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Abstract
The purpose of this paper is to introduce the intuitionistic fuzzy I-convergent sequence spaces and
defined by compact operator and study the fuzzy topology on the above said spaces.
Public Interest Statement
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 are the latest developments in fuzzy topology. Quite recently, V. A. Khan and Yasmeen studied the notion of I-convergence in Intuitionistic Fuzzy Zweier I-convergent Sequence Spaces.
The purpose of this paper is to introduce the intuitionistic fuzzy I-convergent sequence spaces defined by compact operator and study the fuzzy topology on the said spaces.
1. Introduction and preliminaries
After the pioneering work of Zadeh (Citation1965), a huge number of research papers have been 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, Bassanezi, & Tonelli, Citation2000), chaos control (Fradkov & Evans, Citation2005), computer programming (Giles, Citation1980), nonlinear dynamical system (Hong & Sun,Citation2006), 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 (Saddati & Park, Citation2006) and of intuitionistic fuzzy 2-normed space (Mursaleen & Lohani, Citation2009) are the latest developments in fuzzy topology. Recently Khan, Ebadullah, and Yasmeen (Citation2014), Khan and Yasmeen (Citation2015,Citation2014,Citation2016a,Citation2016b) 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 (Alotaibi, Hazarika, & Mohiuddine, Citation2014; Fast, Citation1951; Hazarika & Mohiuddine, Citation2013; Mohiuddine, Alotaibi, & Alsulami, Citation2012; Mohiuddine & Lohani, Citation2009; Mursaleen & Mohiuddine, Citation2009a,Citation2009b,Citation2010; Mursaleen, Mohiuddine, & Edely, Citation2010) was introduced by Kostyrko, Salat and Wilczynski (Citation2000) using the idea of I of subsets of the set of natural numbers and further studied in Nabiev, Pehlivan, and Gürdal (Citation2007). Recently, the notion of statistical convergence of double sequences
has been defined and studied by Mursaleen and Edely (Citation2003), and for fuzzy numbers by Savaş and Mursaleen (Citation2004), Mursaleen, Srivastava and Sharma (Citation2016). Quite recently, Das, Kostyrko, Wilczynski and Malik (Citation2008) studied the notion of I and
-convergence of double sequences in
.
We recall some notations and basic definitions used in this paper.
Definition 1.1
Let be a non-trivial ideal in
. Then a sequence
is said to be I-convergent to a number L if, for every
, the set
Definition 1.2
Let X be a non empty set. Then is said to be a filter on X if and only if
and for each
i.e. to each Ideal I there is a Filter corresponding to I,
Definition 1.3
Let be a non-trivial ideal in
. Then a sequence
is said to be I-Cauchy if, for each
, there exists a number
such that the set
Definition 1.4
(See, Khan, Ebadullah, & Rababah, Citation2016c) The five-tuple is said to be an intuitionistic fuzzy normed space (for short, IFNS) if X is a vector space,
is a continuous t-norm,
is a continuous t-conorm and
,
are fuzzy sets on
satisfying the following conditions for every
and
(a) | |||||
(b) | |||||
(c) |
| ||||
(d) |
| ||||
(e) | |||||
(f) |
| ||||
(g) |
| ||||
(h) |
| ||||
(i) |
| ||||
(j) |
| ||||
(k) | |||||
(l) |
| ||||
(m) |
|
Definition 1.5
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
.
Definition 1.6
Let be an IFNS. Then a sequence
is said to be a Cauchy sequence with respect to the intuitionistic fuzzy norm
if, for every
and
, there exists
such that
and
for all
.
Definition 1.7
Let K be the subset of natural numbers . Then the asymptotic density of K, denoted by
, is defined as
where the vertical bars denotes the cardinality of of the enclosed set.
A number sequence is said to be statistically convergent to a number
if, for each
, the set
has asymptotic density zero, i.e.
In this case we write .
Definition 1.8
A number sequence is said to be statistically Cauchy sequence if, for every
there exists a number
such that
The concepts of statistical convergence and statistical Cauchy for double sequences in intuitionistic fuzzy normed spaces have been studied by Mursaleen and Mohiuddine (Citation2010).
Definition 1.9
Let be a non trivial ideal and
be an IFNS. A sequence
of elements of X is said to be I-convergent to
with respect to the intuitionistic fuzzy norm
if for every
and
, the set
In this case L is called the I-limit of the sequence with respect to the intuitionistic fuzzy norm
and we write
.
Definition 1.10
(See Khan, Shafiq, & Lafuerza-Guillen, Citationin press) 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
The set of all bounded linear operators (Kreyszig, Citation1978) is a normed linear spaces normed by
and is a Banach space if Y is a Banach space.
Definition 1.11
(See Khan et al., Citation2014) 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
(i) | T is linear, | ||||
(ii) | T maps every bounded sequence |
Khan et al. (Citation2015) introduced the following sequence spaces:
In this article, we introduce the following sequence spaces:
We also define an open ball with centre x and radius r with respect to t as follows:
2. Main results
Theorem 2.1
and
are linear spaces.
Proof
We shall prove the result for . The proof for the other space will follow similarly. Let
and
be scalars. Then for a given
, we have
Define the set , so that
. It follows that
is a non empty set in
. We shall show that for each
Let . In this case
and
We have
and
This implies that Hence
is a linear space.
Theorem 2.2
Every open ball is an open set in
Proof
Let be an open ball with centre x and radius r with respect to t. That is
Let . Then
and
Since
there exists
such that
and
Putting
, we have
, there exists
such that
. For
, we have
such that
and
Putting
. Consider the ball
. We prove that
Let
, then
and
.
Therefore
and
Thus and hence
Remark 2.3
is an IFNS.
Define
Then is a topology on
.
Theorem 2.4
The topology on
is first countable.
Proof
is a local base at x, the topology
on
is first countable.
Theorem 2.5
and
are Housdorff spaces.
Proof
We prove the result for . Similarly the proof follows for
. 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
and
which is a contradiction. Hence is Housdorff.
Theorem 2.6
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
,
such that . Then
and
. Hence
and
as
.
Conversely, if for each and
as
, then for
there exists
such that
and
, for all
. It follows that
and
for all
Thus
for all
and hence
.
Theorem 2.7
A sequence is I-convergent if and only if for every
and
there exists a number
such that
Proof
Suppose that and let
and
. 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, define for each
Now we have to show that . Suppose that
. Then there exists
and
. Therefore we have
In particular Therefore we have
which is not possible. On the other hand
In particular Therefore we have
which is not possible. Hence .
implies
.
Acknowledgements
The authors would like to record their gratitude to the reviewer for his careful reading and making some useful corrections which improved the presentation of the paper.
Additional information
Funding
Notes on contributors
Vakeel A. Khan
Vakeel A. Khan received his MPhil and PhD degrees in Mathematics from Aligarh Muslim University, Aligarh, India. Currently he is an associate professor at Aligarh Muslim University, Aligarh, India. A vigorous researcher in the area of Sequence Spaces , he has published a number of research papers in reputed national and international journals, including Numerical Functional Analysis and Optimization (Taylor’s and Francis), Information Sciences (Elsevier), Applied Mathematics Letters Applied Mathematics (Elsevier), A Journal of Chinese Universities (Springer- Verlag, China).
Hira Fatima
Hira Fatima received her MSc degree from Aligarh Muslim University, and is cur rently a Ph.D. scholar at Aligarh Muslim University.
Henna Altaf
Henna Altaf received MSc from Aligarh Muslim University, and is currently a PhD scholar at Aligarh Muslim University.
Q.M. Danish Lohani
Q.M. Danish Lohani is working as an assistant professor in the Department of Mathematics, South Asian University, New Delhi, India.
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