In 1986, Atanassov introduced the concept of intuitionistic fuzzy set theory which is based on the extensions of definitions of fuzzy set theory given by Zadeh. This theory provides a variable model to elaborate uncertainty and vagueness involved in decision making problems. In this chapter, we concentrate our study on the ideal convergence of sequence spaces with respect to intuitionistic fuzzy norm and discussed their topological and algebraic properties.
- intuitionistic fuzzy normed spaces
- Orlicz function
- compact operator
In recent years, the fuzzy theory has emerged as the most active area of research in many branches of mathematics, computer and engineering . After the excellent work of Zadeh , a large number of research work have been done on fuzzy set theory and its applications as well as fuzzy analogues of the classical theories. It has a wide number of applications in various fields such as population dynamics , nonlinear dynamical system , chaos control , computer programming , etc. In 2006, Saadati and Park  introduced the concept of intuitionistic fuzzy normed spaces after that the concept of statistical convergence in intuitionistic fuzzy normed space was studied for single sequence in . The study of intuitionistic fuzzy topological spaces , intuitionistic fuzzy 2-normed space  and intuitionistic fuzzy Zweier ideal convergent sequence spaces  are the latest developments in fuzzy topology.
First, let us recall some notions, basic definitions and concepts which are used in sequel.
Definition 1.1. (See Ref. ). The five-tuple is said to be an intuitionistic fuzzy normed space (for short, IFNS) if is a vector space, is a continuous t-norm, is a continuous t-conorm, and and are fuzzy sets on satisfying the following conditions for every and
(c) if and only if ,
(d) for each
(f) is continuous,
(i) if and only if ,
(j) for each
(l) is continuous,
In this case is called an intuitionistic fuzzy norm.
Example 1.1. Let be a normed space. Denote and for all and let and be fuzzy sets on defined as follows:
for all . Then is an intuitionistic fuzzy normed space.
Definition 1.2. Let be an IFNS. Then a sequence is said to be convergent to with respect to the intuitionistic fuzzy norm if, for every and , there exists such that and for all . In this case we write -.
In 1951, the concept of statistical convergence was introduced by Steinhaus  and Fast  in their papers “Sur la convergence ordinaire et la convergence asymptotique” and “Sur la convergence statistique,” respectively. Later on, in 1959, Schoenberg  reintroduced this concept. It is a very useful functional tool for studying the convergence of numerical problems through the concept of density. The concept of ideal convergence, which is a generalization of statistical convergence, was introduced by Kostyrko et al.  and it is based on the ideal as a subsets of the set of positive integers and further studied in [16, 17, 18, 19, 20].
Let be a non-empty set then a family is said to be an ideal in if , is additive, i.e., for all and is hereditary, i.e., for all . A non empty family of sets is said to be a filter on if for all implies and for all with implies . An ideal is said to be nontrivial if , this non trivial ideal is said to be admissible if and is said to be maximal if there cannot exist any nontrivial ideal containing as a subset. For each ideal , there is a filter called as filter associate with ideal , that is (see ),
2. IF-ideal convergent sequence spaces using compact operator
This section consists of some double sequence spaces with respect to intuitionistic fuzzy normed space and study the fuzzy topology on the said spaces. First we recall some basic definitions on compact operator.
Definition 2.1. (See ). Let and be two normed linear spaces and be a linear operator, where Then, the operator is said to be bounded, if there exists a positive real such that
The set of all bounded linear operators  is a normed linear spaces normed by
and is a Banach space if is a Banach space.
Definition 2.2. (See ). Let and be two normed linear spaces. An operator is said to be a compact linear operator (or completely continuous linear operator), if
(i) is linear,
(ii) maps every bounded sequence in on to a sequence in which has a convergent subsequence.
The set of all compact linear operators is a closed subspace of and is Banach space, if is a Banach space.
In 2015, Khan et al.  introduced the following sequence spaces:
Motivated by this, we introduce the following sequence spaces with the help of compact operator in intuitionistic fuzzy normed spaces:
Here, we also define an open ball with center and radius with respect to as follows:
Now, we are ready to state and prove our main results. This theorem is based on the linearity of new define sequence spaces which is stated as follows.
Theorem 2.1. The sequence spaces and are linear spaces.
Proof. Let and be scalars. Then for a given , we have the sets:
Now, we define the set , so that . It shows that is a non-empty set in . We shall show that for each
Let , in this case
Thus, we have
This implies that
Therefore, the sequence space is a linear space.
Similarly, we can proof for the other space. □
In the following theorems, we discussed the convergence problem in the said sequence spaces. For this, firstly we have to discuss about the topology of this space. Define
Then is a topology on .
Theorem 2.2. Let is an IFNS and is a topology on . Then a sequence if and only if and as .
Proof. Fix . Suppose . Then for , there exists such that for all . So, we have
such that . Then and . Hence and as .
Conversely, if for each and as , then for there exists , such that and , for all . It shows that and for all Therefore for all and hence .
There are some facts that arise in connection with the convergence of sequences in these spaces. Let us proceed to the next theorem on Ideal convergence of sequences in these new define spaces.
Theorem 2.3. A sequence is -convergent if and only if for every and there exists a number such that
Proof. Suppose that and let . For a given , choose such that and Then for each ,
which implies that
Conversely, let us choose . Then
Now, we want to show that there exists a number such that
For this, we define for each
So, we have to show that . Let us suppose that , then there exists and . Therefore, we have
In particular Therefore, we have
which is not possible. On the other hand
In particular So, we have
which is not possible. Hence . which implies .□
3. IF-ideal convergent sequence spaces using Orlicz function
In this section, we have discussed the ideal convergence of sequences in Intuitionistic fuzzy -convergent sequence spaces defined by compact operator and Orlicz function. We shall now define the concept of Orlicz function, which is basic definition in our work.
Definition 3.1. An Orlicz function is a function which is continuous, non-decreasing and convex with for and as . If the convexity of Orlicz function is replaced by , then this function is called modulus function.
Remark 3.1. If is an Orlicz function, then for all with
In 2009, Mohiuddine and Lohani  introduced the concept of statistical convergence in intuitionistic fuzzy normed spaces in their paper published in Chaos, Solitons and Fractals. This motivated us to introduced some sequence spaces defined by compact operator and Orlicz function which are as follows:
We also define an open ball with center and radius with respect to as follows:
We shall now consider some theorems of these sequence spaces and invite the reader to verify the linearity of these sequence spaces.
Theorem 3.1. Every open ball is an open set in .
Proof. Let be an open ball with center and radius with respect to . That is
Let , then and
Since there exists such that and .
Putting , so we have , there exists such that . For , we have such that and Putting . Now we consider a ball . And we prove that
Let , then and . Therefore, we have
Thus and hence, we get
Remark 3.2. is an IFNS.
Then is a topology on .
In the above result we can easily verify that the open sets in these spaces are open ball in the same spaces. This theorem itself will have various applications in our future work.
Theorem 3.2. The topology on is first countable.
Proof. is a local base at , the topology on is first countable.□
Theorem 3.3. and are Hausdorff spaces.
Proof. Let such that . Then and
Putting , and For each there exists and such that and .
Putting and consider the open balls and . Then clearly . For if there exists , then
which is a contradiction. Hence, is Hausdorff. Similarly the proof follows for .□
The concept of defining intuitionistic fuzzy ideal convergent sequence spaces as it generalized the fuzzy set theory and give quite useful and interesting applications in many areas of mathematics and engineering. This chapter give brief introduction to intuitionistic fuzzy normed spaces with some basic definitions of convergence applicable on it. We have also summarized different types of sequence spaces with the help of ideal, Orlicz function and compact operator. At the end of this chapter some theorems and remarks based on these new defined sequence spaces are discussed for proper understanding.
Conflict of interest
The authors declare that they have no competing interests.