The triple entire difference Ideal of fuzzy real numbers over fuzzy p -metric spaces defined by Musielak Orlicz function

Abstract

In this article we introduce the triple entire difference sequence spaces and triple analytic difference sequence of Musielak Orlicz function and study some basic topological and algebraic properties of these spaces. Also we investigate the relations related to these spaces and some of their properties like not solidity, non-monotone, not perfect, dual, not symmetricity, not convergence free etc., and also investigate some inclusion relations related to these spaces.

Keywords

Analytic sequence triple difference sequences Musielak-Orlicz function fuzzy number solid space symmetricity convergence free sequence algebra ideal

1 Introduction

Throughout w, χ and Λ denote the classes of all, gai and analytic scalar valued single sequences, respectively. We write w³ for the set of all complex triple sequences (x_mnk), where $m, n, k \in ℕ$ , the set of positive integers. Then, w³ is a linear space under the coordinate wise addition and scalar multiplication.

We can represent triple sequences by matrix. In case of double sequences we write in the form of a square. In the case of a triple sequence it will be in the form of a box in three dimensional case.

Some initial work on double series is found in Apostol [1] and double sequence spaces is found in Hardy [5], Deepmala et al. [6, 7] and many others. Later on investigated by some initial work on triple sequence spaces is found in Sahiner et al. [8], Esi et al. [2 –4], Subramanian et al. [9], Prakash et al. [10] and many others.

Let (x_mnk) be a triple sequence of real or complex numbers. Then the series $\sum_{m, n, k = 1}^{\infty} x_{mnk}$ is called a triple series. The triple series $\sum_{m, n, k = 1}^{\infty} x_{mnk}$ give one space is said to be convergent if and only if the triple sequence (S_mnk) is convergent, where $S_{mnk} = \sum_{i, j, q = 1}^{m, n, k} x_{ijq} (m, n, k = 1, 2, 3, \dots) .$

A sequence Δx = (Δx_mnk) is said to be triple difference analytic if $\sup_{m, n, k} | Δ x_{mnk} |^{\frac{1}{m + n + k}} < \infty .$

The vector space of all triple difference analytic sequences are usually denoted by Λ^3Δ. A sequence Δx = (Δx_mnk) is called triple difference entire sequence if $| Δ x_{mnk} |^{\frac{1}{m + n + k}} \to 0 as m, n, k \to \infty .$

The vector space of all triple difference entire sequences are usually denoted by Γ^3Δ. The spaces Λ^3Δ and Γ^3Δ are metric spaces with the metric $\begin{matrix} d (x, y) & = & \sup_{m, n, k} {| Δ x_{mnk} - Δ y_{mnk} |^{\frac{1}{m + n + k}} : m, n, \\ k : 1, 2, 3, \dots}, \end{matrix}$ forall Δx = {Δx_mnk} and Δy = {Δy_mnk} in Γ^3Δ. Let φ = {finite sequences} .

Consider a triple sequence x = (x_mnk). The (m, n, k) ^th section x^{[m, n, k]} of the sequence is defined by $x^{[m, n, k]} = \sum_{i, j, q = 0}^{m, n, k} x_{ijq} ℑ_{ijq}$ for all $m, n, k \in ℕ,$ $ℑ_{ijq} = [\begin{matrix} 0 & 0 & \dots 0 & 0 & \dots \\ 0 & 0 & \dots 0 & 0 & \dots \\ . \\ . \\ . \\ 0 & 0 & \dots 1 & 0 & \dots \\ 0 & 0 & \dots 0 & 0 & \dots \\ . & . & \dots & . & \dots \\ . & . & \dots & . & \dots \end{matrix}]$ with 1 in the (i, j, q) ^th position and zero otherwise. The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz [12] as follows $Z (Δ) = {x = (x_{k}) \in w : (Δ x_{k}) \in Z},$ for Z = c, c₀ and ℓ_∞, where Δx_k = x_k - x_k+1 for all $k \in ℕ$ .

Here c, c₀ and ℓ_∞ denote the classes of convergent, null and bounded scalar valued single sequences respectively. The difference sequence space bv_p of the classical space ℓ_p is introduced and studied in the case 1≤ p ≤ ∞ by Basar and Altay [17] and in the case 0 < p < 1 by Altay et al. [18]. The spaces c (Δ) , c₀ (Δ) , ℓ _∞ (Δ) and bv_p are Banach spaces normed by $\begin{matrix} | | x | | & = & | x_{1} | + \sup_{k \geq 1} | Δ x_{k} | and | | x | |_{{bv}_{p}} \\ = & {(\sum_{k = 1}^{\infty} | x_{k} |^{p})}^{1 / p}, (1 \leq p < \infty) . \end{matrix}$

Later on the notion was further investigated by Colak and Altinok [15], Altinok and Yagdiran [16] and many others. We now introduce the following difference double sequence spaces defined by $Z (Δ) = {x = (x_{mn}) \in w^{2} : (Δ x_{mn}) \in Z},$ where Z = Λ², χ² and Δx_mn = (x_mn - x_mn+1) - (x_m+1n - x_m+1n+1) = x_mn - x_mn+1 - x_m+1n + x_m+1n+1 for all $m, n \in ℕ$ . The generalized difference double notion has the following representation: Δ^mx_mn = Δ^m-1x_mn - Δ^m-1x_mn+1 - Δ^m-1x_m+1n + Δ^m-1x_m+1n+1, and also this generalized difference double notion has the following binomial representation: $Δ^{m} x_{mn} = \sum_{i = 0}^{m} \sum_{j = 0}^{m} (- 1)^{i + j} (\overset{m}{i}) (\overset{m}{j}) x_{m + i, n + j} .$

Let w³, χ³ (Δ_mnk) , Λ³ (Δ_mnk) be denote the spaces of all, triple gai difference sequence space and triple analytic difference sequence space respectively and is defined as $\begin{matrix} Δ_{mnk} & = & x_{mnk} - x_{m, n + 1, k} - x_{m, n, k + 1} \\ + x_{m, n + 1, k + 1} - x_{m + 1, n, k} + x_{m + 1, n + 1, k} \\ + x_{m + 1, n, k + 1} - x_{m + 1, n + 1, k + 1} \end{matrix}$ and Δ⁰x_mnk = 〈 x_mnk 〉 .

2 Definitions and preliminaries

2.1 Definition

[see [13]] An Orlicz function is a function M : [0, ∞) → [0, ∞) which is continuous, non-decreasing and convex with M (0) =0, M (x) >0, for x > 0 and M (x)→ ∞ as x→ ∞. If convexity of Orlicz function M is replaced by M (x + y) ≤ M (x) + M (y) , then this function is called modulus function.

Lindenstrauss and Tzafriri [13] used the idea of Orlicz function to construct Orlicz sequence space $ℓ_{M} = {x \in w : \sum_{k = 1}^{\infty} M (\frac{| x_{k} |}{ρ}) < \infty, for some ρ > 0},$

The space ℓ_M with the norm $∥ x ∥ = inf {ρ > 0 : \sum_{k = 1}^{\infty} M (\frac{| x_{k} |}{ρ}) \leq 1},$ becomes a Banach space which is called an Orlicz sequence space. For M (t) = t^p (1 ≤ p < ∞), the spaces ℓ_M coincide with the classical sequence space ℓ_p.

A sequence f = (f_mnk) of Orlicz function is called a Musielak-Orlicz function. A sequence g = (g_mn) defined by $\begin{matrix} g_{mn} (v) & = & \sup {| v | u - (f_{mnk}) (u) : u \geq 0}, m, n, \\ k = 1, 2, \dots \end{matrix}$ is called the complementary function of a Musielak-Orlicz function f. For a given Musielak-Orlicz function f, the Musielak-Orlicz sequence space t_f is defined as follows $\begin{matrix} t_{f} & = & {x \in w^{3} : I_{f} (| x_{mnk} |)^{1 / m + n + k} \\ \to 0 as m, n, k \to \infty}, \end{matrix}$ where I_f is a convex modular defined by $\begin{matrix} I_{f} (x) & = & \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \sum_{k = 1}^{\infty} f_{mnk} (| x_{mnk} |)^{1 / m + n + k}, \\ x = (x_{mnk}) \in t_{f} . \end{matrix}$

We consider t_f equipped with the Luxemburg metric $\begin{matrix} d (x, y) \\ = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \sum_{k = 1}^{\infty} f_{mnk} (\frac{| x_{mnk} |^{1 / m + n + k}}{mnk}) \end{matrix}$ is an exteneded real number.

Let X be a non-empty set, then a family of sets I ⊂ 2^X×X×X (the class of all subsets of X) is called an ideal if and only if for each A, B ∈ I, we have A ∪ B ∈ I and for each A ∈ I and each each B ⊂ A, we have B ∈ I . A non-empty family of sets F ⊂ 2^X×X×X is a filter on X if and only if φ ∉ F, for each A, B ∈ F, we have A ∩ B ∈ F and each A ∈ F and each A ⊂ B, we have B ∈ F . An ideal I is called non-trivial ideal if I ≠ φ and X ∉ I . Clearly I ⊂ 2^X×X×X is a non-trivial ideal if F = F (I) = {X/A : A ∈ I} is a filter on X. A non-trivial ideal I ⊂ 2^X×X×X is called admissible if and only if {{x} : x ∈ X} ⊂ I . Further details on ideals of 2^X×X×X can be found in Kostyrko. The notion was further investigated by Salat, et. al. and others. Throughout the ideals of 2^N×N×N and 2^N×N×N will be denoted by I and I₂ respectively.

A fuzzy real number X is a fuzzy set on R, a mapping X : R × R × R → L × L × L (= [0, 1]) associating each real number t with its grade of membership X (t). The α-level set of a fuzzy real number X, 0 < α < 1 denoted by [X] ^α is defined as [X] ^α = {t ∈ R : X (t) ≥ α} . A fuzzy real number X is called convex if X (t) ≥ X (s) ∧ X (r) ∧ X (ν) = min(X (s) , X (r) , X (ν)) , where s < t < r < v . If there exists t₀ ∈ R such that X (t₀) =1, then the fuzzy real number X is called normal. A fuzzy real 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 L (R).

Throughout a fuzzy real valued triple sequence is denoted by (X_mnk) i.e a triple infinite array of fuzzy real number X_mnk for all $m, n, k \in ℕ .$

Every real number r can express as a fuzzy real number $\bar{r}$ as follows: $\bar{r} = {\begin{matrix} 1, & if t = r; \\ 0, & otherwise \end{matrix}$

Let D be the set of all closed bounded intervals X = [X^L, X^R]. Then X ≤ Y if and only if X^L ≤ Y^L and X^R ≤ Y^R.

Also d (X, Y) = max(|X^L - Y^L|, |X^R - Y^R|) . Then (D, d) is a complete metric space.

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

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

2.2 Definition

A triple difference sequences (ΔX_mnk) is said to be convergent to the fuzzy real number X, if for every ɛ > 0, there exists m₀ = m₀ (ɛ) , n₀ = n₀ (ɛ) , k₀ = k₀ (ɛ) ∈ N such that $\bar{d} (Δ X_{mnk}, X) < ɛ$ for all m ≥ m₀, n ≥ n₀, k ≥ k₀ .

2.3 Definition

A triple difference sequence (ΔX_mnk) is said to be I- convergent to the fuzzy number X₀, if for all ɛ > 0, the set {(m, n, k) ∈ N : d (ΔX_mnk, X₀) ≥ ɛ} ∈ I₃. We write I₃ - lim ΔX_mnk = X₀.

2.4 Definition

A fuzzy real-valued triple difference sequence space E is said to be solid of (ΔY_mnk) ∈ E whenever (ΔX_mnk) ∈ E and |ΔY_mnk| ≤ |ΔX_mnk| for all $m, n, k \in ℕ$ .

Let $K = (m_{i}, n_{i}, k_{i}) : i \in ℕ; m_{1} < m_{2} < m_{3} \dots, n_{1} < n_{2} < n_{3} \dots and k_{1} < k_{2} < k_{3} \dots \subseteq ℕ \times ℕ \times ℕ$ and E be a triple sequence space.

A K- step space of E is a sequence space $λ_{K}^{E} = {(x_{m_{i} n_{i} k_{i}}) \in w^{3 F} : (x_{mnk}) \in E}$ .

A canonical pre-image of a triple difference sequence (Δx_{m
_i
n
_i
k
_i}) ∈ E is a difference sequence (ΔY_mnk) defined as follows: $Δ Y_{mnk} = {\begin{matrix} Δ X_{mnk}, & if (m, n, k) \in K, \\ \bar{0}, & otherwise . \end{matrix}$

A canonical pre-image of a step space $λ_{K}^{E}$ is a set of canonical pre-images of all elements in $λ_{K}^{E}$ .

2.5 Definition

A triple difference sequence E is said to be monotone if E contains the canonical pre-image of all its step spaces.

2.6 Definition

A triple difference sequence E is said to be symmetric if (X_π(m,n,k)) ∈ E, whenever (ΔX_mnk) ∈ E, where π is a permutation of N × N × N .

2.7 Definition

A triple difference sequence E is said to be sequence algebra if (ΔX_mnk ⊗ ΔY_mnk ⊗ ΔZ_mnk) ∈ E, whenever (ΔX_mnk) , (ΔY_mnk) , (ΔZ_mnk) ∈ E .

2.8 Definition

A triple difference sequence E is said to be convergence free if (ΔY_mnk) ∈ E, whenever (ΔZ_mnk), (ΔX_mnk) ∈ E and $Δ X_{mnk} = \bar{0}$ implies $Δ Y_{mnk} = \bar{0}$ implies $Δ Z_{mnk} = \bar{0} .$

Let (ΔX_mnk) be a triple sequence of fuzzy numbers and (p_mnk) be a triple sequence of analytic strictly positive real numbers such that $0 < p_{mnk} \leq \underset{mnk}{supp} < \infty .$

2.9 Definition

Let $n \in ℕ$ and X be a real vector space of dimension m, where n ≤ m . A real valued function d_p (x₁, …, x_n) = ∥ (d₁ (x₁) , …, d_n (x_n)) ∥ _p on X satisfying the following four conditions:

∥ (d₁ (x₁) , …, d_n (x_n)) ∥ _p = 0 if and and only if d₁ (x₁) , …, d_n (x_n) are linearly dependent,

∥ (d₁ (x₁) , …, d_n (x_n)) ∥ _p is invariant under permutation,

$∥ (α d_{1} (x_{1}), \dots, α d_{n} (x_{n})) ∥_{p} = | α | ∥ (d_{1} (x_{1}), \dots, d_{n} (x_{n})) ∥_{p}, α \in ℝ$

d_p ((x₁, y₁) , (x₂, y₂)⋯ (x_n, y_n)) = (d_X (x₁, x₂, ⋯ x_n) ^p + d_Y (y₁, y₂, ⋯ y_n) ^p) ^1/pfor 1 ≤ p < ∞ ; (or)

d ((x₁, y₁) , (x₂, y₂) , ⋯ (x_n, y_n)) : = sup { d_X (x₁, x₂, ⋯ x_n) , d_Y (y₁, y₂, ⋯ y_n) } ,

for x₁, x₂, ⋯ x_n ∈ X, y₁, y₂, ⋯ y_n ∈ Y is called the p product metric of the Cartesian product of n metric spaces is the p norm of the n-vector of the norms of the n subspaces.

A trivial example of p product metric of n metric space is the p norm space is $X = ℝ$ equipped with the following Euclidean metric in the product space is the p norm: $\begin{matrix} {∥ (d_{1} (x_{1}), \dots, d_{n} (x_{n})) ∥}_{E} \\ = \sup (| det (d_{mn} (x_{mn})) |) \\ = \sup (| \begin{matrix} d_{11} (x_{11}) & d_{12} (x_{12}) & \dots & d_{1 n} (x_{1 n}) \\ d_{21} (x_{21}) & d_{22} (x_{22}) & \dots & d_{2 n} (x_{1 n}) \\ . \\ . \\ . \\ d_{n 1} (x_{n 1}) & d_{n 2} (x_{n 2}) & \dots & d_{nn} (x_{nn}) \end{matrix} |) \end{matrix}$ where $x_{i} = (x_{i 1}, \dots x_{in}) \in ℝ^{n}$ for each i = 1, 2, ⋯ n .

If every Cauchy sequence in X converges to some L ∈ X, then X is said to be complete with respect to the p- metric. Any complete p- metric space is said to be p- Banach metric space.

2.10 Lemma

If $I \subset 2^{ℕ \times ℕ \times ℕ}$ is a maximal ideal, then for each $A \subset ℕ$ we have either A ∈ I or $ℕ - A \in I$ .

2.11 Definition

Let d be a mapping from R (I) × R (I) × R (I) into R^* (I) × R^* (I) × R^* (I) and let the mappings L, f : [0, 1] × [0, 1] × [0, 1] → [0, 1] × [0, 1] × [0, 1] be symmetric, non-decreasing Musielak Orlicz in both arguments and satisfy L × L × L (0, 0) = 0 and f × f × f (1, 1, 1) = 1 . Denote [d (X, Y)] _α = [λ_α (X, Y) , (X, Y)] , for X, Y ∈ R (I) × R (I) and 0 < α < 1 .

The (R (I) × R (I) × R (I) , d, L × L × L, f × f × f) is called a fuzzy p- metric space and d a fuzzy translation metric, if

$d (X, Y) = \bar{0}$ if and only if X = Y,

d (X, Y) = d (Y, X) for all X, Y ∈ X,

for all X, Y, Z ∈ R (I) × R (I) , (i) d (X, Y) (s + t) ≥ L × L (d (X, Z) (s) , d (Z, Y) (t)) whenever s ≤ λ₁ (X, Z) , t ≤ λ₁ (Z, Y) and (s + t) ≤ λ₁ (X, Y) ,

(ii) d (X, Y) (s + t) ≤ f × f (d (X, Z) (s) , d (Z, Y) (t)) whenever s ≥ λ₁ (X, Z) , t ≥ λ₁ (Z, Y) and (s + t) ≤ λ₁ (X, Y) ,

The following well-known inequality will be used throughout the article. Let p = (p_mn) be any sequence of positive real numbers with $0 \leq p_{mnk} \leq \sup_{mnk} p_{mnk} = G, D = \max {1, 2 G - 1}$ then $\begin{matrix} {| a_{mnk} + b_{mnk} |}^{p_{mnk}} \\ \leq D ({| a_{mnk} |}^{p_{mnk}} + {| b_{mnk} |}^{p_{mnk}}) \\ for all m, n, k \in ℕ and a_{mnk}, b_{mnk} \in ℂ . \end{matrix}$

Also |a_mnk|^{p
_mnk}≤ max { 1, |a|^G } for all $a \in ℂ$ .

First we procure some known results; those will help in establishing the results of this article.

2.12 Lemma

If a sequence E is solid, then it is monotone. See [14], p. 53.

3 Some new triple difference sequence spaces of fuzzy numbers

The main aim of this article to introduce the following triple difference sequence spaces and examine topological and algebraic properties of the resulting sequence spaces. Let p = (p_mnk) be a sequence of positive real numbers for all $m, n, k \in ℕ$ . f = (f_mnk) be a Musielak-Orlicz function, (ΔX, ∥ (d (x₁) , d (x₂) , ⋯ , d (x_n-1)) ∥ _p) be a triple difference fuzzy p- metric space, and $μ_{mnk} (X) = λ (Δ X_{mnk}^{1 / m + n + k}, \bar{0}) \to \bar{0}$ and $(Δ X_{mnk}^{1 / m + n + k}, \bar{0}) \to \bar{0},$ as m, n, k→ ∞ and $η_{mnk} (X) = \sup_{mnk} λ (Δ X_{mnk}^{1 / m + n + k}, \bar{0}) < \infty$ and $\sup_{mnk} (Δ X_{mnk}^{1 / m + n + k}, \bar{0}) < \infty,$ be a triple difference sequence of fuzzy numbers. Using the concept of fuzzy metric, we introduce the following class of triple difference sequences: $\begin{matrix} [Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}] \\ = (m, n, k) \in ℕ^{3} : \\ [f_{mnk} ({∥ μ_{mnk} (x), (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p})] \in I, \\ [Λ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}] \\ = (m, n, k) \in N^{3} : [f_{mnk} (∥ η_{mnk} (x), (d (x_{1}), \\ d (x_{2}), \dots, d (x_{n - 1})) ∥_{p})] \in I . \end{matrix}$

3.1 Theorem

Let f = (f_mnk) be a Musielak-Orlicz function, the triple difference sequence spaces $[Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}]$ and $[Λ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}]$ are linear spaces.

Proof. It is trivial. Therefore omit the proof.

3.2 Remark

It is easy to verify $[Λ_{f}^{3 I (Δ F)}, ∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥_{p}]$ is a linear space.

3.3 Theorem

The class of triple difference sequences $[Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}]$ isnot solid.

Proof. Let $X = (X_{mnk}) = 1 \in [Γ_{f}^{3 I (Δ F)}, ∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥_{p}] .$ If α_mnk = (- 1) ^m+n+k then $(α_{mnk} x_{mnk}) \notin [Γ_{f}^{3 I (F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}] .$ Hence $[Γ_{f}^{3 I (F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}]$ is not solid.

3.4 Theorem

The class of triple difference sequences $[Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}]$ is not perfect.

3.5 Corollary

The triple difference sequence space of $[Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}]$ isnot monotone.

Proof. By Lemma 2.12, it follows that the space $[Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}]$ is not monotone.

3.6 Theorem

The class of sequences $[Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}]$ is not convergence free.

Proof. Consider a sequence $(Δ X_{mnk}) \in [Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}]$ defined as follows: For m, n, k are even $\begin{matrix} Δ X_{mnk} {(t)}^{1 / m + n + k} \\ = {\begin{matrix} 1 + {(mnk)}^{2} t, for - {(mnk)}^{- 3} \leq t \leq 0, \\ 1 - {(mnk)}^{2} t, for 0 \leq t \leq {(mnk)}^{- 2}, \\ 0, otherwise \end{matrix} \end{matrix}$ and for m, n, k are odd, $Δ X_{mnk} {(t)}^{1 / m + n + k} = \bar{0} .$

Now for α ∈ (0, 1] , $Δ X_{mnk}^{α / m + n + k} = {\begin{matrix} [(α - 1) {(mn)}^{- 3}, (1 - α) {(mnk)}^{- 3}], for m, n, k even \\ [0, 0], for m, n, k odd . \end{matrix}$

Then $λ_{α} (Δ X_{mnk}^{1 / m + n + k}, \bar{0}) = (α - 1) {(mnk)}^{- 3} \to \bar{0},$ as m, n, k → ∞ , and $(Δ X_{mnk}^{1 / m + n + k}, \bar{0}) = (1 - α) {(mnk)}^{- 3} \to \bar{0},$ as m, n, k → ∞ .Thus, $(Δ X_{mnk}) \in [Γ_{f}^{3 I (Δ F)}, ∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥_{p}] .$ Let us define a sequence (ΔY_mnk) as follows For m, n, k are even $\begin{matrix} Δ Y_{mnk} {(t)}^{1 / m + n + k} \\ = {\begin{matrix} 1 + {(mnk)}^{1} t, for - {(mnk)}^{- 1} \leq t \leq 0, \\ 1 - {(mnk)}^{1} t, for 0 \leq t \leq {(mnk)}^{- 1}, \\ 0, otherwise \end{matrix} \end{matrix}$ and for m, n, k are odd, $Δ Y_{mnk} {(t)}^{1 / m + n + k} = \bar{0} .$

Now for α ∈ (0, 1] , $Δ Y_{mnk}^{α / m + n + k} = {\begin{matrix} [(α - 1) {(mnk)}^{- 1}, (1 - α) {(mnk)}^{- 1}], for m, n, k even \\ [0, 0], for m, n, k odd . \end{matrix}$

Then $λ_{α} (Δ Y_{mnk}^{1 / m + n + k}, \bar{0}) = (α - 1) {(mnk)}^{- 3} ↛ \bar{0},$ as m, n, k → ∞ , and $(Δ Y_{mnk}^{1 / m + n + k}, \bar{0}) = (1 - α) {(mnk)}^{- 3} ↛ \bar{0},$ as m, n, k → ∞ . Thus, $(Δ Y_{mnk}) \notin [Γ_{f}^{3 I (Δ F)}, ∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥_{p}] .$ Hence $[Γ_{f}^{3 I (Δ F)}, ∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥_{p}]$ is not convergence free.

3.7 Theorem

The class of sequences $[Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}]$ is not symmetric.

Proof. Under the restrictions on X and λ as given in the proof of Theorem 3.3, consider the sequence x = (mnk) , then $x \in [Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}] .$ Let (y_mnk) be a rearrangement of (x_mnk) , which is defined as follows: $(y_{mnk}) = {x_{111}, x_{231}, x_{495}, x_{578}, x_{678}, x_{943} \dots} .$

Then $(y_{mnk}) \notin [Γ_{f}^{3 I (Δ F)}, ∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥_{p}] .$

3.8 Theorem

The class of triple difference sequences $[Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}]$ is not a sequence algebras.

Proof. Under the restrictions on X and λ as given in the proof of Theorem 3.3, consider the sequence x = (m^-2n^-2k^-2) and y = (m^-2n^-2k^-2) , then $x, y \in [Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}]$ but $x \cdot y \notin [Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}]$ . The other cases can be proved on considering similar examples.

3.9 Theorem

The dual space of $[Γ_{f}^{3 I (Δ F)}, ∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥_{p}]$ is $[Λ_{f}^{3 I (Δ F)}, ∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥_{p}] .$ In other words $\begin{matrix} {[Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}]}^{*} \\ = [Λ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}] \end{matrix}$

Proof: We recall that $Δ x = (Δ x_{mnk}) = (\begin{matrix} 0 & 0 & \dots 0 & 0 & \dots \\ 0 & 0 & \dots 0 & 0 & \dots \\ . \\ . \\ . \\ 0 & 0 & \dots 1 & 0 & \dots \\ 0 & 0 & \dots 0 & 0 & \dots \\ . & . & \dots & . & \dots \end{matrix})$ with 1 in the (m, n, k) th position and zero’s else where, with $Δ x = (Δ x_{mnk}), [Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}] = (\begin{matrix} 0 . & . & . & 0 \\ . \\ . \\ . \\ 0 & f {(1)}^{1 / m + n + k} & . & 0 \\ (m, n, k)^{th} \\ 0 & . & . & 0 \end{matrix})$ which is a p- metric ideal of triple entire difference sequence of fuzzy real numbers. Hence, $Δ x_{mnk} \in [Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}] . f (x) = \sum_{m, n, k = 1}^{\infty} Δ x_{mnk} Δ y_{mnk}$ with $x \in [Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}]$ and $f \in {[Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}]}^{*},$ where ${[Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}]}^{*}$ is the dual space of $[Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}] .$

Take $Δ x = (Δ x_{mnk}) \in [Γ_{f}^{3 I (Δ F)}, ∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥_{p}] .$ Then, $| Δ y_{mnk} | \leq ∥ f ∥ d (Δ x_{mnk}, 0) < \infty \forall m, n, k$ (1)

Thus, (Δy_mnk) is a p- metric ideal of triple analytic difference sequence of fuzzy real numbers and hence an p- metric of triple analytic difference sequence. In other words, $y \in [Λ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}] .$ Therefore ${[Γ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}]}^{*} = [Λ_{f}^{3 I (Δ F)}, {∥ (d (x_{1}), d (x_{2}), \dots, d (x_{n - 1})) ∥}_{p}] .$ This completes the proof.

Competing interests

The authors declare that there is not any conflict of interests regarding the publication of this manuscript.

Footnotes

Acknowledgments

The authors are extremely grateful to the anonymous learned referee(s) for their keen reading, valuable suggestion and constructive comments for the improvement of the manuscript. The authors are thankful to the editor(s) and reviewers of Journal of Intelligent and Fuzzy Systems.

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