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Some results of neutrosophic normed space VIA Tribonacci convergent sequence spaces

Abstract

The concept of Tribonacci sequence spaces by the domain of a regular Tribonacci matrix was introduced by Yaying and Hazarika (Math. Slovaca 70(3):697–706, 2000). In this paper, by using the domain of regular Tribonacci matrix \(T = (t _{ik} )\) and the concept of neutrosophic convergence, we introduce some neutrosophic normed space in Tribonacci convergent spaces and prove some topological and algebraic properties based results with respect to these spaces.

Introduction

The theory of fuzzy sets was generalized from classical sets by Zadeh in 1965 [2], which was further generalized to intuitionistic fuzzy sets by Atanassov [3]. This theory deals with a situation that may be imprecise or vague or uncertain by attributing a degree of membership and a degree of non-membership to a certain object. Several literature works on their corresponding sequence spaces can be found in [46]. In 2004, Park laid the grounds of intuitionistic fuzzy metric space which was later redefined by Saadati [7] and Park [8] as fuzzy norm and intuitionistic fuzzy norm.

The idea of neutrosophic sets was introduced by Smarandache [9] as an extension of the intuitionistic fuzzy set. For the situation when the aggregate of the components is 1, in the wake of satisfying the condition by applying the neutrosophic set operators, different outcomes can be acquired by applying the intuitionistic fuzzy operators, since the operators disregard the indeterminacy, while the neutrosophic operators are taken into the cognizance of the indeterminacy at a similar level as truth-membership and falsehood-nonmembership. Using the idea of neutrosophic sets, the notion of neutrosophic bipolar vague soft set [25] and its application to decision making problems were defined. Further, Smarandache [10, 11] investigated neutroalgebra which is a generalization of partial algebra, neutroalgebraic structures, and antialgebraic structures. Neutrosophic set is a more adaptable and effective tool because it handles, aside from autonomous components, additionally partially independent and dependent information [12, 13]. Summability theory and matrix transformation have been necessary modes in developing the theory of nonconverging sequences. The motivation of it being able to transform the sequence or series which does not converge originally but approaches some number on applying the transformation. An infinite matrix is usually used for this approach, since it is the most natural operator between two sequence spaces. Some work on sequence spaces via matrix transformation can be found in [1].

Recently in [1], the authors defined the matrix corresponding to the Tribonacci sequence in [14, 15]. In this paper we aim to define novel neutrosophic sequence spaces with the help of neutrosophic norm and using the Tribonacci matrix as a mode. Also, we study Tribonacci convergent and Tribonacci Cauchy in neutrosophic normed space by using the Tribonacci matrix T. Prior to the introduction of new spaces of Tribonacci convergent sequence with respect to neutrosophic norm \((\mathsf{P},\mathsf{Q},\mathsf{R})\), we mention the following notions that will be used in the article.

Preliminaries

Let \(\mathbb{R}\) and \(\mathbb{C}\) denote the sets of real and complex numbers respectively. By ω we denote a linear space of sequence of real or complex numbers. Any vector subspace of ω is called a sequence space.

$$\begin{aligned} \omega:= \bigl\{ \vartheta =(\vartheta _{k}),k\in \mathbb{N}| \vartheta =( \vartheta _{k})\in \mathbb{R},\text{ or } \mathbb{C} \bigr\} . \end{aligned}$$
(2.1)

Let \(X_{1}\) and \(X_{2}\) be two sequence spaces and let \(T=(t_{ik})\) be an infinite matrix of real entries. We write \(T_{i}\) to denote the sequence in the nth row of matrix T. Recalling that T defines a matrix mapping from sequence space \(X_{1}\) to \(X_{2}\) if for every sequence \(\vartheta =(\vartheta _{k})\), the W transform of ϑ is defined as \(T\vartheta =\{T_{i}(\vartheta )\}_{i=1}^{\infty }\in X_{2}\), where

$$\begin{aligned} T_{i}(\vartheta )=\sum_{k=1}t_{ik} \vartheta _{k},\quad i \in \mathbb{N}. \end{aligned}$$

For any sequence space E, the sequence space \(E_{T}\) defined by

$$\begin{aligned} E_{T}= \bigl\{ \vartheta =(\vartheta _{k})\in w: T\vartheta \in E \bigr\} \end{aligned}$$

is known as domain of the matrix T.

Definition 2.1

([1, 16])

A matrix \(T=(t_{ik})_{{i,k}\in \mathbb{N}}\) is said to be regular iff the following conditions hold:

(a) There exists \(M>0\) such that for every \(i\in \mathbb{N}\), \(\sum_{k}|t_{ik}|\leq M\),

(b) \(\lim_{i\rightarrow \infty } t_{ik}=0\) for every \(k\in \mathbb{N}\),

(c) \(\lim_{i\rightarrow \infty }\sum_{k}t_{ik}=1\).

First, we give some background about Tribonacci numbers. The studies on Tribonacci numbers were first initiated by a 14-year-old student Mark Feinberg in 1963. In 1963, Mark Feinberg [15, 19] defined the sequence \((t_{n})_{{n}\in \mathbb{N}}\) of Tribonacci numbers given by third recurrence relation

$$\begin{aligned} t_{n}=t_{n-1}+t_{n-2}+t_{n-3}, \quad n\geq 3 \text{ with } t_{1}=t_{2}=1 \text{ and } t_{3}=2. \end{aligned}$$

Thus, the first few numbers of Tribonacci sequence are \(1,1,2,4,7,13,24,44,81,\ldots \) Some basic properties of Tribonacci sequence are:

$$\begin{aligned} \lim_{k\rightarrow \infty }\frac{t_{k}}{t_{k+1}}=0.54368901\cdots \end{aligned}$$

and

$$\begin{aligned} &\sum_{k=1}^{i}t_{k}= \frac{t_{i+2}+t_{i}-1}{2} \\ &\lim_{k\rightarrow \infty }\frac{t_{k+1}}{t_{k}}=1.83929\quad (\text{approx}). \end{aligned}$$

Throughout this paper we use the lower triangular Tribonacci matrix \(T=(t_{ik})\), defined in [1] as follows:

$$\begin{aligned} t_{ik}= \textstyle\begin{cases} \frac{2t_{k}}{t_{i+2}+t_{i}-1}& \text{if $(1\leq k\leq i)$}, \\ 0& \text{otherwise}. \end{cases}\displaystyle \end{aligned}$$

Equivalently,

$$\begin{aligned} T= \begin{bmatrix} 1&0&0&0&0&0&\cdots \\ \frac{1}{2}&\frac{1}{2}&0&0&0&0&\cdots \\ \frac{1}{4}&\frac{1}{4}&\frac{2}{4}&0&0&0&\cdots \\ \frac{1}{8}&\frac{1}{8}&\frac{2}{8}&\frac{4}{8}&0&0&\cdots \\ \frac{1}{15}&\frac{1}{15}&\frac{2}{15}&\frac{4}{15}&\frac{7}{15}&0&\cdots \\ \frac{1}{28}&\frac{1}{28}&\frac{2}{28}&\frac{4}{28}&\frac{7}{28}& \frac{13}{28}&\cdots \\ \vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \end{bmatrix} \end{aligned}$$
(2.2)

It can be easily verified that T is a regular matrix (from Definition 2.1). By using the Tribonacci matrix (2.1), for any sequence \(\vartheta =(\vartheta _{k})\in \omega \), the T− transformation of \((\vartheta _{k})\) is defined as

$$\begin{aligned} T_{i}(\vartheta )=\sum_{k=1}^{i} \frac{2\jmath _{k}}{\jmath _{i+2}+\jmath _{i}-1}\vartheta _{k},\quad i\in \mathbb{N}. \end{aligned}$$
(2.3)

Definition 2.2

([17, 20, 23])

Given a binary operation \(\ast:[0,1]\times [0,1]\longrightarrow [0,1] \) is said to be a continuous t-norm if

(a) is associative and commutative,

(b) is continuous,

(c) \(\vartheta \ast 1=\vartheta \) \(\vartheta \in [0,1]\),

(d) \(\vartheta \ast y \leq w*z\) whenever \(\vartheta \leq w\) and \(y\leq z\) for each \(\vartheta,y,w,z \in [0,1]\).

Example 2.1

For \(\vartheta,y\in [0,1]\), define \(\vartheta \ast y=\vartheta y\) or \(\vartheta \ast y= \min \{ \vartheta,y\}\), then is a continuous t-norm.

Definition 2.3

([17, 20, 24])

Given a binary operation, \(\diamond:[0,1]\times [0,1]\longrightarrow [0,1]\) is said to be a continuous t-conorm if

(a) is associative and commutative,

(b) is continuous,

(c) \(\vartheta \diamond 0 =\vartheta \) \(\vartheta \in [0,1]\),

(d) \(\vartheta \diamond y \leq w \diamond z \) whenever \(\vartheta \leq w\) and \(y \leq z\) for each \(\vartheta,y,w,z \in [0,1]\).

Example 2.2

Let \(\vartheta, y\in [0, 1]\). Define \(\vartheta \diamond y =\min \{\vartheta +y, 1\}\) or \(\vartheta \diamond y =\max \{\vartheta,y\}\), then is continuous t-conorm.

From the above definitions, we note that if we choose \(0<\epsilon _{1},\epsilon _{2}<1\) for \(\epsilon _{1}>\epsilon _{2}\), then there exist \(0<\epsilon _{3},\epsilon _{4}<0,1\) such that \(\epsilon _{1}\ast \epsilon _{3} \geq \epsilon _{2},\epsilon _{1} \geq \epsilon _{4} \diamond \epsilon _{2}\).

Further, if we choose \(\epsilon _{5}\in (0,1)\), then there exist \(\epsilon _{6},\epsilon _{7}\in (0,1)\) such that \(\epsilon _{6}\ast \epsilon _{6}\geq \epsilon _{5}\) and \(\epsilon _{7}\diamond \epsilon _{7}\leq \epsilon _{5}\).

Definition 2.4

([18, 21, 22])

Assume to be a continuous t-norm, to be a continuous t-conorm, and Y to be a linear space over the neutrosophic field \(\mathbb{R}\) or \(\mathbb{C}\), and \(\mathcal{Z}= \{<\vartheta,\mathsf{P}(\vartheta ),\mathsf{Q}( \vartheta ),\mathsf{R}(\vartheta ) >: \vartheta \in Y\}\) to be a normed space such that \(\mathcal{Z}:Y \times (0,\infty ) \rightarrow [0,1]\). The four-tuple \((Y, \mathcal{Z},\star, \diamond )\) is called a neutrosophic normed space \((\mathit{NNS})\) if the subsequent terms hold; for all \(\vartheta, y, z \in Y\) and \(\jmath, s> 0\),

  1. (i)

    \(0\leq \mathsf{P}(\vartheta,\jmath )\leq 1\), \(0\leq \mathsf{Q}(y,\jmath )\leq 1\), \(0\leq \mathsf{R}(z,\jmath )\leq 1\), \(\jmath \in R^{+}\),

  2. (ii)

    \(\mathsf{P}(\vartheta,\jmath )+\mathsf{Q}(\vartheta,\jmath )+ \mathsf{R}(\vartheta,\jmath )\leq 3\) for \(\jmath \in R^{+}\),

  3. (iii)

    \(\mathsf{P}(\vartheta,\jmath )=1\) for \(\jmath >0\) iff \(\vartheta =0\),

  4. (iv)

    \(\mathsf{P}(\lambda \vartheta,\jmath )=\mathsf{P}(\vartheta, \frac{\jmath }{\vert \lambda \vert })\),

  5. (v)

    \(\mathsf{P}(\vartheta,\jmath ) \star \mathsf{P}(y,s)\leq \mathsf{P}( \vartheta +y,\jmath +s)\),

  6. (vi)

    \(\mathsf{P}(\vartheta,\star )\) is a continuous nondecreasing function,

  7. (vii)

    \(\lim_{\jmath \rightarrow \infty } \mathsf{P}(\vartheta, \jmath )=1\),

  8. (viii)

    \(\mathsf{Q}(y,\jmath )=0\) for \(\jmath >0\) iff \(\vartheta =0\),

  9. (ix)

    \(\mathsf{Q}(\lambda y,\rho )=\mathsf{Q}(y, \frac{\jmath }{\vert \lambda \vert })\),

  10. (x)

    \(\mathsf{Q}(y,\jmath ) \diamond \mathsf{Q}(z,\jmath )\geq \mathsf{Q}(y+z, \jmath +s)\),

  11. (xi)

    \(\mathsf{Q}(y, \diamond )\) is a continuous nonincreasing function,

  12. (xii)

    \(\lim_{\jmath \rightarrow \infty } \mathsf{Q}(\vartheta, \jmath )=0\),

  13. (xiii)

    \(\mathsf{R}(\vartheta,\jmath )=0\) for \(\jmath >0\) iff \(\vartheta =0\),

  14. (xiv)

    \(\mathsf{R}(\lambda \vartheta,\jmath )=\mathsf{R}(\vartheta, \frac{\jmath }{\vert \lambda \vert })\),

  15. (xv)

    \(\mathsf{R}(z,\jmath ) \diamond \mathsf{R}(\vartheta,s)\geq \mathsf{R}(z+\vartheta,\jmath +s)\),

  16. (xvi)

    \(\mathsf{R}(z,\diamond )\) is a continuous nonincreasing function,

  17. (xvii)

    \(\lim_{\jmath \rightarrow \infty } \mathsf{R}(z,\jmath )=0\),

  18. (xviii)

    If \(\jmath \leq 0\), then \(\mathsf{P}(\vartheta,\jmath )=0\), \(\mathsf{Q}(y,\jmath )=1\), \(\mathsf{R}(z,\jmath )=1\).

In such a case, \(\mathcal{Z} = (\mathsf{P},\mathsf{Q},\mathsf{R})\) is called a neutrosophic norm \((\mathit{NN})\).

Example 2.3

([18])

Let \((Y,\|\cdot\|)\) be an NNS. Given the operation and as t-norm \(\vartheta \ast y=\vartheta.y\) and t-conorm \(\vartheta \diamond y=\vartheta +y-\vartheta y\) for \(\jmath >\Vert y\Vert \) and \(\jmath >0\)

$$\begin{aligned} \mathsf{P}(\vartheta,\jmath ) = \frac{\jmath }{\jmath + \Vert \vartheta \Vert }, \qquad \mathsf{Q}(\vartheta, \jmath ) = \frac{ \Vert \vartheta \Vert }{\jmath + \Vert \vartheta \Vert } \quad \text{and}\quad \mathsf{R}(\vartheta,\jmath )= \frac{ \Vert \vartheta \Vert }{\jmath } \end{aligned}$$
(2.4)

for all \(\vartheta,y \in Y \). If we take \(\jmath \leq \|\vartheta \|\), then \(\mathsf{P}(\vartheta,\jmath ) = 0,\mathsf{Q}(\vartheta,\jmath )=1, \text{ and } \mathsf{R}(\vartheta,\jmath )=1\). Then \((Y,\mathcal{Z}, \ast,\diamond )\) is an NNS where \(\mathcal{Z}:Y\times \mathbb{R}\mathbbm{^{+}} \rightarrow [0,1]\).

Example 2.4

Let \((Y= \mathbb{R},\|\cdot\|)\) be an NNS where \(\|c\|=|c| \ \forall c\in \mathbb{R}\). Given the operation and as t-norm \(\vartheta \ast y=\min \{\vartheta,y\}\) and t-conorm \(\vartheta \diamond y=\max \{\vartheta,y\}\). \(\forall \vartheta,y\in [0,1]\) and define

$$\begin{aligned} \mathsf{P}(\vartheta,\jmath )= \frac{\jmath }{\jmath +m \Vert \vartheta \Vert },\qquad \mathsf{Q}(\vartheta, \jmath ) = \frac{m \Vert \vartheta \Vert }{\jmath +m \Vert \vartheta \Vert }, \quad\text{and}\quad \mathsf{R}( \vartheta,\jmath )=\frac{m \Vert \vartheta \Vert }{\jmath }, \end{aligned}$$
(2.5)

where \(m>0\). Then \(\mathcal{Z}= \{(\vartheta,\jmath ),\mathsf{P}(\vartheta,t), \mathsf{Q}(\vartheta,t),\mathsf{R}(\vartheta,\jmath ):(\vartheta, \jmath ) \in Y\times \mathbb{R}\mathbbm{^{+}}\}\) is an NNS on Y.

Definition 2.5

Suppose that X is an NNS, the sequence \(b = (b_{i})\) in X is called convergent to \(\xi \in X\) \(N\in \mathbb{N}\), with respect to NN-\(\mathcal{Z}=(\mathsf{P},\mathsf{Q},\mathsf{R})\) if for every \(\epsilon > 0\), \(\jmath > 0\)

$$\begin{aligned} \mathsf{P} \bigl((b_{i}) - \xi, \jmath \bigr) > 1- \epsilon, \qquad\mathsf{Q} \bigl((b_{i}) - \xi, \jmath \bigr) < \epsilon \quad\text{and}\quad \mathsf{R} \bigl((b_{i}) - \xi, \jmath \bigr)< \epsilon \end{aligned}$$
(2.6)

for all \(i \geq N\), i.e.,

$$\lim_{i \rightarrow \infty } \mathsf{P} \bigl((b_{i})-\xi, \jmath \bigr) = 1,\qquad \lim_{i \rightarrow \infty } \mathsf{Q} \bigl((b_{i})- \xi, \jmath \bigr) = 0\quad \text{and}\quad \lim_{i \rightarrow \infty } \mathsf{R} \bigl((b_{i})-\xi, \jmath \bigr) = 0.$$

In such a case, we denote \(\mathcal{Z} -\lim b_{i} = \xi \).

Definition 2.6

([18])

Let \((Y,\mathcal{Z},\ast,\diamond )\) be a neutrosophic normed space. A sequence \(b=(b_{i})\) is called a Cauchy sequence with respect to \(\mathcal{Z}\) if for each \(\epsilon >0\) and \(\jmath >0\), \(\eta \in \mathbb{N}\) such that \(\mathsf{P}(b_{i}- b_{k},\jmath ) > 1-\epsilon \), \(\mathsf{Q}(b_{i}-b_{k}, \jmath ) <\epsilon \), and \(\mathsf{R}(b_{i}-b_{k},\jmath ) < \epsilon \) for all \(i,k \geq \eta \).

Definition 2.7

Consider \((Y,\mathcal{Z},\ast,\diamond )\) to be an NNS. A subset H of X is said to be neutrosophic bounded-\((\mathit{NB})\) if \(\exists, \jmath >0\) and \(0<\epsilon <1\) such that \(\mathsf{P}(\vartheta,\jmath )>1-\epsilon \) and \(\mathsf{Q}(\vartheta,\jmath )<\epsilon \), \(\mathsf{R}(\vartheta,\jmath )<\epsilon \) for each ϑ H .

Definition 2.8

Let \((Y,\mathcal{Z},\ast,\diamond )\) be an NNS. Then \((Y,\mathcal{Z},\ast,\diamond )\) is said to be complete if every Cauchy sequence is convergent with respect to the norms \(\mathcal{Z}\).

By using this T- transformation and notion of neutrosophic convergence, we define some sequence spaces, namely \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\), \(\mathcal{L}_{0(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\), and \(\mathcal{L}_{{\infty }(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\).

Main results

In this section, we introduce the following sequence spaces:

$$\begin{aligned} \begin{aligned}& \mathcal{L_{(\mathsf{P},\mathsf{Q},\mathsf{R})}}(T) \\ &\quad = \bigl\{ \vartheta _{k}\in \omega: \bigl(T_{i}( \vartheta ) \bigr) \text{ is convergent} \bigr\} \\ &\quad= \bigl\{ \vartheta _{k}\in \omega: \text{ for some } \beta \in Y, \forall \epsilon >0, \forall \jmath, \exists n\in \mathbb{N}: \forall i\geq n, \\ &\qquad \mathsf{P} \bigl(T_{i}(\vartheta ) - \beta,\jmath \bigr) > 1 - \epsilon, \mathsf{Q} \bigl(T_{i}(\vartheta )-\beta, \jmath \bigr)< \epsilon \text{ and } \mathsf{R} \bigl(T_{i}(\vartheta )-\beta, \jmath \bigr)< \epsilon \bigr\} \\ &\quad= \bigl\{ \vartheta =(\vartheta _{k})\in \omega: \bigl\{ i \in \mathbb{N}: \text{ for some } \beta \in Y, \forall \epsilon >0,\mathsf{P} \bigl(T_{i}( \vartheta ) - \beta,\jmath \bigr) \leq 1 - \epsilon \text{ or} \\ &\qquad \mathsf{Q} \bigl(T_{i}(\vartheta )-\beta, \jmath \bigr)\geq \epsilon \text{ or }\mathsf{R} \bigl(T_{i}(\vartheta )-\beta, \jmath \bigr)\geq \epsilon \bigr\} \text{ is finite} \bigr\} \end{aligned} \end{aligned}$$
(3.1)
$$\begin{aligned} \begin{aligned} &\mathcal{L}_{0(\mathsf{P},\mathsf{Q},\mathsf{R})}(T) \\ &\quad = \bigl\{ \vartheta _{k}\in \omega: \bigl(T_{i}( \vartheta ) \bigr) \text{ is convergent} \bigr\} \\ &\quad= \bigl\{ \vartheta _{k}\in \omega: \forall \epsilon >0, \forall \jmath, \exists n\in \mathbb{N}:\forall i\geq n, \mathsf{P} \bigl(T_{i}( \vartheta ),\jmath \bigr) > 1 - \epsilon, \mathsf{Q} \bigl(T_{i}( \vartheta ), \jmath \bigr)< \epsilon \text{ and} \\ &\qquad \mathsf{R} \bigl(T_{i}(\vartheta ), \jmath \bigr)< \epsilon \bigr\} \\ &\quad= \bigl\{ \vartheta =(\vartheta _{k})\in \omega: \bigl\{ i \in \mathbb{N}: \forall \epsilon >0,\mathsf{P} \bigl(T_{i}(\vartheta ), \jmath \bigr) \leq 1 - \epsilon \mathsf{Q} \bigl(T_{i}(\vartheta ), \jmath \bigr) \geq \epsilon, \\ &\qquad \text{or }\mathsf{R} \bigl(T_{i}(\vartheta ), \jmath \bigr) \geq \epsilon \bigr\} \text{ is finite} \bigr\} \end{aligned} \end{aligned}$$
(3.2)
$$\begin{aligned} \begin{aligned}& \mathcal{L}_{{\infty }(\mathsf{P},\mathsf{Q},\mathsf{R})}(T) \\ &\quad= \bigl\{ \vartheta _{k}\in \omega: \bigl(T_{i}( \vartheta ) \bigr) \text{ is convergent} \bigr\} \\ &\quad= \bigl\{ \vartheta _{k}\in \omega: \forall \epsilon \in (0,1), \forall \jmath, \exists n\in \mathbb{N}:\forall i\geq n, \mathsf{P} \bigl(T_{i}(\vartheta ),\jmath \bigr) > 1 - \epsilon, \\ &\qquad \mathsf{Q} \bigl(T_{i}(\vartheta ), \jmath \bigr)< \epsilon \text{ and } \mathsf{R} \bigl(T_{i}(\vartheta ), \jmath \bigr)< \epsilon \bigr\} \\ &\quad= \bigl\{ \vartheta =(\vartheta _{k})\in \omega: \bigl\{ i \in \mathbb{N}: \forall \epsilon >0,\mathsf{P} \bigl(T_{i}(\vartheta ), \jmath \bigr) \leq 1 - \epsilon, \mathsf{Q} \bigl(T_{i}(\vartheta ), \jmath \bigr) \geq \epsilon, \\ &\qquad \text{or } \mathsf{R} \bigl(T_{i}(\vartheta ), \jmath \bigr) \geq \epsilon \bigr\} \text{ is finite} \bigr\} . \end{aligned} \end{aligned}$$
(3.3)

We define the open ball and closed ball with the center at ϑ and the radius \(r>0\) with respect to the parameter of neutrosophic \(\epsilon \in (0, 1)\) as follows:

$$\begin{aligned} \begin{aligned} &\mathcalligra{B}_{\vartheta }(r,\epsilon ) (T) \\ &\quad= \bigl\{ y=(y_{k}) \in \omega: \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i}(\vartheta )-T_{i}(y), r \bigr)\leq 1- \epsilon, \mathsf{Q} \bigl(T_{i}(\vartheta )-T_{i}(y),r \bigr) \geq \epsilon, \\ &\qquad\text{or } \mathsf{R} \bigl(T_{i}(\vartheta )-T_{i}(y),r \bigr)\geq \epsilon \bigr\} \text{ is finite} \bigr\} \end{aligned} \end{aligned}$$
(3.4)

and

$$\begin{aligned} \begin{aligned}& \mathcalligra{B}_{\vartheta }[r,\epsilon ](T) \\ &\quad= \bigl\{ y=(y_{k}) \in \omega: \bigl\{ j\in \mathbb{N}: \mathsf{P} (T_{j}\vartheta )-T_{j}(y), r )< 1-\epsilon \text{ or } \mathsf{Q} (T_{j}\vartheta )-T_{j}(y),r )> \epsilon, \\ &\qquad\text{or } \mathsf{R} (T_{j}\vartheta )-T_{j}(y),r )> \epsilon, \bigr\} \text{ is finite} \bigr\} . \end{aligned} \end{aligned}$$
(3.5)

If \((\vartheta _{n})\in \mathcal{L}_{(\mathsf{P}, \mathsf{Q}, \mathsf{R})}(T)\), then \((\vartheta _{n})\) converges to some \(\beta \in Y\), denoted by \(\vartheta _{n} \mathop{\longrightarrow}\limits^{{(\mathsf{P}, \mathsf{Q}, \mathsf{R})}(T)} \beta \).

Lemma 3.1

Consider the space \(\mathcal{L}_{(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T)\). Let \(\vartheta =(\vartheta _{k})\in \mathcal{L}_{(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T)\). Then the following statements are equivalent:

(a) \(\mathcal{Z}_{(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T)\)\(\lim (\vartheta )=\beta \);

(b) For every \(0<\epsilon <1\) and \(\jmath >0\), there exists \(n\in \mathbb{N}\) such that for every \(i\geq n\)

$$\begin{aligned} \mathsf{P} \bigl(T_{i}(\vartheta )-\beta, \jmath \bigr)\leq 1-\epsilon \quad\textit{and}\quad \mathsf{Q} \bigl(T_{i}(\vartheta )-\beta, \jmath \bigr) \geq \epsilon, \qquad\mathsf{R} \bigl(T_{i}(\vartheta )-\beta, \jmath \bigr) \geq \epsilon. \end{aligned}$$

(c) For every \(0<\epsilon <1\) and \(\jmath >0\), the set

$$\begin{aligned} \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i}(\vartheta )-\beta, \jmath \bigr)> 1-\epsilon \textit{ or }\mathsf{Q} \bigl(T_{i}(\vartheta )- \beta, \jmath \bigr)< \epsilon,\mathsf{R} \bigl(T_{i}(\vartheta )- \beta, \jmath \bigr)< \epsilon \bigr\} \end{aligned}$$

is finite.

(d) For every \(\jmath >0\), \(\lim_{i\rightarrow \infty } \mathsf{P} (T_{i}(\vartheta )- \beta, \jmath )=1, \textit{ and } \lim_{i\rightarrow \infty } \mathsf{Q} (T_{i}(\vartheta )-\beta, \jmath )=0, \lim_{i\rightarrow \infty } \mathsf{R} (T_{i}(\vartheta )- \beta, \jmath )=0\).

Theorem 3.1

The inclusion relation \(\mathcal{L}_{{0}(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T) \subset \mathcal{L}_{(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T)\subset \mathcal{L}_{{ \infty }(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T)\) holds.

Proof

It can be easily seen that \(\mathcal{L}_{{0}(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T) \subset \mathcal{L}_{(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T)\). We only show that \(\mathcal{L}_{(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T)\subset \mathcal{L}_{{\infty }(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T)\). Let \(\vartheta =(\vartheta _{k})\in \mathcal{L}_{(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T)\). Then there exists \(\beta \in Y\) such that \(\mathcal{Z}_{(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T)\)\(\lim (\vartheta _{k})=\beta \). Thus, for every \(0<\epsilon <1\) and \(\jmath >0\), the set A is finite

$$\begin{aligned} A= \biggl\{ i\in \mathbb{N}: \mathsf{P} \biggl(T_{i}(\vartheta )-\beta, \frac{\jmath }{2} \biggr)>1-\epsilon \text{ or } \mathsf{Q} \biggl(T_{i}( \vartheta )-\beta,\frac{\jmath }{2} \biggr)< \epsilon, \mathsf{R} \biggl(T_{i}( \vartheta )-\beta,\frac{\jmath }{2} \biggr)< \epsilon \biggr\} . \end{aligned}$$

Let \(\mathsf{P} (\beta,\frac{\jmath }{2} )=p\), \(\mathsf{Q} (\beta,\frac{\jmath }{2} )=q \), and \(\mathsf{R} (\beta,\frac{\jmath }{2} )=r \) for all \(\jmath >0\). Since \(p,q,r\in (0, 1)\) and \(0<\epsilon <1\), there exist \(s_{1}, s_{2},s_{3}\in (0, 1)\) such that \((1-\epsilon )*p>1-s_{1}\), \(\epsilon \diamond q< s_{2}\), and \(\epsilon \diamond r< s_{3}\), and so for \(\jmath \in Y\), we have

$$\begin{aligned} \begin{aligned} \mathsf{P} \bigl(T_{i}(\vartheta ),\jmath \bigr)={} & \mathsf{P} \bigl(T_{i}(\vartheta )-\beta +\beta,\jmath \bigr) \\ \geq {}&\mathsf{P} \biggl(T_{i}(\vartheta )-\beta,\frac{\jmath }{2} \biggr) \ast \mathsf{P} \biggl(\beta,\frac{\jmath }{2} \biggr) \\ > {}&(1-\epsilon )\ast p \\ >{}&1-s_{1}, \end{aligned} \\ \begin{aligned} \mathsf{Q} \bigl(T_{i}(\vartheta ),\jmath \bigr)={} & \mathsf{Q} \bigl(T_{i}(\vartheta )-\beta +\beta,\jmath \bigr) \\ \leq &\mathsf{Q} \biggl(T_{i}(\vartheta )-\beta,\frac{\jmath }{2} \biggr) \ast \mathsf{Q} \biggl(\beta,\frac{\jmath }{2} \biggr) \\ < {}&\epsilon \diamond q \\ < {}&s_{2} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \mathsf{R} \bigl(T_{i}(\vartheta ),\jmath \bigr)={}& \mathsf{R} \bigl(T_{i}(\vartheta )-\beta +\beta,\jmath \bigr) \\ \leq {}&\mathsf{R} \biggl(T_{i}(\vartheta )-\beta,\frac{\jmath }{2} \biggr) \ast \mathsf{R} \biggl(\beta,\frac{\jmath }{2} \biggr) \\ < {}&\epsilon \diamond r \\ < {}&s_{3}. \end{aligned} \end{aligned}$$

Taking \(s=\max \{s_{1}, s_{2}, s_{3}\}\), we have the set

$$\begin{aligned} & \bigl\{ i\in \mathbb{N},\exists s\in (0,1): \mathsf{P} \bigl(T_{i}( \vartheta ),t \bigr)>1-s \text{ or } \mathsf{Q} \bigl(T_{i}(\vartheta )- \beta,\jmath \bigr)< s, \mathsf{R} \bigl(T_{i}(\vartheta )-\beta,t \bigr)< s \bigr\} \\ &\quad \implies \quad \vartheta =(\vartheta _{k})\in \mathcal{L}_{{\infty }(\mathsf{P}, \mathsf{Q},\mathsf{R})}(T)\quad \text{implies } \mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T) \subset \mathcal{L}_{{ \infty }(\mathsf{P},\mathsf{Q},\mathsf{R})}(T). \end{aligned}$$

 □

The converse of the inclusion relation does not hold. We present the following examples in support of our claim.

Example 3.1

Let \((\mathbb{R},\|\cdot\|)\) be an NNS space such that \(\|\vartheta \|=\sup_{k}\mid \vartheta _{k}\mid \). Let \(\vartheta \ast y= \min \{\vartheta, y\}\) and \(\vartheta \diamond y= \max \{\vartheta, y\}\), \(\forall \vartheta,y\in (0, 1)\). Now define norms \((\mathsf{P},\mathsf{Q},\mathsf{R})\) on \(Y^{2}\times (0, \infty )\) as follows:

$$\begin{aligned} \mathsf{P} (\vartheta,\jmath )= \frac{\jmath }{\jmath + \Vert \vartheta \Vert },\qquad \mathsf{Q} (\vartheta, \jmath )=\frac{ \Vert \vartheta \Vert }{\jmath + \Vert \vartheta \Vert } \quad\text{and}\quad \mathsf{R} (\vartheta,\jmath )= \frac{ \Vert \vartheta \Vert }{\jmath }. \end{aligned}$$

Then \((\mathbb{R},\mathcal{Z},\ast,\diamond )\) is an NNS. Consider the sequence \((\vartheta _{k})=\{1\}\). It can be easily seen that \((\vartheta _{k})\in \mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) and \(\vartheta _{k} \mathop{\longrightarrow}\limits^{\mathcal{Z}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)} 1\), but \(\vartheta _{k}\notin \mathcal{L}_{{0}(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T)\).

Example 3.2

Let \((\mathbb{R},\|\cdot\|)\) be the NNS and \((\mathsf{P},\mathsf{Q},\mathsf{R})\) be the neutrosophic norms as defined in the above example. Consider the sequence \((\vartheta _{k})=(-1)^{k}\). Then \((\vartheta _{k})\in \mathcal{L}_{{\infty }(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T)\), but \((\vartheta _{k})\notin \mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\).

Theorem 3.2

The spaces \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) and \(\mathcal{L}_{{0}(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) are linear spaces.

Proof

We shall prove the result for \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\). The proof of linearity of the space \(\mathcal{L}_{{0}(\mathsf{P},\mathsf{Q},\mathsf{R})}{(T)}\) follows similarly. Let \(\vartheta =(\vartheta _{k}), y=(y_{k}) \in \mathcal{L}_{(\mathsf{P}, \mathsf{Q},\mathsf{R})}(T)\). Then there exist \(\beta _{1}, \beta _{2}\in Y\) such that \((y_{k})\) and \((z_{k})\) \(\mathcal{Z}\)—converge to \(\beta _{1}\) and \(\beta _{2}\) respectively. We shall show that for any scalars \(\zeta _{1}\) and \(\zeta _{2}\) the sequence \(\zeta _{1}\vartheta _{k}+\zeta _{2}y_{k}\) \(\mathcal{Z}\)—converges to \(\zeta _{1}\beta _{1}+\zeta _{2}\beta _{2}\). For \(\jmath >0\) and \(0<\epsilon <1\), consider the following finite sets \(\mathcalligra{C}_{1}\) and \(\mathcalligra{C}_{1}\):

$$\begin{aligned} &\mathcalligra{C}_{1}= \biggl\{ i\in \mathbb{N}:\mathsf{P} \biggl(T_{i}( \vartheta )-\beta _{1},\frac{\jmath }{2 \vert \zeta _{1} \vert } \biggr)\leq 1- \epsilon \text{ or } \mathsf{Q} \biggl(T_{i}(\vartheta )- \beta _{1}, \frac{\jmath }{2 \vert \zeta _{1} \vert } \biggr)\geq \epsilon, \\ &\phantom{\mathcalligra{C}_{1}=}\mathsf{R} \biggl(T_{i}( \vartheta )-\beta _{1},\frac{\jmath }{2 \vert \zeta _{1} \vert } \biggr)\geq \epsilon \biggr\} ,\\ &\mathcalligra{C}^{c}_{1}= \biggl\{ i\in \mathbb{N}:\mathsf{P} \biggl(T_{i}( \vartheta )-\beta _{1},\frac{\jmath }{2 \vert \zeta _{1} \vert } \biggr)> 1- \epsilon \text{ or } \mathsf{Q} \biggl(T_{i}(\vartheta )- \beta _{1}, \frac{\jmath }{2 \vert \zeta _{1} \vert } \biggr)< \epsilon, \\ &\phantom{\mathcalligra{C}^{c}_{1}=}\mathsf{R} \biggl(T_{i}( \vartheta )-\beta _{1},\frac{\jmath }{2 \vert \zeta _{1} \vert } \biggr)< \epsilon \biggr\} ,\\ &\mathcalligra{C}_{2}= \biggl\{ i\in \mathbb{N}:\mathsf{P} \biggl(T_{i}(y)- \beta _{2},\frac{\jmath }{2 \vert \zeta _{2} \vert } \biggr)\leq 1- \epsilon \text{ or } \mathsf{Q} \biggl(T_{i}(y)-\beta _{2}, \frac{\jmath }{2 \vert \zeta _{2} \vert } \biggr)\geq \epsilon, \\ &\phantom{\mathcalligra{C}_{2}=}\mathsf{R} \biggl(T_{i}(y)- \beta _{2}, \frac{\jmath }{2 \vert \zeta _{2} \vert } \biggr)\geq \epsilon \biggr\} ,\\ &\mathcalligra{C}^{c}_{2}= \biggl\{ i\in \mathbb{N}:\mathsf{P} \biggl(T_{i}(y)- \beta _{2},\frac{t}{2 \vert \zeta _{2} \vert } \biggr)>1- \epsilon \text{ or } \mathsf{Q} \biggl(T_{i}(y)-\beta _{2}, \frac{\jmath }{2 \vert \zeta _{2} \vert } \biggr)< \epsilon, \\ &\phantom{\mathcalligra{C}^{c}_{2}}\mathsf{R} \biggl(T_{i}(y)-\beta _{2}, \frac{\jmath }{2 \vert \zeta _{2} \vert } \biggr)< \epsilon \biggr\} . \end{aligned}$$

Define the set \(\mathcalligra{C}_{3}= \mathcalligra{C}_{1}\cup \mathcalligra{C}_{2}\) so that \(\mathcalligra{C}_{3}\) is finite. It follows that \(\mathcalligra{C}_{3}^{c} \neq \phi \). We shall show that for each \((\vartheta ),(y)\in \mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\),

$$\begin{aligned} \mathcalligra{C}^{c}_{3}\subset{}& \bigl\{ i\in \mathbb{N}:\mathsf{P} \bigl( \bigl(\zeta _{1}T_{i}(\vartheta )-\zeta _{2}T_{i}(y) \bigr)- (\zeta _{1} \beta _{1}-\zeta _{2}\beta _{2} ),\jmath \bigr)>1- \epsilon \text{ or} \\ &{} \mathsf{Q} \bigl( \bigl(\zeta _{1}T_{i}(\vartheta )- \zeta _{2}T_{i}(y) \bigr)- (\zeta _{1}\beta _{1}-\zeta _{2}\beta _{2} ),\jmath \bigr)< \epsilon, \\ &{} \mathsf{R} \bigl( \bigl(\zeta _{1}T_{i}(\vartheta )- \zeta _{2}T_{i}(y) \bigr)- (\zeta _{1}\beta _{1}-\zeta _{2}\beta _{2} ),\jmath \bigr)< \epsilon \bigr\} . \end{aligned}$$

Let \(q\in \mathcalligra{C}_{3}^{c}\). In this case,

$$\begin{aligned} &\mathsf{P} \biggl(T_{q}(\vartheta )-\beta _{1}, \frac{\jmath }{2 \vert \zeta _{1} \vert } \biggr)> 1-\epsilon \quad \text{or}\quad \mathsf{Q} \biggl(T_{q}(\vartheta )-\beta _{1}, \frac{\jmath }{2 \vert \zeta _{1} \vert } \biggr)< \epsilon, \\ & \mathsf{R} \biggl(T_{q}( \vartheta )-\beta _{1}, \frac{\jmath }{2 \vert \zeta _{1} \vert } \biggr) < \epsilon \end{aligned}$$

and

$$\begin{aligned} &\mathsf{P} \biggl(T_{q}(y)-\beta _{2}, \frac{\jmath }{2 \vert \zeta _{2} \vert } \biggr)> 1-\epsilon \quad \text{or}\quad \mathsf{Q} \biggl(T_{q}(y)-\beta _{2}, \frac{\jmath }{2 \vert \zeta _{2} \vert } \biggr)< \epsilon, \\ & \mathsf{R} \biggl(T_{q}(y)- \beta _{2}, \frac{\jmath }{2 \vert \zeta _{2} \vert } \biggr) < \epsilon ,\\ & \mathsf{P} \bigl( \bigl(\zeta _{1}T_{q}( \vartheta )-\zeta _{2}T_{q}(y) \bigr)-(\zeta _{1} \beta _{1}-\zeta _{2}\beta _{2}),\jmath \bigr) \\ &\quad\geq \mathsf{P} \biggl(\zeta _{1}T_{q}(\vartheta )- \zeta _{1}\beta _{1}, \frac{\jmath }{2} \biggr)\ast \mathsf{P} \biggl(\zeta _{2}T_{q}(y)-\zeta _{2} \beta _{2},\frac{\jmath }{2} \biggr) \\ &\quad=\mathsf{P} \biggl(T_{q}(\vartheta )-\beta _{1}, \frac{\jmath }{2 \vert \zeta _{1} \vert } \biggr)\ast \mathsf{P} \biggl(T_{q}(y)- \beta _{2},\frac{\jmath }{2 \vert \zeta _{2} \vert } \biggr) \\ &\quad>(1-\epsilon )\ast (1-\epsilon ) \\ &\quad=1-\epsilon. \\ &\quad\implies\quad \mathsf{P} \bigl( \bigl(\zeta _{1}T_{q}( \vartheta )-\zeta _{2}T_{q}(y) \bigr)-(\zeta _{1} \beta _{1}-\zeta _{2}\beta _{2}),\jmath \bigr)>1- \epsilon. \end{aligned}$$

In a similar way,

$$\begin{aligned} & \mathsf{Q} \bigl( \bigl(\zeta _{1}T_{q}( \vartheta )-\zeta _{2}T_{q}(y) \bigr)-(\zeta _{1} \beta _{1}-\zeta _{2}\beta _{2}),\jmath \bigr) \\ &\quad\leq \mathsf{Q} \biggl(\zeta _{1}T_{q}(\vartheta )- \zeta _{1}\beta _{1}, \frac{\jmath }{2} \biggr)\diamond \mathsf{Q} \biggl(\zeta _{2}T_{q}(y)- \zeta _{2} \beta _{2},\frac{\jmath }{2} \biggr) \\ &\quad=\mathsf{Q} \biggl(T_{q}(\vartheta )-\beta _{1}, \frac{\jmath }{2 \vert \zeta _{1} \vert } \biggr)\diamond \mathsf{Q} \biggl(T_{q}(y)- \beta _{2},\frac{\jmath }{2 \vert \zeta _{2} \vert } \biggr) \\ &\quad< \epsilon \diamond \epsilon \\ &\quad=\epsilon, \\ &\quad\implies\quad \mathsf{Q} \bigl( \bigl(\zeta _{1}T_{q}( \vartheta )-\zeta _{2}T_{q}(y) \bigr)-(\zeta _{1} \beta _{1}-\zeta _{2}\beta _{2}),\jmath \bigr)< \epsilon, \end{aligned}$$

and

$$\begin{aligned} &\mathsf{R} \bigl( \bigl(\zeta _{1}T_{q}( \vartheta )-\zeta _{2}T_{q}(y) \bigr)-(\zeta _{1} \beta _{1}-\zeta _{2}\beta _{2}),\jmath \bigr) \\ &\quad\leq \mathsf{R} \biggl(\zeta _{1}T_{q}(\vartheta )- \zeta _{1}\beta _{1}, \frac{\jmath }{2} \biggr)\diamond \mathsf{R} \biggl(\zeta _{2}T_{q}(y)- \zeta _{2} \beta _{2},\frac{\jmath }{2} \biggr) \\ &\quad=\mathsf{R} \biggl(T_{q}(\vartheta )-\beta _{1}, \frac{\jmath }{2 \vert \zeta _{1} \vert } \biggr)\diamond \mathsf{R} \biggl(T_{q}(y)- \beta _{2},\frac{\jmath }{2 \vert \zeta _{2} \vert } \biggr) \\ &\quad< \epsilon \diamond \epsilon \\ &\quad=\epsilon, \\ &\quad\implies\quad \mathsf{R} \bigl( \bigl(\zeta _{1}T_{q}( \vartheta )-\zeta _{2}T_{q}(y) \bigr)-(\zeta _{1} \beta _{1}-\zeta _{2}\beta _{2}),\jmath \bigr)< \epsilon. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \mathcalligra{C}^{c}_{3}\subset {}& \bigl\{ q\in \mathbb{N}: \mathsf{P} \bigl( \bigl(\zeta _{1}T_{q}(\vartheta )-\zeta _{2}T_{q}(y) \bigr)-(\zeta _{1} \beta _{1}-\zeta _{2}\beta _{2}),\jmath \bigr)>1- \epsilon, \\ &{}\mathsf{Q} \bigl( \bigl(\zeta _{1}T_{q}(\vartheta )- \zeta _{2}T_{q}(y) \bigr)-(\zeta _{1}\beta _{1}-\zeta _{2}\beta _{2}),\jmath \bigr)< \epsilon \text{ or} \\ &{}\mathsf{R} \bigl( \bigl(\zeta _{1}T_{q}(\vartheta )- \zeta _{2}T_{q}(y) \bigr)-(\zeta _{1}\beta _{1}-\zeta _{2}\beta _{2}),\jmath \bigr)< \epsilon \bigr\} \end{aligned}$$

and

$$\begin{aligned} & \bigl\{ q\in \mathbb{N}:\mathsf{P} \bigl( \bigl(\zeta _{1}T_{q}( \vartheta )-\zeta _{2}T_{q}(y) \bigr)-(\zeta _{1} \beta _{1}-\zeta _{2} \beta _{2}),\jmath \bigr)>1- \epsilon, \\ &\quad\mathsf{Q} \bigl( \bigl(\zeta _{1}T_{q}(\vartheta )- \zeta _{2}T_{q}(y) \bigr)-(\zeta _{1}\beta _{1}-\zeta _{2}\beta _{2}),\jmath \bigr)< \epsilon \text{ or} \\ &\quad\mathsf{R} \bigl( \bigl(\zeta _{1}T_{q}(\vartheta )- \zeta _{2}T_{q}(y) \bigr)-(\zeta _{1}\beta _{1}-\zeta _{2}\beta _{2}),\jmath \bigr)< \epsilon \bigr\} , \end{aligned}$$

which implies that the sequence \((\zeta _{1}\vartheta _{i}+\zeta _{2}y_{i})\) \(\mathcal{Z}\)—converges to \(\zeta _{1}\beta _{1}+\zeta _{2}\beta _{2}\). Therefore, \((\zeta _{1}\vartheta _{i}+\zeta _{2}y_{i}) \in \mathcal{L}_{( \mathsf{P},\mathsf{Q},\mathsf{R})}(T)\). Hence \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) is a linear space. □

Theorem 3.3

Every open ball with the center at ϑ and the radius \(r>0\) with respect to the parameter of fuzziness \(0<\epsilon <1\), i.e., \(\mathcalligra{B}_{\vartheta }(r,\epsilon )(T)\) is an open set in \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\).

Proof

Consider the open ball with center at ϑ and radius \(r>0\) with the parameter of neutrosophic \(0<\epsilon <1\),

$$\begin{aligned} \mathcalligra{B}_{\vartheta } (r,\epsilon ) (T)={}& \bigl\{ \vartheta =(\vartheta _{i})\in \omega: \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i}(\vartheta )-T_{i}(y),r \bigr)\leq 1-\epsilon \text{ or} \\ & \mathsf{Q} \bigl(T_{i}(\vartheta )-T_{i}(y),r \bigr)\geq \epsilon, \\ &\mathsf{R} \bigl(T_{i}(\vartheta )-T_{i}(y),r \bigr)\geq \epsilon \bigr\} \text{ is finite} \bigr\} . \end{aligned}$$

Then

$$\begin{aligned} \mathcalligra{B}^{c} _{\vartheta } (r,\epsilon ) (T)={}& \bigl\{ \vartheta =(\vartheta _{k})\in \omega: \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i}(\vartheta )-T_{i}(y),r \bigr)> 1- \epsilon \text{ or} \\ &\mathsf{Q} \bigl(T_{i}(\vartheta )-T_{i}(y),r \bigr)< \epsilon, \\ &\mathsf{R} \bigl(T_{i}(\vartheta )-T_{i}(y),r \bigr)< \epsilon \bigr\} \bigr\} . \end{aligned}$$

Let \(y=(y_{i})\in \mathcalligra{B}^{c}_{\vartheta } (r,\epsilon )(T)\). Then the set

$$\begin{aligned} & \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i}(\vartheta )-T_{i}(y), r \bigr)> 1-\epsilon \text{ or } \mathsf{Q} \bigl(T_{i}(\vartheta )-T_{j}(y),r \bigr)< \epsilon, \mathsf{R} \bigl(T_{i}(y)-T_{i}(z),r \bigr)< \epsilon \bigr\} , \\ &\mathsf{P} \bigl(T_{i}(\vartheta )-T_{i}(y),r \bigr)> 1- \epsilon, \qquad\mathsf{Q} \bigl(T_{i}(\vartheta )-T_{i}(z), r \bigr)< \epsilon\quad\text{and}\quad \mathsf{R} \bigl(T_{i}(\vartheta )-T_{i}(y), r \bigr)< \epsilon, \end{aligned}$$

there exists \(r_{0}\in (0,r)\) such that

$$\begin{aligned} &\mathsf{P} \bigl(T_{i}(\vartheta )-T_{i}(y), r_{0} \bigr)> 1-\epsilon,\qquad \mathsf{Q} \bigl(T_{i}( \vartheta )-T_{i}(y),r_{0} \bigr)< \epsilon \quad \text{and} \\ & \mathsf{R} \bigl(T_{i}(\vartheta )-T_{i}(y), r_{0} \bigr)< \epsilon. \end{aligned}$$

Put \(\epsilon _{0}=\mathsf{P} (T_{i}(\vartheta )-T_{i}(y), r_{0} ) \implies \epsilon _{0}>1-\epsilon \). Then \(\exists s\in (0,1)\) such that \(\epsilon _{0}>1-s>1-\epsilon \).

For \(\epsilon _{0}>1-s\), we can have \(\epsilon _{1},\epsilon _{2}, \epsilon _{3}\in (0,1)\) such that \(\epsilon _{0}\ast \epsilon _{1}>1-s\), \((1-\epsilon _{0})\diamond (1-\epsilon _{2})< s\),

and

$$\begin{aligned} (1-\epsilon _{0})\diamond (1-\epsilon _{3})< s. \end{aligned}$$

Let \(\epsilon _{4}=\max \{\epsilon _{1},\epsilon _{2},\epsilon _{3}\}\)

Now consider the open ball \(\mathcalligra{B}^{c}_{\vartheta } (r-r_{0}, 1-\epsilon _{4} )(T)\).

We shall show that \(\mathcalligra{B}^{c}_{\vartheta } (r-r_{0}, 1-\epsilon _{4} )(T) \subset \mathcalligra{B}^{c}_{\vartheta } (r,\epsilon )(T)\).

Let \(z=(z_{i})\in \mathcalligra{B}^{c}_{z} (r-r_{0}, 1-\epsilon _{4} )(T)\), then

$$\begin{aligned} & \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i}(\vartheta )-T_{i}(z),r-r_{0} \bigr)>\epsilon _{4} \text{ or} \\ &\quad \mathsf{Q} \bigl(T_{i}(\vartheta )-T_{i}(z),r-r_{0} \bigr)< 1-\epsilon _{4},\mathsf{R} \bigl(T_{i}(\vartheta )-T_{i}(z),r-r_{0} \bigr)< 1-\epsilon _{4} \bigr\} . \end{aligned}$$

Therefore,

$$\begin{aligned} &\mathsf{P} \bigl(T_{i}(\vartheta )-T_{i}(z),r \bigr) \\ &\quad\geq \mathsf{P} \bigl(T_{i}(\vartheta )-T_{i}(z),r_{0} \bigr)\ast \mathsf{P} \bigl(T_{i}(\vartheta )-T_{i}(z),r-r_{0} \bigr) \\ &\quad\geq \epsilon _{0}\ast \epsilon _{4}\geq \epsilon _{0}*\epsilon _{1} \\ &\quad >(1-s)>(1-\epsilon ) \\ &\quad\implies\quad \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i}(\vartheta )-T_{i}(z),r \bigr)>1-\epsilon \bigr\} \\ &\mathsf{Q} \bigl(T_{i}(\vartheta )-T_{i}(z),r \bigr) \\ &\quad \leq \mathsf{Q} \bigl(T_{i}(\vartheta )-T_{i}(z),r_{0} \bigr)\diamond \mathsf{Q} \bigl(T_{i}(\vartheta )-T_{i}(z),r-r_{0} \bigr) \\ &\quad \leq (1-\epsilon _{0})\diamond (1-\epsilon _{4}) \leq (1-\epsilon _{0}) \diamond (1-\epsilon _{2}) \\ &\quad \leq(s)< \epsilon \\ &\quad\implies \quad \bigl\{ i\in \mathbb{N}: \mathsf{Q} \bigl(T_{i}( \vartheta )-T_{i}(z), r \bigr)< \epsilon \bigr\} \end{aligned}$$

and

$$\begin{aligned} & \mathsf{R} \bigl(T_{i}(\vartheta )-T_{i}(z),r \bigr) \\ &\quad\leq \mathsf{R} \bigl(T_{i}(\vartheta )-T_{i}(z),r_{0} \bigr)\diamond \mathsf{R} \bigl(T_{i}(\vartheta )-T_{i}(z),r-r_{0} \bigr) \\ &\quad\leq (1-\epsilon _{0})\diamond (1-\epsilon _{4})\leq (1-\epsilon _{0}) \diamond (1-\epsilon _{3}) \\ &\quad \leq(s)< \epsilon \\ &\quad\implies\quad \bigl\{ i\in \mathbb{N}: \mathsf{R} \bigl(T_{i}( \vartheta )-T_{i}(z), r \bigr)< \epsilon \bigr\} . \end{aligned}$$

Therefore the set

$$\begin{aligned} & \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i}(\vartheta )-T_{i}(z),r-r_{0} \bigr)> 1-\epsilon \text{ or } \mathsf{Q} \bigl(T_{i}(\vartheta )-T_{i}(z),r-r_{0} \bigr)< \epsilon, \\ &\quad\mathsf{R} \bigl(T_{i}(\vartheta )-T_{i}(z),r-r_{0} \bigr)< \epsilon \bigr\} , \\ &\quad\implies\quad \vartheta =(\vartheta _{i})\in \mathcalligra{B}^{c}_{\vartheta } (r,\epsilon ) (T) \\ &\quad\implies\quad \mathcalligra{B}^{c}_{\vartheta } (r-r_{0}, 1-\epsilon _{4} ) (T) \subset \mathcalligra{B}^{c}_{y} (r,\epsilon ) (T). \end{aligned}$$

 □

Remark 3.1

The spaces \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) and \(\mathcal{L}_{{0}(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) are NNS with respect to neutrosophic norms \((\mathsf{P},\mathsf{Q},\mathsf{R})\).

Now define

$$\begin{aligned} \tau _{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)={}& \bigl\{ U\subset \mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T): \text{ for each } \vartheta =(\vartheta _{k})\in U \text{ there exist } r>0 \text{ and} \\ & \epsilon \in (0,1) \text{ such that } \mathcalligra{B}_{\vartheta }(r, \epsilon ) \subset U \bigr\} . \end{aligned}$$

Then \(\tau _{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) defines a topology on the sequence space \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\). The collection defined by \(\mathcal{B}= \{\mathcalligra{B}_{\vartheta }(r,\epsilon ):\vartheta \in \mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T),r>0\text{ and } \epsilon \in (0,1) \}\) is a base for the topology \(\tau _{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) on the space \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\).

Theorem 3.4

The topology \(\tau _{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) on the space \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) is first countable.

Proof

For each \(\vartheta =(\vartheta _{i})\in \mathcal{L}_{(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T)\), consider the set \(\mathcal{B}= \{\mathcalligra{B}_{\vartheta }{(\frac{1}{n},\frac{1}{n})}: n= 1,2,3,4,\ldots \}\), which is a countable local base at ϑ. Therefore the topology \(\tau _{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) on the space \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) is first countable. □

Theorem 3.5

The spaces \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) and \(\mathcal{L}_{{0}(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) are Hausdorff spaces.

Proof

We shall prove the result only for \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\), and the other one follows similarly. Let \(\vartheta =(\vartheta _{i})\) and \(y=(y_{i})\in \mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) such that \(\vartheta \neq y\). Then, for each \(i\in \mathbb{N}\) and \(r>0\), this implies

$$\begin{aligned} \begin{aligned} &0< \mathsf{P} \bigl(T_{i}(\vartheta )-T_{i}(y),r \bigr)< 1, \qquad 0< \mathsf{Q} \bigl(T_{i}(\vartheta )-T_{i}(y),r \bigr)< 1\quad \text{and} \\ & 0< \mathsf{Q} \bigl(T_{i}( \vartheta )-T_{i}(y),r \bigr)< 1. \end{aligned} \end{aligned}$$
(3.6)

Put

$$\begin{aligned} \begin{aligned} &\epsilon _{1}=\mathsf{P} \bigl(T_{i}( \vartheta )-T_{i}(y),r \bigr),\qquad \epsilon _{2}=\mathsf{Q} \bigl(T_{i}(\vartheta )-T_{i}(y),r \bigr), \\ & \epsilon _{3}=\mathsf{R} \bigl(T_{i}(\vartheta )-T_{i}(y), r \bigr)\quad \text{and}\quad \epsilon =\max \{\epsilon _{1},1- \epsilon _{2}, 1- \epsilon _{3}\}. \end{aligned} \end{aligned}$$
(3.7)

Then for each \(\epsilon _{0}>\epsilon \) there exist \(\epsilon _{4}, \epsilon _{5}, \epsilon _{6}\in (0,1)\) such that

$$\begin{aligned} \epsilon _{4}\ast \epsilon _{4}\geq \epsilon _{0},\qquad (1-\epsilon _{5}) \diamond (1-\epsilon _{5})\leq (1-\epsilon _{0}) \quad \text{and} \quad (1- \epsilon _{6}) \diamond (1-\epsilon _{6})\leq (1-\epsilon _{0}). \end{aligned}$$

Again putting \(\epsilon _{7}=\max \{\epsilon _{4},\epsilon _{5},\epsilon _{6}\}\), consider the open balls \(\mathcalligra{B}_{\vartheta }{ (1-\epsilon _{7},\frac{r}{2} )}(T)\) and \(\mathcalligra{B}_{y}{ (1-\epsilon _{7},\frac{r}{2} )}(T)\) centered at ϑ and y respectively. We show that \(\mathcalligra{B}_{\vartheta }{ (1-\epsilon _{7},\frac{r}{2} )}(T)\cap \mathcalligra{B}_{y}{ (1-\epsilon _{7},\frac{r}{2} )}(T)= \phi \).

If possible, let \(\vartheta =(\vartheta _{i})\in \mathcalligra{B}_{\vartheta }{ (1-\epsilon _{7}, \frac{r}{2} )}(T)\cap \mathcalligra{B}_{y}{ (1-\epsilon _{7}, \frac{r}{2} )}(T)\).

Then, for the set \(\{k\in \mathbb{N}\}\), we have

$$\begin{aligned} \begin{aligned} \epsilon _{1}={}&\mathsf{P} \bigl(T_{k}(\vartheta )-T_{k}(y),r \bigr) \\ \geq {}&\mathsf{P} \biggl(T_{k}(\vartheta )-T_{k}(z), \frac{r}{2} \biggr) \ast \mathsf{P} \biggl(T_{k}(z)-T_{k}(y), \frac{r}{2} \biggr) \\ > {}& \epsilon _{7}\ast \epsilon _{7}\geq \epsilon _{4}\ast \epsilon _{4} \geq \epsilon _{0}> \epsilon _{1} ,\end{aligned} \end{aligned}$$
(3.8)
$$\begin{aligned} \begin{aligned} \epsilon _{2}={}&\mathsf{Q} \bigl(T_{k}(\vartheta )-T_{k}(y),r \bigr) \\ \leq {}& \mathsf{Q} \biggl(T_{k}(\vartheta )-T_{k}(z), \frac{r}{2} \biggr) \diamond \mathsf{Q} \biggl(T_{k}(z)-T_{k}(y), \frac{r}{2} \biggr) \\ < {}&(1-\epsilon _{7})\diamond (1-\epsilon _{7})\leq (1- \epsilon _{5}) \diamond (1-\epsilon _{5}) \\ < {}& (1-\epsilon _{0})< \epsilon _{2} \end{aligned} \end{aligned}$$
(3.9)

and

$$\begin{aligned} \begin{aligned} \epsilon _{3}={}&\mathsf{R} \bigl(T_{k}(\vartheta )-T_{k}(y),r \bigr) \\ \leq {}& \mathsf{R} \biggl(T_{k}(\vartheta )-T_{j}(z), \frac{r}{2} \biggr) \diamond \mathsf{R} \biggl(T_{k}(z)-T_{k}(y), \frac{r}{2} \biggr) \\ < {}&(1-\epsilon _{7})\diamond (1-\epsilon _{7})\leq (1- \epsilon _{6}) \diamond (1-\epsilon _{6}) \\ < {}&(1-\epsilon _{0})< \epsilon _{3}. \end{aligned} \end{aligned}$$
(3.10)

From equations (3.8), (3.9), and (3.10) we have a contradiction.

Therefore, \(\mathcalligra{B}_{\vartheta }{ (1-\epsilon _{7},\frac{r}{2} )}(T)\cap \mathcalligra{B}_{y}{ (1-\epsilon _{7},\frac{r}{2} )}(T)= \phi \). Hence the space \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) is a Hausdorff space. □

On the Tribonacci sequence \({T_{n}}\)

Definition 4.1

A sequence \(\vartheta =(\vartheta _{n})\in \omega \) is said to be Tribonacci convergent to \(\beta \in Y\) if for every \(\epsilon >0\) the set \(B_{1}\) is finite, where

$$\begin{aligned} B_{1}= \bigl\{ i\in \mathbb{N}: \bigl\vert T_{i}(\vartheta )-\beta \bigr\vert \geq \epsilon \bigr\} . \end{aligned}$$

Definition 4.2

A sequence \(\vartheta =(\vartheta _{k})\in \omega \) is said to be neutrosophic Tribonacci convergent to \(\beta \in Y\) with respect to neutrosophic norms- \((\mathsf{P},\mathsf{Q},\mathsf{R})\), denoted by \(\vartheta _{i}\rightarrow \beta \), if for every \(\epsilon \in (0,1)\) and \(\jmath >0\), the set \(T_{1}\) is finite, where

$$\begin{aligned} T_{1}= \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i}( \vartheta )-\beta, \jmath \bigr)\leq 1-\epsilon \text{ or } \mathsf{Q} \bigl(T_{i}( \vartheta )-\beta,\jmath \bigr)\geq \epsilon, \mathsf{R} \bigl(T_{i}( \vartheta )-\beta,\jmath \bigr)\geq \epsilon \bigr\} , \end{aligned}$$

and we write \(\mathcal{Z}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\)\(\lim ( \vartheta _{i})=\beta \).

Definition 4.3

A sequence \(\vartheta =(\vartheta _{i})\in \omega \) is said to be Tribonacci Cauchy if for every \(\epsilon >0\) there exists \(k=k(\epsilon )\in \mathbb{N}\) such that the set \(B_{2}\) is finite, where

$$\begin{aligned} B_{2}= \bigl\{ i\in \mathbb{N}: \bigl\vert T_{i}(\vartheta )-T_{k}(\vartheta ) \bigr\vert \geq \epsilon \bigr\} . \end{aligned}$$

Definition 4.4

A sequence \(\vartheta =(\vartheta _{i})\in \omega \) is said to neutrosophic Tribonacci Cauchy with respect to neutrosophic norms-\((\mathsf{P}, \mathsf{Q},\mathsf{R})\) if for every \(\epsilon \in (0,1)\) and \(\jmath >0\) there exists \(k\in \mathbb{N}\) such that the set \(T_{2}\) is finite, where

$$\begin{aligned} T_{2}={}& \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i}( \vartheta )-T_{k}( \vartheta ),\jmath \bigr)\leq 1-\epsilon \text{ or } \mathsf{Q} \bigl(T_{i}( \vartheta )-T_{k}(\vartheta ),\jmath \bigr)\geq \epsilon, \\ &\mathsf{R} \bigl(T_{i}(\vartheta )-T_{k}( \vartheta ), \jmath \bigr)\geq \epsilon \bigr\} . \end{aligned}$$

Definition 4.5

A sequence \(\vartheta =(\vartheta _{i})\in \omega \) is said to be Tribonacci bounded if there exists \(M>0\) such that the set

$$\begin{aligned} B_{3}= \bigl\{ i\in \mathbb{N}: \bigl\vert T_{i}(\vartheta ) \bigr\vert >M \bigr\} . \end{aligned}$$

Definition 4.6

A subset D of ω is said to be neutrosophic Tribonacci bounded with respect to neutrosophic norms \((\mathsf{P},\mathsf{Q},\mathsf{R})\) if \(\forall \vartheta \in D\) there exist \(0<\epsilon <1\) and \(\jmath >0\) such that the set

$$\begin{aligned} \bigl\{ i\in \mathbb{N}:\mathsf{P} \bigl(T_{i}(\vartheta ),\jmath \bigr) \leq 1-\epsilon \text{ or } \mathsf{Q} \bigl(T_{i}(\vartheta ), \jmath \bigr)\geq \epsilon, \mathsf{R} \bigl(T_{i}(\vartheta ),\jmath \bigr) \geq \epsilon \bigr\} . \end{aligned}$$
(4.1)

Theorem 4.1

If a sequence \(\vartheta =(\vartheta _{i})\in \omega \) is neutrosophic Tribonacci convergent, then the \(\mathcal{Z}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\)-limit is unique.

Proof

Suppose \(\vartheta =(\vartheta _{i})\in \omega \) such that \((\vartheta _{i})\) is neutrosophic Tribonacci convergent.

Let \(\mathcal{Z}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T) \lim (\vartheta _{i})= \beta _{1}\) and \(\mathcal{Z}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)-\lim (\vartheta _{i})= \beta _{2}\). We show that \(\beta _{1}=\beta _{2}\). Now, for given \(\epsilon \in (0, 1)\), there exists \(\epsilon _{1} \in (0, 1)\) such that \((1-\epsilon _{1})*(1-\epsilon _{1})>1-\epsilon \) and \(\epsilon _{1}\diamond \epsilon _{1}<\epsilon \). Therefore the sets \(\mathcalligra{C}_{1}\) and \(\mathcalligra{C}_{2}\) are finite, where

$$\begin{aligned} &\mathcalligra{C}_{1}= \biggl\{ i\in \mathbb{N}: \mathsf{P} \biggl(T_{i}( \vartheta )-\beta _{1},\frac{\jmath }{2} \biggr)\leq 1-\epsilon _{1} \text{ or } \mathsf{Q} \biggl(T_{i}(\vartheta )-\beta _{1}, \frac{\jmath }{2} \biggr)\geq \epsilon _{1}, \\ &\phantom{\mathcalligra{C}_{1}=} \mathsf{R} \biggl(T_{i}( \vartheta )-\beta _{1},\frac{\jmath }{2} \biggr)\geq \epsilon _{1} \biggr\} , \\ &\mathcalligra{C}_{1}= \biggl\{ i\in \mathbb{N}: \mathsf{P} \biggl(T_{i}( \vartheta )-\beta _{2},\frac{\jmath }{2} \biggr)\leq 1-\epsilon _{1} \text{ or } \mathsf{Q} \biggl(T_{i}(\vartheta )-\beta _{2}, \frac{\jmath }{2} \biggr)\geq \epsilon _{1}, \\ &\phantom{\mathcalligra{C}_{1}=} \mathsf{R} \biggl(T_{i}( \vartheta )-\beta _{2},\frac{\jmath }{2} \biggr)\geq \epsilon _{1} \biggr\} , \\ &\quad\implies \\ &\mathcalligra{C}^{c}_{1}= \biggl\{ i\in \mathbb{N}: \mathsf{P} \biggl(T_{i}( \vartheta )-\beta _{1},\frac{\jmath }{2} \biggr)> 1-\epsilon _{1} \text{ or } \mathsf{Q} \biggl(T_{i}( \vartheta )-\beta _{1}, \frac{\jmath }{2} \biggr)< \epsilon _{1}, \\ &\phantom{\mathcalligra{C}^{c}_{1}=}\mathsf{R} \biggl(T_{i}( \vartheta )-\beta _{1},\frac{\jmath }{2} \biggr)< \epsilon _{1} \biggr\} , \\ &\mathcalligra{C}^{c}_{2}= \biggl\{ i\in \mathbb{N}: \mathsf{P} \biggl(T_{i}( \vartheta )-\beta _{2},\frac{\jmath }{2} \biggr)> 1-\epsilon _{1} \text{ or } \mathsf{Q} \biggl(T_{i}( \vartheta )-\beta _{2}, \frac{\jmath }{2} \biggr)< \epsilon _{1}, \\ &\phantom{\mathcalligra{C}^{c}_{2}=}\mathsf{R} \biggl(T_{i}( \vartheta )-\beta _{2},\frac{\jmath }{2} \biggr)< \epsilon _{1} \biggr\} . \end{aligned}$$

Then \(\mathcalligra{C}^{c}_{1}\cap \mathcalligra{C}^{c}_{2}\neq \phi \). Taking \(i\in \mathcalligra{C}^{c}_{1}\cap \mathcalligra{C}^{c}_{2}\), we have

$$\begin{aligned} \begin{aligned} \mathsf{P} (\beta _{1}-\beta _{2}, \jmath )\geq {}& \mathsf{P} \biggl(T_{i}(\vartheta )-\beta _{1},\frac{\jmath }{2} \biggr) \ast \mathsf{P} \biggl(T_{i}( \vartheta )-\beta _{2},\frac{\jmath }{2} \biggr) \\ >{}&(1-\epsilon _{1})\ast (1-\epsilon _{1}) \\ >{}&(1-\epsilon ), \end{aligned} \\ \begin{aligned} \mathsf{Q} (\beta _{1}-\beta _{2}, \jmath )\leq {}& \mathsf{Q} \biggl(T_{i}(\vartheta )-\beta _{1},\frac{\jmath }{2} \biggr) \diamond \mathsf{Q} \biggl(T_{i}(\vartheta )-\beta _{2}, \frac{\jmath }{2} \biggr) \\ < {}&\epsilon _{1}\diamond \epsilon _{1} \\ < {}&\epsilon \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \mathsf{R} (\beta _{1}-\beta _{2}, \jmath )\leq {}& \mathsf{R} \biggl(T_{i}(\vartheta )-\beta _{1},\frac{\jmath }{2} \biggr) \diamond \mathsf{R} \biggl(T_{i}(\vartheta )-\beta _{2}, \frac{\jmath }{2} \biggr) \\ < {}&\epsilon _{1}\diamond \epsilon _{1} \\ < {}&\epsilon. \end{aligned} \end{aligned}$$

Since \(\epsilon \in (0, 1)\) is arbitrary, therefore \(\mathsf{P} (\beta _{1}-\beta _{2},\jmath )=1\), \(\mathsf{Q} (\beta _{1}-\beta _{2},\jmath )=0\), and \(\mathsf{R} (\beta _{1}-\beta _{2},\jmath )=0\) for all \(\jmath >0\). Hence \(\beta _{1}-\beta _{2}=0\). Thus \(\mathcal{Z}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\)- limit is unique. □

Theorem 4.2

A sequence \(\vartheta =(\vartheta _{i})\in \omega \) is neutrosophic Tribonacci convergent with respect to neutrosophic norms \((\mathsf{P},\mathsf{Q},\mathsf{R})\) iff it is neutrosophic Tribonacci Cauchy with respect to the same norms.

Proof

Suppose that \(\vartheta =(\vartheta _{i})\in \omega \) is neutrosophic Tribonacci convergent with respect to neutrosophic norms \((\mathsf{P},\mathsf{Q},\mathsf{R})\) such that \(\mathcal{Z}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)-\lim (\vartheta _{i})= \beta \). For given \(\epsilon \in (0,1)\), there exists \(\epsilon _{1} \in (0,1)\) such that \((1-\epsilon _{1})*(1-\epsilon _{1})>1-\epsilon \) and \(\epsilon _{1}\diamond \epsilon _{1}<\epsilon \). Since \(\mathcal{Z}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\lim (\vartheta _{i})= \beta \), therefore for all \(\jmath >0\)

$$\begin{aligned} &\mathcalligra{C}= \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i}(\vartheta )- \beta, \jmath \bigr)\leq 1-\epsilon _{1} \text{ or } \mathsf{Q} \bigl(T_{i}( \vartheta )-\beta, \jmath \bigr)\geq \epsilon _{1}, \mathsf{R} \bigl(T_{i}( \vartheta )-\beta,\jmath \bigr)\geq \epsilon _{1} \bigr\} \\ & \text{is finite,} \end{aligned}$$

which implies

$$\begin{aligned} \mathcalligra{C}^{c}={}& \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i}( \vartheta )-\beta, \jmath \bigr)> 1-\epsilon _{1} \quad\text{or}\quad \mathsf{Q} \bigl(T_{i}(\vartheta )- \beta, \jmath \bigr)< \epsilon _{1}, \\ & \mathsf{R} \bigl(T_{i}(\vartheta )-\beta,\jmath \bigr)< \epsilon _{1} \bigr\} . \end{aligned}$$

For \(i\in \mathcalligra{C}^{c}\), we have

$$\begin{aligned} \mathsf{P} \bigl(T_{i}(\vartheta )-\beta, \jmath \bigr)>1-\epsilon _{1} \quad\text{or}\quad \mathsf{Q} \bigl(T_{i}(\vartheta )- \beta, \jmath \bigr)< \epsilon _{1}, \qquad\mathcalligra{R} \bigl(T_{j}(y)-\beta, t \bigr)< \epsilon _{1}. \end{aligned}$$

For fixed \(k\in {\mathcalligra{C}^{c}}\), let

$$\begin{aligned} A= {}& \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i}(\vartheta )-T_{k}( \vartheta ),\jmath \bigr)\leq 1-\epsilon \text{ or } \mathsf{Q} \bigl(T_{i}( \vartheta )-T_{k}(\vartheta ),\jmath \bigr)\geq \epsilon, \\ &\mathsf{R} \bigl(T_{i}(\vartheta )-T_{k}( \vartheta ), \jmath \bigr)\geq \epsilon \bigr\} \quad\text{be finite }. \end{aligned}$$

We show that \(A\subset \mathcalligra{C}\). Let \(i\in A\), we have

$$\begin{aligned} &\mathsf{P} \bigl(T_{i}(\vartheta )-T_{k}(\vartheta ), \jmath \bigr)\leq 1- \epsilon \quad\text{or}\quad \mathsf{Q} \bigl(T_{i}( \vartheta )-T_{k}(\vartheta ), \jmath \bigr)\geq \epsilon, \\ & \mathsf{R} \bigl(T_{i}(\vartheta )-T_{k}( \vartheta ), \jmath \bigr)\geq \epsilon. \end{aligned}$$

We have two possible cases, firstly consider \(\mathsf{P} (T_{i}(\vartheta )-T_{k}(\vartheta ),\jmath )\leq 1- \epsilon \). Then \(\mathsf{P} (T_{i}(\vartheta )-\beta, \frac{\jmath }{2} )\leq 1- \epsilon _{1}\). If possible, let \(\mathsf{P} (T_{i}(\vartheta )-\beta, \frac{\jmath }{2} )> 1- \epsilon _{1}\). Then

$$\begin{aligned} \begin{aligned} 1-\epsilon \geq {}& \mathsf{P} \bigl(T_{i}( \vartheta )-T_{k}( \vartheta ), \jmath \bigr) \\ \geq {}& \mathsf{P} \biggl(T_{i}(\vartheta )-\beta, \frac{\jmath }{2} \biggr)\ast \mathsf{P} \biggl(T_{k}(\vartheta )-\beta, \frac{\jmath }{2} \biggr) \\ >{}& (1-\epsilon _{1})\ast (1-\epsilon _{1}) \\ >{}& (1-\epsilon ), \end{aligned} \end{aligned}$$

which is a contradiction. \(\implies \mathsf{P} (T_{i}(\vartheta )-\beta, \frac{\jmath }{2} )\leq 1-\epsilon _{1}\).

Similarly, consider \(\mathsf{Q} (T_{i}(\vartheta )-T_{k}(\vartheta ), \jmath ) \geq \epsilon \), then \(\mathsf{Q} (T_{i}(\vartheta )-\beta, \frac{\jmath }{2} )\geq \epsilon _{1}\).

If possible, suppose \(\mathsf{Q} (T_{i}(\vartheta )-\beta, \frac{\jmath }{2} )< \epsilon _{1}\). Hence

$$\begin{aligned} \begin{aligned} \epsilon \leq {}& \mathsf{Q} \bigl(T_{i}( \vartheta )-T_{k}( \vartheta ), \jmath \bigr) \\ \leq{} & \mathsf{Q} \biggl(T_{i}(\vartheta )-\beta, \frac{\jmath }{2} \biggr)\diamond \mathsf{Q} \biggl(T_{k}(\vartheta )-\beta, \frac{\jmath }{2} \biggr) \\ < {}& \epsilon _{1}\diamond \epsilon _{1} \\ < {}& \epsilon, \end{aligned} \end{aligned}$$

which is again a contradiction. \(\implies \mathsf{Q} (T_{i}(\vartheta )-\beta, \frac{\jmath }{2} )\geq \epsilon _{1}\).

Similarly,

$$\mathsf{R} \bigl(T_{i}(\vartheta )-T_{k}(\vartheta ), \jmath \bigr) \geq \epsilon , \quad \text{then } \mathsf{R} \biggl(T_{i}( \vartheta )-\beta, \frac{t}{2} \biggr)\geq \epsilon _{1}. $$

If possible, suppose \(\mathsf{R} (T_{i}(\vartheta )-\beta, \frac{\jmath }{2} )< \epsilon _{1}\). Hence

$$\begin{aligned} \begin{aligned} \epsilon \leq {}& \mathsf{R} \bigl(T_{i}( \vartheta )-T_{k}( \vartheta ), \jmath \bigr) \\ \leq {}& \mathsf{R} \biggl(T_{i}(\vartheta )-\beta, \frac{\jmath }{2} \biggr)\diamond \mathsf{R} \biggl(T_{k}(\vartheta )-\beta, \frac{\jmath }{2} \biggr) \\ < {}& \epsilon _{1}\diamond \epsilon _{1} \\ < {}& \epsilon, \end{aligned} \end{aligned}$$

which is again a contradiction

$$\implies \mathsf{R} \biggl(T_{i}(\vartheta )-\beta, \frac{\jmath }{2} \biggr)\geq \epsilon _{1}.$$

Therefore, for \(i\in A\), we have

$$\begin{aligned} &\mathsf{P} \biggl(T_{i}(\vartheta )-\beta, \frac{\jmath }{2} \biggr) \leq (1- \epsilon _{1}),\qquad \mathsf{Q} \biggl(T_{i}(\vartheta )- \beta, \frac{\jmath }{2} \biggr)\geq \epsilon _{1}\quad \text{and}\\ & \mathsf{R} \biggl(T_{i}( \vartheta )-\beta,\frac{\jmath }{2} \biggr) \geq \epsilon _{1}\\ &\quad\implies\quad i\in \mathcalligra{C}. \end{aligned}$$

Hence, \(A\subset \mathcalligra{C}\). Since \(\mathcalligra{C}\) is finite, so the sequence \(\vartheta =(\vartheta _{i})\) is neutrosophic Tribonacci Cauchy with respect to the norms \((\mathsf{P},\mathsf{Q},\mathsf{R})\).

Conversely, suppose that the sequence \(\vartheta =(\vartheta _{i})\in \omega \) is neutrosophic Tribonacci Cauchy with respect to the norms \((\mathsf{P},\mathsf{Q},\mathsf{R})\). Let on the contrary the sequence \(\vartheta =(\vartheta _{i})\) be not neutrosophic Tribonacci convergent. Then there exists \(i\in \mathbb{N}\) such that

$$\begin{aligned} T_{1}={}& \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i}( \vartheta )-T_{k}( \vartheta ),\jmath \bigr)\leq 1-\epsilon \text{ or } \mathsf{Q} \bigl(T_{i}( \vartheta )-T_{k}(\vartheta ), \jmath \bigr)\geq \epsilon, \\ &\mathsf{R} \bigl(T_{i}(\vartheta )-T_{k}( \vartheta ), \jmath \bigr)\geq \epsilon \bigr\} \end{aligned}$$

but

$$\begin{aligned} &T_{2}= \biggl\{ i\in \mathbb{N}: \mathsf{P} \biggl(T_{i}( \vartheta )-\beta, \frac{\jmath }{2} \biggr)> 1-\epsilon _{1} \text{ or } \mathsf{Q} \biggl(T_{i}( \vartheta )-\beta, \frac{\jmath }{2} \biggr)< \epsilon _{1}, \\ &\phantom{T_{2}=} \mathsf{R} \biggl(T_{j}(y)-\beta, \frac{\jmath }{2} \biggr)< \epsilon _{1} \biggr\} , \\ &\quad \implies \\ &1-\epsilon \geq \mathsf{P} \bigl(T_{i}(\vartheta )-T_{k}( \vartheta ), \jmath \bigr) \\ &\phantom{1-\epsilon}\geq \mathsf{P} \biggl(T_{i}(\vartheta )-\beta, \frac{\jmath }{2} \biggr)\ast \mathsf{P} \biggl(T_{k}(\vartheta )- \beta, \frac{\jmath }{2} \biggr) \\ &\phantom{1-\epsilon}> (1-\epsilon _{1})\ast (1-\epsilon _{1}) \\ &\phantom{1-\epsilon}> 1-\epsilon, \end{aligned}$$

which is a contradiction.

Now,

$$\begin{aligned} \epsilon \leq {}& \mathsf{Q} \bigl(T_{i}(\vartheta )-T_{k}( \vartheta ), \jmath \bigr) \\ \leq {}& \mathsf{Q} \biggl(T_{i}(\vartheta )-\beta, \frac{\jmath }{2} \biggr)\diamond \mathsf{Q} \biggl(T_{k}(\vartheta )-\beta, \frac{\jmath }{2} \biggr) \\ < {}& \epsilon _{1}\diamond \epsilon _{1} \\ < {}& \epsilon, \end{aligned}$$

which is again a contradiction; and

$$\begin{aligned} \epsilon \leq {}& \mathsf{R} \bigl(T_{i}(\vartheta )-T_{k}( \vartheta ), \jmath \bigr) \\ \leq {}& \mathsf{R} \biggl(T_{i}(\vartheta )-\beta, \frac{\jmath }{2} \biggr)\diamond \mathsf{R} \biggl(T_{k}(\vartheta )-\beta, \frac{\jmath }{2} \biggr) \\ < {}& \epsilon _{1}\diamond \epsilon _{1} \\ < {}& \epsilon, \end{aligned}$$

which is again a contradiction.

Therefore \(T_{2}\in \mathcal{Z}\) and hence \(\vartheta =(\vartheta _{i})\) is neutrosophic Tribonacci convergent. □

Theorem 4.3

Every subset D of \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) is neutrosophic Tribonacci bounded.

Proof

The proof of this theorem follows from Theorem 3.1 and can be verified on similar grounds. □

Theorem 4.4

Let NNS \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) and \(\tau _{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) be the topology on \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\). Let \((\vartheta ^{n})=(\vartheta ^{n}_{k})_{n=1}^{\infty }\) be a sequence of points in \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\). Then \(\vartheta ^{n}\longrightarrow \vartheta \) as \(n\longrightarrow \infty \) iff

$$\begin{aligned} &\mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}(\vartheta ),\jmath \bigr) \longrightarrow 1,\qquad \mathsf{Q} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ),\jmath \bigr) \longrightarrow 0\quad\textit{and} \\ & \mathsf{R} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ),\jmath \bigr) \longrightarrow 0\quad\textit{as }i\longrightarrow \infty. \end{aligned}$$

Proof

Let \(\vartheta ^{n}\longrightarrow \vartheta \) as \(n\longrightarrow \infty \). Fix \(r>0\). Then, for \(0<\epsilon <1\), there exists \(k\in \mathbb{N}\) such that \((\vartheta ^{n})\in \mathcalligra{B}_{\vartheta }(r,\epsilon )(T)\) for all \(n\geq k\). Then the set A is finite, where

$$\begin{aligned} A= {}& \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ), r \bigr)\leq 1-\epsilon \text{ or } \mathsf{Q} \bigl(T_{i} \bigl( \vartheta ^{n} \bigr)-T_{i}(\vartheta ), r \bigr)\geq \epsilon, \\ & \mathsf{R} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ), r \bigr)\geq \epsilon \bigr\} , \end{aligned}$$

which is equivalent to

$$\begin{aligned} A^{c}={}& \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i} \bigl( \vartheta ^{n} \bigr)-T_{i}( \vartheta ), r \bigr)> 1- \epsilon \text{ or } \mathsf{Q} \bigl(T_{i} \bigl( \vartheta ^{n} \bigr)-T_{i}(\vartheta ), r \bigr)< \epsilon, \\ & \mathsf{R} \bigl(T_{i} \bigl( \vartheta ^{n} \bigr)-T_{i}( \vartheta ), r \bigr)< \epsilon \bigr\} . \end{aligned}$$

For \(\{i\in \mathbb{N} \}\subseteq A^{c}\),

$$\begin{aligned} &\mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}(\vartheta ), r \bigr)>1- \epsilon \\ &\quad\implies\quad 1- \mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ), r \bigr)>\epsilon,\qquad \mathsf{Q} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ), r \bigr)< \epsilon \quad \text{and} \\ &\mathsf{R} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}(\vartheta ),r \bigr)< \epsilon . \end{aligned}$$

Therefore, for \(n\longrightarrow \infty\),

$$\begin{aligned} &\implies\quad 1-\mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}(\vartheta ), r \bigr)\longrightarrow 0,\qquad \mathsf{Q} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ),r \bigr) \longrightarrow 0\quad \text{and} \\ & \mathsf{R} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ),r \bigr) \longrightarrow 0, \\ &\implies \quad \mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}(\vartheta ), r \bigr) \longrightarrow 1,\qquad \mathsf{Q} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ),r \bigr) \longrightarrow 0\quad \text{and} \\ & \mathsf{R} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ),r \bigr) \longrightarrow 0\quad \text{as } n \longrightarrow \infty . \end{aligned}$$

Conversely, suppose that for each \(i>0,\mathsf{P} (T_{i}(\vartheta ^{n})-T_{i}(\vartheta ), r ) \longrightarrow 1,\mathsf{Q} (T_{i}(\vartheta ^{n})-T_{i}( \vartheta ), r ) \longrightarrow 0\), and \(\mathsf{R} (T_{i}(\vartheta ^{n})-T_{i}(\vartheta ), r ) \longrightarrow 0\) as \(n\longrightarrow \infty \). Then for each \(\epsilon \in (0,1)\) there exists \(k\in \mathbb{N}\) such that

$$\begin{aligned} &\implies\quad 1-\mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}(\vartheta ), r \bigr) \longrightarrow 0,\qquad \mathsf{Q} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ), r \bigr)\longrightarrow 0\quad \text{and} \\ & \mathsf{R} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ), r \bigr) \longrightarrow 0 \end{aligned}$$

for all \(n\geq k\),

$$\begin{aligned} &\mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}(\vartheta ),r \bigr) >1- \epsilon \quad\implies\quad 1- \mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ), r \bigr)>\epsilon, \\ & \mathsf{Q} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ), r \bigr)< \epsilon \quad \text{and} \\ &\mathsf{R} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}(\vartheta ), r \bigr)< \epsilon \quad \text{for all }n\geq k. \end{aligned}$$

Consider that the neutrosophic-\(\mathcal{Z}\) generated by the set \(\{t\in \mathbb{N}: t< k\}\) implies the collection of sets generated by the set \(\{i\in \mathbb{N}: i\geq k\}\).

Thus

$$\begin{aligned} & \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ), r \bigr)>1-\epsilon \text{ or } \mathsf{Q} \bigl(T_{i} \bigl( \vartheta ^{n} \bigr)-T_{i}(\vartheta ), r \bigr)< \epsilon, \\ &\quad\mathsf{R} \bigl(T_{i} \bigl( \vartheta ^{n} \bigr)-T_{i}( \vartheta ), r \bigr)< \epsilon \bigr\} \\ &\quad \implies\quad \bigl(\vartheta ^{n} \bigr)\in \mathcalligra{B}_{\vartheta }(r,\epsilon ) (T)\quad \text{for all }n\geq k. \end{aligned}$$

Hence \(\vartheta ^{n}\longrightarrow \vartheta \) as \(n\longrightarrow \infty \). □

Theorem 4.5

Every closed ball with the center at ϑ and the radius \(r>0\) with respect to the parameter of fuzziness \(0<\epsilon <1\), i.e., \(\mathcalligra{B}_{\vartheta }[r,\epsilon ](T)\) is a closed set in \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\).

Proof

Let \(\vartheta =(\vartheta _{k})\in \omega \) be such that \(\vartheta \in \overline{\mathcalligra{B}_{\vartheta }[r,\epsilon ](T)}\). Then there exists a sequence \((\vartheta ^{n})=(\vartheta ^{n}_{k})\in {\mathcalligra{B}_{\vartheta }[r, \epsilon ](T)}\) such that \(\vartheta ^{n}\longrightarrow \vartheta \) as \(n\longrightarrow \infty \). This implies the set

$$\begin{aligned} X={}& \bigl\{ i\in \mathbb{N}:\mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ), r \bigr)\geq 1-\epsilon \text{ or } \mathsf{Q} \bigl(T_{i} \bigl( \vartheta ^{n} \bigr)-T_{i}(\vartheta ), r \bigr)\leq \epsilon, \\ &\mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ), r \bigr)\leq \epsilon \bigr\} . \end{aligned}$$

Since \(\vartheta ^{n}\longrightarrow \vartheta \) as \(n\longrightarrow \infty \), by Theorem 4.4,

$$\begin{aligned} &\mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}(\vartheta ), r \bigr) \longrightarrow 1,\qquad \mathsf{Q} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ), r \bigr)\longrightarrow 0 \quad\text{and} \\ & \mathsf{R} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ), r \bigr) \longrightarrow 0\quad \text{for all }t>0\text{ as }j \longrightarrow \infty . \end{aligned}$$

Hence for \(i\in X\)

$$\begin{aligned} \mathsf{P} \bigl(T_{i}(\vartheta )-T_{i}(y), \jmath +r \bigr)\geq {}&\lim_{i\rightarrow \infty }\mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ),\jmath \bigr)\ast \mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ),r \bigr) \\ \geq {}& 1\ast (1-\epsilon ) \\ ={}&1-\epsilon, \\ \mathsf{Q} \bigl(T_{i}(\vartheta )-T_{i}(y), \jmath +r \bigr)\leq {}&\lim_{i\rightarrow \infty }\mathsf{Q} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ),\jmath \bigr)\diamond \mathsf{Q} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ),r \bigr) \\ \leq {}& 0\diamond \epsilon \\ ={}&\epsilon \end{aligned}$$

and

$$\begin{aligned} \mathsf{R} \bigl(T_{i}(\vartheta )-T_{i}(y), \jmath +r \bigr)\leq {}&\lim_{i\rightarrow \infty }\mathsf{R} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ),\jmath \bigr)\diamond \mathsf{R} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ),r \bigr) \\ \leq {}& 0\diamond \epsilon \\ ={}&\epsilon. \end{aligned}$$

In particular, for \(k\in \mathbb{N}\), take \(\jmath =\frac{1}{k}\). Then

$$\begin{aligned} &\mathsf{P} \bigl(T_{i}(\vartheta )-T_{i}(y),r \bigr)= \lim _{k \rightarrow \infty }\mathsf{P} \biggl(T_{i}(\vartheta )-T_{i}(y),r+ \frac{1}{k} \biggr)\geq 1-\epsilon, \\ &\mathsf{Q} \bigl(T_{i}(\vartheta )-T_{i}(y),r \bigr)= \lim _{k \rightarrow \infty }\mathsf{Q} \biggl(T_{i}(\vartheta )-T_{i}(y),r+ \frac{1}{k} \biggr)\leq \epsilon \end{aligned}$$

and

$$\begin{aligned} &\mathsf{R} \bigl(T_{i}(\vartheta )-T_{i}(y),r \bigr)= \lim _{k \rightarrow \infty }\mathsf{R} \biggl(T_{i}(\vartheta )-T_{i}(y),r+ \frac{1}{k} \biggr)\leq \epsilon \\ &\implies\quad \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i}( \vartheta )-T_{i}(y),r \bigr)\geq 1-\epsilon \text{ or }\mathsf{Q} \bigl(T_{i}(\vartheta )-T_{i}(y),r \bigr)\leq \epsilon, \\ &\phantom{\implies\quad}\mathsf{R} \bigl(T_{i}(\vartheta )-T_{i}(y),r \bigr) \leq \epsilon \bigr\} .\\ &\implies\quad y\in {\mathcalligra{B}_{\vartheta }[r,\epsilon ](T).} \end{aligned}$$

Therefore \({\mathcalligra{B}_{\vartheta }[r,\epsilon ](T)}\) is a closed set. □

Theorem 4.6

Let \(\vartheta =(\vartheta _{k})\in \mathcal{L}_{(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T)\). Then, for some \(\beta \in Y\), \(\vartheta _{k}\rightarrow \beta \) if and only if for every \(\epsilon \in (0,1)\) and \(\jmath >0\) there exist positive integers \(N=N(\vartheta,\epsilon,\jmath )\) such that

$$\begin{aligned} \biggl\{ N\in \mathbb{N}: \mathsf{P} \biggl(T_{N}(\vartheta )-\beta, \frac{\jmath }{2} \biggr)>1-\epsilon \textit{ or } \mathsf{Q} \biggl(T_{N}( \vartheta )-\beta, \frac{\jmath }{2} \biggr)< \epsilon,\mathsf{R} \biggl(T_{N}( \vartheta )-\beta, \frac{\jmath }{2} \biggr)< \epsilon \biggr\} . \end{aligned}$$

Proof

Suppose \(\vartheta _{k}\rightarrow \beta \) for some \(\beta \in Y\). For given \(\epsilon \in (0,1)\), there exists \(r \in (0,1)\) such that \((1-\epsilon )\ast (1-\epsilon )>1-r\) and \(\epsilon \diamond \epsilon < r\). Since \(\vartheta _{k}\rightarrow \beta \), for all \(\jmath >0\),

$$\begin{aligned} \mathcal{S}= {}&\biggl\{ i\in \mathbb{N}:\mathsf{P} \biggl(T_{i}(\vartheta )- \beta,\frac{\jmath }{2} \biggr)\leq 1-\epsilon \text{ or } \mathsf{Q} \biggl(T_{i}(\vartheta )-\beta, \frac{\jmath }{2} \biggr)\geq \epsilon,\\ & \mathsf{R} \biggl(T_{i}(\vartheta )-\beta, \frac{\jmath }{2} \biggr) \geq \epsilon \biggr\} , \end{aligned}$$

which implies that

$$\begin{aligned} \mathcal{S}^{c}= \biggl\{ i\in \mathbb{N}:\mathsf{P} \biggl(T_{i}( \vartheta )-\beta,\frac{\jmath }{2} \biggr)> 1-\epsilon \text{ or } \mathsf{Q} \biggl(T_{i}(\vartheta )-\beta, \frac{\jmath }{2} \biggr)< \epsilon,\mathsf{R} \biggl(T_{i}(\vartheta )-\beta, \frac{\jmath }{2} \biggr)< \epsilon \biggr\} . \end{aligned}$$

Conversely, let us choose \(N\in \mathcal{S}^{c}\). Then

$$\begin{aligned} \mathsf{P} \biggl(T_{N}(\vartheta )-\beta,\frac{\jmath }{2} \biggr)> 1- \epsilon \quad\text{or}\quad \mathsf{Q} \biggl(T_{N}(\vartheta )- \beta, \frac{\jmath }{2} \biggr)< \epsilon,\qquad \mathsf{R} \biggl(T_{N}( \vartheta )- \beta, \frac{\jmath }{2} \biggr)< \epsilon. \end{aligned}$$

We show that there exists a positive integer \(N=N(\vartheta,\epsilon,\jmath )\) such that

$$\begin{aligned} & \bigl\{ k\in \mathbb{N}:\mathsf{P} \bigl(T_{k}(\vartheta )-T_{N}( \vartheta ), \jmath \bigr)\leq 1-r \text{ or }\mathsf{Q} \bigl(T_{k}( \vartheta )-T_{N}(\vartheta ), \jmath \bigr)\geq r, \\ &\quad\mathsf{R} \bigl(T_{k}( \vartheta )-T_{N}(\vartheta ), \jmath \bigr)\geq r \bigr\} . \end{aligned}$$

So, for \(\vartheta =(\vartheta _{k})\in \mathcal{L}_{(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T)\), define

$$\begin{aligned} \mathcalligra{C}={}& \bigl\{ k\in \mathbb{N}:\mathsf{P} \bigl(T_{k}( \vartheta )-T_{N}( \vartheta ), \jmath \bigr)\leq 1-r \text{ or } \mathsf{Q} \bigl(T_{k}( \vartheta )-T_{N}(\vartheta ), \jmath \bigr)\geq r, \\ & \mathsf{R} \bigl(T_{k}( \vartheta )-T_{N}(\vartheta ), \jmath \bigr)\geq r \bigr\} . \end{aligned}$$

We shall show that \(\mathcalligra{C}\subseteq Y\). Let on the contrary \(\mathcalligra{C}\nsubseteq Y\), i.e., there exists \(m\in \mathcalligra{C}\) such that \(m\notin Y\). Then

$$\begin{aligned} \mathsf{P} \bigl(T_{m}(\vartheta )-T_{N}(\vartheta ), \jmath \bigr)\leq 1-r\quad \text{or}\quad\mathsf{P} \biggl(T_{m}( \vartheta )-\beta,\frac{\jmath }{2} \biggr)> 1-\epsilon. \end{aligned}$$

In particular,

$$\begin{aligned} \mathsf{P} \biggl(T_{N}(\vartheta )-\beta,\frac{\jmath }{2} \biggr)> 1- \epsilon. \end{aligned}$$

Therefore we have

$$\begin{aligned} 1-r\geq {}& \mathsf{P} \bigl(T_{m}(\vartheta )-T_{N}( \vartheta ), \jmath \bigr) \\ \geq {}& \mathsf{P} \biggl(T_{m}(\vartheta )-\beta, \frac{\jmath }{2} \biggr)\ast \mathsf{P} \biggl(T_{N}(\vartheta )-\beta, \frac{\jmath }{2} \biggr) \\ \geq {}& (1-\epsilon )\ast (1-\epsilon ) \\ >{}& 1-r, \end{aligned}$$

which is a contradiction. Similarly,

$$\begin{aligned} \mathsf{Q} \bigl(T_{m}(\vartheta )-T_{N}(\vartheta ), \jmath \bigr)\geq r \quad\text{or}\quad \mathsf{Q} \biggl(T_{m}( \vartheta )- \beta,\frac{\jmath }{2} \biggr)< \epsilon. \end{aligned}$$

In particular,

$$\begin{aligned} \mathsf{Q} \biggl(T_{N}(\vartheta )-\beta,\frac{\jmath }{2} \biggr)< \epsilon. \end{aligned}$$

Therefore we have

$$\begin{aligned} r\leq {}& \mathsf{Q} \bigl(T_{m}(\vartheta )-T_{N}( \vartheta ), \jmath \bigr) \\ \leq {}& \mathsf{Q} \biggl(T_{m}(\vartheta )-\beta, \frac{\jmath }{2} \biggr)\diamond \mathsf{Q} \biggl(T_{N}(\vartheta )-\beta, \frac{\jmath }{2} \biggr) \\ \leq {}& \epsilon \diamond \epsilon \\ < {}& r. \end{aligned}$$

Similarly, in the other way,

$$\begin{aligned} \mathsf{R} \bigl(T_{m}(\vartheta )-T_{N}( \vartheta ), \jmath \bigr)\geq r \quad\text{or}\quad\mathsf{R} \biggl(T_{m}(\vartheta )-\beta, \frac{\jmath }{2} \biggr)< \epsilon. \end{aligned}$$

In particular,

$$\begin{aligned} &\mathsf{R} \biggl(T_{N}(\vartheta )-\beta,\frac{\jmath }{2} \biggr)< \epsilon, \\ &r\leq \mathsf{R} \bigl(T_{m}(\vartheta )-T_{N}( \vartheta ), \jmath \bigr) \\ &\phantom{r}\leq \mathsf{R} \biggl(T_{m}(\vartheta )-\beta, \frac{\jmath }{2} \biggr)\diamond \mathsf{R} \biggl(T_{N}(\vartheta )- \beta, \frac{\jmath }{2} \biggr) \\ &\phantom{r}\leq \epsilon \diamond \epsilon \\ &\phantom{r}< r, \end{aligned}$$

which is again a contradiction.

Hence \(\mathcalligra{C}\subseteq Y\) since \(Y\in \mathcal{Z}\) implies \(\mathcalligra{C}\in \mathcal{Z}\). □

Definition 4.7

Consider \(D\subseteq \mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\). Then D is compact if every open cover of D by the open set of \(\tau _{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) has a finite subcover.

Theorem 4.7

Every finite subset D of \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) is compact.

Proof

Let \(D= \{\vartheta _{1}, \vartheta _{2}, \vartheta _{3},\ldots, \vartheta _{n} \}\) be the finite subset of \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\). For \(r>0\) and \(0<\epsilon <1\),

let us assume that \(\{\mathcalligra{B}_{\vartheta }(r,\epsilon )(T):\vartheta \in D \}\) is an open cover of D. Then \(D\subseteq \bigcup_{\vartheta \in D} \mathcalligra{B}_{\vartheta }(r, \epsilon )(T)\).

Now, for all \(\vartheta _{i} \in D\), \(i= 1,2,3,\ldots n\), we have \(\vartheta _{i} \in \bigcup_{\vartheta _{i}\in D} \mathcalligra{B}_{ \vartheta _{i}}(r,\epsilon )(T)\). That implies \(\vartheta _{i} \in \mathcalligra{B}_{\vartheta _{i}}(r,\epsilon )(T)\) for some \(i\in \{1,2,3,\ldots,n \}\). Then \(\{\mathcalligra{B}_{\vartheta _{i}}(r,\epsilon )(T): i= 1,2,3,\ldots,n \}\) is a finite subcover of D.

Therefore D is compact. □

Theorem 4.8

Let \(D\subseteq \mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\). Then D is compact iff every sequence in D has a convergent subsequence.

Proof

Suppose that D is a compact subset of \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\). Let \((\vartheta ^{n}_{k})=(\vartheta ^{n})_{n=1}^{\infty }\) be a sequence in D. For given \(0<\epsilon <1\) and \(r>0\), let \(\{\mathcalligra{B}_{\vartheta }(\frac{r}{3},\epsilon )(T):\vartheta = ( \vartheta _{k})\in D \}\) be an open cover of S.

This implies \((\vartheta ^{n})\in \bigcup_{\vartheta \in D}\mathcalligra{B} _{\vartheta }( \frac{r}{3},\epsilon )(T)\). Then there exists some \(\vartheta =(\vartheta _{k})\in D\) such that \((\vartheta ^{n})\in \mathcalligra{B}_{\vartheta }(\frac{r}{3},\epsilon )(T)\). Therefore the set

$$\begin{aligned} Y_{1}={}& \biggl\{ i\in \mathbb{N}:\mathsf{P} \biggl(T_{i} \bigl( \vartheta ^{n} \bigr)-T_{i}( \vartheta ),\frac{r}{3} \biggr)>1-\epsilon \text{ or } \mathsf{Q} \biggl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ), \frac{r}{3} \biggr)< \epsilon, \\ &\mathsf{R} \biggl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}(\vartheta ), \frac{r}{3} \biggr)< \epsilon \biggr\} . \end{aligned}$$

Since D is compact, there exists a finite subcover \(\{\mathcalligra{B}_{\vartheta _{i}}(\frac{r}{3},\epsilon )(T): \vartheta _{i}\in D \text{ and } i=1,2,3,\ldots.m \}\) of D such that \(D\subseteq \bigcup_{i=1}^{m} \mathcalligra{B}_{\vartheta _{i}}( \frac{r}{3},\epsilon )(T)\).

Let \((\vartheta ^{n_{p}})\) be a subsequence of \((\vartheta ^{n})\). Then \((\vartheta ^{n_{p}}) \in \bigcup_{i=1}^{m} \mathcalligra{B}_{ \vartheta _{i}}(\frac{r}{3},\epsilon )(T)\) implies \((\vartheta ^{n_{p}}) \in \mathcalligra{B}_{\vartheta _{i}}(\frac{r}{3}, \epsilon )(T)\) for some \(\vartheta _{i}\in D\). Therefore the set

$$\begin{aligned} Y_{2}={}& \biggl\{ i\in \mathbb{N}: \mathsf{P} \biggl(T_{i} \bigl( \vartheta ^{n_{p}} \bigr)-T_{i}( \vartheta _{i}), \frac{r}{3} \biggr)>1- \epsilon \text{ or } \mathsf{Q} \biggl(T_{i} \bigl(\vartheta ^{n_{p}} \bigr)-T_{i}( \vartheta _{i}), \frac{r}{3} \biggr)< \epsilon, \\ &\mathsf{R} \biggl(T_{i} \bigl(\vartheta ^{n_{p}} \bigr)-T_{i}(\vartheta _{i}), \frac{r}{3} \biggr)< \epsilon \biggr\} . \end{aligned}$$

Now, for \(k\in Y_{1}\cap Y_{2}\),

$$\begin{aligned} & \mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n_{p}} \bigr)-T_{i}( \vartheta ),r \bigr) \\ &\quad\geq \mathsf{P} \biggl(T_{i} \bigl(\vartheta ^{n_{p}} \bigr)-T_{i}(\vartheta _{i}), \frac{r}{3} \biggr) \\ &\qquad{}\ast \mathsf{P} \biggl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}(\vartheta _{i}), \frac{r}{3} \biggr) \\ &\qquad{}\ast \mathsf{P} \biggl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}(\vartheta ), \frac{r}{3} \biggr) \\ &\quad >(1-\epsilon )\ast (1-\epsilon )\ast (1-\epsilon ) \\ &\quad =(1-\epsilon ), \\ &\quad \implies\quad \mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n_{p}} \bigr)-T_{i}(y),r \bigr)>1- \epsilon, \\ & \mathsf{Q} \bigl(T_{i} \bigl(\vartheta ^{n_{p}} \bigr)-T_{i}( \vartheta ),r \bigr) \\ &\quad\leq \mathsf{Q} \biggl(T_{i} \bigl(\vartheta ^{n_{p}} \bigr)-T_{i}(\vartheta _{i}), \frac{r}{3} \biggr) \\ &\qquad{}\diamond \mathsf{Q} \biggl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}(\vartheta _{i}), \frac{r}{3} \biggr) \\ &\qquad{}\diamond \mathsf{Q} \biggl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}(\vartheta ), \frac{r}{3} \biggr) \\ &\quad< \epsilon \diamond \epsilon \diamond \epsilon \\ &\quad =\epsilon, \end{aligned}$$

and

$$\begin{aligned} &\mathsf{R} \bigl(T_{i} \bigl(\vartheta ^{n_{p}} \bigr)-T_{i}( \vartheta ),r \bigr) \\ &\quad\leq\mathsf{R} \biggl(T_{i} \bigl(\vartheta ^{n_{p}} \bigr)-T_{i}(\vartheta _{i}), \frac{r}{3} \biggr) \\ &\qquad\diamond \mathsf{R} \biggl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}(\vartheta _{i}), \frac{r}{3} \biggr) \\ &\qquad\diamond \mathsf{R} \biggl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}(\vartheta ), \frac{r}{3} \biggr) \\ &\quad< \epsilon \diamond \epsilon \diamond \epsilon \\ &\quad =\epsilon. \end{aligned}$$

Take \(\epsilon =\frac{1}{p}\). Then

$$\begin{aligned} &\lim_{p\rightarrow \infty }\mathsf{P} \bigl(T_{i} \bigl( \vartheta ^{n_{p}} \bigr)-T_{i}(\vartheta ),r \bigr)=\lim _{p\rightarrow \infty }1- \frac{1}{p}=1, \\ & \lim_{p\rightarrow \infty } \mathsf{Q} \bigl(T_{k} \bigl(\vartheta ^{n_{p}} \bigr)-T_{k}(\vartheta ),r \bigr)= \lim _{p\rightarrow \infty }\frac{1}{p}=0 \quad\text{and} \\ &\lim_{p\rightarrow \infty }\mathsf{R} \bigl(T_{k} \bigl(\vartheta ^{n_{p}} \bigr)-T_{k}( \vartheta ),r \bigr)=\lim _{p\rightarrow \infty } \frac{1}{p}=0. \end{aligned}$$

Hence, by Theorem (4.4), \(\vartheta ^{n_{p}}\rightarrow \vartheta \), as \(p\rightarrow \infty \).

Conversely, suppose that \((\vartheta ^{n_{p}})\) is a subsequence of the sequence \((\vartheta ^{n})\) in D such that \((\vartheta ^{n_{p}})\rightarrow \vartheta \) in D. Let on the contrary D be not a compact subset of \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\).

Suppose that \(\{\mathcalligra{B}_{\vartheta }(r,\epsilon )(T):\vartheta \in D \}\) is an open cover of D \(D\subseteq \bigcup_{\vartheta \in d}\mathcalligra{B}_{\vartheta }(r, \epsilon )(T)\).

Therefore the set

$$\begin{aligned} & \bigl\{ i\in \mathbb{N}:\mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n} \bigr)-T_{i}( \vartheta ), r \bigr)>1-\epsilon \text{ or } \mathsf{Q} \bigl(T_{i} \bigl( \vartheta ^{n} \bigr)-T_{i}(\vartheta ), r \bigr)< \epsilon, \\ &\quad\mathsf{R} \bigl(T_{i} \bigl( \vartheta ^{n} \bigr)-T_{i}( \vartheta ),r \bigr)< \epsilon \bigr\} . \end{aligned}$$

Since D is not compact, there exists a finite subcover \(\{\mathcalligra{B}_{\vartheta _{i}}(r,\epsilon )(T):\vartheta _{i}\in D, i= 1,2,3,\ldots, m \}\) such that \(D\nsubseteq \bigcup_{\vartheta _{i}\in D}\mathcalligra{B}_{\vartheta _{i}}(r, \epsilon )(T)\), which implies that the set

$$\begin{aligned} & \bigl\{ i\in \mathbb{N}:\mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n_{p}} \bigr)-T_{i}( \vartheta _{i}), r \bigr)>1- \epsilon \text{ or } \mathsf{Q} \bigl(T_{i} \bigl( \vartheta ^{n_{p}} \bigr)-T_{i}(\vartheta _{i}), r \bigr)< \epsilon, \\ &\phantom{\quad\implies\quad} \mathsf{R} \bigl(T_{i} \bigl(\vartheta ^{n_{p}} \bigr)-T_{i}(\vartheta _{i}), r \bigr)< \epsilon \bigr\} \\ &\quad\implies\quad \bigl\{ i\in \mathbb{N}:\mathsf{P} \bigl(T_{i} \bigl(\vartheta ^{n_{p}} \bigr)-T_{i}( \vartheta ),r \bigr)>1-\epsilon \text{ or }\mathsf{Q} \bigl(T_{i} \bigl( \vartheta ^{n_{p}} \bigr)-T_{i}(\vartheta ), r \bigr)< \epsilon, \\ &\phantom{\quad\implies\quad}\mathsf{R} \bigl(T_{i} \bigl(\vartheta ^{n_{p}} \bigr)-T_{i}(\vartheta _{i}), r \bigr)< \epsilon \bigr\} \\ &\quad\implies\quad \text{For any }0< \epsilon < 1\text{ and }r>0, \bigl(\vartheta ^{n_{p}} \bigr)\notin \mathcalligra{B}_{\vartheta }(r,\epsilon ) \\ &\quad\implies\quad \vartheta ^{n_{p}}\nrightarrow \vartheta ,\quad \text{which is a contradiction}. \end{aligned}$$

Hence D is compact. □

Theorem 4.9

Let D be the compact subset of \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) such that \(\vartheta =(\vartheta _{k})\notin D\). Then there exist two open sets U and V in \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) such that \(D\subseteq V,\vartheta \in U\), and \(U\cap V=\phi \).

Proof

Let D be a compact subset of \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) and \(\vartheta \notin D\). Then, for any \(s\in D\), we have \(\vartheta \neq s\). Since \(\mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) is a Hausdorff space, then for some \(r>0\) and \(0<\epsilon <1\) there exist two open balls \(U=\mathcalligra{B}_{\vartheta }(r,\epsilon )(T)\) and \(V=\mathcalligra{B}_{s}(r,\epsilon )(T)\) such that \(\vartheta \in U\), \(s\in V\) and \(U\cap V=\phi \).

Consider the open cover \(V_{s}= \{\mathcalligra{B}_{s}(r,\epsilon )(T): s\in D \}\) of D and D is compact.

Therefore there exists a finite subcover \(V_{s_{i}}= \{\mathcalligra{B}_{s_{i}}(r,\epsilon )(T):s_{i}\in D \text{ and } i=1,2,3,\ldots,n \}\) such that \(D\subseteq \bigcup_{i=1}^{n}V_{s_{i}}\). Taking \(V=\bigcap_{i=1}^{n}V_{s_{i}}\), we have \(\vartheta \notin D\).

Hence U and V are open sets such that \(D\subseteq V\) and \(U\cap V=\phi \). □

Theorem 4.10

Consider the NNS \(\mathcalligra{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\). Let \(r>0\) and \(\epsilon _{1}, \epsilon _{2}\in (0,1)\) such that \((1-\epsilon _{1})\ast (1-\epsilon _{1})\geq (1-\epsilon _{2})\) and \(\epsilon _{1}\diamond \epsilon _{1}\leq \epsilon _{2}\). Then, for any \(\vartheta =(\vartheta _{k})\in \mathcal{L}_{(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T), \overline{\mathcalligra{B}_{\vartheta }(\frac{r}{2},\epsilon _{1})(T)} \subseteq \mathcalligra{B}_{\vartheta }(r,\epsilon _{2})(T)\).

Proof

Let \(\ell =(\ell _{k})\in \overline{\mathcalligra{B}_{\vartheta }(\frac{r}{2},\epsilon _{1})(T)}\) and \(\mathcalligra{B}_{\ell }(\frac{r}{2},\epsilon _{1})(T)\) be an open ball with the center at and the radius \(\epsilon _{1}\). Hence \(\mathcalligra{B}_{\ell }(\frac{r}{2},\epsilon _{1})(T)\cap \mathcalligra{B}_{ \vartheta }(\frac{r}{2},\epsilon _{1})(T)\neq \phi \).

Suppose \(z=(z_{k})\in \mathcalligra{B}_{\ell }(\frac{r}{2},\epsilon _{1})(T)\cap \mathcalligra{B}_{\vartheta }(\frac{r}{2},\epsilon _{1})(T)\). Then the sets

$$\begin{aligned} Y_{1}={}& \biggl\{ i\in \mathbb{N}:\mathsf{P} \biggl(T_{i}( \ell )-T_{i}(z), \frac{r}{2} \biggr)>1-\epsilon _{1} \text{ or } \mathsf{Q} \biggl(T_{i}( \ell )-T_{i}(z), \frac{r}{2} \biggr)< \epsilon _{1}, \\ &\mathsf{R} \biggl(T_{i}( \ell )-T_{i}(z),\frac{r}{2} \biggr)< \epsilon _{1} \biggr\} \\ Y_{2}={}& \biggl\{ i\in \mathbb{N}: \mathsf{P} \biggl(T_{i}( \vartheta )-T_{i}(z), \frac{r}{2} \biggr)>1-\epsilon _{1} \text{ or } \mathsf{Q} \biggl(T_{i}( \vartheta )-T_{i}(z),\frac{r}{2} \biggr)< \epsilon _{1}, \\ &\mathsf{R} \biggl(T_{i}( \vartheta )-T_{i}(z), \frac{r}{2} \biggr)< \epsilon _{1} \biggr\} \end{aligned}$$

Consider \(k\in Y_{1}\cap Y_{2}\). Then

$$\begin{aligned} \mathsf{P} \bigl(T_{k}(\vartheta )-T_{k}(\ell ),r \bigr) \geq{} & \mathsf{P} \biggl(T_{k}(\vartheta )-T_{k}(z),\frac{r}{2} \biggr) \ast \mathsf{P} \biggl(T_{k}(\ell )-T_{k}(z), \frac{r}{2} \biggr) \\ >{}& (1-\epsilon _{1})\ast (1-\epsilon _{1}) \\ \geq {}& (1-\epsilon _{2}) \\ \mathsf{Q} \bigl(T_{k}(\vartheta )-T_{k}(\ell ),r \bigr) \leq{} & \mathsf{Q} \biggl(T_{k}(\vartheta )-T_{k}(z),\frac{r}{2} \biggr) \diamond \mathsf{Q} \biggl(T_{k}(\ell )-T_{k}(z), \frac{r}{2} \biggr) \\ < {}& \epsilon _{1}\diamond \epsilon _{1} \\ \leq {}& \epsilon _{2} \end{aligned}$$

and

$$\begin{aligned} \mathsf{R} \bigl(T_{k}(\vartheta )-T_{k}(\ell ),r \bigr) \leq {}& \mathsf{R} \biggl(T_{k}(\vartheta )-T_{k}(z),\frac{r}{2} \biggr) \diamond \mathsf{R} \biggl(T_{k}(\ell )-T_{k}(z), \frac{r}{2} \biggr) \\ < {}& \epsilon _{1}\diamond \epsilon _{1} \\ \leq{} & \epsilon _{2}. \end{aligned}$$

Therefore the set

$$\begin{aligned} & \bigl\{ i\in \mathbb{N}:\mathsf{P} \bigl(T_{i}(\vartheta )-T_{i}(\ell ),r \bigr)>1-\epsilon _{2} \text{ or } \mathsf{Q} \bigl(T_{i}(\vartheta )-T_{i}( \ell ), r \bigr)< \epsilon _{2},\mathsf{R} \bigl(T_{i}(\vartheta )-T_{i}( \ell ),r \bigr)< \epsilon _{2} \bigr\} . \\ &\quad \implies \quad\ell =(\ell _{k})\in \mathcalligra{B}_{\vartheta }(r,\epsilon _{2}) (T). \end{aligned}$$

Hence \(\overline{\mathcalligra{B}_{\vartheta }(\frac{r}{2},\epsilon _{1})(T)} \subseteq \mathcalligra{B}_{\vartheta }(r,\epsilon _{2})(T)\). □

Theorem 4.11

Let \(\vartheta =(\vartheta _{k})\in \omega \). If there exists a sequence \(y=(y_{k})\in \mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\) such that \(T_{i}(\vartheta )=T_{i}(y)\) for almost all i relative to neutrosophic-\(\mathcal{Z}\), then \(\vartheta \in \mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\).

Proof

Suppose \(T_{i}(\vartheta )=T_{i}(y)\) for almost all i relative to \(\mathcal{Z}\). Then \(\{i\in \mathbb{N}:T_{i}(\vartheta )\neq T_{i}(y) \}\in \mathcal{Z}\implies \{i\in \mathbb{N}: T_{i}(\vartheta )= T_{i}(y) \}\). Therefore, for all \(\jmath >0\),

$$\begin{aligned} &\mathsf{P} \biggl(T_{i}(\vartheta )-T_{i}(y), \frac{\jmath }{2} \biggr)=1 \quad\text{and}\quad \mathsf{Q} \biggl(T_{i}( \vartheta )-T(y), \frac{\jmath }{2} \biggr)=0, \\ & \mathsf{R} \biggl(T_{i}(\vartheta )-T_{i}(y), \frac{\jmath }{2} \biggr)=0. \end{aligned}$$

Since \((y_{k})\in \mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\), let \(\lim (z_{k})=\beta \). Then, for every \(\epsilon \in (0, 1)\) and \(\jmath >0\),

$$\begin{aligned} Y_{1}= \biggl\{ i\in \mathbb{N}: \mathsf{P} \biggl(T_{i}(y)- \beta, \frac{\jmath }{2} \biggr)>1-\epsilon \text{ or } \mathsf{Q} \biggl(T_{i}(y)- \beta,\frac{\jmath }{2} \biggr)< \epsilon, \mathsf{R} \biggl(T_{i}(y)- \beta,\frac{\jmath }{2} \biggr)< \epsilon \biggr\} . \end{aligned}$$

Consider the set

$$\begin{aligned} Y_{2}= \bigl\{ i\in \mathbb{N}: \mathsf{P} \bigl(T_{i}( \vartheta )-\beta, \jmath \bigr)>1-\epsilon \text{ or } \mathsf{Q} \bigl(T_{i}(\vartheta )- \beta,\jmath \bigr)< \epsilon, \mathsf{R} \bigl(T_{i}(\vartheta )- \beta,\jmath \bigr)< \epsilon \bigr\} . \end{aligned}$$

We show that \(Y_{1}\subset Y_{2}\). So for \(j\in Y_{1}\) we have

$$\begin{aligned} \mathsf{P} \bigl(T_{i}(\vartheta )-\beta,\jmath \bigr) \geq {}& \mathsf{P} \biggl(T_{i}(\vartheta )-T_{i}(y), \frac{\jmath }{2} \biggr)\ast \mathsf{P} \biggl(T_{i}(y)-\beta, \frac{\jmath }{2} \biggr) \\ >{}& 1\ast (1-\epsilon ) \\ ={}& 1-\epsilon, \\ \mathsf{Q} \bigl(T_{i}(\vartheta )-\beta,\jmath \bigr) \leq {}& \mathsf{P} \biggl(T_{i}(\vartheta )-T_{i}(y), \frac{\jmath }{2} \biggr)\diamond \mathsf{Q} \biggl(T_{i}(y)-\beta, \frac{\jmath }{2} \biggr) \\ < {}& 0\diamond \epsilon \\ ={}& \epsilon, \end{aligned}$$

and

$$\begin{aligned} \mathsf{R} \bigl(T_{i}(\vartheta )-\beta,\jmath \bigr) \leq {}& \mathsf{R} \biggl(T_{i}(\vartheta )-T_{i}(y), \frac{\jmath }{2} \biggr)\diamond \mathsf{R} \biggl(T_{i}(y)-\beta, \frac{\jmath }{2} \biggr) \\ < {}& 0\diamond \epsilon \\ ={}& \epsilon \end{aligned}$$

\(\implies i\in Y_{2}\) and hence \(Y_{1}\subset Y_{2}\).

Hence \(\vartheta =(\vartheta _{k})\in \mathcal{L}_{(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T)\). □

Conclusion

Tribonacci numbers have been studied by several authors in the past who investigated Tribonacci identities, recurrence relations, and generalized Tribonacci numbers. However, in this paper, we focus on different directions by introducing a Tribonacci sequence space with the aid of a neutrosophic sequence space. We expect that our results might be a reference for further studies in this field. We have defined the Tribonacci matrix from neutrosophic convergence of sequence spaces and examine some topological and algebraic properties.

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Acknowledgements

The authors would like to thank the referees and the editor for their careful reading and their valuable comments.

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This work is financially supported by Aligarh Muslim University, Aligarh, India.

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VAK carried out the mathematical studies, participated in the sequence alignment, and drafted the manuscript. MA and MDK participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.

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Correspondence to Vakeel A. Khan.

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Khan, V.A., Arshad, M. & Khan, M.D. Some results of neutrosophic normed space VIA Tribonacci convergent sequence spaces. J Inequal Appl 2022, 42 (2022). https://doi.org/10.1186/s13660-022-02775-3

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Keywords

  • Tribonaaci matrix and sequence
  • Regular Tribonacci matrix
  • Convergence
  • t-norm
  • t-conorm
  • Neutrosophic normed space
  • Tribonacci convergence