Fibonacci Ideal Convergence on Intuitionistic Fuzzy Normed Linear Spaces

The main goal of this article is to present the notion of Fibonacci I-convergence of sequences on intuitionistic fuzzy normed linear space. To accomplish this goal, we mainly investigate some fundamental properties of the newly introduced notion. Then, we examine the Fibonacci I-Cauchy sequences and Fibonacci I completeness in the construction of an intuitionistic fuzzy normed linear space. Some intuitionistic fuzzy Fibonacci ideal convergent spaces have been established. Further, we prove on some algebraic and topological features of these convergent sequence spaces.


Introduction and Background
The initial work on the statistical convergence of sequences was carried out by Fast [1]. Schoenberg [2] validated a number of elementary properties of statistical convergence and represented this notion as a method of summability.
The notion of I-convergence initially originated in the study of Kostyrko et al. [3]. Kostyrko et al. [4] proposed and proved some new properties of I-convergence and introduced extremal I-limit points. Further, the study was extended bySalát et al. [5], Tripathy and Hazarika [6] and many others.
Fibonacci sequences were published by Fibonacci in the book 'Liber Abaci'. The Fibonacci sequences were earlier stated as Virahanka numbers by Indian mathematics [7]. The sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .) is known as the Fibonacci sequence [8]. The Fibonacci numbers may be given by the following relation: The first application of Fibonacci sequence in the sequence spaces was given by Kara and Başarır [9]. Then, Kara [10] obtained the Fibonacci difference matrix F via Fibonacci sequence (f n ) for n ∈ {1, 2, 3, . . .}, and studied some new sequence spaces in this connection. The definition of statistical convergence using the Fibonacci sequence was introduced in [11]. Some works on spaces connected Fibonacci sequence can be found in [12][13][14][15]. Kara [10] defined the infinite matrix F = ( f kn ) by where f k is the kth Fibonacci number for every k ∈ N. The Fibonacci sequence of numbers and the associated 'Golden Ratio' are observed in nature. We examine that various natural things follow the Fibonacci sequence. It appears in biological settings such as branching in trees, the flowering of an artichoke and the arrangement of a pine cone's bracts etc. Nowadays Fibonacci numbers play a very significant role in coding theory. Fibonacci numbers in different forms are extensively applied in constructing security coding. The Fibonacci Numbers are also applied in Pascal's Triangle. Amazing applications can be examined in [16].
Fuzzy set theory has found large-scale applications in many fields of science and engineering, such as computer programming [24], non-linear operators [25], population changes [26], control of chaos [27], and quantum physics [28].
The intuitionistic fuzzy sets were focused on by Atanassov [29], and it has been utilized in decision-making problems [30], E-infinity theory of high-energy physics [31]. In intuitionistic fuzzy sets (IFSs) the 'degree of non-belongingness' is not independent but it is dependent on the 'degree of belongingness'. Fuzzy sets (FSs) can be thought as a remarkable case of an IFS where the 'degree of non-belongingness' of an element is absolutely equal to '1-degree of belongingness'. Uncertainty is based on the belongingness degree in IFSs. An intuitionistic fuzzy metric space was considered by Park [32]. Saadati and Park [33] obtained an intuitionistic fuzzy normed linear space (IFNLS for short). Karakuş et al. [34] studied statistical convergence in IFNLS and Mursaleen et al. [35] studied the statistical convergence of double sequences in IFNLS. Some works related to the convergence of sequences in a few IFNLS can be found in [36][37][38][39][40][41][42][43][44].
Recently, Kirişci [45] studied the Fibonacci statistical convergence on IFNLS. He defined the Fibonacci statistically Cauchy sequences in an IFNLS and investigated the Fibonacci statistical completeness of the space.

Main Results
In this section, we give the Fibonacci I-convergence in an IFNLS.

Definition 2.1:
Let (X , φ, ψ, * , ♦) be an IFNLS and I ⊂ P(N) be a nontrivial ideal. A sequence x = (x k ) in X is said to be Fibonacci I-convergence with regards to the intuitionistic fuzzy norm (IFN) (φ, ψ) (briefly, FIC-IFN), if there is a number ξ ∈ X such that for every p > 0 and ε ∈ (0, 1), the set

Example 2.4:
Suppose (X = R, . ) be a normed space and (φ, ψ) be the IFN as determined in the above example. Examine the sequence (

Lemma 2.1:
Let (X , φ, ψ, * , ♦) be an IFNLS. For all ε > 0 and p > 0, the following statements are equivalent: Proof: It is easy to demonstrate the equivalence of (a)-(d). Here, we just prove the equivalence of (b) and (e). Let (b) holds. For every ε > 0 and p > 0, we get In a similar way, for all ε > 0 and p > 0, So, for any p > 0, we determine the following: .

Proof: It is clear that
It gives Since K ∈ F(I), it concludes that k ∈ N : φ Fx k , p > 1 − ε and ψ Fx k , p < ε ∈ F(I).

Proof:
Examine the open ball B I x (p, ε)( F) with center at x and radius p w.r.t. parameter of fuzziness 0 < ε < 1, Therefore the set So z = (z k ) ∈ (B I y ) c (p, ε)( F). As a result, we get

Proof: It is clear that c
We have to prove the result for only c I (φ,ψ) ( F). Assume x = (x k ), y = (y k ) ∈ c I (φ,ψ) ( F) such that x = y. At that time, for all p ∈ N, we get Presume centered at x and y respectively. Then, we demonstrate that If possible assume Then, we obtain From the above equations we obtain a contradiction. So, As a result, the space c I (φ,ψ) ( F) is a Hausdorff space.

Definition 2.2:
Let (X , φ, ψ, * , ♦) be an IFNLS and I ⊂ P(N) be a nontrivial ideal. A sequence x = (x k ) in X is named Fibonacci I-Cauchy with regards to the IFN (φ, ψ) or I FI (φ,ψ) -Cauchy sequence if, for all ε > 0 and p > 0, there exists a positive integer N so that for all p > 0, which implies that which is impossible. Hence, B( F) ⊂ K( F). Thus, in all cases, we get B( F) ⊂ K( F). By (2) B( F) ∈ I. This shows that (x k ) in X Fibonacci I-Cauchy sequence.
Sufficiency. Let x = (x k ) in X Fibonacci I-Cauchy with respect to the IFN (φ, ψ) but not Fibonacci I-convergent with regards to the IFN (φ, ψ). Then there exists r such that 2 and ψ( Fx k − ξ , p 2 ) < ε 2 , respectively, we have A c (ε,p) ( F) ∈ I, and so A (ε,p) ( F) ∈ F(I), which is a contradiction, as x = (x k ) was Fibonacci I-Cauchy with respect to the IFN (φ, ψ). Hence, x = (x k ) must be Fibonacci I-convergent with regards to the IFN (φ, ψ).

Conclusion
In the current study, using the concept of Fibonacci sequence, we have introduced the new notion of Fibonacci ideal convergent sequence in IFNLS. We have shown that these sequences follow many properties similar to that of classical real-valued sequences. Further, Fibonacci I-Cauchy sequences have been introduced and the Fibonacci I-completeness of an IFNLS has been established. Finally, the concept of Fibonacci I * -convergence, which is stronger than Fibonacci ideal convergence, has been investigated. Several intuitionistic fuzzy Fibonacci ideal convergent spaces have been established and significant features of these spaces have been obtained.

Disclosure statement
No potential conflict of interest was reported by the author(s).