On generalized statistical and ideal convergence of metric-valued sequences
We consider the notion of generalized density, namely, natural density of weight g recently introduced in [Balcerzak M., Das P., Filipczak M., Swaczyna J. Generalized kinds of density and the associated ideals // Acta Math. Hung. – 2015. –147, № 1. – P. 97 – 115] and primarily study some sufficient...
Gespeichert in:
| Datum: | 2016 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2016
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1946 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507845020090368 |
|---|---|
| author | Das, P. Savaş, E. Дас, П. Саваш, Є. |
| author_facet | Das, P. Savaş, E. Дас, П. Саваш, Є. |
| author_sort | Das, P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:32:42Z |
| description | We consider the notion of generalized density, namely, natural density of weight g recently introduced in [Balcerzak M.,
Das P., Filipczak M., Swaczyna J. Generalized kinds of density and the associated ideals // Acta Math. Hung. – 2015. –147, № 1. – P. 97 – 115] and primarily study some sufficient and almost converse necessary conditions for the generalized statistically convergent sequence under which the subsequence is also generalized statistically convergent. Some results
are also obtained in more general form using the notion of ideals. The entire investigation is performed in the setting of
general metric spaces extending the recent results of Kucukaslan M., Deger U., Dovgoshey O. On statistical convergence
of metric valued sequences, see Ukr. Math. J. – 2014. – 66, № 5. – P. 712 – 720. |
| first_indexed | 2026-03-24T02:15:47Z |
| format | Article |
| fulltext |
UDC 517.5
P. Das (Jadavpur Univ., West Bengal, India),
E. Savas (Istanbul Commerce Univ., Turkey)
ON GENERALIZED STATISTICAL AND IDEAL CONVERGENCE
OF METRIC-VALUED SEQUENCES
ПРО УЗАГАЛЬНЕНУ СТАТИСТИЧНУ ТА IДЕАЛЬНУ ЗБIЖНIСТЬ
МЕТРИЧНОЗНАЧНИХ ПОСЛIДОВНОСТЕЙ
We consider the notion of generalized density, namely, natural density of weight g recently introduced in [Balcerzak M.,
Das P., Filipczak M., Swaczyna J. Generalized kinds of density and the associated ideals // Acta Math. Hung. – 2015. –
147, № 1. – P. 97 – 115] and primarily study some sufficient and almost converse necessary conditions for the generalized
statistically convergent sequence under which the subsequence is also generalized statistically convergent. Some results
are also obtained in more general form using the notion of ideals. The entire investigation is performed in the setting of
general metric spaces extending the recent results of Kücükaslan M., Deger U., Dovgoshey O. On statistical convergence
of metric valued sequences, see Ukr. Math. J. – 2014. – 66, № 5. – P. 712 – 720.
Ми розглядаємо поняття узагальненої щiльностi, тобто натуральної щiльностi з вагою g, нещодавно введеної в статтi
[Balcerzak M., Das P., Filipczak M., Swaczyna J. Generalized kinds of density and the associated ideals // Acta Math.
Hung. – 2015. – 147, № 1. – P. 97 – 115], та переважно вивчаємо деякi достатнi та майже протилежнi необхiднi умови
для узагальненої статистично збiжної послiдовностi, за яких пiдпослiдовнiсть також є узагальненою та статистично
збiжною. Деякi результати також отримано в бiльш загальному виглядi за допомогою поняття iдеалiв. Наше
дослiдження виконано в постановцi загальних метричних просторiв i узагальнює нещодавнi результати, отриманi
у статтi Kücükaslan M., Deger U., Dovgoshey O. On statistical convergence of metric valued sequences (див. Укр. мат.
журн. – 2014. – 66, № 5. – С. 712 – 720).
1. Introduction. In recent years there have been rapid developments of the analytical studies in metric
spaces which can be seen in [21, 27]. In [16] Dovgoshey and Martio introduced a new approach to
the introduction of smooth structures for general metric spaces (one can see also [4, 5, 14, 15] where
more references can be found). In the language of [16] this new approach is completely based on the
convergence of metric-valued sequences but it is not a priori clear that the ordinary convergence is
the best possible way to obtain smooth structures for arbitrary metric spaces.
From the beginnings of 1800’s several methods have been introduced to make a divergent real or
complex sequence convergent (for example Česaro, Nörlund, weighted mean, Abel etc.) but most of
these convergence methods are dependent on the algebraic structures of the spaces of reals or complex
numbers. It should be noted that in general metric spaces do not have algebraic structures. However
if one considers the notion of statistical convergence introduced in [18, 29] and its extensions like
statistical convergence of order \alpha [3, 6] or more generally the notion of ideal convergence [22], it is
clear that they can be readily extended to arbitrary metric spaces.
On the other direction the study of statistical convergence and its many extensions and in particular
ideal convergence and its applications has been one of the most active areas of research in summability
theory over the last 15 years.
Naturally it seems that the studies of these generalized methods of convergence may provide
a natural foundation for the upbuilding of various tangent spaces to general metric spaces. The
construction of tangent spaces in [4, 5, 14 – 16] is primarily based on the fundamental fact that for
c\bigcirc P. DAS, E. SAVAS, 2016
1598 ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
ON GENERALIZED STATISTICAL AND IDEAL CONVERGENCE OF METRIC-VALUED SEQUENCES 1599
a convergent sequence (xn) in a metric space, each of its subsequence (xn(k)) is also convergent.
However this is generally not true for the generalized methods of convergence mentioned above.
Very recently following the line of investigation of [25], in [23] conditions were studied for the
density of a subsequence of a statistically convergent sequence under which the subsequence is also
statistically convergent in metric space settings.
As a natural consequence, in this paper we continue the investigation proposed in [23] and
investigate similar problems for metric-valued sequences by considering the notion of natural density
of weight g which was very recently introduced in [1] as also for certain results we use the most
general notion of ideals.
2. Basic facts and definitions. Let \BbbN denote the set of all positive integers. By \mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(A) we
denote the cardinality of a set A. The natural density of a set A \subset \BbbN is defined as follows: The lower
and the upper densities of A are given by the formulas
d(A) = \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(A \cap [1, n])
n
,
d(A) = \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(A \cap [1, n])
n
.
If d(A) = d(A), we say that the natural density of A exists and it is denoted by d(A). The notion of
statistical convergence was introduced by Fast [18] (see also [29]) using this notion of natural density.
Now recall that a family \scrI \subset 2Y of subsets of a nonempty set Y is said to be an ideal in Y if
(i) A, B \in \scrI implies A\cup B \in \scrI , (ii) A \in \scrI , B \subset A implies B \in \scrI , while an admissible ideal \scrI of
Y further satisfies \{ x\} \in \scrI for each x \in Y. Such ideals are also called free ideals. If \scrI is a proper
ideal in Y (i.e., Y /\in \scrI , \scrI \not = \{ \varnothing \} ), then the family of sets \scrF (\scrI ) = \{ M \subset Y : there exists A \in \scrI :
M = Y \setminus A\} is a filter in Y. It is called the filter associated with the ideal \scrI . Throughout the paper \scrI
will stand for a proper admissible ideal of \BbbN . We denote the ideal of all finite subsets of \BbbN by \scrI fin.
For more example of different ideals see [22].
An admissible ideal \scrI is said to satisfy the condition (AP) (or is called a P -ideal or sometimes
AP -ideal) if for every countable family of mutually disjoint sets (A1, A2, . . .) \in \scrI there exists a
countable family of sets (B1, B2, . . .) such that Aj\bigtriangleup Bj is finite for each j \in \BbbN and
\infty \bigcup
k=1
Bk \in \scrI .
Several examples of P -ideals can be found in [1].
It is known that the density ideal
\scrI d = \{ A \subset \BbbN : d(A) = 0\}
is an F\sigma \delta P-ideal on \BbbN . It is also an example of a so-called Erdős – Ulam ideal (for further information
see [17]).
In [3] the authors proposed a modified version of density. Namely, for 0 < \alpha \leq 1 and A \subset \BbbN ,
they put
d\alpha (A) = \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}(A \cap [1, n])
n\alpha
and d\alpha (A) is defined analogously. It has been very recently observed in [1] that these density
functions also generate P -ideals.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
1600 P. DAS, E. SAVAS
In this connection it can be mentioned that Kostyrko et al. [22] considered arbitrary ideals \scrI on \BbbN
and defined the notion of \scrI -convergence of sequences extending the idea of statistical convergence.
Following the general line of [22], ideals were used to study sequences in topological spaces [9, 24],
to study nets in topological and uniform spaces [11, 12]. More recent applications of ideals can be
found in [8, 10, 13] where many more references can be found.
We now start our main discussions. In [1] the notion of natural density (as also natural density of
order \alpha ) has further been extended as follows. Let g : \BbbN \rightarrow [0,\infty ) be a function with \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty g(n) =
= \infty . The upper density of weight g was defined in [1] by the formula
dg(A) = \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d} (A \cap [1, n])
g(n)
for A \subset \BbbN . Then the family
\scrI g =
\bigl\{
A \subset \BbbN : dg(A) = 0
\bigr\}
forms an ideal. It has been observed in [1] that \BbbN \in \scrI g iff
n
g(n)
\rightarrow 0. So we additionally assume
that n/g(n) \nrightarrow 0 so that \BbbN /\in \scrI g and in has been observed in [1] that \scrI g is a proper admissible
P -ideal of \BbbN . The collection of all such functions g satisfying the above mentioned properties will
be denoted by G. As a natural consequence we can introduce the following definition.
Throughout (X, \rho ) will stand for a metric space and \widetilde X will denote the set of all sequences of
points of X.
Definition 2.1. A metric-valued sequence \widetilde x = (xn) \in \widetilde X is said to be dg -statistically convergent
to a \in X if for any \epsilon > 0 we have dg(A(\varepsilon )) = 0 where A(\varepsilon ) =
\bigl\{
n \in \BbbN : \rho (xn, a) \geq \varepsilon
\bigr\}
.
Below some more basic definitions are given which will be needed throughout the paper.
Definition 2.2. A set K \subset \BbbN is called dg -dense subset of \BbbN if dg(Kc) = 0.
Definition 2.3 (see [22]). A metric-valued sequence x = (xn) \in \widetilde X is \rho - \scrI -convergent to
a \in X if for any \varepsilon > 0, A (\varepsilon ) = \{ n \in N : \rho (xn, a) \geq \varepsilon \} \in \scrI .
Definition 2.4. A set K \subset \BbbN is called \scrI -dense subset of \BbbN if K \in F (\scrI ).
Definition 2.5. If
\bigl(
n(k)
\bigr)
is an infinite strictly increasing sequence of natural numbers and
x = (xn) \in \widetilde X, then we write \widetilde x\prime = (xn(k)) and K\~x\prime =
\bigl\{
n(k) : k \in \BbbN
\bigr\}
. \widetilde x\prime is called an \scrI -dense
subsequence of \~x if K\~x is an \scrI -dense subset of \BbbN .
Definition 2.6. Two sequences \~x = (xn) \in \~X and \~y = (yn) \in \~X are \scrI -equivalent, \~x \asymp \~y if
there is an \scrI -dense set M \subset N such that xn = yn for every n \in M.
The following definitions are special cases of the above two definitions.
Definition 2.7. If (n(k)) is an infinite strictly increasing sequence of natural numbers and
x = (xn) \in \widetilde X, then we write \widetilde x\prime = (xn(k)) and K\~x\prime = \{ n(k) : k \in N\} . \widetilde x\prime is called dg -dense
subsequence of \~x if K\~x\prime is dg -dense in \BbbN .
Definition 2.8. Two sequences \~x = (xn) \in \~X and \~y = (yn) \in \~X are dg -statistically equivalent,
\~x \asymp \~y (dg -statistically) if there is an dg -dense set M \subset N such that xn = yn for every n \in M.
3. Main results. The first result given below extends Theorem 2.1 [23] and shows that there is
a one to one correspondence between metrizable topologies on X and the subsets of \widetilde X consisting of
all \scrI -convergent sequences for certain special types of ideals.
Theorem 3.1. Let (X, \rho 1) and (X, \rho 2) be two metric spaces. Let \scrI be a P -ideal which is not
maximal. Then the following statements are equivalent:
(i) The set of all \rho 1 - \scrI -convergent sequences coincides with the set of all \rho 2 - \scrI -convergent
sequences.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
ON GENERALIZED STATISTICAL AND IDEAL CONVERGENCE OF METRIC-VALUED SEQUENCES 1601
(ii) The set of all sequences convergent in (X, \rho 1) coincides with the set of all sequences
convergent in (X, \rho 2).
(iii) The metrics \rho 1 and \rho 2 induce one and the same topology on X.
Proof. (ii) \Leftarrow \Rightarrow (iii). The result is well known.
(ii) \Rightarrow (i). Let \~x = (xn) be \rho 1 - \scrI -convergent. Since \scrI is a P -ideal so \~x is \rho 1 - \scrI \ast -convergent,
i.e., there is a set M \in F (\scrI ) such that (\~x)M is \rho 1-convergent (see [22]). By (ii), (\~x)M is \rho 2-
convergent and so \~x is \rho 2 - \scrI \ast -convergent which consequently implies that \~x is \rho 2 - \scrI -convergent
(see [22]).
(i) \Rightarrow (iii). Assume that (i) holds. But on the contrary assume that the topologies induced by the
metrics \rho 1 and \rho 2 are distinct. Then there is a x0 \in X and \varepsilon 0 > 0 such that\bigl\{
x \in X : \rho 1(x, x0) < \varepsilon 0
\bigr\}
\not \supset
\bigl\{
x \in X : \rho 2(x, x0) < \delta
\bigr\}
(3.1)
for all \delta > 0 or \bigl\{
x \in X : \rho 2(x, x0) < \varepsilon 0
\bigr\}
\not \supset
\bigl\{
x \in X : \rho 1(x, x0) < \delta
\bigr\}
for all \delta > 0. Without any loss of generality assume that (3.1) holds. For each n \in \BbbN we can then
choose xn \in X such that
\rho 2(xn, x0) <
1
n
and \rho 1(xn, x0) \geq \varepsilon 0 (3.2)
for each n \in \BbbN . Choose a set K \subset \BbbN such that K /\in \scrI as well as Kc /\in \scrI (since \scrI is not maximal).
Define a sequence \~y = (yn) \in \widetilde X by
yn =
\left\{ xn if n \in K,
x0 if n /\in K.
Clearly \bigl\{
n \in \BbbN : \rho 1(yn, x0) \geq \varepsilon 0
\bigr\}
= K /\in \scrI . (3.3)
Now observe that the sequence \~y = (yn) is convergent to x0 in (X, \rho 2) and so is \rho 2 - \scrI -convergent.
By (i), \~y = (yn) is then also \rho 1 - \scrI -convergent. Observe that \~y must be \rho 1 - \scrI -convergent to x0
for otherwise if \~y is \rho 1 - \scrI -convergent to y0 \not = x0 then taking 0 < \varepsilon < \rho 1(x0, y0) we have\bigl\{
n \in \BbbN : \rho 1(yn, y0) \geq \varepsilon
\bigr\}
\supset Kc.
Since Kc /\in \scrI so
\bigl\{
n \in \BbbN : \rho 1(yn, y0) \geq \varepsilon
\bigr\}
/\in \scrI which is a contradiction to the fact that \~y = (yn) is
\rho 1 - \scrI -convergent to y0. But if \~y is \rho 1 - \scrI -convergent to x0 then we must have\bigl\{
n \in \BbbN : \rho 1(yn, x0) \geq \varepsilon 0
\bigr\}
= K \in \scrI
which contradicts (3.3). This proves that (i) \Rightarrow (iii) holds.
Lemma 3.1. Let (X, \rho ) be a metric space, x0 \in X and \~x = (xn) \in \widetilde X. Then \~x is \rho - \scrI -
convergent to x0 in X if and only if the sequence (\rho (xn, x0)) is \scrI -convergent to 0 in \BbbR .
The proof is straightforward and so is omitted.
Lemma 3.2. Let (X, \rho ) be a metric space, x0 \in X and let \~x = (xn) \in \widetilde X be \rho - \scrI -convergent
to x0. Then there is \~y = (yn) \in \widetilde X such that \~y \asymp \~x and \~y is convergent to x0 in (X, \rho ) provided \scrI
is a P -ideal.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
1602 P. DAS, E. SAVAS
The result again follows from the fact that \~x = (xn) is \rho - \scrI \ast -convergent to x0 as \scrI is a P -ideal
[22] and consequently we get the required sequence \~y.
We now start our discussions on subsequences. The first natural question that arises is that if a
sequence is dg -statistically convergent which of its subsequences are also dg -statistically convergent
to the same limit. It is also natural to ask when the converse is also true. We prove the next two
results in this direction.
Theorem 3.2. Let (X, \rho ) be a metric space, \~x = (xn) \in \widetilde X and \widetilde x\prime = (xn(k)) be a subsequence
of \~x such that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
g
\bigl( \bigm| \bigm| K\~x(n)
\bigm| \bigm| \bigr)
g(n)
> 0.
If \~x is dg -statistically convergent to x0 \in X, then \widetilde x\prime is also dg -statistically convergent to x0.
Proof. Assume that \~x is dg -statistically convergent to x0. Now clearly\bigl\{
n(k) : n(k) \leq n, d(xn(k), x0) \geq \varepsilon
\bigr\}
\subseteq
\bigl\{
m : m \leq n, d(xm, x0) \geq \varepsilon
\bigr\}
for all n \in \BbbN where \varepsilon > 0 is given. Then we get
1
g (| K\~x(n)| )
\bigm| \bigm| \bigl\{ n(k) : n (k) \leq n, \rho
\bigl(
xn(k), x0
\bigr)
\geq \varepsilon
\bigr\} \bigm| \bigm| \leq
\leq | \{ m : m \leq n, \rho (xm, x0) \geq \varepsilon \} |
g (| K\~x (n)| )
. (3.4)
In order to prove that \widetilde x\prime is dg -statistically convergent to x0 we need to show that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
\bigm| \bigm| \bigl\{ n (k) : n (k) \leq n, \rho
\bigl(
xn(k), x0
\bigr)
\geq \varepsilon
\bigr\} \bigm| \bigm|
g (| K\~x (n)| )
= 0.
Recall that for any two sequences (cn) and (dn) of nonnegative real numbers with
0 \not = \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}n\rightarrow \infty cn < \infty we have (see [2])
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
cn \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
dn \leq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
cndn. (3.5)
In (3.5) let us take
cn =
g (| K\~x (n)| )
g(n)
\mathrm{a}\mathrm{n}\mathrm{d} dn =
| \{ m : m \leq n, \rho (xm, x0) \geq \varepsilon \} |
g (| K\~x (n)| )
so that
cndn =
| \{ m : m \leq n, \rho (xm, x0) \geq \varepsilon \} |
g(n)
.
Therefore we get from (3.5)
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
g (| K\~x (n)| )
g (n)
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
| \{ m : m \leq n, \rho (xm, x0) \geq \varepsilon \} |
g (| K\~x (n)| )
\leq
\leq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
| \{ m : m \leq n, \rho (xm, x0) \geq \varepsilon \} |
g(n)
.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
ON GENERALIZED STATISTICAL AND IDEAL CONVERGENCE OF METRIC-VALUED SEQUENCES 1603
Since \~x is dg -statistically convergent to x0 so the right-hand side of the above inequality is zero.
Since by our assumption \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}n\rightarrow \infty
g (| K\~x (n)| )
g(n)
> 0 so it follows that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
| \{ m : m \leq n, \rho (xm, x0) \geq \varepsilon 0\} |
g (| K\~x (n)| )
= 0
and the result now follows from (3.4).
Theorem 3.3. Let (X, \rho ) be a metric space and \~x \in \widetilde X. Then the following statements are
equivalent:
(a) \~x is dg -statistically convergent;
(b) every subsequence \widetilde x\prime of \~x with
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
g (| K\~x (n)| )
g (n)
> 0
is dg -statistically convergent;
(c) every dg -statistically dense subsequence \widetilde x\prime of \~x is dg -statistically convergent provided g \in G
is such that 0 < \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}n\rightarrow \infty
n
g(n)
< \infty .
Proof. From Theorem 3.2 it follows that (a) \Rightarrow (b). Since evidently \~x itself is a dg -dense
subsequence of itself so (c) \Rightarrow (a).
(b) \Rightarrow (c). Let \widetilde x\prime be a dg -statistically dense subsequence of \~x. This means that K\widetilde x\prime has the
property that dg
\bigl(
Kc\widetilde x\prime
\bigr)
= 0, i.e.,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
g
\bigl( \bigm| \bigm| Kc\widetilde x\prime (n)
\bigm| \bigm| \bigr)
g(n)
= 0.
Since | K\widetilde x\prime (n)| + | Kc\widetilde x\prime (n)| = n, consequently we have
| K\widetilde x\prime (n)|
g(n)
+
| K\widetilde x\prime c(n)|
g(n)
=
n
g(n)
\forall n \in \BbbN .
It now follows that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
| K\widetilde x\prime (n)|
g(n)
+ \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
\bigm| \bigm| Kc\widetilde x\prime (n)
\bigm| \bigm|
g(n)
\geq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
n
g(n)
which implies that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
| K\widetilde x\prime (n)|
g(n)
\geq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
n
g (n)
> 0
is a finite positive number. Finally we obtain
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
g(| K\widetilde x\prime (n)| )
g(n)
\geq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
g(| K\widetilde x\prime (n)| )
| K\widetilde x\prime (n)|
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
| K\widetilde x\prime (n)|
g(n)
> 0
in view of the fact that | K\widetilde x\prime (n)| \rightarrow \infty as n \rightarrow \infty . By (b) it now readily follows that \widetilde x\prime is
dg -statistically convergent. This completes the equivalence of the three statements.
The next three results are given in the most general version in terms of ideals. As a consequence,
one should note that these results hold for natural density of weight g also.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
1604 P. DAS, E. SAVAS
Lemma 3.3. Let (X, \rho ) be a metric space with | X| > 2. Let \~x = (xn) \in \widetilde X and \widetilde x\prime = (xn(k))
be an infinite subsequence of \~x such that K\widetilde x\prime \in \scrI . Then there exists a sequence \~y \in \widetilde X and a
subsequence \widetilde y\prime of \~y such that K\widetilde x\prime = K\widetilde y\prime where \widetilde y\prime is not \scrI -convergent provided \scrI is not a maximal
ideal.
Proof. Choose two distinct elements a and b from X. Choose a subset M \subset \BbbN such that M /\in \scrI
as well as M /\in \scrF (\scrI ). Let us define a sequence \~y = (yn) \in \widetilde X by
yn =
\left\{
xn if n \in \BbbN \setminus K\widetilde x\prime ,
a if n = n(k) \in K\widetilde x\prime , where k \in M,
b if n = n(k) \in K\widetilde x\prime , where k /\in M.
Since K\~x\prime \in \scrI so \BbbN \setminus K\widetilde x\prime \in \scrF (\scrI ) which shows that \~x
\scrI \asymp \~y. Obviously taking \widetilde y\prime = \bigl(
yn(k)
\bigr)
we see
that K\widetilde x\prime = K\widetilde y\prime . Since for any c \in X, taking 0 < \varepsilon < \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
\rho (a, c), \rho (b, c)
\bigr\}
we observe that\bigl\{
k : \rho (yn(k), c) \geq \varepsilon
\bigr\}
\supset M or M c
and so cannot belong to \scrI . This shows that \widetilde y\prime is not \scrI -convergent.
Lemma 3.4. Let (X, \rho ) be a metric space. Let a \in X, \~x = (xn) and \~y = (yn) belong to \widetilde X. If
\~x is \scrI -convergent to a and \~x
\scrI \asymp \~y, then \~y is also \scrI -convergent to a.
Proof. Since \~x
\scrI \asymp \~y so there is M \in \scrF (\scrI ) such that xn = yn for all n \in M. Clearly for any
\varepsilon > 0, \bigl\{
n : \rho (yn, a) \geq \varepsilon
\bigr\}
\subset M c \cup
\bigl\{
n : \rho (xn, a) \geq \varepsilon
\bigr\}
.
Since \~x is \scrI -convergent to a so the set on the right-hand side belong to \scrI which implies that\bigl\{
n : \rho (yn, a) \geq \varepsilon
\bigr\}
\in \scrI and \~y is also \scrI -convergent to a.
Theorem 3.4. Let (X, \rho ) be a metric space with | X| > 2, a \in X and \scrI be not maximal. Let
\~x = (xn) be \scrI -convergent to a. Then for every infinite subsequence \widetilde x\prime of \~x with K\widetilde x\prime \in \scrI , there
exist a sequence \~y \in \widetilde X and a subsequence \widetilde y\prime of \~y such that:
(i) \~y
\scrI \asymp \~x and K\widetilde x\prime = K\widetilde y\prime ,
(ii) \~y is \scrI -convergent to a,
(iii) \widetilde y\prime is not \scrI -convergent.
The result follows from Lemmas 3.3 and 3.4.
Lemma 3.5. Let (X, \rho ) be a metric space. Let \~x, \~y \in \widetilde X and \~x \asymp \~y (dg -statistically). If K is
a subset of \BbbN such that
0 < \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
g
\bigl( \bigm| \bigm| K(n)
\bigm| \bigm| \bigr)
g(n)
, (3.6)
if \widetilde x\prime = (xn(k)) and \widetilde y\prime = (yn(k)) are subsequences of \~x and \~y respectively such that K\widetilde x\prime = K\widetilde y\prime = K,
then the relation \widetilde y\prime \asymp \widetilde x\prime (dg -statistically) is true.
Proof. We have to show that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\bigm| \bigm| \bigl\{ n (k) \in K : xn(k) \not = yn(k), n (k) \leq m
\bigr\} \bigm| \bigm|
g
\bigl(
| K(m)|
\bigr) = 0. (3.7)
Observe that for any m \in \BbbN we obtain
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
ON GENERALIZED STATISTICAL AND IDEAL CONVERGENCE OF METRIC-VALUED SEQUENCES 1605\bigl\{
n(k) \in K : xn(k) \not = yn(k), n(k) \leq m
\bigr\}
\subset
\bigl\{
n \in \BbbN : xn \not = yn, n \leq m
\bigr\}
.
Consequently we get
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
\bigm| \bigm| \bigl\{ n (k) \in K : xn(k) \not = yn(k), n (k) \leq m
\bigr\} \bigm| \bigm|
g (| K (m)| )
\leq
\leq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
| \{ n \in N : xn \not = yn, n \leq m\} |
g (| K (m)| \leq )
\leq
\leq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
g (m)
g (| K (m)| )
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
| \{ n \in \BbbN : xn \not = yn, n \leq m\} |
g (m)
=
= \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
| \{ n \in N : xn \not = yn, n \leq m\} |
g (m)
\biggl(
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
m\rightarrow \infty
g (| K (m)| )
g (m)
\biggr) - 1
. (3.8)
From (3.6) it follows that
0 \leq
\biggl(
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
m\rightarrow \infty
g (| K (m)| )
g (m)
\biggr) - 1
< \infty .
Also as \~x \asymp \~y (dg -statistically) we have
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
m\rightarrow \infty
| \{ n \in \BbbN : xn \not = yn, n \leq m\} |
g (m)
= 0.
Now (3.7) readily follows that (3.8) and this completes the proof.
Theorem 3.5. Let (X, \rho ) be a metric space, a \in X and \~x = (xn) is dg -statistically convergent
to a. Suppose that \widetilde x\prime = (xn(k)) is a subsequence of \~x for which there are \~y = (yn) \in \widetilde X and \widetilde y\prime
such that:
(i) \~y \asymp \~x (dg -statistically) and K\widetilde x\prime = K\widetilde y\prime ,
(ii) \widetilde y\prime is not dg -statistically convergent,
then \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}n\rightarrow \infty
| K\~x\prime (n)|
g(n)
= 0, provided g : \BbbN \rightarrow [0,\infty ) satisfies 0 < \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}n\rightarrow \infty
n
g (n)
and
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}n\rightarrow \infty
n
g(n)
< \infty .
Proof. On the contrary suppose that
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
| K\widetilde x\prime (n)|
g(n)
> 0.
Then
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
g (| K\widetilde x\prime (n)| )
g (n)
\geq
\geq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
g
\bigl( \bigm| \bigm| K\widetilde x\prime (n)
\bigm| \bigm| \bigr)
| K\widetilde x\prime (n)|
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
\bigm| \bigm| K\widetilde x\prime (n)
\bigm| \bigm|
g(n)
> 0.
Let \~y \in \widetilde X and \widetilde y\prime be a subsequence of \~y such that (i) and (ii) hold. Then we have K\widetilde x\prime = K\widetilde y\prime and
\~x \asymp \~y (dg -statistically). Then from Lemma 3.5 it follows that \widetilde x\prime \asymp \widetilde y\prime (dg -statistically). Applying
Theorem 3.2 we observe that \widetilde x\prime is dg -statistically convergent to a. Since \widetilde x\prime \asymp \widetilde y\prime (dg -statistically)
so by Lemma 3.4 \widetilde y\prime is also dg -statistically convergent to a which contradicts (ii).
Acknowledgement. The first author is thankful to TUBITAK for granting Visiting Scien-
tist position for one month and to SERB, DST, New Delhi for granting a research project No.
SR/S4/MS:813/13 during the tenure of which this work was done.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
1606 P. DAS, E. SAVAS
References
1. Balcerzak M., Das P., Filipczak M., Swaczyna J. Generalized kinds of density and the associated ideals // Acta Math.
Hung. – 2015. – 147, № 1. – P. 97 – 115.
2. Baranenkov G. S., Demidovich B. P., Efimenko V. A. etc. Problems in mathematical analysis. – Moskow: Mir, 1976.
3. Bhunia S., Das P., Pal S. K. Restricting statistical convergence // Acta Math. Hung. – 2012. – 134, № 1-2. –
P. 153 – 161.
4. Bilet V. Geodesic spaces tangent to metric spaces // Ukr. Math. J. – 2013. – 62, № 11. – P. 1448 – 1456.
5. Bilet V., Dovgoshey O. Isometric embeddings of pretangent spaces in En // Bull. Belg. Math. Soc. Simon Stevin. –
2013. – 20. – P. 91 – 110.
6. Colak R. Statistical convergence of order \alpha // Modern Methods in Analysis and its Applications. – New Delhi, India:
Anamaya Publ., 2010. – P. 121 – 129.
7. Connor J. The statistical and and strong p-Cesaro convergence of sequences // Analysis. – 1998. – 8. – P. 207 – 212.
8. Das P. Certain types of open covers and selection principles using ideals // Houston J. Math. – 2013. – 39, № 2. –
P. 637 – 650.
9. Das P. Some further results on ideal convergence in topological spaces // Topology and Appl. – 2012. – 159. –
P. 2621 – 2625.
10. Das P., Chandra D. Some further results on \scrI - \gamma and \scrI - \gamma k -covers // Topology and Appl. – 2013. – 16. –
P. 2401 – 2410.
11. Das P., Ghosal S. K. On \scrI -Cauchy nets and completeness // Topology and Appl. – 2010. – 157. – P. 1152 – 1156.
12. Das P., Ghosal S. K. When \scrI -Cauchy nets in complete uniform spaces are \scrI -convergent // Topology and Appl. –
2011. – 158. – P. 1529 – 1533.
13. Das P., Savas E. Some further results on ideal summability of nets in (\ell )- groups // Positivity. – 2015. – 19, № 1. –
P. 53 – 63.
14. Dovgoshey O. Tangent spaces to metric spaces and to their subspaces // Ukr. Mat. Visn. – 2008. – 5. – P. 468 – 485.
15. Dovgoshey O., Abdullayev F. G., Kücükaslan M. Compactness and boundedness of tangent spaces to metric spaces //
Baitr. Algebra Geom. – 2010. – 51. – P. 100 – 113.
16. Dovgoshey O., Martio O. Tangent spaces to metric spaces // Repts Math. Helsinki Univ. – 2008. – 480.
17. Farah I. Analytic quotients. Theory of lifting for quotients over analytic ideals on integers // Mem. Amer. Math.
Soc. – 2000. – 148.
18. Fast H. Sur la convergence statistique // Colloq. math. – 1951. – 2. – P. 41 – 44.
19. Freedman A. R., Sember J. J. On summing sequences of 0’s and 1’s // Rocky Mountain J. Math. – 1981. – 11. –
P. 419 – 425.
20. Fridy J. A. On statistical convergence // Analysis. – 1985. – 5. – P. 301 – 313.
21. Heinonen J. Lectures on analysis on metric spaces. – Springer, 2001.
22. Kostyrko P., Šalát T., Wilczyński W. \scrI -convergence // Real Anal. Exchange. – 2000/2001. – 26. – P. 669 – 685.
23. Kücükaslan M., Deger U., Dovgoshey O. On the statistical convergence of metric-valued sequences // Ukr. Math. J. –
2014. – 66, № 5. – P. 712 – 720.
24. Lahiri B. K., Das P. \scrI and \scrI \ast -convergence in topological spaces // Math. Bohemica. – 2005. – 130. – P. 153 – 160.
25. Mačaj M., Šalát T. Statistical convergence of subsequences of a given sequence // Math. Bohemica. – 2001. – 126. –
P. 191 – 208.
26. Miller H. I. A measure theoretic subsequence characterization of statistical convergence // Trans. Amer. Math. Soc. –
1995. – 347. – P. 1811 – 1819.
27. Papadopoulos A. Metric spaces, convexity and nonpositive curvature // Eur. Math. Soc. – 2005.
28. Šalát T. On statistically convergent sequences of real numbers // Math. Slovaca. – 1980. – 30. – P. 139 – 150.
29. Steinhaus H. Sur la convergence ordinaire et la convergence asymptotique // Colloq. math. – 1951. – 2. – P. 73 – 74.
Received 31.07.15
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 12
|
| id | umjimathkievua-article-1946 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:15:47Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/47/5d26d120a0c5845b1cea4fc4ff0d5047.pdf |
| spelling | umjimathkievua-article-19462019-12-05T09:32:42Z On generalized statistical and ideal convergence of metric-valued sequences Про узагальнену статистичну та iдеальну збiжнiсть метричнозначних послiдовностей Das, P. Savaş, E. Дас, П. Саваш, Є. We consider the notion of generalized density, namely, natural density of weight g recently introduced in [Balcerzak M., Das P., Filipczak M., Swaczyna J. Generalized kinds of density and the associated ideals // Acta Math. Hung. – 2015. –147, № 1. – P. 97 – 115] and primarily study some sufficient and almost converse necessary conditions for the generalized statistically convergent sequence under which the subsequence is also generalized statistically convergent. Some results are also obtained in more general form using the notion of ideals. The entire investigation is performed in the setting of general metric spaces extending the recent results of Kucukaslan M., Deger U., Dovgoshey O. On statistical convergence of metric valued sequences, see Ukr. Math. J. – 2014. – 66, № 5. – P. 712 – 720. Ми розглядаємо поняття узагальненої щiльностi, тобто натуральної щiльностi з вагою g, нещодавно введеної в статтi [Balcerzak M., Das P., Filipczak M., Swaczyna J. Generalized kinds of density and the associated ideals // Acta Math. Hung. – 2015. – 147, № 1. – P. 97 – 115], та переважно вивчаємо деякi достатнi та майже протилежнi необхiднi умови для узагальненої статистично збiжної послiдовностi, за яких пiдпослiдовнiсть також є узагальненою та статистично збiжною. Деякi результати також отримано в бiльш загальному виглядi за допомогою поняття iдеалiв. Наше дослiдження виконано в постановцi загальних метричних просторiв i узагальнює нещодавнi результати, отриманi у статтi Kucukaslan M., Deger U., Dovgoshey O. On statistical convergence of metric valued sequences (див. Укр. мат. журн. – 2014. – 66, № 5. – С. 712 – 720). Institute of Mathematics, NAS of Ukraine 2016-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1946 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 12 (2016); 1598-1606 Український математичний журнал; Том 68 № 12 (2016); 1598-1606 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1946/928 Copyright (c) 2016 Das P.; Savaş E. |
| spellingShingle | Das, P. Savaş, E. Дас, П. Саваш, Є. On generalized statistical and ideal convergence of metric-valued sequences |
| title | On generalized statistical and ideal convergence of metric-valued sequences |
| title_alt | Про узагальнену статистичну та iдеальну збiжнiсть
метричнозначних послiдовностей |
| title_full | On generalized statistical and ideal convergence of metric-valued sequences |
| title_fullStr | On generalized statistical and ideal convergence of metric-valued sequences |
| title_full_unstemmed | On generalized statistical and ideal convergence of metric-valued sequences |
| title_short | On generalized statistical and ideal convergence of metric-valued sequences |
| title_sort | on generalized statistical and ideal convergence of metric-valued sequences |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1946 |
| work_keys_str_mv | AT dasp ongeneralizedstatisticalandidealconvergenceofmetricvaluedsequences AT savase ongeneralizedstatisticalandidealconvergenceofmetricvaluedsequences AT dasp ongeneralizedstatisticalandidealconvergenceofmetricvaluedsequences AT savašê ongeneralizedstatisticalandidealconvergenceofmetricvaluedsequences AT dasp prouzagalʹnenustatističnutaidealʹnuzbižnistʹmetričnoznačnihposlidovnostej AT savase prouzagalʹnenustatističnutaidealʹnuzbižnistʹmetričnoznačnihposlidovnostej AT dasp prouzagalʹnenustatističnutaidealʹnuzbižnistʹmetričnoznačnihposlidovnostej AT savašê prouzagalʹnenustatističnutaidealʹnuzbižnistʹmetričnoznačnihposlidovnostej |