On paranorm intuitionistic fuzzy I-convergent sequence spaces defined by compact operator

Article history: Received 29 December 2016 Received in revised form 19 March 2017 Accepted 23 March 2017 The purpose of this paper is to introduce paranorm intuitionistic fuzzy Iconvergent sequence spaces defined by compact operator and study the fuzzy topology on the said spaces. We defined more general type of paranorm intuitionistic fuzzy I-convergent sequence S(μ,ν) I = (T)(p) and S(0,ν) I = (T)(p) spaces by using compact operators. Moreover, we established some topological properties concerning with those spaces.


Introduction
*After the pioneering work of Zadeh (1965), a huge number of research papers have appeared on fuzzy theory and its applications as well as fuzzy analogues of the classical theories. Fuzzy set theory is a powerful hand set for modelling uncertainty and vagueness in various problems arising in field of science and engineering. It has a wide range of applications in various fields: population dynamics (Barros et al., 2000), chaos control (Fradkov and Evans, 2005), computer programming (Giles, 1980), nonlinear dynamical system (Hong and Sun, 2006), etc. Fuzzy topology is one of the most important and useful tools and it proves to be very useful for dealing with such situations where the use of classical theories breaks down. The concept of intuitionistic fuzzy normed space (Saadati and Park, 2006) and of intuitionistic fuzzy 2-normed space (Mursaleen and Lohani, 2009) is the latest developments in fuzzy topology. Khan et al. (2014Khan et al. ( , 2015Khan et al. ( , 2017 and Khan and Yasmeen (2016a,b,c) studied the intuitionistic fuzzy zweier I-convergent sequence spaces defined by paranorm, modulus function and Orlicz function.
The notion of statistical convergence is a very useful functional tool for studying the convergence problems of numerical problems/matrices (double sequences) through the concept of density. The notion of I-convergence, which is a generalization of statistical convergence (Fast, 1951;Esi and Özdemir, 2016;Mursaleen and Mohiuddine, 2009a;Hazarika and Mohiuddine, 2013;Alotaibi et al., 2014;Mohiuddine and Lohani, 2009; was introduced by Kostyrko et al. (2000) by using the idea of I of subsets of the set of natural numbers ℕ and further studied in (Nabiev et al., 2007). Recently, the notion of statistical convergence of double sequences x = ( ) has been defined and studied by Edely (2003), and for fuzzy numbers by Savas (2004), Mursaleen et al. (2016). Quite recently, Das et al. (2008) studied the notion of I and Iconvergence of double sequences in ℝ.
We recall some notations and basic definitions used in this paper. ∈ , ⊆ ⇒ ∈ .
The above properties are called additivity and hereditary respectively. An Ideal is called nontrivial if ≠ .
Definition 1.7: Let be a non-empty set. A fuzzy set in is characterized by its membership function: and ( ) is called as the degree of membership of element in fuzzy set A for each in . In this case ( , ) is called an intuitionistic fuzzy norm.
Definition 1.9: Let ( , , , * ,⋄) be IFNS then the sequence = ( ) is said to be convergent to continuous ∈ with respect to the intuitionistic fuzzy norm ( , ) if, for every > 0 and > 0, there exists 0 ∈ ℕ such that ( − , ) > 1 − and ( − , ) < for all ≥ 0 . In this case we write ( , ) − lim = . The concepts of statistical convergence and statistical Cauchy for double sequences in intuitionistic fuzzy normed spaces have been studied by . Definition 1.14: Let ⊆ 2ℕ be a non-trivial ideal and ( , , , * ,⋄) be an IFNS then the sequence = ( ) of elements of X is said to be I-convergent to ∈ with respect to the intuitionistic fuzzy norm ( , ) if for every > 0 and > 0, the set In this case L is called the I-limit of the sequence ( ) with respect to the intuitionistic fuzzy norm ( , ) and ( , ) − lim( ) = .
Definition 1.15: Let X and Y be two normed linear spaces and : ( ) → be a linear operator, where ⊂ Then the operator T is said to be bounded, if there exists a positive real k such that (Khan et al., 2015) ‖ ‖ ≤ ‖ ‖, ∈ ( ).
The set of all bounded linear operators B(X; Y) (Kreyszig, 1989) is a normed linear space normed by Definition 1.16: Let X and Y be two normed linear spaces. An operator ∶ → is said to be a compact linear operator (or completely continuous linear operator) if (Khan et al., 2015), 1. T is linear, 2. T maps every bounded sequence ( ) in X on to a sequence ( ( )) in Y which has a convergent subsequence.
The set of all compact linear operators ( , ) is a closed subspace of ( , ) and ( , ) is Banach space, if Y is a Banach space.
Definition 1.17: Let X is a vector space. A function ∶ → ℝ is said to be a paranorm if satisfies the following conditions: In this article we introduce the following sequence spaces: We also define an open ball with center x and radius r with respect to t as follows:

Proof:
We shall prove the result for ( , ) ( )( ). The proof for the other space will follow similarly.

Conclusion
Fuzzy set theory is a powerful hand set for modelling uncertainty and vagueness in various problems arising in field of science and engineering. It has a wide range of applications in various fields. The concept of intuitionistic fuzzy normed space and of intuitionistic fuzzy 2-normed space is the latest developments in fuzzy topology. In the present paper we studied a more general type of paranorm intuitionistic fuzzy I-convergent sequence spaces defined by compact operator and study the fuzzy topology on the said spaces. These results provide new tools to deal with the I-convergence in intuitionistic fuzzy problems of sequences occurring in many branches of science and engineering.