Some results of neutrosophic normed space VIA Tribonacci convergent sequence spaces

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=(tik)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T = (t _{ik} )$\end{document} 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 [4][5][6]. 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 falsehoodnonmembership. 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 non-converging 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 (P, Q, R), we mention the following notions that will be used in the article.

Preliminaries
Let R and 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.
ω := ϑ = (ϑ k ), k ∈ N|ϑ = (ϑ k ) ∈ R, or C . (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 ϑ = (ϑ k ), the W transform of ϑ is defined as For any sequence space E, the sequence space E T defined by is known as domain of the matrix T. k∈N is said to be regular iff the following conditions hold: (a) There exists M > 0 such that for every i ∈ N, k |t ik | ≤ M, (b) lim i→∞ t ik = 0 for every k ∈ N, (c) lim i→∞ 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∈N of Tribonacci numbers given by third recurrence relation t n = t n-1 + t n-2 + t n-3 , n ≥ 3 with t 1 = t 2 = 1 and t 3 = 2.
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 R or C, and The four-tuple (Y , Z, , ) is called a neutrosophic normed space (NNS) if the subsequent terms hold; for all ϑ, y, z ∈ Y and j , s > 0, In such a case, Z = (P, Q, R) is called a neutrosophic norm (NN).

Definition 2.5
Suppose that X is an NNS, the sequence b = (b i ) in X is called convergent to ξ ∈ X ⇐⇒ ∃ N ∈ N, with respect to NN-Z = (P, Q, R) if for every > 0, j > 0 In such a case, we denote Zlim b i = ξ .
is called a Cauchy sequence with respect to Z if for each > 0 and j > 0, ∃ η ∈ N such that
(3.6) Then, for the set {k ∈ N}, we have

On the Tribonacci sequence T n
Definition 4.1 A sequence ϑ = (ϑ n ) ∈ ω is said to be Tribonacci convergent to β ∈ Y if for every > 0 the set B 1 is finite, where

Definition 4.3
A sequence ϑ = (ϑ i ) ∈ ω is said to be Tribonacci Cauchy if for every > 0 there exists k = k( ) ∈ N such that the set B 2 is finite, where

Definition 4.4 A sequence ϑ = (ϑ i )
∈ ω is said to neutrosophic Tribonacci Cauchy with respect to neutrosophic norms-(P, Q, R) if for every ∈ (0, 1) and j > 0 there exists k ∈ N such that the set T 2 is finite, where

Definition 4.6
A subset D of ω is said to be neutrosophic Tribonacci bounded with respect to neutrosophic norms (P, Q, R) if ∀ϑ ∈ D there exist 0 < < 1 and j > 0 such that the set
Therefore the set ⇒ P T i ϑ n p -T i (y), r > 1 -, Take = 1 p . Then Hence, by Theorem (4.4), ϑ n p → ϑ, as p → ∞. Conversely, suppose that (ϑ n p ) is a subsequence of the sequence (ϑ n ) in D such that (ϑ n p ) → ϑ in D. Let on the contrary D be not a compact subset of L (P,Q,R) (T).
Hence D is compact. Hence U and V are open sets such that D ⊆ V and U ∩ V = φ.

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.