 X_{nlk} 〉 \in_{3} (m_{0}^{I (F)}) (Δ, p)$ and 〈Y_nlk〉 be such that $\bar{d} (Y_{nlk}, \bar{0}) \leq \bar{d} (X_{nlk}, \bar{0})$ for all n, l, k ∈ N .

Let ɛ > 0 be given. Then the solidness of $_{3} (m_{0}^{I (F)}) (Δ, p)$ follows from the following inclusion relation: $\begin{matrix} {(n, l, k) \in N \times N \times N : {[\bar{d} (Δ X_{nlk}, \bar{0})]}^{p_{nlk}} \\ \geq ɛ} \supseteq {(n, l, k) \in N \times N \times N : {[\bar{d} (Δ Y_{nlk}, \bar{0})]}^{p_{nlk}} \geq ɛ} . \end{matrix}$

Now using Lemma, the classes of sequences $_{3} (m_{0}^{I (F)}) (Δ, p)$ is monotone. ■

Theorem 5.T he classes of sequences $_{3} (c^{I (F)}) (Δ, p),_{3} (c_{0}^{I (F)}) (Δ, p),_{3} (m^{I (F)}) (Δ, p)$ and $_{3} (m_{0}^{I (F)}) (Δ, p)$ are sequence algebras.

Proof. We prove the result for the space $_{3} (c_{0}^{I (F)}) (Δ, p) .$

Let $〈 X_{nlk} 〉, 〈 Y_{nlk} 〉 \in_{3} (c_{0}^{I (F)}) (Δ, p) and 0 < ɛ < 1 .$

Then the result follows from the following inclusion relation: $\begin{matrix} {(n, l, k) \in N \times N \times N : {[\bar{d} (Δ X_{nlk} \otimes Δ Y_{nlk}, \bar{0})]}^{p_{nlk}} < ɛ} \\ \supset {(n, l, k) \in N \times N \times N : {[\bar{d} (Δ X_{nlk}, \bar{0})]}^{p_{nk}} < ɛ} \\ \cap {(n, l, k) \in N \times N \times N : {[\bar{d} (Δ Y_{nlk}, \bar{0})]}^{p_{nk}} < ɛ} . \end{matrix}$

Similarly the other cases can be established. ■

Theorem 6.T he classes of sequences $_{3} (c^{I (F)}) (Δ, p),_{3} (c_{0}^{I (F)}) (Δ, p),_{3} (m^{I (F)}) (Δ, p)$ and $_{3} (m_{0}^{I (F)}) (Δ, p)$ are not symmetric.

Proof. The results followed by the following example:

Example 4. Let I₃ = I₃ (ρ) and $p_{nlk} = {\begin{matrix} 1, & for n even and all l, k \in N \\ 3, & otherwise \end{matrix}$

We consider the triple sequence 〈X_nlk〉 defined by:

For n = i², i ∈ N and for all l, k ∈ N, $X_{nlk} = n, foralln, l, k \in N .$

Clearly $〈 Δ X_{nlk} 〉 \in_{3} (c^{I (F)}) (Δ, p),_{3} (c_{0}^{I (F)}) (Δ, p),_{3} (m^{I (F)}) (Δ, p),_{3} (m_{0}^{I (F)}) (Δ, p) .$

We consider the rearrangement (Y_nk) of (X_nk) defined by: $Y_{nlk} = {\begin{matrix} n + 1, for & n = l, k is even \\ n - 1, for & n = l + 1, k is even \\ n, & otherwise \end{matrix}$

Then clearly $〈 Δ Y_{nlk} 〉 \notin_{3} (c^{I (F)}) (Δ, p),_{3} (c_{0}^{I (F)}) (Δ, p),_{3} (m^{I (F)}) (Δ, p),_{3} (m_{0}^{I (F)}) (Δ, p) .$

Hence the sequence spaces $_{3} (c^{I (F)}) (Δ, p),_{3} (c_{0}^{I (F)}) (Δ, p),_{3} (m^{I (F)}) (Δ, p)$ and $_{3} (m_{0}^{I (F)}) (Δ, p)$ are not symmetric. ■

Theorem 7.The classes of sequences $_{3} (c^{I (F)}) (Δ, p),_{3} (c_{0}^{I (F)}) (Δ, p),_{3} (m^{I (F)}) (Δ, p)$ and $_{3} (m_{0}^{I (F)}) (Δ, p)$ are not convergence free.

Proof. The result follows from the following example:

Example 5. We provide an example to prove the result.

Let A ∈ I₃, $p_{nlk} = \frac{1}{3},$ for all n, l, k ∈ N .

We consider the triple sequence 〈X_nlk〉 defined by: $X_{nlk} = {\begin{matrix} 0, & if k = 1, for all n, l \in N \\ - 2, & otherwise \end{matrix}$

Clearly $〈 Δ X_{nlk} 〉 \in_{3} (c^{I (F)}) (Δ, p),_{3} (c_{0}^{I (F)}) (Δ, p),_{3} (m^{I (F)}) (Δ, p),_{3} (m_{0}^{I (F)}) (Δ, p) .$

Let the sequence 〈Y_nlk〉 be defined by: $Y_{nlk} = {\begin{matrix} 0, & if k is odd, for all n, l \in N \\ X_{nlk}, & otherwise \end{matrix}$

Then clearly $〈 Δ Y_{nlk} 〉 \notin_{3} (c^{I (F)}) (Δ, p),_{3} (c_{0}^{I (F)}) (Δ, p),_{3} (m^{I (F)}) (Δ, p),_{3} (m_{0}^{I (F)}) (Δ, p) .$

So $_{3} (c^{I (F)}) (Δ, p),_{3} (c_{0}^{I (F)}) (Δ, p),_{3} (m^{I (F)}) (Δ, p)$ and $_{3} (m_{0}^{I (F)}) (Δ, p)$ are not convergence free. ■

Theorem 8.The classes of sequences $_{3} (m_{0}^{I (F)}) (Δ, p)$ and₃ (m^I(F)) (Δ, p) are nowhere dense subsets of $_{3} (ℓ_{\infty}^{F}) (Δ, p) .$

Proof. By Theorem 2, the sequence spaces $_{3} (m_{0}^{I (F)}) (Δ, p)$ and₃ (m^I(F)) (Δ, p) are proper subspaces of $_{3} (ℓ_{\infty}^{F}) (Δ, p) .$ Hence the result follows from Theorem 3. ■

4 Conclusion

We have introduced and studied the notions of of null, convergent and bounded triple difference sequence spacesof fuzzy real numbers based on the difference operator Δ. We have verified some algebraic and topological properties. Further generalizations may be possible based on the difference operator Δ^m . The introduced notion can be applied for further investigations from different aspects.

Footnotes

Acknowledgments

The authors would like to record their gratitude to the reviewers for their careful reading and making some useful corrections which improved the presentation of the article.

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