1. Introduction
Intuitionistic fuzzy set (IFNS) is one of the generalizations of fuzzy set theory [
1]. Out of several higher-order fuzzy sets, IFNS, first introduced by Atanassov [
2], have been found to be compatible to deal with vagueness. The conception of IFS can be viewed as an appropriate and alternative approach in cases where available information is not sufficient to define the impreciseness by the conventional fuzzy set theory. In fuzzy sets, only the degree of acceptance is considered. However, IFNS is characterized by a membership function and a non-membership function such that the sum of both values is less than one. Presently, intuitionistic fuzzy sets are being studied and used in different fields of science and engineering, e.g., population dynamics, chaos control, computer programming, nonlinear dynamical systems, fuzzy physics, fuzzy topology, etc. The concept of an intuitionistic fuzzy metric space was introduced by Park [
3]. Furthermore, Saadati and Park [
4] gave the notion of an intuitionistic fuzzy normed space.
In recent years, fuzzy topology has proven to be a very useful tool to deal with such situations where the use of classical theories breaks down. The most popular application of fuzzy topology in quantum particle physics aries in the string and
-theory of El-Nashie [
5] who presented the relation of fuzzy Kähler interpolation of
to the recent work on cosmo-topology. In [
6], E. Savas introduced
λ-double sequence spaces of fuzzy real numbers defined by Orlicz function.
The term “
statistical convergence” was first defined by Fast [
7] and Schoenberg [
8] independently, which is a generalization of the concept of ordinary convergence. Statistical convergence appears in many fields such as in the theory of Fourier analysis, ergodic theory and number theory. Later on it was further investigated from the sequence space point of view and linked with summability theory by Fridy [
9], S̆alát [
10], Cakalli [
11], Di Maio and Kocinac [
12] and many others.
Definition 1. Let K be a subset of , the set of natural numbers. Then the asymptotic density of K denoted by (see, [13] is defined aswhere the vertical bars denote the cardinality of the enclosed set. Definition 2. A number sequence is said to be statistically convergent to the number L if for each , the set has asymptotic density zero, i.e., In this case we write (see, [7,9]). Note that every convergent sequence is statistically convergent to the same limit, but the converse need not be true.
On the other hand, Mursaleen [
14] introduced the concept of
-statistical convergence as a generalization of the statistical convergence and studied its relation to statistical convergence, Cesàro summability and strong
-summability. In [
15] we used ideals to introduce the concept of
-statistical convergence and study their some properties.
Furthermore, in [
16,
17] a different direction was given to the study of these important summability methods where the notions of statistical convergence of order
α and
λ-statistical convergence of order
α were introduced and studied.
In [
18], P. Kostyrko et al. introduced and investigated
-convergence of sequences in a metric space which is an interesting generalization of statistical convergence and studied some properties of such convergence. Subsequently, more investigations and more applications of ideals were introduced and studied in different directions, for instance, see [
15,
19,
20,
21,
22,
23,
24,
25].
Also, in [
26] Mohiuddine and Lohani introduced the notion of the generalized statistical convergence in intuitionistic fuzzy normed spaces. Some works related to the convergence of sequences in several normed linear spaces in a fuzzy setting can be found in [
26,
27].
Quite recently, -double statistical convergence has been established as a better study than double statistical convergence. It is very interesting that some results on sequences, series and summability can be proved by replacing the double statistical convergence by -double statistical convergence. Also, it should be noted that the results of -double statistical convergence in an intuitionistic fuzzy normed linear space happen to be stronger than those proved for λ-double statistical convergence in an intuitionistic fuzzy normed linear space.
In this paper, we intend to use ideals to introduce the concept of -double statistical convergence of order α with respect to the intuitionistic fuzzy normed space , and study some of its consequences.
It should be noted that throughout the paper, will denote the set of all natural numbers.
The following definitions and notions will be needed in the sequel.
Definition 3. A triangular norm (t-norm) is a continuous mapping such that is an abelian monoid with unit one and if and for all [28]. Definition 4. A binary operation is said to be a continuous t-conorm if it satisfies the following conditions [28]:- (i)
◊ is associate and commutative,
- (ii)
◊ is continuous,
- (iii)
for all
- (iv)
whenever and for each
For example, we can give and for all
Using the continuous
t-norm and
t-conorm, Saadati and Park [
4] has recently introduced the concept of intuitionistic fuzzy normed space as follows.
Definition 5. 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 v are fuzzy sets on satisfying the following conditions for every , and [4]:- (a)
- (b)
- (c)
if and only if
- (d)
for each
- (e)
- (f)
is continuous,
- (g)
and
- (h)
- (i)
if and only if
- (j)
for each
- (k)
- (l)
is continuous,
- (m)
and
In this case is called an intuitionistic fuzzy norm. As a standard example, we can give the following:
Let
be a normed space, and let
and
for all
For all
and every
consider
Then observe that
is an intuitionistic fuzzy normed space.
We also recall that the concept of double convergence in an intuitionistic fuzzy normed space is studied in [
29].
Definition 6. Let be an IFNS. Then, a sequence is said to be convergent to L with respect to the intuitionistic fuzzy norm if, for every and there exists such that and for all and . It is denoted by or as [29]. 2. -Double Statistical Convergence on IFNS
In this section we deal with the relation between these two new methods as also the relation between -double statistical convergence and -double statistical convergence in an intuitionistic fuzzy normed space . Before proceeding further, we should recall some notation on the -double statistical convergence and ideal convergence.
The idea of
λ-statistical convergence of single and double sequences of fuzzy numbers has been studied by Savas [
6,
30] respectively .
Statistical convergence of double sequences
has been defined and studied by Karakus, S., Demirci, K. and Duman, O. [
31]; and for fuzzy numbers by Savas and Mursaleen [
32].
Now, it would be helpful to give some definitions.
Let
be a two-dimensional set of positive integers and let
be the numbers of
in
K such that
and
. Then the two-dimensional analogue of natural density can be defined as follows [
33].
The
lower asymptotic density of the set
is defined as
In case the sequence
has a limit then we say that
K has a
double natural density and is defined as
Definition 7. A real double sequence is said to be statistically convergent to the number ℓ if for each , the set [33]has double natural density zero. In this case we write . We define the concept of double λ-density:
Let and be two non-decreasing sequences of positive real numbers both of which tends to ∞ as m and n approach ∞, respectively. Also let and . The collection of such sequence will be denoted by
Let
. The number
where
and
and
, is said to be the
λ-
density of
K, provided the limit exists.
The family
of subsets a nonempty set
Y is said to be an ideal in
Y if (i) Ø
; (ii)
imply
; (iii)
,
imply
, while an admissible ideal
of
Y further satisfies
for each
. If
is an ideal in
Y then the collection
forms a filter in
Y which is called the filter associated with
Let
be a nontrivial ideal in
. Then a sequence
in
X is said to be
-convergent to
, if for each
the set
belongs to
(see [
18]).
Throughout will stand for a proper admissible ideal in .
Definition 8. A sequence is said to be - double statistically convergent of order α to L or -convergent to L, where , if for each and [24]or equivalently if for each where and In this case we write The class of all - double statistically convergent of order α sequences will be denoted by simply
Remark 1. For : A is a finite}, -convergence coincides with double statistical convergence of order α. For an arbitrary ideal and for it coincides with -double statistical convergence [24]. When and it becomes only double statistical convergence [33]. Definition 9. Let be a nontrivial admissible ideal in Let be an IFNS. Then, a sequence is said to be -double statistically convergent of order α to , where , with respect to the intuitionistic fuzzy normed space , if for every , and every and , In this case we write .
Remark 2. For -convergence coincides with statistical convergence of order α, with respect to the intuitionistic fuzzy normed space . For an arbitrary ideal and for it coincides with -double statistical convergence, with respect to the intuitionistic fuzzy normed space (see, [27]). When and it becomes only double statistical convergence with respect to the intuitionistic fuzzy normed space ,34]. We write the generalized double de la Valée-Poussin mean by
A sequence
is said to be
-summable to a number
, if
i.e., for any
Throughout this paper we shall denote by and by .
We are now ready to obtain our main results.
Definition 10. Let be an IFNS. We say that a sequence is said to be -summable of order α to with respect to the intuitionistic fuzzy normed space if for any and In this case we write
Definition 11. Let be an IFNS. A sequence is said to be -double statistically convergent of order α or -convergent to with respect to the intuitionistic fuzzy normed space if for every and In this case we write or
Remark 3. For -convergence with respect to the intuitionistic fuzzy normed space again coincides with λ-double statistical convergence of order α, with respect to the intuitionistic fuzzy normed space . For an arbitrary ideal and for it coincides with -double statistical convergence with respect to the intuitionistic fuzzy normed space . Finally, for and it becomes λ-double statistical convergence with respect to the intuitionistic fuzzy normed space [29]. Also note that taking we get Definition 9 from Definition 11. We denote by and the collections of all -convergent of order α and -convergent of order α sequences respectively.
We now have
Theorem 1. Let be an IFNS. Let Then .
Proof. By hypothesis, for every
and
let
We have
Then observe that
which implies
Since , we immediately see that , this completed the proof of the theorem. ☐
Theorem 2. Let be an IFNS. if
Proof. For given
and every
we have
If
then from definition
is finite. For every
Since is admissible, the set on the right-hand side belongs to and this completed the proof of the theorem. ☐
It is easy to check that both and are linear subspaces of the space of real sequences. Next we present a topological characterization of these spaces. As both the proofs are similar, we give the detailed proof for the class only.
Theorem 3. Let be an IFNS such that and If X is a Banach space then is a closed subset of , where stands for the space of all double bounded sequences of intuitionistic fuzzy norm .
Proof. We first assume that , , is a convergent sequence and it converges to We need to show that Suppose that for all . Take a sequence of strictly decreasing positive numbers converging to zero. We can find an such that for all and
Write
belongs to
and
belongs to
Since
and Ø
, we can choose
. Then
Since
and
is infinite, we can actually choose the above
so that
. Hence there must exist a
for which we have simultaneously,
or
and
or
For a given
choose
such that
and
Then it follows that
and
Hence we have
and similarly we have
This implies that
is a Cauchy sequence in
X and let
as
We shall prove that
. For any
and
, choose
such that
,
,
or
. Now since
and similarly
it follows that
for any given
Hence we have
This completes the proof of the theorem. ☐