- Research
- Open access
- Published:
Some results of neutrosophic normed space VIA Tribonacci convergent sequence spaces
Journal of Inequalities and Applications volume 2022, Article number: 42 (2022)
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.
1 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 [4–6]. 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 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.
2 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.
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
For any sequence space E, the sequence space \(E_{T}\) defined by
is known as domain of the matrix T.
Definition 2.1
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
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:
and
Throughout this paper we use the lower triangular Tribonacci matrix \(T=(t_{ik})\), defined in [1] as follows:
Equivalently,
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
Definition 2.2
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
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
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 \(\jmath, s> 0\),
-
(i)
\(0\leq \mathsf{P}(\vartheta,\jmath )\leq 1\), \(0\leq \mathsf{Q}(y,\jmath )\leq 1\), \(0\leq \mathsf{R}(z,\jmath )\leq 1\), \(\jmath \in R^{+}\),
-
(ii)
\(\mathsf{P}(\vartheta,\jmath )+\mathsf{Q}(\vartheta,\jmath )+ \mathsf{R}(\vartheta,\jmath )\leq 3\) for \(\jmath \in R^{+}\),
-
(iii)
\(\mathsf{P}(\vartheta,\jmath )=1\) for \(\jmath >0\) iff \(\vartheta =0\),
-
(iv)
\(\mathsf{P}(\lambda \vartheta,\jmath )=\mathsf{P}(\vartheta, \frac{\jmath }{\vert \lambda \vert })\),
-
(v)
\(\mathsf{P}(\vartheta,\jmath ) \star \mathsf{P}(y,s)\leq \mathsf{P}( \vartheta +y,\jmath +s)\),
-
(vi)
\(\mathsf{P}(\vartheta,\star )\) is a continuous nondecreasing function,
-
(vii)
\(\lim_{\jmath \rightarrow \infty } \mathsf{P}(\vartheta, \jmath )=1\),
-
(viii)
\(\mathsf{Q}(y,\jmath )=0\) for \(\jmath >0\) iff \(\vartheta =0\),
-
(ix)
\(\mathsf{Q}(\lambda y,\rho )=\mathsf{Q}(y, \frac{\jmath }{\vert \lambda \vert })\),
-
(x)
\(\mathsf{Q}(y,\jmath ) \diamond \mathsf{Q}(z,\jmath )\geq \mathsf{Q}(y+z, \jmath +s)\),
-
(xi)
\(\mathsf{Q}(y, \diamond )\) is a continuous nonincreasing function,
-
(xii)
\(\lim_{\jmath \rightarrow \infty } \mathsf{Q}(\vartheta, \jmath )=0\),
-
(xiii)
\(\mathsf{R}(\vartheta,\jmath )=0\) for \(\jmath >0\) iff \(\vartheta =0\),
-
(xiv)
\(\mathsf{R}(\lambda \vartheta,\jmath )=\mathsf{R}(\vartheta, \frac{\jmath }{\vert \lambda \vert })\),
-
(xv)
\(\mathsf{R}(z,\jmath ) \diamond \mathsf{R}(\vartheta,s)\geq \mathsf{R}(z+\vartheta,\jmath +s)\),
-
(xvi)
\(\mathsf{R}(z,\diamond )\) is a continuous nonincreasing function,
-
(xvii)
\(\lim_{\jmath \rightarrow \infty } \mathsf{R}(z,\jmath )=0\),
-
(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 \(\jmath >0\)
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
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 \(\epsilon > 0\), \(\jmath > 0\)
for all \(i \geq N\), i.e.,
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 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 .
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 \(\mathcal{L}_{{\infty }(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\).
3 Main results
In this section, we introduce the following sequence spaces:
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:
and
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 \(i\geq n\)
(c) For every \(0<\epsilon <1\) and \(\jmath >0\), the set
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
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
and
Taking \(s=\max \{s_{1}, s_{2}, s_{3}\}\), we have the set
□
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:
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 \(\mathcalligra{C}_{1}\):
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 \((\vartheta ),(y)\in \mathcal{L}_{(\mathsf{P},\mathsf{Q},\mathsf{R})}(T)\),
Let \(q\in \mathcalligra{C}_{3}^{c}\). In this case,
and
In a similar way,
and
Therefore, we have
and
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 \(0<\epsilon <1\),
Then
Let \(y=(y_{i})\in \mathcalligra{B}^{c}_{\vartheta } (r,\epsilon )(T)\). Then the set
there exists \(r_{0}\in (0,r)\) such that
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
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
Therefore,
and
Therefore the set
□
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
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
Put
Then for each \(\epsilon _{0}>\epsilon \) there exist \(\epsilon _{4}, \epsilon _{5}, \epsilon _{6}\in (0,1)\) such that
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
and
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. □
4 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
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
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
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
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
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
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
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
and
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 \(\jmath >0\)
which implies
For \(i\in \mathcalligra{C}^{c}\), we have
For fixed \(k\in {\mathcalligra{C}^{c}}\), let
We show that \(A\subset \mathcalligra{C}\). Let \(i\in A\), we have
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
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
which is again a contradiction. \(\implies \mathsf{Q} (T_{i}(\vartheta )-\beta, \frac{\jmath }{2} )\geq \epsilon _{1}\).
Similarly,
If possible, suppose \(\mathsf{R} (T_{i}(\vartheta )-\beta, \frac{\jmath }{2} )< \epsilon _{1}\). Hence
which is again a contradiction
Therefore, for \(i\in A\), we have
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
but
which is a contradiction.
Now,
which is again a contradiction; and
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
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
which is equivalent to
For \(\{i\in \mathbb{N} \}\subseteq A^{c}\),
Therefore, for \(n\longrightarrow \infty\),
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
for all \(n\geq k\),
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
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
Since \(\vartheta ^{n}\longrightarrow \vartheta \) as \(n\longrightarrow \infty \), by Theorem 4.4,
Hence for \(i\in X\)
and
In particular, for \(k\in \mathbb{N}\), take \(\jmath =\frac{1}{k}\). Then
and
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
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 \(\jmath >0\),
which implies that
Conversely, let us choose \(N\in \mathcal{S}^{c}\). Then
We show that there exists a positive integer \(N=N(\vartheta,\epsilon,\jmath )\) such that
So, for \(\vartheta =(\vartheta _{k})\in \mathcal{L}_{(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T)\), define
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
In particular,
Therefore we have
which is a contradiction. Similarly,
In particular,
Therefore we have
Similarly, in the other way,
In particular,
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
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
Now, for \(k\in Y_{1}\cap Y_{2}\),
and
Take \(\epsilon =\frac{1}{p}\). Then
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
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
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 ℓ 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
Consider \(k\in Y_{1}\cap Y_{2}\). Then
and
Therefore the set
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 \(\jmath >0\),
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 \(\jmath >0\),
Consider the set
We show that \(Y_{1}\subset Y_{2}\). So for \(j\in Y_{1}\) we have
and
\(\implies i\in Y_{2}\) and hence \(Y_{1}\subset Y_{2}\).
Hence \(\vartheta =(\vartheta _{k})\in \mathcal{L}_{(\mathsf{P},\mathsf{Q}, \mathsf{R})}(T)\). □
5 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.
Availability of data and materials
Not applicable.
References
Yaying, T., Hazarika, B.: On sequence spaces defined by the domain of a regular Tribonacci matrix. Math. Slovaca 70(3), 697–706 (2000)
Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)
Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986)
Tripathy, B.C., Sen, M.: On fuzzy I-convergent difference sequence spaces. J. Intell. Fuzzy Syst. 25(3), 643–647 (2013)
Tripathy, B.C., Baruah, A.: New type of difference sequence spaces of fuzzy real numbers. Math. Model. Anal. 14(3), 391–397 (2009)
Khan, V.A., Fatima, H., Altaf, H., Danish Lohani, Q.M., et al.: Intuitionistic fuzzy I-convergent sequence spaces defined by compact operator. Cogent Math. Stat. 3(1), 1267940 (2016)
Park, J.H.: Intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 22, 1039–1046 (2004)
Saadati, R., Park, J.H.: On the intuitionistic fuzzy topological spaces. Chaos Solitons Fractals 27, 331–344 (2006)
Smarandache, F.: Neutrosophic set, a generalisation of the intuitionistic fuzzy sets. Int. J. Pure Appl. Math. 24, 287–297 (2005)
Smarandache, F.: Neutrosophic set is a generalization of intuitionistic fuzzy set, inconsistent intuitionistic fuzzy set (picture fuzzy set, ternary fuzzy set), Pythagorean fuzzy set, spherical fuzzy set, and q-rung orthopair fuzzy set, while neutrosophication is a generalization of regret theory, grey system theory, and three-ways decision (revisited). J. New Theory 29, 1–31 (2019)
Smarandache, F.: NeutroAlgebraic structures and AntiAlgebraic structures. In: Advances of Standard and Nonstandard Neutrosophic Theories, vol. 6, pp. 240–265. Pons Publishing House, Brussels (2019)
Khan, V.A., Esi, A., Ahmad, M., Khan, M.D.: Continuous and bounded linear operators in neutrosophic normed spaces. J. Intell. Fuzzy Syst. 40, 11063–11070 (2021)
Khan, V.A., Ahmad, M., Fatima, H., Khan, M.F.: On some results in intuitionistic fuzzy ideal convergence double sequence spaces. Adv. Differ. Equ. 1, 1–10 (2019)
Tan, B., Wen, Z.Y.: Some properties of the Tribonacci sequence. Eur. J. Comb. 28(6), 1703–1719 (2007)
Feinberg, M.: Fibonacci-Tribonacci. Fibonacci Q. 1(1), 71–74 (1963)
Wilansky, A.: Summability Through Functional Analysis. Elsevier, Amsterdam (2000)
Khan, V.A., Khan, M.D.: Some topological character of neutrosophic normed spaces. Neutrosophic Sets Syst. 47(9), 397–410 (2021)
Kirişci, M., Şimşek, N.: Neutrosophic normed spaces and statistical convergence. J. Anal. 28, 1059–1073 (2020)
Spikerman, W.R.: Binet’s formula for the Tribonacci sequence. Fibonacci Q. (1982)
Khan, V.A., Rahaman, S.K.A.: Intuitionistic fuzzy Tribonacci I-convergent sequence spaces. Math. Slovaca (2022, to appear)
Khan, V.A., Khan, M.D., Ahmad, M.: Some new type of lacunary statistically convergent sequences in neutrosophic normed space. Neutrosophic Sets Syst. 42, 1–14 (2021)
Khan, V.A., Khan, M.D., Ahmad, M.: Some results of neutrosophic normed spaces via Fibonacci matrix. UPB Sci. Bull., Ser. A 83(2), 1–12 (2021)
Talo, O., Yavuz, E.: Cesaro summability of sequences in intuitionistic fuzzy normed spaces and related Tauberian theorems. Soft Comput. 25(3), 2315–2323 (2021)
Yavuz, E.: On the logarithmic summability of sequences in intuitionistic fuzzy normed spaces. Fundam. J. Math. Appl. 3(2), 101–108 (2020)
Mukherjee, A., Das, R.: Neutrosophic bipolar vague soft set and its application to decision making problems. Neutrosophic Sets Syst. 32, 410–424 (2020)
Acknowledgements
The authors would like to thank the referees and the editor for their careful reading and their valuable comments.
Funding
This work is financially supported by Aligarh Muslim University, Aligarh, India.
Author information
Authors and Affiliations
Contributions
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.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-022-02775-3