$I_\lambda$-Double statistical convergence of order $α$ in topological groups
We introduce а new notion, namely, $I_\lambda$-double statistical convergence of order \alpha in topological groups. Consequently, we investigate some inclusion relations between $I$ -double statistical and $I_\lambda$ -double statistical convergence of order $\alpha$ in topological groups. We als...
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Institute of Mathematics, NAS of Ukraine
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507807874285568 |
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| author | Savaş, E. Саваш, Є. |
| author_facet | Savaş, E. Саваш, Є. |
| author_sort | Savaş, E. |
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| description | We introduce а new notion, namely, $I_\lambda$-double statistical convergence of order \alpha in topological groups. Consequently,
we investigate some inclusion relations between $I$ -double statistical and $I_\lambda$ -double statistical convergence of order $\alpha$ in
topological groups. We also study many other related concepts. |
| first_indexed | 2026-03-24T02:15:11Z |
| format | Article |
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UDC 519.21
Ekrem Savaş (Istanbul Commerce Univ., Turkey)
\bfscrI \bfitlambda -DOUBLE STATISTICAL CONVERGENCE
OF ORDER \bfitalpha IN TOPOLOGICAL GROUPS
\bfscrI \bfitlambda -DOUBLE STATISTICAL CONVERGENCE
OF ORDER \bfitalpha IN TOPOLOGICAL GROUPS
We introduce а new notion, namely, \scrI \lambda -double statistical convergence of order \alpha in topological groups. Consequently,
we investigate some inclusion relations between \scrI -double statistical and \scrI \lambda -double statistical convergence of order \alpha in
topological groups. We also study many other related concepts.
Введено нове поняття, а саме поняття \scrI \lambda -подвiйної статистичної збiжностi порядку \alpha в топологiчних групах.
Таким чином, вивчено деякi вiдношеняя включення мiж \scrI -подвiйною статистичною збiжнiстю та \scrI \lambda –подвiйною
статистичною збiжнiстю порядку \alpha в топологiчних групах. Також вивчено багато iнших подiбних понять.
1. Introduction. The notion of statistical convergence, which is an extension of the idea of usual
convergence, was introduced by Fast [9] and Schoenberg [30] and its topological consequences were
studied first by Fridy [10], S̆alát [19] (also later by Maddox [15]). Recently Di Maio and KoЎcinac
[16] introduced the concept of statistical convergence in topological spaces and statistical Cauchy
condition in uniform spaces and established the topological nature of this convergence. This notion
was used by Kolk in [11] to extended the statistical convergence to normed spaces. Also, in [2]
Cakalli extended this notion to topological Hausdorff groups.
Later on Cakalli and Savaş studied the statistical convergence of double sequences to topological
groups (see [3]). Also Savaş [28], introduced Lacunary statistical convergence of double sequences
in topological groups. Quite recently, Savaş [29] introduced \scrI \lambda -statistical convergence of double
sequences in topological groups where more references can be found.
The notion of \scrI -convergence (\scrI denotes the ideal of subsets of, the set of positive integers),
which is a generalization of statistical convergence, was introduced by Kostyrko et al. [12] and Das
et al. [4] continued with this study and extended these ideas from single to double sequences. Further
it was studied by many other authors (see [1, 5, 8, 13, 21 – 25]).
In [7], we used ideals to introduce the concepts of \scrI -statistical convergence and \scrI -lacunary
statistical convergence of order \alpha which naturally extend the notions of mentioned convergence
in [8].
The concept of statistical convergence depends on the density of subsets of the set N of natural
numbers. If K \subset \BbbN , then K(m,n) denotes the cardinality of the set K \cap [m,n]. The upper and
lower natural density of the subset K is defined by
d(K) = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
K(1, n)
n
and d(K) = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{i}\mathrm{n}\mathrm{f}
K(1, n)
n
.
If d(K) = d(K), then we say that the natural density of K exists and it is denoted simply by d(K).
Clearly d(K) = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
K(1, n)
n
.
A sequence (xk) of real numbers is said to be statistically convergent to L if for arbitrary \epsilon > 0,
the set K(\epsilon ) = \{ k \in \BbbN : | xk - L| \geq \epsilon \} has natural density zero.
c\bigcirc EKREM SAVAŞ, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 9 1251
1252 EKREM SAVAŞ
Throughout the paper, \BbbN will denote the set of all natural numbers.
By X, we will denote an Abelian topological Hausdorff group, written additively, which satisfies
the first axiom of countability. In [2], a sequence x = (xk) in X is called to be statistically convergent
to an element L of X if for each neighbourhood U of 0,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
n
\bigm| \bigm| \{ k \leq n : xk - L /\in U\}
\bigm| \bigm| = 0,
where the vertical bars indicate the number of elements in the enclosed set. The set of all statistically
convergent sequences in X is denoted by st(X).
2. Preliminaries. We now quote the following definitions and notions which will be needed in
the sequel.
Definition 1. A family \scrI \subset 2\BbbN is said to be an ideal of \BbbN if the following conditions hold:
(a) A,B \in \scrI implies A \cup B \in \scrI ,
(b) A \in \scrI , B \subset A implies B \in \scrI .
Definition 2. A nonempty family F \subset 2\BbbN is said to be an filter of \BbbN if the following conditions
hold:
(a) \phi /\in F,
(b) A,B \in F implies A \cap B \in F,
(c) A \in F, A \subset B implies B \in F.
If \scrI is a proper ideal of \BbbN ( i.e., \BbbN /\in I ), then the family of sets F (\scrI ) = \{ M \subset \BbbN : \exists A \in \scrI :
M = \BbbN \setminus A\} is a filter of \BbbN . It is called the filter associated with the ideal.
Definition 3. A proper ideal \scrI is said to be admissible if \{ n\} \in \scrI for each n \in \BbbN .
Definition 4 (see [12]). Let I \subset 2\BbbN be a proper admissible ideal in \BbbN . Then the sequence
(xk) of elements of real numbers is said to be \scrI -convergent to L \in \BbbR , if for each \epsilon > 0 the set
A(\epsilon ) = \{ k \in \BbbN : | xk - L| \geq \epsilon \} \in \scrI .
By the convergence of a double sequence we mean the convergence in Pringsheims sense [18].
A double sequence x = (xkl) of real numbers is said to be convergent in the Pringsheim’s sense
or P -convergent if for each \epsilon > 0 there exists N \in \BbbN such that | xkl - L| < \epsilon whenever k, l \geq N
and L is called Pringsheim limit (denoted by P - \mathrm{l}\mathrm{i}\mathrm{m}x = L ).
Let K \subseteq \BbbN \times \BbbN be a two dimensional set of positive integers and Km,n be the numbers of (i, j)
in K such that i \leq n and j \leq m. Then the lower asymptotic density of K is defined as
P - \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
m,n
Km,n
mn
= \delta 2(K).
In the case when the sequence
\biggl(
Km,n
mn
\biggr) \infty ,\infty
m,n=1,1
has a limit then we say that K has a natural density
and is defined as
P - \mathrm{l}\mathrm{i}\mathrm{m}
m,n
Km,n
mn
= \delta 2(K).
For example, let K =
\bigl\{
(i2, j2) : i, j \in \BbbN
\bigr\}
, where \BbbN is the set of natural numbers. Then
\delta 2(K) = P - \mathrm{l}\mathrm{i}\mathrm{m}
m,n
Km,n
mn
\leq P - \mathrm{l}\mathrm{i}\mathrm{m}
m,n
\surd
m
\surd
n
mn
= 0
(i.e., the set K has double natural density zero).
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 9
\scrI \lambda -DOUBLE STATISTICAL CONVERGENCE OF ORDER \alpha IN TOPOLOGICAL GROUPS 1253
The concept of statistical convergence of double sequences was first introduced by Mursaleen
and Edeley (see [17]) who have given main definition for double sequences and proved some related
results supporting by some interesting example.
Throughout \scrI 2 will stand for a proper admissible ideal in \BbbN \times \BbbN .
A double sequence x = (xkl) of real number is said to be convergent to the number L with
respect to the ideal \scrI 2, if for each \epsilon > 0
A(\epsilon ) =
\bigl\{
(k, l) \in \BbbN \times \BbbN : | xkl - L| \geq \epsilon
\bigr\}
\in \scrI 2.
In this case we write \scrI - \mathrm{l}\mathrm{i}\mathrm{m}kl xkl = L
Note that, if we take Id =
\bigl\{
A \subset \BbbN \times \BbbN : \delta 2(A) = 0
\bigr\}
, then \scrI d-convergence becomes statistical
convergence for double sequences. We now define the concept of double \lambda -density:
Let \lambda = (\lambda m) and \mu = (\mu n) be two nondecreasing sequences of positive real numbers both of
which tends to \infty as m and n approach \infty , respectively. Also let \lambda m+1 \leq \lambda m + 1, \lambda 1 = 0 and
\mu n+1 \leq \mu n + 1, \mu 1 = 0. The collection of such sequence (\lambda , \mu ) will be denoted by \Delta .
Let K \subseteq \BbbN \times \BbbN . The number
\delta \lambda (K) = \mathrm{l}\mathrm{i}\mathrm{m}
mn
1
\lambda mn
\bigm| \bigm| \{ k \in In, l \in Jm : (k, l) \in K\}
\bigm| \bigm| ,
where Im = [m - \lambda m+1,m] and Jn = [n - \mu n+1, n] and \lambda mn = \lambda m\mu n, is said to be the \lambda -density
of K, provided the limit exists.
Throughout this paper we shall denote (k \in Im, l \in Jn) by (k, l) \in Imn.
In this paper, we introduce the concept of \scrI \lambda -double statistical convergence of order \alpha in
topological groups and investigate some of its consequences.
3. \bfscrI \bfitlambda -double convergence of order \bfitalpha . The order of double statistical convergence of a
sequence of real numbers was given by Savaş in [27] as follows: Let \lambda = (\lambda mn) \in \Delta and 0 < \alpha \leq 1
be given. The sequences x = (xkl) is said to be \lambda -double statistically convergent of order \alpha if there
is a complex number L such that\biggl\{
n \in \BbbN :
1
\lambda \alpha
mn
\bigm| \bigm| \{ (k, l) \in Imn : | xkl - L| \geq \epsilon \}
\bigm| \bigm| = 0
\biggr\}
,
where \lambda \alpha
mn denote the \alpha th power (\lambda mn)
\alpha of \lambda mn.
In this section, we introduce and study \scrI \lambda -double statistical convergence of order \alpha for sequence
in topological groups.
We now have the following definition.
Definition 5. A sequence x = (xkl) of points in a topological group X is called to be statisti-
cally convergent of order \alpha to L of X if for each neighbourhood U of 0,
\mathrm{l}\mathrm{i}\mathrm{m}
m,n\rightarrow \infty
1
(mn)\alpha
\bigm| \bigm| \{ k \leq m and l \leq m : xkl - L /\in U\}
\bigm| \bigm| = 0,
where the vertical bars indicate the number of elements in the enclosed set.
The set of all double statistically convergent of order \alpha sequences in X is denoted by S\alpha (X)2.
Also we define \lambda -double statistical convergence of order \alpha in topological groups as follows:
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 9
1254 EKREM SAVAŞ
Definition 6. A sequence x = (xkl) of points in a topological group X, is said to be S\alpha
\lambda (X)2 -
convergent of order \alpha to L (or \lambda -double statistically convergent of order \alpha to L) if for each
neighborhood U of 0,
\mathrm{l}\mathrm{i}\mathrm{m}
mn
1
\lambda \alpha
mn
\bigm| \bigm| (k, l) \in Imn : xkl - L /\in U\}
\bigm| \bigm| = 0.
In this case, we define
S\alpha
\lambda (X)2 =
\biggl\{
x = (xkl) : for some L, S\alpha
\lambda (X)2 - \mathrm{l}\mathrm{i}\mathrm{m}
k,l\rightarrow \infty
xkl = L
\biggr\}
.
If we take \lambda mn = mn, S\alpha
\lambda (X)2 reduce to S\alpha (X)2.
We now introduce our main definitions. Throughout \scrI 2 will stand for a proper admissible ideal
in \BbbN \times \BbbN .
Definition 7. A double sequence x = (xkl) of points in a topological group X, is said to be
\scrI 2- double statistically convergent of order \alpha to L or S\alpha (\scrI 2)-convergent to L, where 0 < \alpha \leq 1, if
for each \epsilon > 0 and \delta > 0\biggl\{
(m,n) \in \BbbN \times \BbbN :
1
(mn)\alpha
\bigm| \bigm| \{ k \leq m and l \leq m : xkl - L /\in U\}
\bigm| \bigm| \geq \delta
\biggr\}
\in \scrI 2.
In this case we write xkl \rightarrow L(S\alpha (\scrI 2)). The set of all \scrI 2-double statistically convergent sequences
will be denoted by simply S\alpha (\scrI 2)(X).
Remark 1. For \scrI 2 = \scrI 2fin = \{ A \subset \BbbN \times \BbbN , A is a finite}, then S(\scrI 2)-convergence coincides
with double statistical convergence in a topological group X which is studied by Cakalli and Savaş [2].
Definition 8. A sequences x = (xkl) of points in a topological group X, is said to be \scrI \lambda
2 -
double statistically convergent of order \alpha to L or S\alpha
\lambda (\scrI 2)-convergent of order \alpha to L, where
0 < \alpha \leq 1, if for any \delta > 0 and for each neighbourhood U of 0,\biggl\{
(m,n) \in \BbbN \times \BbbN :
1
\lambda \alpha
mn
\bigm| \bigm| \{ (k, l) \in Imn : xkl - L /\in U\}
\bigm| \bigm| \geq \delta
\biggr\}
\in \scrI 2.
In this case, we write
S\alpha
\lambda (\scrI 2)(X) =
\biggl\{
x = xkl : for some L, S\alpha
\lambda (\scrI 2) - \mathrm{l}\mathrm{i}\mathrm{m}
k,l\rightarrow \infty
xkl = L
\biggr\}
and, if we take \alpha = 1, we have
S\lambda (\scrI 2)(X) =
\biggl\{
x = (xkl) : for some L, S\lambda (\scrI 2) - \mathrm{l}\mathrm{i}\mathrm{m}
k,l\rightarrow \infty
xkl = L
\biggr\}
.
Remark 2. For \scrI 2 = \scrI 2fin, \scrI 2-double statistical convergence of order \alpha becomes double
statistical convergence of order \alpha in topological groups which has not been study till now. Finally,
for \scrI 2 = \scrI 2fin, \lambda mn = mn, and \alpha = 1 it becomes statistical convergence which is studied in [3].
It is obvious that every \scrI \lambda
2 - double statistically convergent sequence has only one limit, that is,
if a sequence is \scrI \lambda
2 double statistically convergent to L1 and L2, then L1 = L2.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 9
\scrI \lambda -DOUBLE STATISTICAL CONVERGENCE OF ORDER \alpha IN TOPOLOGICAL GROUPS 1255
4. Inclusion theorems. In this section we prove some inclusion theorems.
Theorem 1. Let 0 < \alpha \leq \beta \leq 1. Then S\alpha
\lambda (\scrI 2)(X) \subset S\beta
\lambda (\scrI 2)(X).
Proof. Let 0 < \alpha \leq \beta \leq 1. Then
| \{ (k, l) \in Imn : xkl - L /\in U\} |
\lambda \beta
mn
\leq | \{ (k, l) \in Imn : xkl - L /\in U\} |
\lambda \alpha
mn
and so for any \delta > 0 and any neighbourhood U of 0\biggl\{
(m,n) \in N \times N :
| \{ (k, l) \in Imn : xkl - L /\in U\} |
\lambda \beta
mn
\geq \delta
\biggr\}
\subset
\subset
\biggl\{
(m,n) \in \BbbN \times \BbbN :
| \{ (k, l) \in Imn : xkl - L /\in U\} |
\lambda \alpha
mn
\geq \delta
\biggr\}
.
Hence if the set on the right-hand side belongs to the ideal \scrI then obviously the set on the left
hand-side also belongs to \scrI . This shows that S\alpha
\lambda (\scrI 2)(X) \subset S\beta
\lambda (\scrI 2)(X).
Theorem 1 is proved.
Corollary 1. If a sequence is \scrI \lambda
2 -double statistically convergent of order \alpha to L for some
0 < \alpha \leq 1, then it is \scrI \lambda
2 - double statistically convergent to L, i.e., S\alpha
\lambda (\scrI 2) \subset S\lambda (\scrI 2).
Similarly we can show that the following theorem.
Theorem 2. Let 0 < \alpha \leq \beta \leq 1. Then
(i) S\alpha (\scrI 2) \subset S\beta (\scrI 2),
(ii) In particular S\alpha (\scrI 2) \subset S(\scrI 2).
Theorem 3. S(\scrI 2)(X) \subset S\alpha
\lambda (\scrI 2)(X) if \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
\lambda \alpha
mn
mn
\alpha
> 0.
Proof. Let us take any neighbourhood U of 0. Then
1
(mn)\alpha
\bigm| \bigm| \{ k \leq m, l \leq n : xkl - L /\in U\}
\bigm| \bigm| \geq 1
(mn)\alpha
\bigm| \bigm| \{ (k, l) \in Imn : xkl - L /\in U\}
\bigm| \bigm| \geq
\geq \lambda \alpha
mn
(mn)\alpha
1
\lambda \alpha
mn
\bigm| \bigm| \{ (k, l) \in Imn : xkl - L /\in U\}
\bigm| \bigm| .
If \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
mn\rightarrow \infty
\lambda \alpha
mn
mn
\alpha
= a, then from definition
\biggl\{
(m,n) \in N \times N :
\lambda \alpha
mn
mn
\alpha
<
a
2
\biggr\}
is finite. For \delta > 0,
and any neighbourhood U of 0,\biggl\{
(m,n) \in \BbbN \times \BbbN :
1
\lambda \alpha
mn
\bigm| \bigm| \bigl\{ (k, l) \in Imn : xkl - L /\in U
\bigr\} \bigm| \bigm| \geq \delta
\biggr\}
\subset
\subset
\biggl\{
(k, l) \in Imn :
1
(mn)\alpha
\bigm| \bigm| \bigl\{ k \leq m, l \leq n : xkl - L /\in U
\bigr\} \bigm| \bigm| \geq a
2
\delta
\biggr\}
\cup
\cup
\biggl\{
(m,n) \in \BbbN \times \BbbN :
\lambda \alpha
mn
(mn)\alpha
<
a
2
\biggr\}
.
The set on the right-hand side belongs to \scrI 2 and this completed the proof.
Theorem 4. Let \lambda = (\lambda mn) and \mu = (\mu mn) be two sequences in \bigtriangleup such that \lambda mn \leq \mu mn for
all (m,n) \in \BbbN \times \BbbN and let \alpha and \beta be fixed real numbers such that 0 < \alpha \leq \beta \leq 1.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 9
1256 EKREM SAVAŞ
(i) If
\mathrm{l}\mathrm{i}\mathrm{m}
mn\rightarrow \infty
\mathrm{i}\mathrm{n}\mathrm{f}
\lambda \alpha
mn
\mu \beta
mn
> 0, (4.1)
then S\beta
\mu (\scrI 2)(X) \subseteq S\alpha
\lambda (\scrI 2)(X),
(ii) If
\mathrm{l}\mathrm{i}\mathrm{m}
mn\rightarrow \infty
\mu mn
\lambda \beta
mn
= 1, (4.2)
then S\alpha
\lambda (\scrI 2) \subseteq S\beta
\mu (\scrI 2)(X).
Proof. (i) Suppose that \lambda mn \leq \mu mn for all (m,n) \in \BbbN \times \BbbN and let 4.1 be satisfied. For
neighbourhood U of 0, we have\bigl\{
(k, l) \in Jmn : xkl - L /\in U
\bigr\}
\supseteq
\bigl\{
(k, l) \in Imn : xkl - L /\in U
\bigr\}
.
Therefore we can write
1
\mu \beta
mn
\bigm| \bigm| \bigl\{ (k, l) \in Jmn : xkl - L /\in U
\bigr\} \bigm| \bigm| \geq \lambda \alpha
mn
\mu \beta
mn
1
\lambda \alpha
mn
\bigm| \bigm| \bigl\{ (k, l) \in Imn : xkl - L /\in U
\bigr\} \bigm| \bigm|
and so for all (m,n) \in \BbbN \times \BbbN we have\biggl\{
(m,n) \in \BbbN \times \BbbN :
1
\lambda \alpha
mn
| \{ (k, l) \in Imn : xkl - L /\in U\} | \geq \delta
\biggr\}
\subseteq
\subseteq
\biggl\{
(m,n) \in \BbbN \times \BbbN :
1
\mu mn
\bigm| \bigm| \bigl\{ (k, l) \in Jmn : xkl - L /\in U
\bigr\} \bigm| \bigm| \geq \delta
\lambda \alpha
mn
\mu \beta
mn
\biggr\}
\in \scrI .
Hence S\beta
\mu (\scrI 2)(X) \subseteq S\alpha
\lambda (\scrI 2)(X).
(ii) Let x = (xkl) \in S\alpha
\lambda (\scrI 2)(X) and (4.2) be satisfied. Since Imn \subset Jmn, for neighbourhood U
of 0, we may write
1
\mu \beta
mn
| \{ (k, l) \in Jmn : xkl - L /\in U\} | =
=
1
\mu \beta
mn
| \{ m - \mu m + 1 < k \leq m - \lambda m, n - \mu n + 1 < k \leq n - \lambda n : xkl - L /\in U\} | +
+
1
\mu \beta
mn
| \{ (k, l) \in Imn : xkl - L /\in U\} | \leq
\leq \mu mn - \lambda mn
\mu \beta
mn
+
1
\lambda \beta
mn
| \{ (k, l) \in Imn : xkl - L /\in U\} | \leq
\leq
\Biggl(
\mu mn - \lambda \beta
mn
\lambda \beta
mn
\Biggr)
+
1
\lambda \alpha
mn
| \{ (k, l) \in Imn : xkl - L /\in U\} | \leq
\leq
\biggl(
\mu mn
\lambda \beta
mn
- 1
\biggr)
+
1
\lambda \alpha
mn
| \{ (k, l) \in Imn : xkl - L /\in U\} |
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 9
\scrI \lambda -DOUBLE STATISTICAL CONVERGENCE OF ORDER \alpha IN TOPOLOGICAL GROUPS 1257
for all (m,n) \in \BbbN \times \BbbN . Hence we have\biggl\{
(m,n) \in \BbbN \times \BbbN :
1
\mu \beta
mn
| \{ (k, l) \in Jmn : xkl - L /\in U\} | \geq \delta
\biggr\}
\subseteq
\subseteq
\biggl\{
(m,n) \in \BbbN \times \BbbN :
1
\lambda \alpha
mn
| \{ (k, l) \in Imn : xkl - L /\in U\} | \geq \delta
\biggr\}
\in \scrI .
This implies that S\alpha
\lambda (\scrI 2)(X) \subseteq S\beta
\mu (\scrI 2)(X).
Theorem 4 is proved.
From Theorem 2 we have the following corollary.
Corollary 2. Let \lambda = (\lambda mn) and \mu = (\mu mn) be two sequences in \bigtriangleup such that \lambda mn \leq \mu mn for
all (m,n) \in \BbbN \times \BbbN . If (4.1) holds, then
(i) S\alpha
\mu (\scrI 2)(X) \subseteq S\alpha
\lambda (\scrI 2)(X),
(ii) S\mu (\scrI 2)(X) \subseteq S\alpha
\lambda (\scrI 2)(X),
(iii) S\mu (\scrI 2)(X) \subseteq S\lambda (\scrI 2)(X).
Corollary 3. Let \lambda = (\lambda mn) and \mu = (\mu mn) be two sequences in \Lambda such that \lambda mn \leq \mu mn for
all (m,n) \in \BbbN \times \BbbN . If (4.2) holds, then
(i) S\alpha
\lambda (\scrI 2)(X) \subseteq S\alpha
\mu (\scrI 2)(X),
(ii) S\alpha
\lambda (\scrI 2)(X) \subseteq S\mu (\scrI 2)(X) ,
(iii) S\lambda (\scrI 2)(X) \subseteq S\mu (\scrI 2)(X).
Theorem 5. If \lambda \in \bigtriangleup be such that \mathrm{l}\mathrm{i}\mathrm{m}m,n
\lambda mn
mn
= 1, then S\alpha
\lambda (\scrI 2)(X) \subset S\alpha (\scrI 2)(X).
Proof. Let \delta > 0 be given. Since \mathrm{l}\mathrm{i}\mathrm{m}mn
\lambda mn
mn
= 1, we can choose (r, s) \in N \times N such that\bigm| \bigm| \bigm| \bigm| \lambda mn
mn
- 1
\bigm| \bigm| \bigm| \bigm| < \delta
2
, for all m \geq r, n \geq s. Let us take any neighbourhood U of 0. Now observe that,
1
(mn)\alpha
\bigm| \bigm| \bigl\{ k \leq m, l \leq n : xkl - L /\in U
\bigr\} \bigm| \bigm| =
=
1
mn
\bigm| \bigm| \bigl\{ k \leq m - \lambda m, l \leq n - \lambda n : xkl - L /\in U
\bigr\} \bigm| \bigm| + 1
(mn)\alpha
\bigm| \bigm| \bigl\{ (k, l) \in Imn : xkl - L /\in U
\bigr\} \bigm| \bigm| \leq
\leq mn - \lambda mn
(mn)\alpha
+
1
mn
\bigm| \bigm| \bigl\{ (k, l) \in Imn : xkl - L /\in U
\bigr\} \bigm| \bigm| =
=
\delta
2
+
1
(mn)\alpha
\bigm| \bigm| \bigl\{ (k, l) \in Imn : xkl - L /\in U
\bigr\} \bigm| \bigm|
for all m \geq r, n \geq s. Hence, for \delta > 0 and any neighbourhood U of 0\biggl\{
n \in N :
1
(mn)\alpha
| \{ k \leq m, l \leq n : xkl - L /\in U\} | \geq \delta
\biggr\}
\subset
\subset
\biggl\{
n \in N :
1
\lambda \alpha
mn
| \{ (k, l) \in Imn : xkl - L /\in U\} | \geq \delta
2
\biggr\}
\cup A,
where A is the union of the first m0 rows and the first n0 columns of the double sequence.
If S\alpha
\lambda (\scrI 2) - \mathrm{l}\mathrm{i}\mathrm{m}x = L, then the set on the right-hand side belongs to \scrI 2 and so the set on the
left-hand side also belongs to \scrI 2. This shows that x = (xkl) is \scrI 2- statistically convergent to L.
Theorem 5 is proved.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 9
1258 EKREM SAVAŞ
Remark 3. We do not know whether the condition in Theorem 5 is necessary and leave it as
an open problem.
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Received 24.02.16
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 9
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| id | umjimathkievua-article-1918 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:15:11Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/94/71f39cfdc04763298aaf04048cdaac94.pdf |
| spelling | umjimathkievua-article-19182019-12-05T09:31:35Z $I_\lambda$-Double statistical convergence of order $α$ in topological groups $I_\lambda$- double statistical convergence of order $\alpha$ in topological groups Savaş, E. Саваш, Є. We introduce а new notion, namely, $I_\lambda$-double statistical convergence of order \alpha in topological groups. Consequently, we investigate some inclusion relations between $I$ -double statistical and $I_\lambda$ -double statistical convergence of order $\alpha$ in topological groups. We also study many other related concepts. Введено нове поняття, а саме поняття $I_\lambda$ -подвiйної статистичної збiжностi порядку $\alpha$ в топологiчних групах. Таким чином, вивчено деякi вiдношеняя включення мiж $I$-подвiйною статистичною збiжнiстю та $I_\lambda$–подвiйною статистичною збiжнiстю порядку $\alpha$ в топологiчних групах. Також вивчено багато iнших подiбних понять. Institute of Mathematics, NAS of Ukraine 2016-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1918 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 9 (2016); 1251-1258 Український математичний журнал; Том 68 № 9 (2016); 1251-1258 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1918/900 Copyright (c) 2016 Savaş E. |
| spellingShingle | Savaş, E. Саваш, Є. $I_\lambda$-Double statistical convergence of order $α$ in topological groups |
| title | $I_\lambda$-Double statistical convergence of order $α$ in topological groups |
| title_alt | $I_\lambda$- double statistical convergence of order $\alpha$ in topological groups |
| title_full | $I_\lambda$-Double statistical convergence of order $α$ in topological groups |
| title_fullStr | $I_\lambda$-Double statistical convergence of order $α$ in topological groups |
| title_full_unstemmed | $I_\lambda$-Double statistical convergence of order $α$ in topological groups |
| title_short | $I_\lambda$-Double statistical convergence of order $α$ in topological groups |
| title_sort | $i_\lambda$-double statistical convergence of order $α$ in topological groups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1918 |
| work_keys_str_mv | AT savase ilambdadoublestatisticalconvergenceoforderaintopologicalgroups AT savašê ilambdadoublestatisticalconvergenceoforderaintopologicalgroups AT savase ilambdadoublestatisticalconvergenceoforderalphaintopologicalgroups AT savašê ilambdadoublestatisticalconvergenceoforderalphaintopologicalgroups |