Rough convergence of Bernstein fuzzy I -convergent of χ f ( Δ,p ) 3 I ( F ) space defined by Orlicz function

Abstract

There are several notions of convergence of fuzzy number sequences in the literature. The aim of this paper is to introduce and study a new concept of the rough fuzzy ideal convergent triple sequences defined by Orlicz function. Also, some topological properties of the resulting sequence spaces of rough fuzzy numbers were examined.

Keywords

Triple sequences rough convergence fuzzy numbers Orlicz function difference sequence duals integral sequence 40F05 40J05 40G05

1. Introduction

The notion of $I$ -convergence was introduced by Kostyrko, et al. [17] as a generalization of statistical convergence which is based on the structure of the ideal $I$ of subset of positive integers $ℕ$ . A family of sets $I \subset 2^{ℕ}$ is called an ideal if and only if (i) $\emptyset \in I$ , (ii) For each $A, B \in I$ we have $A \cup B \in I$ , (iii) For each $A \in I$ and each B ⊂ A we have $B \in I$ . An ideal is called nontrivial if $ℕ \notin I$ and nontrivial ideal is called admissible if ${n} \in I$ for each $n \in ℕ$ . A family of sets $F \subset 2^{ℕ}$ is called a filter if and only if (i) $\emptyset \in F$ , (ii) For each $A, B \in F$ we have $A \cap B \in F$ , (iii) For each $A \in F$ and each B ⊃ A we have $B \in F$ . Clearly $I$ is a nontrivial ideal in $ℕ$ if and only if $F (I) = {M \subset ℕ : (\exists A \in I) (M = ℕ ∖ A)}$ is a filter in $ℕ$ . A nontrivial ideal $I$ is maximal if there cannot exist any nontrivial ideal $J \neq I$ containing $I$ as a subset. Further details on ideals can be found in Kostyrko, et al. (see [17]). Recall that a sequence x = (x_k) of points in $ℝ$ is said to be $I$ -convergent to a real number ℓ if ${k \in ℕ : | x_{k} - ℓ | \geq ɛ} \in I$ for every ɛ > 0. In this case we write $I - \lim x_{k} = ℓ$ .

If we take $I = I_{fin} = {A \subseteq ℕ : A is a finite$ subset}. Then, $I_{fin}$ is a non-trivial admissible ideal of $ℕ$ and the corresponding convergence coincides with the usual convergence. If we take $I = I_{δ} = {A \subseteq ℕ : δ (A) = 0}$ where δ (A) denote the asymptotic density of the set A, then $I_{δ}$ is a nontrivial admissible ideal of $ℕ$ and the corresponding convergence coincides with the statistical convergence.

The idea of rough convergence was first introduced by Phu [21 –23] in finite dimensional normed spaces. He showed that the set ${LIM}_{x}^{r}$ is bounded, closed and convex; and he introduced the notion of rough Cauchy sequence. He also investigated the relations between rough convergence and other convergence types and the dependence of ${LIM}_{x}^{r}$ on the roughness of degree r.

Throughout the paper, let r be a nonnegative real number. The sequence x = (x_i) is said to be r-convergent to x_∗, denoted by x_i → ^rx_∗ provided that $\forall ϵ > 0 \exists i_{ϵ} \in ℕ : i \geq i_{ϵ} \Rightarrow ‖ x_{i} - x_{*} ‖ < r + ϵ .$ The set ${LIM}^{r} x : = {x_{*} \in ℝ^{n} {: x}_{i} \to^{r} x_{*}}$ is called the r-limit set of the sequence x = (x_i). A sequence x = (x_i) is said to be r-convergent if LIM^rx ≠ ∅. In this case, r is called the convergence degree of the sequence x = (x_i). For r = 0, we get the ordinary convergence. There are several reasons for this interest.

A sequence x = (x_i) is said to be rough $I$ -convergent to x_∗, denoted by $x_{i} \to^{r - I} x_{*}$ provided that ${i \in ℕ : ‖ x_{i} - x_{*} ‖ \geq r + ϵ}$ belongs to $I$ for every ɛ > 0; or equivalently, if the condition $ℐ - \lim \sup ‖ x_{i} - x_{*} ‖ \leq r$ is satisfied. In addition, we can write $x_{i} \to^{r - I} x_{*}$ iff the inequality $‖ x_{i} - x_{*} ‖ < r + ϵ$ holds for everyɛ > 0 and almost all i.

Aytar [2] studied rough statistical convergence and defined the set of rough statistical limit points of a sequence and obtained two statistical convergence criteria associated with this set and prove that this set is closed and convex. Also, Aytar [3] studied that the r-limit set of the sequence is equal to intersection of these sets and that r-core of the sequence is equal to the union of these sets. Dündar and Çakan [11] investigated rough ideal convergence and defined the set of rough ideal limit points of a sequence. The notion of $I_{3}$ -convergence of a triple sequence spaces which is based on the structure of the ideal $I_{3}$ of subsets of $ℕ \times ℕ \times ℕ$ is a natural generalization of the notion of convergence and statistical convergence. For more details on $I$ -convergence and rough convergence, one can refer to Dündar and Talo [13], Dündar and Çakan [12], Dündar [9], Kişi and Dündar [16], Arslan and Dündar [1], Dündar and Altay [10], Ulusu et al. [28, 29] and Nuray et al. [20].

Let K be a subset of the set $ℕ \times ℕ \times ℕ$ and let us denote the set K_ijℓ = {(m, n, k) ∈ K : m ≤ i, n ≤ j, k ≤ ℓ}. Then, the natural density of K is given by $δ (K) = \lim_{i, j, ℓ \to \infty} \frac{| K_{i j ℓ} |}{i j ℓ},$ where $| K_{i j ℓ} |$ denotes the number of elements in K_ijℓ.

The Bernstein operator of order (r, s, t) is given by $\begin{matrix} B_{rst} (f, x) = \sum_{m = 0}^{r} \sum_{n = 0}^{s} \sum_{k = 0}^{t} f (\frac{mnk}{rst}) (\overset{r}{m}) (\overset{s}{n}) \\ (\overset{t}{k}) x^{m + n + k} {(1 - x)}^{(m - r) + (n - s) + (k - t)} \end{matrix}$ where f is a continuous (real or complex valued) function defined on [0, 1].

Throughout the paper, $ℝ^{3}$ denotes the real of three dimensional space with metric (X, d). Consider a triple sequence of Bernstein polynomials (B_mnk (f, x)) such that $(B_{mnk} (f, x)) \in ℝ$ , $m, n, k \in ℕ$ .

Let f be an Orlicz function defined on the closed interval [0, 1]. A triple sequence of Bernstein polynomials (B_mnk (f, x)) is said to be statistically convergent to $0 \in ℝ$ , written as st - lim x = 0, provided that the set $K_{ϵ} : = {(m, n, k) \in ℕ^{3} : | B_{m n k} (f, x) - f (x) | \geq ϵ}$ has natural density zero for any ɛ > 0. In this case, 0 is called the statistical limit of the triple sequence of Bernstein polynomials, i.e., δ (K_ɛ) = 0. That is, $\begin{matrix} \lim_{r, s, \to \infty} \frac{1}{rst} | {(m, n, k) \leq (r, s, t) : \\ | B_{mnk} (f, x) - (f, x) | \geq ɛ} | = 0 . \end{matrix}$

In this case, we write δ - lim B_mnk (f, x) = f (x) or B_mnk (f, x) → ^{S
_B}f (x).

Throughout the paper, χ_A-the characteristic function of $A \subset ℕ$ . A subset A of $ℕ$ is said to have asymptotic density d (A) if $d (A) = \lim_{i, j, ℓ \to \infty} \frac{1}{ij ℓ} \sum_{m = 1}^{i} \sum_{n = 1}^{j} \sum_{k = 1}^{ℓ} χ_{A} (K) .$

A triple sequence (real or complex) can be defined as a function $x : ℕ \times ℕ \times ℕ \to ℝ (ℂ)$ , where $ℕ$ , $ℝ$ and $ℂ$ denote the set of natural numbers, real numbers and complex numbers, respectively. The different types of notions of triple sequence was introduced and investigated by Şahiner et al. [25, 26], Esi et al. [4 –7], Dutta et al. [14], Subramanian et al. [27], Debnath et al. [8], Potucek [24] and many others.

The set of fuzzy real numbers is denoted by $f (x) (ℝ)$ and d denotes the supremum metric on $f (X) (ℝ^{3})$ . Now let r be nonnegative real number. A triple sequence space of Bernstein polynomials of (B_mnk (f, X)) of fuzzy numbers is r-convergent to a fuzzy number f (X) and we write $B_{mnk} (f, X) \to^{r} f (X) as m, n, k \to \infty,$ provided that for every ɛ > 0 there is an integer m_ɛ, n_ɛ, k_ɛ so that $\begin{matrix} d (B_{mnk} (f, X), f (X)) < r + ɛ whenever \\ m \geq m_{ɛ}, n \geq n_{ɛ}, k \geq k_{ɛ} . \end{matrix}$

The set $\begin{matrix} {LIM}^{r} B_{mnk} (f, X) : = {f (X) \in f (X) (ℝ^{3}) : \\ B_{mnk} (f, X) \to^{r} f (X), as m, n, k \to \infty} \end{matrix}$ is called the r-limit set of the triple sequence space of Bernstein polynomials of (B_mnk (f, X)).

A triple sequence space of Bernstein polynomials of fuzzy numbers which is divergent can be convergent with a certain roughness degree. For instance, let us define $B_{m n k} (f, X) = {\begin{array}{l} η (X), & if m + n + k is odd integer \\ μ (X), & otherwise \end{array},$ where $η (X) = {\begin{array}{l} X, & if X \in [0, 1] \\ - X + 2, & if X \in [1, 2] \\ 0, & otherwise \end{array}$ and $μ (X) = {\begin{array}{l} X - 3, & if X \in [3, 4] \\ - X + 5, & if X \in [4, 5] \\ 0, & otherwise \end{array} .$ Then, we have ${LIM}^{r} B_{m n k} (f, X) = {\begin{array}{l} ϕ & , & if r < \frac{3}{2} \\ [μ - r_{1}, η + r_{1}] & , & otherwise \end{array}$ where r₁ is nonnegative real number with $\begin{matrix} [μ - r_{1}, η + r_{1}] : = {B_{mnk} (f, X) \in f (X) (ℝ^{3}) : \\ μ - r_{1} \leq B_{mnk} (f, X) \leq η + r_{1}} . \end{matrix}$ The ideal of rough convergence of a triple sequence space of Bernstein polynomials can be interpreted as follows:

Let (B_mnk (f, Y)) be a convergent triple sequence space of Bernstein polynomials of fuzzy numbers. Assume that (B_mnk (f, Y)) cannot be determined exactly for every $(m, n, k) \in ℕ^{3}$ . That is, (B_mnk (f, Y)) cannot be calculated so we can use approximate value of (B_mnk (f, Y)) for simplicity of calculation. We only know that (B_mnk (f, Y)) ∈ [μ_mnk, λ_mnk], where d (μ_mnk, λ_mnk) ≤ r for every $(m, n, k) \in ℕ^{3}$ . Then, the triple sequence of Bernstein polynomials of (B_mnk (f, X)) may not be convergent, but the inequality

\begin{matrix} d (B_{mnk} (f, X), f (X)) & \leq d (B_{mnk} (f, X), B_{mnk} (f, Y)) \\ + d (B_{mnk} (f, Y), f (Y)) \\ \leq r + d (B_{mnk} (f, Y), f (Y)) \end{matrix}

implies that the triple sequence space of Bernstein polynomials of (B_mnk (f, X)) is r-convergent.

A sequence x = (x_mnk) is called triple gai sequence if ${((m + n + k)! | x_{m n k} |)}^{\frac{1}{m + n + k}} \to 0$ as m, n, k→ ∞. The triple gai sequences will be denoted by χ³.

Definition 1.1. An Orlicz function (see [19]) 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 (see [18]) used the idea of Orlicz function to construct Orlicz sequence space.

A sequence g = (g_mn) defined by $\begin{array}{l} g_{m n} (v) = \sup {| v | u - (f_{m n k}) (u) : u \geq 0}, \\ m, n, k = 1, 2, \dots \end{array}$ is called the complementary function of a Musielak-Orlicz function f. For a given Musielak-Orlicz function f (see [19]), the Musielak-Orlicz sequence space t_f is defined as follows $\begin{array}{l} t_{f} = {x \in w^{3} : ℐ_{f} {(| x_{m n k} |)}^{1 / (m + n + k)} \to 0 \\ as m, n, k \to \infty}, \end{array}$ where $I_{f}$ is a convex modular defined by $\begin{array}{l} ℐ_{f} (x) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \sum_{k = 1}^{\infty} f_{m n k} {(| x_{m n k} |)}^{1 / (m + n + k)}, \\ x = (x_{m n k}) \in t_{f} . \end{array}$ We consider t_f equipped with the Luxemburg metric $d (x, y) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \sum_{k = 1}^{\infty} f_{m n k} ({| x_{m n k} |}^{1 / (m + n + k)})$ A sequence x = (x_mnk) is said to be triple analytic if $\sup_{m, n, k} {| x_{m n k} |}^{\frac{1}{m + n + k}} < \infty .$ The vector space of all triple analytic sequences are usually denoted by Λ³. A sequence x = (x_mnk) is called triple entire sequence if ${| x_{m n k} |}^{\frac{1}{m + n + k}} \to 0 a s m, n, k \to \infty .$ The vector space of all triple entire sequences are usually denoted by Γ³.

Let w³, χ³ (Δ_mnk), Λ³ (Δ_mnk) be denote the spaces of all triple sequence space, 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 $Δ^{0} x_{m n k} = 〈 x_{m n k} 〉,$ respectively.

In this paper, we first define the concept of rough convergence of Bernstein polynomials of fuzzy $I_{3}$ -convergent of triple sequence defined by Orlicz function and also discuss some topological properties of fuzzy numbers.

2. Definitions and Preliminaries

A fuzzy number X is a fuzzy subset of the real $ℝ^{3}$ , which is normal fuzzy convex, upper semi-continuous and the X⁰ is bounded where $X^{0} = * cl {x \in ℝ^{3} : X (x) > 0}$ and *cl is the closure operator. These properties imply that for each α ∈ (0, 1], the α-level set X^α defined by $X^{α} = {x \in ℝ^{3} : X (x) \geq α} = [{\underline{X}}^{α}, {\bar{X}}^{α}]$ is a non empty compact convex subset of $ℝ^{3}$ .

The supremum metric d on the set $L (ℝ^{3})$ is defined by $d (X, Y) = sup_{α \in [0, 1]} max (| {\underline{X}}^{α} - {\underline{Y}}^{α} |, | {\bar{X}}^{α} - {\bar{Y}}^{α} |) .$ Now, given $X, Y \in L (ℝ^{3})$ , we define X ≤ Y if ${\underline{X}}^{α} \leq {\underline{Y}}^{α}$ and ${\bar{X}}^{α} \leq {\bar{Y}}^{α}$ for each α ∈ [0, 1].

We write X ≤ Y if X ≤ Y and there exists an α₀ ∈ [0, 1] such that ${\underline{X}}^{α_{0}} \leq {\underline{Y}}^{α_{0}}$ or ${\bar{X}}^{α_{0}} \leq {\bar{Y}}^{α_{0}}$ .

A subset E of $L (ℝ^{3})$ is said to be bounded above if there exists a fuzzy number μ, called an upper bound of E, such that X ≤ μ for every X ∈ E. μ is called the least upper bound of E if μ is an upper bound and μ ≤ μ′ for all upper bounds μ′.

A lower bound and the greatest lower bound are defined similarly. E is said to be bounded if it is both bounded above and below.

The notions of least upper bound and the greatest lower bound have been defined only for bounded sets of fuzzy numbers. If the set $E \subset L (ℝ^{3})$ is bounded, then its supremum and infimum exist.

The limit infimum and limit supremum of a triple sequence spaces (X_mnk) is defined by $\begin{matrix} \lim_{m, n, k \to \infty} inf X_{mnk} & : = inf A_{X}, \\ \lim_{m, n, k \to \infty} sup X_{mnk} & : = inf B_{X}, \end{matrix}$ where $\begin{matrix} A_{X} & : = {μ \in L (ℝ^{3}) : The set {(m, n, k) \in ℕ^{3} : \\ X_{mnk} < μ} is infinite}, \\ B_{X} & : = {μ \in L (ℝ^{3}) : The set {(m, n, k) \in ℕ^{3} : \\ X_{mnk} > μ} is infinite} . \end{matrix}$ Now, given two fuzzy numbers $X, Y \in L (ℝ^{3})$ , we define their sum as Z = X + Y, where ${\underline{Z}}^{α} : = {\underline{X}}^{α} + {\underline{Y}}^{α}$ and ${\bar{Z}}^{α} : = {\bar{X}}^{α} + {\bar{Y}}^{α}$ for all α ∈ [0, 1].

To any real number $a \in ℝ^{3}$ , we can assign a fuzzy number $a_{1} \in L (ℝ^{3})$ , which is defied by $a_{1} (x) = {\begin{matrix} 1 & , & if x = a \\ 0 & , & otherwise \end{matrix} .$

An order interval in $L (ℝ^{3})$ is defined by $[X, Y] : = {Z \in L (ℝ^{3}) : X \leq Z \leq Y}$ , where $X, Y \in L (ℝ^{3})$ .

A set E of fuzzy numbers is called convex if λμ₁ + (1 - λ) μ₂ ∈ E for all λ ∈ [0, 1] and μ₁, μ₂ ∈ E.

Definition 2.1. Let f be an Orlicz function defined on the closed interval [0, 1]. A rough triple sequence of Bernstein polynomials of (B_mnk (f, X)) of real numbers is said to be convergent to the fuzzy real number B_mnk (f, X), if for every ɛ > 0, there exists m₀ = m₀ (ɛ), n₀ = n₀ (ɛ), $k_{0} = k_{0} (ɛ) \in ℕ$ such that $\bar{d} (B_{mnk} (f, X), f (X)) < r + ɛ$ for all m ≥ m₀, n ≥ n₀, k ≥ k₀.

Definition 2.2. Let f be an Orlicz function defined on the closed interval [0, 1]. A rough triple sequence of Bernstein polynomials of (B_mnk (f, X)) of real numbers is said to be $I_{3}$ -convergent to the fuzzy number B_mnk (f (X₀)), if for all ϵ > 0, the set ${(m, n, k) \in ℕ^{3} : \bar{d} (B_{m n k} (f, X), f (X_{0})) \geq r + ϵ} \in ℐ_{3}$ . We write $I_{3} - \lim B_{mnk} (f, X) = f (X_{0})$ .

Definition 2.3. Let f be an Orlicz function defined on the closed interval [0, 1]. A rough triple sequence of Bernstein polynomials of (B_mnk (f, X)) of real numbers is said to be solid of (B_mnk (f, Y)) ∈ E^F whenever (B_mnk (f, X)) ∈ E^F and $| B_{m n k} (f, Y) | \leq | B_{m n k} (f, X) |$ for all $m, n, k \in ℕ$ .

Let $K = {(m_{i}, n_{i}, k_{i}) : i \in ℕ; m_{1} < m_{2} < m_{3} \dots, n_{1} < n_{2} < n_{3} \dots and k_{1} < k_{2} < k_{3} \dots} \subseteq ℕ^{3}$ and E^F be a triple sequence space. A K-step space of E^F is a sequence space $λ_{K}^{E} = {(X_{m_{i} n_{i} k_{i}}) \in w^{3 F} : (B_{mnk} (f, X)) \in E^{F}} .$ Let f be an Orlicz function defined on the closed interval [0, 1]. A triple sequence of Bernstein polynomials of (B_mnk (f, X)) of real numbers of a canonical pre-image of a sequence (B_{m
_i
n
_i
k
_i} (f, X)) ∈ E^F is a sequence (B_mnk (f, Y)) defined as follows: $B_{mnk} (f, Y) = {\begin{matrix} B_{mnk} (f, X) & , & if (m, n, k) \in K \\ \bar{0} & , & otherwise \end{matrix}$ Let f be an Orlicz function defined on the closed interval [0, 1]. A triple sequence of Bernstein polynomials of (B_mnk (f, X)) of real numbers of a canonical pre-image of a step space $λ_{K}^{E}$ is a set of canonical pre-images of all elements in $λ_{K}^{E}$ .

Definition 2.4. Let f be an Orlicz function defined on the closed interval [0, 1]. A triple sequence of Bernstein polynomials of (B_mnk (f, X)) of real numbers is said to be monotone if E^F contains the canonical pre-image of all its step spaces.

Definition 2.5. Let f be an Orlicz function defined on the closed interval [0, 1]. A triple sequence of Bernstein polynomials of (B_mnk (f, X)) of real numbers is said to be symmetric if (B_{π(m),π(n),π(k)}) (f, X) ∈ E^F, whenever (B_mnk (f, X)) ∈ E^F, where π is a permutation of $ℕ^{3}$ .

Definition 2.6. Let f be an Orlicz function defined on the closed interval [0, 1]. A triple sequence of Bernstein polynomials of (B_mnk (f, X)) of real numbers is said to be sequence algebra if B_mnk (X ⊗ Y ⊗ Z) ∈ E^F, whenever (X_mnk), (Y_mnk), (Z_mnk) ∈ E^F.

Definition 2.7. Let f be an Orlicz function defined on the closed interval [0, 1]. A triple sequence of Bernstein polynomials of (B_mnk (f, X)) of real numbers is said to be convergence free if (B_mnk (f, Y)) ∈ E^F, whenever (Z_mnk), (B_mnk (f, X)) ∈ E^F and $B_{mnk} (f, X) = \bar{0}$ implies $B_{mnk} (f, Y) = \bar{0}$ implies $B_{mnk} (f, Z) = \bar{0}$ .

Let f be an Orlicz function defined on the closed interval [0, 1]. A triple sequence of Bernstein polynomials of (B_mnk (f, X)) of real numbers and (p_mnk) be a triple sequence of analytic strictly positive real numbers such that 0< p_mnk ≤ sup p_mnk < ∞.

We introduce the following sequence spaces: $\begin{array}{l} χ_{f (Δ, p)}^{3 ℐ (F)} = { (m, n, k) \in ℕ^{3} : \\ {[\bar{d} ( B_{m n k} {(f, (m + n + k)! Δ X)}^{\frac{1}{m + n + k}} , f (\bar{0}) ) ]}^{p_{m n k}} \\ \geq r + ϵ } \in ℐ_{3}, \\ for every ϵ > 0. \end{array}$ $\begin{matrix} Λ_{f (Δ, p)}^{3 F} = {X = (X_{mnk}) : sup_{m, n, k} \\ {[\bar{d} (B_{mnk} {(f, Δ X)}^{\frac{1}{m + n + k}}, f (\bar{0}))]}^{p_{mnk}} < \infty} . \end{matrix}$ Also, we write $Λ_{f (Δ, p)}^{3 I (F)} = χ_{f (Δ, p)}^{3 I (F)} ⋂ Λ_{f (Δ, p)}^{3 F}$ .

Lemma 2.1. Let f be an Orlicz function defined on the closed interval [0, 1]. A triple sequence of Bernstein polynomials of (B_mnk (f, X)) of real numbers is solid, then it is monotone (see [15, p. 53]).

3. Main results

Proposition 3.1. Let f be an Orlicz function defined on the closed interval [0, 1]. If (B_mnk (f, X)) is a rough triple sequence of real numbers, then $Λ_{f (Δ, p)}^{3 I (F)}$ and $Λ_{f (Δ, p)}^{3 F}$ are linear spaces.

Proof. It is easy. Therefore, omit the proof.□

Proposition 3.2. Let f be an Orlicz function defined on the closed interval [0, 1]. If (B_mnk (f, X)) is a rough triple sequence of real numbers, then $χ_{f (Δ, p)}^{3 I (F)} \subseteq Λ_{f (Δ, p)}^{3 F}$ and the inclusion are strict.

Proof. The inclusion $χ_{f (Δ, p)}^{3 I (F)} \subseteq Λ_{f (Δ, p)}^{3 F}$ is obvious. For establishing that the inclusion is proper, consider the following example.□

Example 3.1. We prove the result for the case $χ_{f (Δ, p)}^{3 I (F)} \subseteq Λ_{f (Δ, p)}^{3 F}$ , the other case similar.

Let B_mnk (f, ΔX) = f (ΔX). Let the sequence B_mnk ((f, ΔX) , f (X)) be defined by for m > n > k,

$\begin{matrix} B_{mnk} (f, Δ X (t), f (Δ X)) \\ = {\begin{matrix} \frac{{(mt - m - 1)}^{(m + n + k)} {(m - 1)}^{- (m + n + k)}}{(m + n + k)!}, & for 1 + \frac{1}{m} \leq t \leq 3 \\ \frac{{(3 - t)}^{(m + n + k)}}{(m + n + k)!}, & for 2 < t \leq 4 \\ 0, & otherwise \end{matrix} \end{matrix}$ and for m < n < k,

$\begin{matrix} B_{mnk} (f, Δ X (t), f (Δ X)) \\ = {\begin{matrix} \frac{{(mt - 1)}^{(m + n + k)} {(m - 1)}^{- (m + n + k)}}{(m + n + k)!} & , & for \frac{1}{m} \leq t \leq 2 \\ \frac{{(- t + 2)}^{(m + n + k)}}{(m + n + k)!} & , & for 1 \leq t \leq 3 \\ 0 & , & otherwise \end{matrix} \end{matrix}$

Then, $B_{mnk} (f, Δ X (t), f (Δ X)) \in Λ_{f (Δ, p)}^{3 F}$ but $B_{mnk} (f, Δ X (t), f (Δ X)) \notin χ_{f (Δ, p)}^{3 I (F)}$ .

Proposition 3.3. Let f be an Orlicz function defined on the closed interval [0, 1]. If (B_mnk (f, X)) is a rough triple sequence of real numbers, then the triple difference sequence space $Λ_{f (Δ, p)}^{3 I (F)}$ is not solid.

Proof. Let $(B_{mnk} (f, X), f (X)) = (1) \in Λ_{f (Δ, p)}^{3 I (F)}$ and α_mnk = (-1) ^m+n+k, then (B_mnk (f, α_mnkX, $f (X))) \notin Λ_{f (Δ, p)}^{3 I (F)}$ . Hence, $Λ_{f (Δ, p)}^{3 I (F)}$ is not solid.□

Corollary 3.1. Let f be an Orlicz function defined on the closed interval [0, 1], the triple difference sequence space $Λ_{f (Δ, p)}^{3 I (F)}$ is not monotone.

Proof. By Lemma 2.1, it follows that the space $Λ_{f (Δ, p)}^{3 I (F)}$ is not monotone.□

Proposition 3.4. Let f be an Orlicz function defined on the closed interval [0, 1]. If (B_mnk (f, X)) is a rough triple sequence of real numbers, then the triple difference sequence space of $χ_{f (Δ, p)}^{3 I (F)}$ is sequence algebra.

Proof. Let $(X_{mnk}), (Y_{mnk}) \in χ_{f (Δ, p)}^{3 I (F)}$ and 0 < ɛ < 1. Then, the result follows from the following inclusion relation:

$\begin{array}{l} {(m, n, k) \in ℕ^{3} : \\ {[\bar{d} ( B_{mnk} {(f, (m+n+k)! (Δ X_{mnk} \otimes Δ Y_{mnk} \otimes Δ Z_{mnk}))}^{\frac{1}{m+n+k}}, f (\bar{0}))]}^{p_{mnk}} < r+ ϵ} \\ \supseteq {(m, n, k) \in ℕ^{3} : {[\bar{d} (B_{m n k} {(f, (m + n + k)! Δ X_{m n k})}^{\frac{1}{m + n + k}}, f (\bar{0}))]}^{p_{m n k}} < r + ϵ} \\ \cap {(m, n, k) \in ℕ^{3} : {[\bar{d} (B_{m n k} {(f, (m + n + k)! Δ Y_{m n k})}^{\frac{1}{m + n + k}}, f (\bar{0}))]}^{p_{m n k}} < r + ϵ} \\ \cap {(m, n, k) \in ℕ^{3} : {[\bar{d} (B_{m n k} {(f, (m + n + k)! Δ Z_{m n k})}^{\frac{1}{m + n + k}}, f (\bar{0}))]}^{p_{m n k}} < r + ϵ} . \end{array}$ Similarly we can prove the result for other cases.□

Proposition 3.5. Let f be an Orlicz function defined on the closed interval [0, 1]. If (B_mnk (f, X)) is a rough triple sequence of real numbers, then the triple difference sequence space of $χ_{f (Δ, p)}^{3 I (F)}$ is complete metric space with respect to the metric ρ defined by $\begin{matrix} ρ (X, Y, Z) = & sup_{m} {[\bar{d} (B_{mnk} {(f, (m + 1 + k)! (X_{m 1 k} Y_{m 1 k} Z_{m 1 k}))}^{\frac{1}{m + 1 + k}}, f (\bar{0}))]}^{p_{mnk}} + \\ sup_{n} {[\bar{d} (B_{mnk} {(f, (1 + n + k)! (X_{1 nk} Y_{1 nk} Z_{1 nk}))}^{\frac{1}{1 + n + k}}, f (\bar{0}))]}^{p_{mnk}} + \\ sup_{k} {[\bar{d} (B_{mnk} {(f, (m + n + 1)! (X_{mn 1} Y_{mn 1} Z_{mn 1}))}^{\frac{1}{m + 1 + k}}, f (\bar{0}))]}^{p_{mnk}} + \\ sup_{mnk} {[\bar{d} (B_{mnk} {(f, (m + n + k)! (Δ X Δ Y Δ Z))}^{\frac{1}{m + n + k}}, f (\bar{0}))]}^{p_{mnk}} \end{matrix}$ where X = (X_mnk), Y = (Y_mnk), $Z = (Z_{mnk}) \in χ_{f (Δ, p)}^{3 I (F)}$ , where $\begin{matrix} Δ X = & x_{mnk} - x_{m, n + 1, k} - x_{m, n, k + 1} + x_{m, n + 1, k + 1} \\ - x_{m + 1, n, k} + x_{m + 1, n + 1, k} + x_{m + 1, n, k + 1} \\ - x_{m + 1, n + 1, k + 1} \end{matrix}$

and $Δ^{0} x_{m n k} = 〈 x_{m n k} 〉$ for all $(m, n, k) \in ℕ^{3}$ .

Proposition 3.6. Let f be an Orlicz function defined on the closed interval [0, 1]. If (B_mnk (f, X)) is a rough triple sequence of real numbers, then the triple difference sequence space of $Λ_{f (Δ, p)}^{3 I (F)}$ is nowhere dense subsets of $Λ_{f (Δ, p)}^{3 F}$ .

Proof. By Proposition 3.1, the sequence space $Λ_{f (Δ, p)}^{3 I (F)}$ are proper subspace of $Λ_{f (Δ, p)}^{3 F}$ . Hence, by Proposition 3.5 the result follows.□

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

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