Abstract
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.
1. Introduction and Background
The initial work on the statistical convergence of sequences was carried out by Fast [Citation1]. Schoenberg [Citation2] validated a number of elementary properties of statistical convergence and represented this notion as a method of summability.
The notion of -convergence initially originated in the study of Kostyrko et al. [Citation3]. Kostyrko et al. [Citation4] proposed and proved some new properties of -convergence and introduced extremal -limit points. Further, the study was extended by alát et al. [Citation5], Tripathy and Hazarika [Citation6] 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 [Citation7]. The sequence is known as the Fibonacci sequence [Citation8]. The Fibonacci numbers may be given by the following relation: for some integers
Some properties of Fibonacci numbers are given by The first application of Fibonacci sequence in the sequence spaces was given by Kara and Başarır [Citation9]. Then, Kara [Citation10] obtained the Fibonacci difference matrix via Fibonacci sequence for and studied some new sequence spaces in this connection. The definition of statistical convergence using the Fibonacci sequence was introduced in [Citation11]. Some works on spaces connected Fibonacci sequence can be found in [Citation12–15].
Kara [Citation10] defined the infinite matrix by where is the kth Fibonacci number for every .
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 [Citation16].
After the advent of fuzzy set theory by Zadeh [Citation17], fuzzy logic has found its applications in some subbranches of mathematics like topological spaces [Citation18–20], theory of functions [Citation21,Citation22] and approximation theory [Citation23].
Fuzzy set theory has found large-scale applications in many fields of science and engineering, such as computer programming [Citation24], non-linear operators [Citation25], population changes [Citation26], control of chaos [Citation27], and quantum physics [Citation28].
The intuitionistic fuzzy sets were focused on by Atanassov [Citation29], and it has been utilized in decision-making problems [Citation30], E-infinity theory of high-energy physics [Citation31]. 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 [Citation32]. Saadati and Park [Citation33] obtained an intuitionistic fuzzy normed linear space (IFNLS for short). Karakuş et al. [Citation34] studied statistical convergence in IFNLS and Mursaleen et al. [Citation35] 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 [Citation36–44].
Recently, Kirişci [Citation45] 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 [Citation3,Citation33,Citation41,Citation45].
2. Main Results
In this section, we give the Fibonacci -convergence in an IFNLS.
Definition 2.1
Let be an IFNLS and be a nontrivial ideal. A sequence in is said to be Fibonacci -convergence with regards to the intuitionistic fuzzy norm (IFN) (briefly, FC-IFN), if there is a number such that for every p>0 and , the set We write . The set of FC-IFN will be demonstrated by .
Example 2.1
Taking , is an admissible ideal in and so Fibonacci -convergence coincides with Fibonacci statistical convergence in an IFNLS.
Example 2.2
Let be a normed space and and . Any and p>0, consider Then, be an IFNLS. Define the Since as and , then . Consider for and for all p>0. When k becomes sufficiently large, the quantity becomes less than and similarly the quantity becomes greater than ε. So, for and p>0, .
Now, we investigate the sequence spaces in IFNLS as the sets of sequences whose -transforms are in the spaces , and . In addition, we put forward some inclusion theorems and obtain various topological and algebraic features from these results. Assume that a sequence and is an admissible ideal of a subset of . We identify
Theorem 2.1
Let be an IFNLS. The inclusion relation supplies.
Proof.
It can be observed that . We only denote that . Take . Then, there is so that . So, for all p>0 and , the set and for all p>0. As and , there exist such that and . As a result, for p>0 and , we obtain and Taking , we get the set Hence, implies
The converse of the inclusion relation does not supply. We establish the following example to prove our claim.
Example 2.3
Assume be a normed space such that . Suppose and for each . Identify the norm on as follows Then, is an IFNS. Define the sequence , it can be easily observed that and , but .
Example 2.4
Suppose be a normed space and be the IFN as determined in the above example. Examine the sequence . Then , but .
Lemma 2.1
Let be an IFNLS. For all and p>0, the following statements are equivalent:
(a) | |||||
(b) | and ; | ||||
(c) | |||||
(d) | and | ||||
(e) | and |
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 and p>0, we get and for every the set , it follows together with that . Hence, we have . In a similar way, for all and p>0, and , implies that Also, it is clear that (e) implies (b).
Theorem 2.2
Let be an IFNLS. If is Fibonacci -convergent with regards to the IFN , then is unique.
Proof.
Assume that there exist two distinct elements such that and Given , choose such that and . So, for any , we determine the following: and Since and , all the sets and belongs to . This implies that its complement is a non-empty set in . Let . Then we have or .
Case (i): Suppose that . Then we have and therefore Since is arbitrary, we get for all p>0, which yields
Case (ii): Suppose that . Then, we have , and therefore Since arbitrary , we get for all p>0. This occurs that . So, we conclude that is unique.
Theorem 2.3
Suppose be an IFNLS, and , be two sequences in X.
(a) | If , then | ||||
(b) | If and , then | ||||
(c) | If and α be any real number, then |
Proof.
(a) As , so for each and p>0 there exists such that and for all . The set is contained in , then since is admissible. This shows that .
(b) Let be given. Choose such that and . For any p>0, give and Since and , so for p>0, and belongs to . So, is a non-empty set in . We show that Let . Then, we get Now, we have and This shows that Since . Hence
(c) Case (i): When , for all and and . It gives us , and by part , we get
Case (ii): When . As , for each and (1) (1) To show the result it is enough to prove that for each and Let . Then, we get and Now, and Hence, we have But (Equation1(1) (1) ) shows that
Before the next theorem, we recall the following:
Let be an IFNLS. The open ball with center at x and radius p w.r.t. parameter of fuzziness is given as where p>0. A subset of is called -bounded if there exists p>0 and such that and for all .
Let denotes the space of all -bounded sequences whereas by we denote the space of all IF-bounded and -convergent sequences in . Now, we have the following theorem.
Theorem 2.4
Let be an IFNLS. Then is a closed linear space of .
Proof.
It is clear that is a subspace of . Next, we prove the closedness of . As provides, so we show that . Let . Then, , for each open ball centered at x and radius p w.r.t. parameter of fuzziness . Taking , p>0 and . Choosing such that and . As , there exists a subset such that and for all we get , , , . But for all , we get and It gives Since , it concludes that Therefore, we get .
Theorem 2.5
All open ball with center at x and radius p w.r.t. parameter of fuzziness , i.e. is an open set in
Proof.
Examine the open ball with center at x and radius p w.r.t. parameter of fuzziness , Then Assume Then, the set For there is a so that Taking means . Then, there exists so that . For , we get such that and . Take . Consider the open ball . We have to denote .
Assume , then So hence and hence Therefore the set So . As a result, we get We prove is an open set in
Theorem 2.6
The spaces and are Hausdorff spaces.
Proof.
It is clear that . We have to prove the result for only . Assume , such that . At that time, for all , we get Presume and . Afterwards, for all there are so that , . Again suppose and contemplate the open balls and 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 is a Hausdorff space.
Definition 2.2
Let be an IFNLS and be a nontrivial ideal. A sequence in is named Fibonacci -Cauchy with regards to the IFN or -Cauchy sequence if, for all and p>0, there exists a positive integer N so that
Theorem 2.7
Let be an IFNLS. Then a sequence in X Fibonacci -convergent with regards to the IFN iff it is Fibonacci -Cauchy with regards to the IFN .
Proof.
Necessity. Let in Fibonacci -convergent to ξ with regards to the IFN , i.e. . For a given , choose such that and . Since , we get (2) (2) for all p>0, which implies that Let . But for we have or . Taking to show the result it is sufficient to prove is contained in . Let , then we have or , for p>0. We have two possible cases.
Case (i): We consider . So, we have and then, As otherwise i.e. if , then we have which is impossible. Hence, .
Case (ii): If , we have and therefore As otherwise i.e. if , we get which is impossible. Hence, . Thus, in all cases, we get . By (Equation2(2) (2) ) . This shows that in X Fibonacci -Cauchy sequence.
Sufficiency. Let in Fibonacci -Cauchy with respect to the IFN but not Fibonacci -convergent with regards to the IFN . Then there exists r such that and equivalently, . Since and If and , respectively, we have , and so , which is a contradiction, as was Fibonacci -Cauchy with respect to the IFN . Hence, must be Fibonacci -convergent with regards to the IFN .
Definition 2.3
Assume that is an IFNLS. A sequence in X is called Fibonacci -convergent to with regards to IFN if there exists a subset of such that and . The element ξ is called the Fibonacci -limit of the sequence with regards to IFN and it is demonstrated by .
Theorem 2.8
Let be an IFNLS and be a nontrivial ideal. If then
Proof.
Suppose that . Then such that . For all and p>0 there exists an integer N>0 such that and for all Since Hence, for all and p>0. As a result, we conclude that
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 -Cauchy sequences have been introduced and the Fibonacci -completeness of an IFNLS has been established. Finally, the concept of Fibonacci -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).
Additional information
Notes on contributors
Ömer Kişi
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 Mathematics, Bartın, Turkey; his area of expertise includes summability theory, sequences spaces, and fuzzy sequence spaces through functional analysis.
Pradip Debnath
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. He has published over 50 papers in various journals of international repute. He is a reviewer for more than 20 international journals including Elsevier, Springer, IOS Press, Taylor and Francis and Wiley. He has successfully guided Ph.D. students in the areas of Fuzzy Logic, Soft Computing and Fixed-Point Theory. At present, he is working on a major Basic Science Research Project funded by UGC, Government of India. He received a gold medal in his postgraduation from Assam University, Silchar and qualified several national level examinations in mathematics.
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