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FUZZY INFORMATION AND ENGINEERING https://doi.org/10.1080/16168658.2022.2160226 Fibonacci Ideal Convergence on Intuitionistic Fuzzy Normed Linear Spaces a b Ömer Kişi and Pradip Debnath a b Deparment of Mathematics, Bartın University, Bartın, Turkey; Department of Applied Science and Humanities, Assam University, Silchar, Assam, India ABSTRACT ARTICLE HISTORY Received 10 July 2019 The main goal of this article is to present the notion of Fibonacci Revised 17 September 2022 I-convergence of sequences on intuitionistic fuzzy normed linear Accepted 11 December 2022 space. To accomplish this goal, we mainly investigate some funda- mental properties of the newly introduced notion. Then, we examine KEYWORDS the Fibonacci I-Cauchy sequences and Fibonacci I completeness in Fibonacci I-convergence; the construction of an intuitionistic fuzzy normed linear space. Some Fibonacci I-Cauchy intuitionistic fuzzy Fibonacci ideal convergent spaces have been sequence; intuitionistic fuzzy normed linear space established. Further, we prove on some algebraic and topological features of these convergent sequence spaces. 1991 MATHEMATICS SUBJECT CLASSIFICATIONS 11B39; 41A10; 41A25; 41A36; 40A30; 40G15 1. 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 intro- duced extremal I-limit points. Further, the study was extended by Salát 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: f = f − f n n+1 n−2 for some integers n ≥ 2. CONTACT Ömer Kişi okisi@bartin.edu.tr © 2022 The Authors. Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creative commons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 Ö. KİŞİ AND P. DEBNATH Some properties of Fibonacci numbers are given by f 1 + 5 n+1 lim = = α, (Golden ratio) n→∞ f 2 f = f − 1 (n ∈ N) , k n+2 k=0 converges, 2 n+1 f f − f = (−1) , n ≥ 1. (Cassini formula) n−1 n+1 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 ) for n ∈{1, 2, 3, ...}, and studied some new sequence spaces in this connec- tion. The definition of statistical convergence using the Fibonacci sequence was introduced in [11]. Some works on spaces connected Fibonacci sequence can be found in [12–15]. Kara [10] defined the infinite matrix F = (f ) by kn ⎪ k+1 − , n = k − 1 ⎨ f f = k kn , n = k k+1 0, 0 ≤ n < k − 1or n > k, where f 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 arrange- ment 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 construct- ing security coding. The Fibonacci Numbers are also applied in Pascal’s Triangle. Amazing applications can be examined in [16]. After the advent of fuzzy set theory by Zadeh [17], fuzzy logic has found its applications in some subbranches of mathematics like topological spaces [18–20], theory of functions [21,22] and approximation theory [23]. 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 intu- itionistic 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 remark- able 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. FUZZY INFORMATION AND ENGINEERING 3 [34] studied statistical convergence in IFNLS and Mursaleen et al. [35] studied the statisti- cal convergence of double sequences in IFNLS. Some works related to the convergence of sequences in a few IFNLS can be found in [36–44]. 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. Firstly, some basic definitions of this paper can be seen in [3,33,41,45]. 2. 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 ) inX is said to be FibonacciI-convergence with regards to the intuition- istic fuzzy norm (IFN) (φ, ψ) (briefly, FIC-IFN), if there is a number ξ ∈ X such that for every p >0and ε ∈ (0, 1), the set K (F) := k ∈ N : φ Fx − ξ, p ≤ 1 − ε or ψ Fx − ξ, p ≥ ε ∈ I. k k We write I − lim x = ξ.ThesetofFIC-IFN will be demonstrated by I(F) . FI k IFN (φ,ψ) Example 2.1: TakingI ={A ⊂ N : δ(A) = 0},I is an admissible ideal inN and so Fibonacci I-convergence coincides with Fibonacci statistical convergence in an IFNLS. Example 2.2: Let (X , ) be a normed space and k ∗ l = kl and k♦l = min{k + l,1}, k, l ∈ [0, 1]. Any x ∈ X and p > 0, consider φ (x, p) := , ψ (x, p) := . p + p + 2 2 2 2 2 Then, (X , φ, ψ, ∗,♦) be an IFNLS. Define the Fx = (f ) = (1, 2 ,3 ,5 , ...).Since f → k+1 k+1 ∞ as k →∞ and Fx = (1, 0, 0, ...), then Fx ∈ I(F) . Consider IFN A (ε, p) := k ∈ N : φ Fx , p ≤ 1 − ε or ψ Fx , p ≥ ε k k k for ε ∈ (0, 1) and for all p > 0. When k becomes sufficiently large, the quantity φ(Fx − ξ, p) becomes less than 1 − ε and similarly the quantity ψ(Fx − ξ, p) becomes greater than ε. So, for ε> 0and p > 0, A (F) ∈ I. Now, we investigate the sequence spaces in IFNLS as the sets of sequences whose F- I I I transforms are in the spaces c (φ, ψ), c (φ, ψ) and l (φ, ψ). In addition, we put forward some inclusion theorems and obtain various topological and algebraic features from these results. Assume that a sequence x = (x ) ∈ ω and I is an admissible ideal of a subset of N. We identify c F = x = (x ) ∈ ω : k ∈ N : φ Fx , p ≤ 1 − ε or ψ Fx , p ≥ ε ∈ I , k k k 0(φ,ψ ) x = (x ) ∈ ω : k ∈ N : φ Fx − ξ, p ≤ 1 − εor k k c F = , (φ,ψ ) ψ Fx − ξ, p ≥ ε for some ξ ∈ R ∈ I x = (x ) ∈ ω : ∃M > 0sothat k ∈ N : φ Fx , p ≤ 1 − M or k k l F = . ∞(φ,ψ ) ψ Fx , p ≥ M ∈ I k 4 Ö. KİŞİ AND P. DEBNATH I I Theorem 2.1: Let (X , φ, ψ, ∗,♦) be an IFNLS. The inclusion relation c (F) ⊂ c (F) ⊂ 0(φ,ψ) (φ,ψ) l (F) supplies. ∞(φ,ψ) I I I Proof: It canbeobservedthat c (F) ⊂ c (F). We only denote that c (F) ⊂ 0(φ,ψ) (φ,ψ) (φ,ψ) I I l (F).Take x = (x ) ∈ c (F). Then, there is ξ ∈ X so that I − lim x = ξ.So, k FI k ∞(φ,ψ) (φ,ψ) (φ,ψ) for all p >0and ε ∈ (0, 1), the set p p K = k ∈ N : φ Fx − ξ, > 1 − ε and ψ Fx − ξ, <ε ∈ F (I) . k k 2 2 p p φ(ξ, ) = s and ψ(ξ, ) = t for all p > 0. As s, t ∈ (0, 1) and ε ∈ (0, 1), there exist u , u ∈ 1 2 2 2 (0, 1) such that (1 − ε) ∗ s > 1 − u and ε♦t < u .Asaresult,for p >0and ε ∈ (0, 1),we 1 2 obtain p p φ Fx , p = φ Fx + ξ − ξ, p ≥ φ Fx − ξ, ∗ φ ξ, k k k 2 2 > (1 − ε) ∗ s > 1 − u and p p ψ Fx , p = ψ Fx + ξ − ξ, p ≤ ψ Fx − ξ, ♦ψ ξ, k k k 2 2 <ε♦t < u . Taking u = max{u , u },wegettheset 1 2 x = (x ) ∈ ω : ∃u > 0sothat k ∈ N : φ Fx , p > 1 − u and k k ψ Fx , p < u ∈ F (I) . I I I Hence, x = (x ) ∈ l (F) implies c (F) ⊂ l (F). ∞(φ,ψ) (φ,ψ) ∞(φ,ψ) The converse of the inclusion relation does not supply. We establish the following example to prove our claim. Example 2.3: Assume (X = R, ) be a normed space such that = sup |x |. Suppose k k k ∗ l = min{k, l} and k♦l = max{k, l} for each k, l ∈ [0, 1]. Identify the norm (φ, ψ) on X × (0, ∞) as follows φ (x, p) := , ψ (x, p) := . p + p + Then, (X , φ, ψ, ∗,♦) is an IFNS. Define the sequence Fx = (1, 0, 0, ...), it can be easily I I observed that (x ) ∈ c (F) and I − lim x = 1, but (x )/ ∈ c (F). k FI k k (φ,ψ) (φ,ψ) 0(φ,ψ) Example 2.4: Suppose (X = R, ) be a normed space and (φ, ψ) be the IFN as deter- k I mined in the above example. Examine the sequence (x ) = (−1) . Then (x ) ∈ l (F), k k ∞(φ,ψ) but (x )/ ∈ c (F). (φ,ψ) Lemma 2.1: Let (X , φ, ψ, ∗,♦) be an IFNLS. For all ε> 0 and p > 0, the following statements are equivalent: FUZZY INFORMATION AND ENGINEERING 5 (a) I − lim x = ξ; FI k (φ,ψ) (b) {k ∈ N : φ(Fx − ξ, p) ≤ 1 − ε}∈ I and {k ∈ N : ψ(Fx − ξ, p) ≥ ε}∈ I; k k (c) {k ∈ N : φ(Fx − ξ, p)> 1 − εandψ(Fx − ξ, p)<ε}∈ F (I), k k (d) {k ∈ N : φ(Fx − ξ, p)> 1 − ε}∈ F (I) and {k ∈ N : ψ(Fx − ξ, p)<ε}∈ F (I) and k k (e) I − lim φ(Fx − ξ, p) = 1 andI − lim ω(Fx − ξ, p) = 0. FI k FI k (φ,ψ) (φ,ψ) Proof: It is easy to demonstrate the equivalence of (a)–(d). Here, we just prove the equiva- lence of (b) and (e). Let (b) holds. For every ε> 0and p > 0, we get k ∈ N : φ Fx − ξ, p − 1 ≥ ε = k ∈ N : φ Fx − ξ, p ≥ 1 + ε ∪ k ∈ N : φ Fx − ξ, p ≤ 1 − ε k k and for everyε> 0 the set {k ∈ N : φ(Fx − ξ, p) ≥ 1 + ε}=∅∈ I, it follows together with (b) that {k ∈ N : |φ(Fx − ξ, p) − 1|≥ ε}∈ I. Hence, we haveI − lim φ(Fx − ξ, p) = k FI k (μ,v) 1. In a similar way, for all ε> 0and p > 0, k ∈ N : ψ Fx − ξ, p − 0 ≥ ε = k ∈ N : ψ Fx − ξ, p ≥ ε ∪ k ∈ N : ψ Fx − ξ, p ≤−ε k k and {k ∈ N : ψ(Fx − ξ, p) ≤−ε}=∅∈ I, implies that I − lim ψ(Fx − ξ, p) = 0. k FI k (μ,v) Also, it is clear that (e) implies (b). Theorem 2.2: Let (X , φ, ψ, ∗,♦) be an IFNLS. If (x ) is FibonacciI-convergent with regards to the IFN (φ, ω), thenI − lim x is unique. FI (μ,v) Proof: Assume that there exist two distinct elements ξ , ξ ∈ X such thatI − lim x = 1 2 FI k (φ,ψ) ξ and I − lim x = ξ .Given ε ∈ (0, 1), choose γ> 0suchthat (1 − γ) ∗ (1 − γ) > 1 FI k 2 (φ,ψ) 1 − ε and γ♦γ< ε.So,forany p > 0, we determine the following: K (γ , p) = k ∈ N : φ Fx − ξ , ≤ 1 − γ , φ,1 k 1 K (γ , p) = k ∈ N : ψ Fx − ξ , ≥ γ , ψ,1 k 1 K (γ , p) = k ∈ N : φ Fx − ξ , ≤ 1 − γ , φ,2 k 2 K (γ , p) = k ∈ N : ψ Fx − ξ , ≥ γ . ψ,2 2 and K (γ , p) = K (γ , p) ∪ K (γ , p) ∩ K (γ , p) ∪ K (γ , p) . φ,ψ φ,1 φ,2 ψ,1 ψ,2 Since I − lim x = ξ and I − lim x = ξ ,allthesets K (γ , p), K (γ , p), FI k 1 FI k 2 φ,1 ψ,1 (φ,ψ) (φ,ψ) K (γ , p), K (γ , p) and K (γ , p) belongs to I. This implies that its complement φ,2 ψ,2 φ,ψ c c c K (γ , p) is a non-empty set in F (I).Let m ∈ K (γ , p). Then we have m ∈ K (γ , p) ∩ φ,ψ φ,ψ φ,1 c c c K (γ , p) or m ∈ K (γ , p) ∩ K (γ , p). φ,2 ψ,1 ψ,2 6 Ö. KİŞİ AND P. DEBNATH c c Case (i): Suppose that m ∈ K (γ , p) ∩ K (γ , p). Then we have φ(Fx − ξ , )> 1 − r, m 1 φ,1 φ,2 2 φ(Fx − ξ , )> 1 − r and therefore m 2 p p φ (ξ − ξ , p) ≥ φ Fx − ξ , ∗ φ Fx − ξ , 1 2 m 1 m 2 2 2 > (1 − γ ) ∗ (1 − γ ) > 1 − ε. Since ε> 0 is arbitrary, we get φ(ξ − ξ , p) = 1 for all p > 0, which yields ξ = ξ . 1 2 1 2 c c Case (ii): Suppose that m ∈ K (γ , p) ∩ K (γ , p). Then, we have ψ(Fx − ξ , )<γ , m 1 ψ,1 ψ,2 2 ψ(Fx − ξ , )<γ and therefore m 2 p p ψ (ξ − ξ , p) <ψ Fx − ξ , ♦ψ Fx − ξ , 1 2 m 1 m 2 2 2 <γ♦γ< ε. Since arbitrary ε> 0, we get ψ(ξ − ξ , p) = 0 for all p > 0. This occurs that ξ = ξ .So, we 1 2 1 2 conclude that I − lim x is unique. FI (φ,ψ) Theorem 2.3: Suppose (X , φ, ψ, ∗,♦) be an IFNLS, and x = (x ),y = (y ) be two sequences k k in X. (a) If (φ, ψ) − lim x = ξ, thenI − lim Fx = ξ. k FI k (φ,ψ) (b) If I − lim Fx = ξ and I − lim Fy = ξ , then I − lim(Fx + Fy ) = FI k 1 FI k 2 FI k k (φ,ψ) (φ,ψ) (φ,ψ) (ξ + ξ ); 1 2 (c) IfI − lim Fx = ξ and α be any real number, thenI − lim αFx = αξ. FI k FI k (φ,ψ) (φ,ψ) Proof: (a) As (φ, ψ) − lim x = ξ,sofor each ε> 0and p > 0 there exists r ∈ N such that φ(x − ξ, p)> 1 − ε and ψ(x − ξ, p)<ε for all k ≥ r . The set k k A = {k ∈ N : φ (x − ξ, p) ≤ 1 − ε or ψ (x − ξ, p) ≥ ε} k k is contained in {1, 2, ... , r − 1}, then k ∈ N : φ Fx − ξ, p ≤ 1 − ε or ψ Fx − ξ, p ≥ ε ∈ I, k k since I is admissible. This shows that I − lim x = ξ. FI k (φ,ψ) (b) Let ε> 0 be given. Chooseγ> 0suchthat (1 − γ) ∗ (1 − γ) > 1 − ε and γ♦γ< ε. For any p > 0, give K (γ , p) = k ∈ N : φ Fx − ξ , ≤ 1 − γ , φ,1 1 K (γ , p) = k ∈ N : ψ Fx − ξ , ≥ γ , ψ,1 k 1 K (γ , p) = k ∈ N : φ Fy − ξ , ≤ 1 − γ , φ,2 k 2 K γ , p = k ∈ N : ψ Fy − ξ , ≥ γ ( ) ψ,2 k 2 and K (γ , p) = K (γ , p) ∪ K (γ , p) ∪ K (γ , p) ∪ K (γ , p) . φ,ψ φ,1 φ,2 ψ,1 ψ,2 FUZZY INFORMATION AND ENGINEERING 7 Since I − lim x = ξ and I − lim y = ξ ,sofor p > 0, K (γ , p), K (γ , p), FI k 1 FI k 2 φ,1 ψ,1 (φ,ψ) (φ,ψ) K (γ , p), K (γ , p) and K (γ , p) belongs to I.So, K (γ , p) is a non-empty set in φ,2 ψ,2 φ,ψ φ,ψ F (I). We show that k ∈ N : φ F x + y − ξ + ξ , p > 1 − ε and ( ) ( ) k k 1 2 K (γ , p) ⊂ . φ,ψ ψ F (x + y ) − (ξ + ξ ) , p <ε k k 1 2 Let m ∈ K (γ , p). Then, we get φ,ψ p p φ Fx − ξ , > 1 − γ , φ Fy − ξ , > 1 − γ m 1 m 2 2 2 p p ψ Fx − ξ , <γ , ψ Fy − ξ , <γ . m 1 m 2 2 2 Now, we have p p φ F (x + y ) − (ξ + ξ ) , p ≥ φ Fx − ξ, ∗ φ Fy − ξ , m m 1 2 m m 2 2 2 > (1 − γ ) ∗ (1 − γ ) > 1 − ε and p p ψ F (x + y ) − (ξ + ξ ) , p ≤ ψ Fx − ξ, ♦ψ Fy − ξ , m m 1 2 m m 2 2 2 <γ♦γ< ε. This shows that k ∈ N : φ F (x + y ) − (ξ + ξ ) , p > 1 − ε and 1 2 c k k K (γ , p) ⊂ . φ,ψ ψ F (x + y ) − (ξ + ξ ) , p <ε 1 2 k k Since K (γ , p) ∈ F (I). Hence I − lim(x + y ) = (ξ + ξ ). FI k k 1 2 φ,ψ (φ,ψ) (c) Case (i): When α = 0, for all ε> 0and p > 0, φ(F0x − 0ξ, p) = φ(0, p) = 1 > 1 − ε and ψ(F0x − 0ξ, p) = ω(0, p) = 0 <ε.Itgivesus (φ, ψ) − lim 0x = θ, and by part (i),we k k get I − lim F0x = θ. FI k (φ,ψ) Case (ii): When α = 0. As I − lim x = ξ,for each ε> 0and p > 0, FI k (φ,ψ) A = k ∈ N : φ Fx − ξ, p > 1 − ε and ψ Fx − ξ, p <ε ∈ F (I).(1) k k To show the result it is enough to prove that for each ε> 0and p > 0, A ⊂ k ∈ N : φ αFx − αξ, p > 1 − ε and ψ αFx − αξ, p <ε . k k Let m ∈ A. Then, we get φ(Fx − ξ, p)> 1 − ε and ψ(Fx − ξ, p)<ε. Now, m m p p φ αFx − αξ, p = φ Fx − ξ , ≥ φ Fx − ξ, p ∗ φ 0, − p m m m |α| |α| = φ Fx − ξ, p ∗ 1 = φ Fx − ξ, p > 1 − ε m m and p p ψ αFx − αξ, t = ψ Fx − ξ , ≤ ψ Fx − ξ, p ♦ψ 0, − p m m m |α| |α| = ψ Fx − ξ, p ♦0 = ψ Fx − ξ, p <ε m m 8 Ö. KİŞİ AND P. DEBNATH Hence, we have A ⊂ k ∈ N : φ αFx − αξ, p > 1 − ε and ψ αFx − αξ, p <ε . k k But (1) shows that I − lim αFx = αξ. FI k (φ,ψ) Before the next theorem, we recall the following: Let (X , φ, ψ, ∗,♦) be an IFNLS. The open ball B (p, ε)(F) with center at x and radius p w.r.t. parameter of fuzziness 0 <ε < 1isgivenas B (p, ε) F = y = (y ) ∈ X : φ F (x) − F (y) , p ≤ 1 − ε or ψ F (x) − F (y) , p ≥ ε ∈ I where p > 0. A subset A of X is called IF-bounded if there exists p >0and 0 <ε < 1such that φ(Fy, p)> 1 − ε and ψ(Fy, p)<ε for all y ∈ A. ∞ ∞ Let l (X ) denotes the space of all IF-bounded sequences whereas by I (X ) we (φ,ψ) (φ,ψ) denote the space of all IF-bounded andI-convergent sequences in (X , φ, ω, ∗,♦). Now, we have the following theorem. Theorem 2.4: Let (X , φ, ψ, ∗,♦) be an IFNLS. Then I (X ) is a closed linear space of FI (φ,ψ) l (X ). (φ,ψ) ∞ ∞ Proof: It is clear that I (X ) is a subspace of l (X ). Next, we prove the closedness (φ,ψ) (φ,ψ) ∞ ∞ ∞ ∞ ∞ of I (X ).As I (X ) ⊂ I (X ) provides, so we show that I (X ) ⊂ I (X ). (φ,ψ) (φ,ψ) (φ,ψ) (φ,ψ) (φ,ψ) ∞ I ∞ I Let x ∈ I (X ). Then,B (p, ε) ∩ I (X ) =∅, for each open ballB (p, ε) centered at x x x (φ,ψ) (φ,ψ) I ∞ and radius p w.r.t. parameter of fuzziness 0 <ε < 1. Taking y ∈ B (p, ε) ∩ I (X ), p > 0 (φ,ψ) and ε ∈ (0, 1). Choosing γ ∈ (0, 1) such that (1 − γ) ∗ (1 − γ) > 1 − ε and γ♦γ< ε.As I ∞ y ∈ B (p, ε) ∩ I (X ), there exists a subset K ⊂ N such that K ∈ F (I) and for all k ∈ (φ,ψ) p p p p K,weget φ(Fx − Fy , )> 1 − γ , ψ(Fx − Fy , )<γ , φ(Fy , )> 1 − γ , ψ(Fy , )<γ . k k k k k k 2 2 2 2 But for all k ∈ K,weget φ Fx , p = φ Fx − Fy + Fy , p k k k k p p ≥ φ Fx − Fy , ∗ φ Fy , > (1 − γ ) ∗ (1 − γ ) > 1 − ε k k k 2 2 and ψ Fx , p = ψ Fx − Fy + Fy , p k k k k p p ≤ ψ Fx − Fy , ♦ψ Fy , <γ♦γ< ε. k k k 2 2 It gives K ⊂ k ∈ N : φ Fx , p > 1 − ε and ψ Fx , p <ε . k k Since K ∈ F (I), it concludes that k ∈ N : φ Fx , p > 1 − ε and ψ Fx , p <ε ∈ F (I). k k Therefore, we get x ∈ I (X ). (φ,ψ) FUZZY INFORMATION AND ENGINEERING 9 Theorem 2.5: All open ball with center at x and radius p w.r.t. parameter of fuzziness 0 <ε < I I 1, i.e.B (p, ε)(F) is an open set in c (F). (φ,ψ) Proof: Examine the open ball B (p, ε)(F) with center at x and radius p w.r.t. parameter of fuzziness 0 <ε < 1, B (p, ε) F = y = (y ) ∈ X : φ F (x) − F (y) , p ≤ 1 − ε or ψ F (x) − F (y) , p ≥ ε ∈ I. Then B (p, ε) F = y = (y ) ∈ X : φ F (x) − F (y) , p > 1 − ε and ψ F (x) − F (y) , p <ε ∈ F (I) . I c Assume y = (y ) ∈ (B ) (p, ε)(F). Then, the set y = (y ) ∈ X : φ F (x) − F (y) , p > 1 − ε and ψ F (x) − F (y) , p <ε ∈ F (I) . For φ F (x) − F (y) , p > 1 − ε and ψ F (x) − F (y) , p <ε there is a p ∈ (0, p) so that φ F (x) − F (y) , p > 1 − ε and ψ F (x) − F (y) , p <ε. 0 0 Taking ε = φ(F(x) − F(y), p ) means ε > 1 − ε. Then, there exists u ∈ (0, 1) so that ε > 0 0 0 0 1 − u > 1 − ε.For ε > 1 − u,weget ε , ε ∈ (0, 1) such that ε ∗ ε > 1 − u and (1 − 0 1 2 0 1 ε )♦(1 − ε )< u.Take ε = max{ε , ε }. Consider the open ball B (p − p ,1 − ε )(F).We 0 0 3 1 2 0 3 I I have to denote B (p − p ,1 − ε )(F) ⊂ B (p, ε)(F). 0 3 y x I c Assume z = (z ) ∈ (B ) (p − p ,1 − ε )(F), then 0 3 k ∈ N : φ F (x ) − F (z ) , p − p >ε and ψ F (x ) − F (z ) , p − p < 1 − ε ∈ F (I) . k k 0 3 k k 0 3 So φ F (x) − F (z) , p ≥ φ F (x) − F (y) , p ∗ φ F (y) − F (z) , p − p 0 0 ≥ ε ∗ ε ≥ ε ∗ ε > 1 − u > 1 − ε, 0 3 0 1 hence k ∈ N : φ F (x ) − F (z ) , p > 1 − ε ∈ F (I) , k k and ψ F (x) − F (z) , p ≤ ψ F (x) − F (y) , p ♦ψ F (y) − F (z) , p − p 0 0 ≤ (1 − ε )♦ (1 − ε ) ≤ (1 − ε )♦ (1 − ε ) < u <ε, 0 3 0 2 hence k ∈ N : ψ F (x) − F (z) , p <ε ∈ F (I) . Therefore the set N : φ F x − F z , p > 1 − ε and ψ F x − F z , p <ε ∈ F I . k ∈ ( ) ( ) ( ) ( ) ( ) k k k k I c I c I c So z = (z ) ∈ (B ) (p, ε)(F). As a result, we get (B ) (p − p ,1 − ε )(F) ⊂ (B ) (p, ε)(F). k 0 3 y y y I I We prove B (p, ε)(F) is an open set in c (F). x (φ,ψ) 10 Ö. KİŞİ AND P. DEBNATH I I Theorem 2.6: The spaces c (F) and c (F) are Hausdorff spaces. (φ,ψ) 0(φ,ψ) I I I Proof: It is clear that c (F) ⊂ c (F).Wehavetoprovetheresultforonly c (F). 0(φ,ψ) (φ,ψ) (φ,ψ) Assume x = (x ), y = (y ) ∈ c (F) such that x = y. At that time, for all p ∈ N,weget k k (φ,ψ) 0 <φ Fx − Fy , p < 1, 0 <ψ Fx − Fy , p < 1. k k k k Presume ε = φ Fx − Fy , p , ε = ψ Fx − Fy , p , 1 k k 2 k k and ε = max{ε ,1 − ε }. Afterwards, for all ε >ε there are ε , ε ∈ (0, 1) so that ε ∗ ε ≥ 1 2 0 3 4 3 3 ε , (1 − ε )♦(1 − ε ) ≤ (1 − ε ). Again suppose ε = max{ε , ε } and contemplate the 0 4 4 0 5 3 4 p p I I open balls B (1 − ε , )(F) and B (1 − ε , )(F) centered at x and y respectively. Then, 5 5 x y 2 2 we demonstrate that p p I I B 1 − ε , F ∩ B 1 − ε , F =∅. 5 5 x y 2 2 If possible assume p p I I z = (z ) ∈ B 1 − ε , F ∩ B 1 − ε , F . k 5 5 x y 2 2 Then, we obtain ε = φ Fx − Fy , p 1 k k p p ≥ φ Fx − Fz , ∗ φ Fz − Fy , k k k k 2 2 >ε ∗ ε 5 5 ≥ ε ∗ ε 3 3 ≥ ε >ε , 0 1 ε = ψ Fx − Fy , p k k ≤ ψ Fx − Fz , p ♦ψ Fz − Fy , k k k k < (1 − ε ) ♦ (1 − ε ) 5 5 ≤ (1 − ε ) ♦ (1 − ε ) 4 4 < (1 − ε ) <ε . 0 2 From the above equations we obtain a contradiction. So, p p I I B 1 − ε , F ∩ B 1 − ε , F =∅. 5 5 x y 2 2 As a result, the space c (F) is a Hausdorff space. (φ,ψ) Definition 2.2: Let (X , φ, ψ, ∗,♦) be an IFNLS and I ⊂ P(N) be a nontrivial ideal. A sequence x = (x ) in X is named Fibonacci I-Cauchy with regards to the IFN (φ, ψ) or I -Cauchy sequence if, for all ε> 0and p > 0, there exists a positive integer N so that FI (φ,ψ) K (F) := k ∈ N : φ Fx − Fx , p ≤ 1 − ε or ψ Fx − Fx , p ≥ ε ∈ I. ε k N k N FUZZY INFORMATION AND ENGINEERING 11 Theorem 2.7: Let (X , φ, ψ, ∗,♦) be an IFNLS. Then a sequence x = (x ) in X Fibonacci I- convergent with regards to the IFN (φ, ψ) iff it is Fibonacci I-Cauchy with regards to the IFN (φ, ψ). Proof: Necessity.Let x = (x ) in X Fibonacci I-convergent to ξ with regards to the IFN (φ, ψ), i.e. I − lim x = ξ. For a given ε> 0, choose γ> 0suchthat (1 − γ) ∗ (1 − FI k (φ,ψ) γ) > 1 − ε and γ♦γ< ε.Since I − lim x = ξ,weget FI k (φ,ψ) K(F) = k ∈ N : φ Fx − ξ, p ≤ 1 − γ or ψ Fx − ξ, p ≥ γ ∈ I (2) k k for all p > 0, which implies that ∅ = K (F) = k ∈ N : φ Fx − ξ, p > 1 − γ or ψ Fx − ξ, p <γ ∈ F (I). k k Let m ∈ K (F). But for p > 0, we have φ(Fx − ξ, p)> 1 − γ or ψ(Fx − ξ, p)<γ . Taking m m B(F) = k ∈ N : φ Fx − Fx , p ≤ 1 − ε or ψ Fx − Fx , p ≥ ε ; p > 0, k m k m to show the result it is sufficient to prove B(F) is contained in K(F).Let k ∈ B(F), then p p we have φ(Fx − Fx , ) ≤ 1 − γ or ψ(Fx − Fx , ) ≥ γ,for p > 0. We have two possible k m k m 2 2 cases. Case (i): We consider φ(Fx − Fx , p) ≤ 1 − ε.So, we have φ(Fx − ξ, ) ≤ 1 − γ and k m k then, k ∈ K(F). As otherwise i.e. if φ(Fx − ξ, )> 1 − γ , then we have p p 1 − ε ≥ φ Fx − Fx , p ≥ φ Fx − ξ, ∗ φ Fx − ξ, k m k m 2 2 > (1 − γ ) ∗ (1 − γ ) > 1 − ε; which is impossible. Hence, B(F) ⊂ K(F). Case (ii):If ψ(Fx − Fx , p) ≥ ε,wehave ψ(Fx − ξ, )>γ and therefore k ∈ K(F).As k m k otherwise i.e. if ψ(Fx − ξ, )<γ,weget p p ε ≤ ψ Fx − Fx , p ≥ ψ Fx − ξ, ♦ψ Fx − ξ, k m k m 2 2 <γ♦γ< ε; which is impossible. Hence, B(F) ⊂ K(F). Thus, in all cases, we get B(F) ⊂ K(F).By(2) B(F) ∈ I.Thisshowsthat (x ) in X Fibonacci I-Cauchy sequence. Sufficiency.Let x = (x ) 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 A (F) := k ∈ N : φ Fx − Fx , p ≤ 1 − ε or ψ Fx − Fx , p ≥ ε ∈ I (ε,p) k r k r and p p B (F) = k ∈ N : φ Fx − ξ, > 1 − ε or ψ Fx − ξ, <ε ∈ I (ε,p) k k 2 2 equivalently, B (F) ∈ F (I).Since (ε,p) φ Fx − Fx , p ≥ 2φ Fx − ξ, > 1 − ε, k r k 2 12 Ö. KİŞİ AND P. DEBNATH and ψ Fx − Fx , p ≤ 2ψ Fx − ξ, <ε, k r k p (1−ε) p ε c If φ(Fx − ξ, )> and ψ(Fx − ξ, )< , respectively, we have A (F) ∈ I,and so k k 2 2 2 2 (ε,p) A (F) ∈ F (I), which is a contradiction, as x = (x ) was FibonacciI-Cauchy with respect (ε,p) k to the IFN (φ, ψ). Hence, x = (x ) must be Fibonacci I-convergent with regards to the IFN (φ, ψ). Definition 2.3: Assume that (X, φ, ψ, ∗,♦) is an IFNLS. A sequence x = (x ) in X is called Fibonacci I -convergent to ξ ∈ X with regards to IFN (φ, ψ) if there exists a subset M = {k , k , ... : k < k < ··· } 1 2 1 2 ofN such thatM ∈ F (I) and φ, ψ − lim x = ξ. The element ξ is called the Fibonacci n→∞ k ∗ ∗ I -limit of the sequence (x ) with regards to IFN (φ, ψ) and it is demonstrated byI − FI (φ,ψ) lim x = ξ. Theorem 2.8: Let (X , φ, ψ, ∗,♦) be an IFNLS and I ⊂ P(N) be a nontrivial ideal. If I − FI (φ,ψ) lim x = ξ thenI − lim x = ξ. k FI k (φ,ψ) Proof: Suppose that I − lim x = ξ. Then M ={k , k , ... : k < k < ···} ∈ F (I) k 1 2 1 2 FI (φ,ψ) x = ξ. For allε> 0and p > 0 there exists an integer N >0such such that (φ, ψ) − lim n→∞ k that φ(x − ξ, p)> 1 − ε and ψ(x − ξ, p)<ε for all n > N.Since k k n n n ∈ N : φ x − ξ, p > 1 − ε or ψ x − ξ, p <ε ∈ I. k k n n Hence, k ∈ N : φ Fx − ξ, p > 1 − ε or ψ Fx − ξ, p <ε k k ⊆ H ∪ {k < k < ··· < ··· < k } ∈ I. 1 2 N−1 for all ε> 0and p > 0. As a result, we conclude that I − lim x = ξ. FI k (φ,ψ) 3. 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). FUZZY INFORMATION AND ENGINEERING 13 Notes on contributors Ömer Kişi received the BSc degree from Cumhuriyet University, Sivas, Turkey in 2007; MS from Cumhuriyet University, Sivas, Turkey in 2010 and PhD degree in mathematical analysis worked with Fatih Nuray from Afyon Kocatepe University, Afyonkarahisar, Turkey in 2014. Started his career as an research assistant at Cumhuriyet University in 2009; then assistant professor at Bartın University, Bartın, Turkey in 2014. Since 2019, he has been a associate professor at the Department of Mathe- matics, Bartın, Turkey; his area of expertise includes summability theory, sequences spaces, and fuzzy sequence spaces through functional analysis. Pradip Debnath is an assistant professor (in mathematics) in the Department of Applied Science and Humanities of Assam University, Silchar (a central university), India. His research interests include Fuzzy Logic, Fuzzy Graphs, Fuzzy Decision Making, Soft Computing and Fixed-Point Theory. 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Fuzzy Information and Engineering – Taylor & Francis
Published: Jul 3, 2022
Keywords: Fibonacci -convergence; Fibonacci -Cauchy sequence; intuitionistic fuzzy normed linear space; 11B39; 41A10; 41A25; 41A36; 40A30; 40G15
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