$A$-cluster points via ideals
Following the line of the recent work by Sava¸s et al., we apply the notion of ideals to $A$-statistical cluster points. We get necessary conditions for the two matrices to be equivalent in a sense of $A^I$ -statistical convergence. In addition, we use Kolk’s idea to define and study $B^I$ -statis...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507539535298560 |
|---|---|
| author | Gurdal, M. Savaş, E. Гюрдал, М. Саваш, Є. |
| author_facet | Gurdal, M. Savaş, E. Гюрдал, М. Саваш, Є. |
| author_sort | Gurdal, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:24:16Z |
| description | Following the line of the recent work by Sava¸s et al., we apply the notion of ideals to $A$-statistical cluster points. We get
necessary conditions for the two matrices to be equivalent in a sense of $A^I$ -statistical convergence. In addition, we use
Kolk’s idea to define and study $B^I$ -statistical convergence. |
| first_indexed | 2026-03-24T02:10:55Z |
| format | Article |
| fulltext |
UDC 519.21
M. Gürdal (Suleyman Demirel Univ., Isparta, Turkey),
E. Savaş (Istanbul Ticaret Univ., Turkey)
\bfitA -CLUSTER POINTS VIA IDEALS
\bfitA -КЛАСТЕРНI ТОЧКИ В ТЕРМIНАХ IДЕАЛIВ
Following the line of the recent work by Savaş et al., we apply the notion of ideals to A-statistical cluster points. We get
necessary conditions for the two matrices to be equivalent in a sense of A\scrI -statistical convergence. In addition, we use
Kolk’s idea to define and study \scrB \scrI -statistical convergence.
Ми продовжуємо дослiдження, розпочате в нещодавнiй роботi Саваша та iн., i застосовуємо поняття iдеалiв до
A-статистичних кластерних точок. Отримано необхiднi умови для того, щоб двi матрицi були еквiвалентними в
сенсi A\scrI -статистичної збiжностi. Крiм того, ми застосовуємо iдею Колка для того, щоб визначити i вивчити поняття
\scrB \scrI -статистичної збiжностi.
1. Introduction and background. In [10], Fridy and Orhan introduced the concepts of statistical
limit superior and inferior. In [2], Connor and Kline extended the concept of a statistical limit (cluster)
point of a number sequence to a A-statistical limit (cluster) point where A is a nonnegative regular
summability matrix. In [4], Demirci extended the concepts of statistical limit superior and inferior
to A-statistical limit superior and inferior and given some A-statistical analogue of properties of
statistical limit superior and inferior for a sequence of real numbers. In [4], Kolk generalized the
idea of A-statistical convergence to \scrB -statistical convergence by using the idea of \scrB -summability
(or F -convergence) due to Steiglitz [30]. More works on matrix summability can be seen from [6],
where many references can be found.
On the other hand, the notion of ideal convergence was introduced first by P. Kostyrko et al. [18]
as an interesting generalization of statistical convergence [7, 31]. More recent applications of ideals
can be seen from [3, 13 – 15, 22 – 24, 26, 27] where more references can be found.
Naturally the purpose of this paper is to unify the above approaches and present the idea of
A-summability with respect to ideal concept and make certain observations. Further, we produce
\scrB -analogues via ideals of the results of Mursaleen and Edely [21].
First, we introduce some notation. Let A = (ank) denote a summability matrix which transforms a
number sequence x = (xk) into the sequence Ax whose nth term is given by (Ax)n =
\sum \infty
k=1
ankxk.
The notion of a statistically convergent sequence can be defined using the asymptotic density of
subsets of the set of positive integers \BbbN = \{ 1, 2, . . .\} . For any K \subseteq \BbbN and n \in \BbbN we denote
K (n) := | K \cap \{ 1, 2, . . . , n\} |
and we define lower and upper asymptotic density of the set K by the formulas
\delta (K) := \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
n\rightarrow \infty
K (n)
n
, \delta (K) := \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
K (n)
n
.
If \delta (K) = \delta (K) =: \delta (K), then the common value \delta (K) is called the asymptotic density of the set
K and
\delta (K) = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
K (n)
n
.
c\bigcirc M. GÜRDAL, E. SAVAŞ, 2017
324 ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3
A-CLUSTER POINTS VIA IDEALS 325
Obviously, the density \delta (K) (if it exist) lie in the unit interval [0, 1]
\delta (K) = \mathrm{l}\mathrm{i}\mathrm{m}
n
1
n
K (n) = \mathrm{l}\mathrm{i}\mathrm{m}
n
(C1\chi K)n = \mathrm{l}\mathrm{i}\mathrm{m}
n
1
n
n\sum
k=1
\chi K (k) ,
if it exists, where C1 is the Cesaro mean of order one and \chi K is the characteristic function of the set
K [8].
The notion of statistical convergence was originally defined for sequences of numbers in the paper
[7] and also in [29]. We say that a number sequence x = (xk)k\in \BbbN statistically converges to a point
L if for each \varepsilon > 0 we have \delta (K (\varepsilon )) = 0, where K (\varepsilon ) = \{ k \in \BbbN : | xk - L| \geq \varepsilon \} and in such
situation we will write L = \mathrm{s}\mathrm{t}-\mathrm{l}\mathrm{i}\mathrm{m}xk.
Statistical convergence can be generalized by using a regular nonnegative summability matrix A
in place of C1. Following Freedman and Sember [8], we say that a set K \subseteq \BbbN has A-density if
\delta A (K) = \mathrm{l}\mathrm{i}\mathrm{m}
n
\sum
k\in K
ank = \mathrm{l}\mathrm{i}\mathrm{m}
n
\infty \sum
k=1
ank\chi K (k) = \mathrm{l}\mathrm{i}\mathrm{m}
n
(A\chi K)n
exists where A is a nonnegative regular summability matrix.
The number sequence x = (xk)k\in \BbbN is said to be A-statistically convergent to L if for every
\varepsilon > 0, \delta A (\{ k \in \BbbN : | xk - L| \geq \varepsilon \} ) = 0. In this case it is denoted as \mathrm{s}\mathrm{t}A-\mathrm{l}\mathrm{i}\mathrm{m}xk = L [2, 20].
For i = 1, 2, . . . , let \scrB i = (bnk (i)) be an infinite matrix of complex (or real) numbers. Let \scrB
denote the sequence of matrices (\scrB i) . Then a sequence x \in \ell \infty , the space of bounded sequences,
is said to be F\scrB -convergent or \scrB -summable to some number L if
\sum \infty
k=1
bnk (i)xk converges to L
as n tends to \infty uniformly for i = 1, 2, . . . . L is said to be the \scrB -limit of x, written \scrB -\mathrm{l}\mathrm{i}\mathrm{m}x = L
(denotes the generalized limit) or (\scrB ix) \rightarrow L, and we say (\scrB ix) is convergent to L.
A sequence of matrices \scrB = (\scrB i) is regular (cf. [1, 30]) if and only if
(i) for each k = 1, 2, . . . , \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty bnk (i) = 0 uniformly for i = 1, 2, . . . ;
(ii) \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
\sum
k
bnk (i) = 1 uniformly for i = 1, 2, . . . ;
(iii) for each n, i = 1, 2, . . . ,
\sum
k=1
| bnk (i)| < \infty , and there exists integers N,M such that\sum
k=1
| bnk (i)| < M for n \geq N and all i = 1, 2, . . . .
In [17], Kolk introduced the following:
An index set K is said to have \scrB -density \delta \scrB (K) equal to d, if the characteristic sequence of K
is \scrB -summable to d, i.e.,
\mathrm{l}\mathrm{i}\mathrm{m}
n
\sum
k\in K
bnk (i) = d, uniformly in i,
where by an index set we mean a set K = \{ ki\} \subset \BbbN , ki < ki+1 for all i. For \scrB = \scrB 1, it is reduced
to uniform statistical convergence [25].
Let \scrR + denote the set of all regular methods \scrB with bnk (i) \geq 0 for all n, k and i.
Let \scrB \in \scrR +. A sequence x = (xk) is called \scrB -statistically convergent to the number L, if for
every \varepsilon > 0
\delta \scrB (\{ k \in \BbbN : | xk - L| \geq \varepsilon \} ) = 0
and we write \mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m}x = L.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3
326 M. GÜRDAL, E. SAVAŞ
The notion of statistical convergence was further generalized in the paper [18, 19] using the notion
of an ideal of subsets of the set \BbbN . We say that a nonempty family of sets \scrI \subset \scrP (\BbbN ) is an ideal on
\BbbN if \scrI is hereditary (i.e., B \subseteq A \in \scrI \Rightarrow B \in \scrI ) and additive (i.e., A,B \in \scrI \Rightarrow A \cup B \in \scrI ). An
ideal \scrI on \BbbN for which \scrI \not = \scrP (\BbbN ) is called a proper ideal. A proper ideal \scrI is called admissible if
\scrI contains all finite subsets of \BbbN . If not otherwise stated in the sequel \scrI will denote an admissible
ideal.
Recall the generalization of statistical convergence from [18, 19].
Let \scrI be an admissible ideal on \BbbN and x = (xk)k\in \BbbN be a sequence of points in a metric space
(X, \rho ). We say that the sequence x is \scrI -convergent (or \scrI -converges) to a point \xi \in X, and we
denote it by \scrI -\mathrm{l}\mathrm{i}\mathrm{m}x = \xi , if for each \varepsilon > 0 we have
A (\varepsilon ) = \{ k \in \BbbN : \rho (xk, \xi ) \geq \varepsilon \} \in \scrI .
This generalizes the notion of usual convergence, which can be obtained when we take for \scrI the ideal
\scrI f of all finite subsets of \BbbN . A sequence is statistically convergent if and only if it is \scrI \delta -convergent,
where \scrI \delta := \{ K \subset \BbbN : \delta (K) = 0\} is the admissible ideal of the sets of zero asymptotic density.
The concept of A\scrI -statistically convergent was studied in [28] and the following definition was
given:
Definition 1. Let A = (ank) be a nonnegative regular matrix. A sequence (xk)k\in \BbbN is said to
be A\scrI -statistically convergent to L if for any \varepsilon > 0 and \delta > 0\left\{ n \in \BbbN :
\sum
k\in K(\varepsilon )
ank \geq \delta
\right\} \in \scrI ,
where K (\varepsilon ) = \{ k \in \BbbN : | xk - L| \geq \varepsilon \} . In this case we write L = \scrI -\mathrm{s}\mathrm{t}A-\mathrm{l}\mathrm{i}\mathrm{m}xk.
By \scrI -\mathrm{s}\mathrm{t}A we denote the set of all A\scrI -statistically convergent sequences.
We say that a set K \subseteq \BbbN has A\scrI -density if
\delta A\scrI (K) := \scrI - \mathrm{l}\mathrm{i}\mathrm{m}
n
\sum
k\in K
ank = \scrI - \mathrm{l}\mathrm{i}\mathrm{m}
n
\infty \sum
k=1
ank\chi K (k) = \scrI - \mathrm{l}\mathrm{i}\mathrm{m}
n
(A\chi K)n ,
exists where A is a nonnegative regular summability matrix. Then a sequence x = (xk)k\in \BbbN is said
to be A\scrI -statistically convergent to L if for each \varepsilon > 0 the set K (\varepsilon ) has A\scrI -density zero, where
K (\varepsilon ) = \{ k \in \BbbN : | xk - L| \geq \varepsilon \} .
Let \scrI f be the family of all finite subsets of \BbbN . Then \scrI f is an admissible ideal in \BbbN and
A\scrI -statistically convergent is the A-statistical convergence introduced by [2, 20]. Also A\scrI -density
coincides with usual A-density in [8].
2. Consistency of \bfitA \bfscrI -statistical convergence. In this section we study the concepts of
A\scrI -statistical cluster points. The result are analogues to those given by Demirci [5]. These notions
generalize the notions of A-statistical cluster points. Also we get necessary conditions on the matrices
A and B so that A and B are equivalent in the A\scrI -statistical convergence sense.
Following the line of Savaş et al. [28] we now introduce the following definition using ideals.
Definition 2. Let \scrI be an ideal of \scrP (\BbbN ) . A number L is said to be an A\scrI -statistical clus-
ter point of the number sequence x = (xk) if for each \varepsilon > 0, \delta A\scrI (K\varepsilon ) \not = 0, where K\varepsilon =
= \{ k \in \BbbN : | xk - L| < \varepsilon \} . We denote the set of all A\scrI -statistically cluster points of x by \Gamma A\scrI (x) .
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3
A-CLUSTER POINTS VIA IDEALS 327
Note that the statement \delta A\scrI (K\varepsilon ) \not = 0 means that either \delta A\scrI (K\varepsilon ) > 0 or K\varepsilon fails to have
A\scrI -density.
Remark 1. If \scrI = \scrI f and A = (C1) , then the above Definition 2 yields the usual definition of
A-statistical cluster point of the number sequence introduced by [9].
Definition 3. If \scrI -\mathrm{s}\mathrm{t}A \supset \scrI -\mathrm{s}\mathrm{t}B, A is said to be stronger than B in the \scrI -statistical convergence
sense.
Definition 4. Matrices A and B are called consistent in the \scrI -statistical convergence sense if
\scrI -\mathrm{s}\mathrm{t}A-\mathrm{l}\mathrm{i}\mathrm{m}x = \scrI -\mathrm{s}\mathrm{t}B -\mathrm{l}\mathrm{i}\mathrm{m}x whenever x \in \scrI -\mathrm{s}\mathrm{t}A\cap \scrI -\mathrm{s}\mathrm{t}B. If A is stronger than B in the \scrI -statistical
convergence sense and consistent with B in the \scrI -statistical convergence sense, then write A
\scrI -st
\supset B.
If A
\scrI -st
\supset B and B
\scrI -st
\supset A, are called equivalent in the \scrI -statistical convergence sense. In this case it
is denoted as A
\scrI -st\thicksim B (see [12]).
Throughout this section A = (ank) and B = (bnk) will denote nonnegative regular summability
matrices.
Theorem 1. If the condition
\scrI - \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n
\infty \sum
k=1
| ank - bnk| = 0 (1)
holds, then \delta A\scrI (K) = 0 if and only if \delta B\scrI (K) = 0 for every K \subseteq \BbbN .
Proof. If \delta A\scrI (K) = 0, then \Biggl\{
n \in \BbbN :
\sum
k\in K
ank \geq \delta
\Biggr\}
\in \scrI
for any \delta > 0. Since
| (A\chi K )n - (B\chi K )n| \leq
\sum
k\in K
| ank - bnk| \leq
\infty \sum
k=1
| ank - bnk| ,
we have \scrI -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}n | (A\chi K )n - (B\chi K )n| = 0 by (1), which implies
\delta B\scrI (K) = \scrI - \mathrm{l}\mathrm{i}\mathrm{m}
n
\sum
k\in K
bnk = 0.
Sufficiency follows from the symmetry.
Hence we can get the following results from Theorem 1.
Theorem 2. If A and B satisfy the condition (1), then
(i) \scrI -\mathrm{s}\mathrm{t}A = \scrI -\mathrm{s}\mathrm{t}B,
(ii) \Gamma A\scrI (x) = \Gamma B\scrI (x) for a real number sequence x.
The \scrI -statistical limits in (i) of Theorem 2 agree (i.e., \scrI -\mathrm{s}\mathrm{t}B -\mathrm{l}\mathrm{i}\mathrm{m}x = L implies \scrI -\mathrm{s}\mathrm{t}A-\mathrm{l}\mathrm{i}\mathrm{m}x =
= L). Therefore, if A and B satisfy condition (1) of Theorem 1, then A and B are consistent in the
\scrI -statistical convergence sense.
Note that the support sets generated by nonnegative summability methods A and B can be used to
determine when, if a sequence x is both A\scrI -statistically convergent and B\scrI -statistically convergent,
the A\scrI -statistical and B\scrI -statistical limits of x agree. In [2], Connor and Kline, using the „\beta \BbbN
program” have shown that A and B assign the same statistical limit to x if KA \cap KB \not = \varnothing , where
the sets KA and KB are the support sets of the nonnegative regular summability matrices A and B.
The next corollary shows that we have the same result under different conditions.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3
328 M. GÜRDAL, E. SAVAŞ
Corollary 1. If A and B satisfy the conditions (1) of Theorem 1, then A
\scrI -st\thicksim B .
Definition 5. The real number sequence x = (xk) is said to be A\scrI -statistically bounded if there
is a number K such that \delta A\scrI (\{ k \in \BbbN : | xk| > K\} ) = 0.
Recall that A\scrI -statistically boundedness of real number sequences implies that \scrI -\mathrm{s}\mathrm{t}A-\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}x
and \scrI -\mathrm{s}\mathrm{t}A-\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} x are finite and \scrI -\mathrm{s}\mathrm{t}A-\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}x and \scrI -\mathrm{s}\mathrm{t}A-\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} x are the greatest and least
A\scrI -statistically cluster point of such an x [16].
For any complex number sequence x = (xk) the A-statistical core of x is given by
\mathrm{s}\mathrm{t}A-\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e} \{ x\} =
\bigcap
H\in \scrH (x)
H,
where \scrH (x) is the collection of all closed half-planes H that satisfy \delta A (\{ k \in \BbbN : xk \in H\} ) = 1
(see [4]).
From Theorem 6 in [4], it is shown that for every A-statistically bounded complex number
sequence x = (xk)
\mathrm{s}\mathrm{t}A-\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e} \{ x\} =
\bigcap
z\in \BbbC
Bx (z) ,
where Bx (z) = \{ w \in \BbbC : | w - z| \leq \scrI -\mathrm{s}\mathrm{t}A- \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}k | xk - z| \} . When A = C1 we shall simply
write \mathrm{s}\mathrm{t}-\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e} instead of \mathrm{s}\mathrm{t}C1 -\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e} (see [11]).
Recall that the core of any A-statistically bounded real number sequence x, that is, \mathrm{s}\mathrm{t}A-\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e} \{ x\} ,
is the interval [\mathrm{s}\mathrm{t}A- \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} x, \mathrm{s}\mathrm{t}A- \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}x] [4]. In analogy to the \mathrm{s}\mathrm{t}A-\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e} \{ x\} we first give a
definition of A\scrI -core of bounded real number sequence x as follows.
Definition 6. If x is any A\scrI -statistically bounded real number sequence, then we define its
A\scrI -core by \Bigl[
\scrI -\mathrm{s}\mathrm{t}A- \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} x, \scrI -\mathrm{s}\mathrm{t}A- \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}x
\Bigr]
.
We use \scrI -\mathrm{s}\mathrm{t}A-\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e} (x) to denote A\scrI -core of real number sequence x.
Hence we can get the following from (ii) of Theorem 2.
Corollary 2. If A and B satisfy the conditions (1) of Theorem 1, then \scrI -\mathrm{s}\mathrm{t}A-\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e} \{ x\} = \scrI -\mathrm{s}\mathrm{t}B -
\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e} \{ x\} for every bounded real sequence x.
Let \scrI = \scrI f . Then all these results imply the similar theorems for A-statistical cluster points
which are investigated in [5].
3. \scrB -statistical convergence via ideals. In this section, we produce \scrB -analogues via ideals of
the results of Fridy and Orhan [10].
We give some analogue definitions for the method \scrB .
Definition 7. A sequence x = (xk)k\in \BbbN is called \scrB \scrI -statistically convergent to the number L, if
for any \varepsilon > 0 and \delta > 0\left\{ n \in \BbbN :
\sum
k\in K(\varepsilon )
bnk (i) \geq \delta for all i = 1, 2, . . .
\right\} \in \scrI ,
where K (\varepsilon ) = \{ k \in \BbbN : | xk - L| \geq \varepsilon \} . In this case we write L = \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m}xk.
We say that a set K \subseteq \BbbN has \scrB \scrI -density if
\delta \scrB \scrI (K) := \scrI - \mathrm{l}\mathrm{i}\mathrm{m}
n
\sum
k\in K
bnk (i) = \scrI - \mathrm{l}\mathrm{i}\mathrm{m}
n
\infty \sum
k=1
bnk (i)\chi K (k) =
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3
A-CLUSTER POINTS VIA IDEALS 329
= \scrI - \mathrm{l}\mathrm{i}\mathrm{m}
n
(\scrB \chi K)n , uniformly for i = 1, 2, . . .
exists. Then a sequence x = (xk)k\in \BbbN is said to be \scrB \scrI -statistically convergent to L if for each \varepsilon > 0
the set K (\varepsilon ) has \scrB \scrI -density zero, where K (\varepsilon ) = \{ k \in \BbbN : | xk - L| \geq \varepsilon \} .
Throughout the paper by \delta \scrB \scrI (K) \not = 0 we mean that either \delta \scrB \scrI (K) > 0 or K fails to have
\scrB -density.
Let \scrI f be the family of all finite subsets of \BbbN . Let \scrI = \scrI f , then \scrB \scrI -statistically convergent
is the \scrB -statistical convergence introduced by [21]. In particular, if \scrI = \scrI f and \scrB = (C1) , then
\scrB \scrI -statistical convergence is reduced the usual statistical convergence. For \scrB = (A) , it is reduced
to A\scrI -statistical cluster point [16].
Definition 8. Let \scrI be an ideal of \scrP (\BbbN ) . The number \zeta is said to be \scrB \scrI -statistical cluster point
of a sequence x = (xk) if for each \varepsilon > 0, \delta \scrB \scrI (K\varepsilon ) \not = 0, where K\varepsilon = \{ k \in \BbbN : | xk - \zeta | < \varepsilon \} . We
denote the set of all \scrB \scrI -statistically cluster points of x by \Gamma \scrB \scrI (x) .
Note that for \scrB =(A) in Definition 8, we get A\scrI -statistical cluster point [16]. For \scrB = (C1) and
\scrI = \scrI f , these are reduced to the usual statistical cluster point [9]. For a number sequence x = (xk) ,
we write
Mg = \{ g \in \BbbR : \delta \scrB \scrI \{ k : xk > g\} \not = 0\} and Mf = \{ f \in \BbbR : \delta \scrB \scrI \{ k : xk < f\} \not = 0\} .
Then we define the \scrB -statistical limit superior and \scrB -statistical limit inferior of x as follows:
\scrI -\mathrm{s}\mathrm{t}\scrB - \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}x =
\Biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}Mg, Mg \not = \varnothing ,
- \infty , Mg = \varnothing ,
and
\scrI -\mathrm{s}\mathrm{t}\scrB - \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} x =
\Biggl\{
\mathrm{i}\mathrm{n}\mathrm{f}Mf , Mf \not = \varnothing ,
+\infty , Mf = \varnothing .
Definition 9. The real number sequence x = (xk) is said to be \scrB \scrI -statistically bounded if there
is a number K such that
\delta \scrB \scrI (\{ k \in \BbbN : | xk| > K\} ) = 0.
The next statement is an analogue of Theorem 2.7 of [21].
Theorem 3. (a) If \beta = \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}x is finite, then for each \varepsilon > 0
\delta \scrB \scrI (\{ k \in \BbbN : xk > \beta - \varepsilon \} ) \not = 0 and \delta \scrB \scrI (\{ k \in \BbbN : xk > \beta + \varepsilon \} ) = 0. (2)
Conversely, if (2) holds for each \varepsilon > 0 then \beta = \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}x.
(b) If \alpha = \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} x is finite, then for each \varepsilon > 0,
\delta \scrB \scrI (\{ k \in \BbbN : xk < \alpha + \varepsilon \} ) \not = 0 and \delta \scrB \scrI (\{ k \in \BbbN : xk < \alpha - \varepsilon \} ) = 0. (3)
Conversely, if (3) holds for each \varepsilon > 0, then \alpha = \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} x.
By Definition 8 we see that Theorem 3 can be interpreted by saying that \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}x and \scrI
-\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} x are the greatest and the least \scrB \scrI -statistically cluster points of x.
The next theorem reinforces this observation.
Theorem 4. For every real sequence x,
\scrI -\mathrm{s}\mathrm{t}\scrB - \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} x \leq \scrI -\mathrm{s}\mathrm{t}\scrB - \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}x.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3
330 M. GÜRDAL, E. SAVAŞ
Proof. First consider the case in which \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}x = - \infty . Hence we have Mg = \varnothing , so for
every g \in \BbbR , \delta \scrB \scrI \{ k : xk > g\} = 0 which implies that \delta \scrB \scrI \{ k : xk \leq g\} = 1, so for every f \in \BbbR ,
\delta \scrB \scrI \{ k : xk < f\} \not = 0. Hence, \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} x = - \infty .
The case in which \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}x = +\infty needs no proof, so we next assume that \beta = \scrI -\mathrm{s}\mathrm{t}\scrB -
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}x is finite, and let \alpha = \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} x. Given \varepsilon > 0 we show that \beta + \varepsilon \in Mf , so that
\alpha \leq \beta +\varepsilon . By Theorem 3(a), \delta \scrB \scrI
\Bigl\{
k : xk > \beta +
\varepsilon
2
\Bigr\}
= 0, since \beta = \mathrm{s}\mathrm{u}\mathrm{p} \{ g \in \BbbR : \delta \scrB \scrI \{ k : xk > g\} \not =
\not = 0\} . This implies \delta \scrB \scrI
\Bigl\{
k : xk \leq \beta +
\varepsilon
2
\Bigr\}
= 1, which, in turn, gives \delta \scrB \scrI \{ k : xk < \beta + \varepsilon \} = 1.
Hence \beta + \varepsilon \in Mf , and since \varepsilon is arbitrary this proves that \alpha \leq \beta .
Remark 2. If \scrI -\mathrm{s}\mathrm{t}A-\mathrm{l}\mathrm{i}\mathrm{m}x exists, then a sequence x is A\scrI -statistically bounded.
Note that \scrB \scrI -statistical boundedness of real number sequences implies that \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p} and
\scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} are finite, so that properties (a) and (b) of Theorem 3 hold good.
Theorem 5. The \scrB \scrI -statistically bounded sequence x is \scrB \scrI -statistically convergent if and only
if \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} x = \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}x.
Proof. We prove the necessity first. Let L = \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m}x and \varepsilon > 0. Then
\delta \scrB \scrI (\{ k \in \BbbN : xk > L+ \varepsilon \} ) = 0 and \delta \scrB \scrI (\{ k \in \BbbN : xk < L - \varepsilon \} ) = 0.
So for any g \geq L + \varepsilon and f < L - \varepsilon , the sets \delta \scrB \scrI (Mg) = 0 and \delta \scrB \scrI
\bigl(
Mf
\bigr)
= 0. We conclude
\mathrm{s}\mathrm{u}\mathrm{p} \{ g : \delta \scrB \scrI (Mg) \not = 0\} \leq L + \varepsilon and \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
f : \delta \scrB \scrI
\bigl(
Mf
\bigr)
\not = 0
\bigr\}
\geq L - \varepsilon . Combining with Theorem
4, we conclude that L = \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} x = \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}x.
To prove sufficiency, suppose that L = \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} x = \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}x and x be \scrB \scrI -statistical
bounded. Then for \varepsilon > 0, by (2) and (3), we have
\delta \scrB \scrI
\Bigl( \Bigl\{
k : xk > L+
\varepsilon
2
\Bigr\} \Bigr)
= 0 and \delta \scrB \scrI
\Bigl( \Bigl\{
k : xk < L - \varepsilon
2
\Bigr\} \Bigr)
= 0.
We conclude that L = \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m}x.
We state the following result without proof, since the result can be established using same the
technique applied for the Theorems 3.3 and 3.4 of [21].
Theorem 6. (i) If number sequence x is bounded from above and \scrB -summable to the number
L = \scrI -\mathrm{s}\mathrm{t}\scrB -\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}x, then x is \scrB \scrI -statistical convergent to L.
(ii) If number sequence x is bounded from below and \scrB -summable to the number L = \scrI -\mathrm{s}\mathrm{t}\scrB -
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} x, then x is \scrB \scrI -statistical convergent to L.
Let \scrI = \scrI f . Then all these results in Section 3 imply the similar theorems for \scrB -statistical
convergence which are investigated in [21].
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Received 04.10.13,
after revision — 23.11.16
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3
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| id | umjimathkievua-article-1699 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:10:55Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2c/ef44e24e82b535e1a23a3700bb13822c.pdf |
| spelling | umjimathkievua-article-16992019-12-05T09:24:16Z $A$-cluster points via ideals $ {A}$ -кластернi точки в термiнах iдеалiв Gurdal, M. Savaş, E. Гюрдал, М. Саваш, Є. Following the line of the recent work by Sava¸s et al., we apply the notion of ideals to $A$-statistical cluster points. We get necessary conditions for the two matrices to be equivalent in a sense of $A^I$ -statistical convergence. In addition, we use Kolk’s idea to define and study $B^I$ -statistical convergence. Ми продовжуємо дослiдження, розпочате в нещодавнiй роботi Саваша та iн., i застосовуємо поняття iдеалiв до $A$-статистичних кластерних точок. Отримано необхiднi умови для того, щоб двi матрицi були еквiвалентними в сенсi $A^I$ -статистичної збiжностi. Крiм того, ми застосовуємо iдею Колка для того, щоб визначити i вивчити поняття $B^I$ -статистичної збiжностi. Institute of Mathematics, NAS of Ukraine 2017-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1699 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 3 (2017); 324-331 Український математичний журнал; Том 69 № 3 (2017); 324-331 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1699/681 Copyright (c) 2017 Gurdal M.; Savaş E. |
| spellingShingle | Gurdal, M. Savaş, E. Гюрдал, М. Саваш, Є. $A$-cluster points via ideals |
| title | $A$-cluster points via ideals |
| title_alt | $ {A}$ -кластернi точки в термiнах iдеалiв |
| title_full | $A$-cluster points via ideals |
| title_fullStr | $A$-cluster points via ideals |
| title_full_unstemmed | $A$-cluster points via ideals |
| title_short | $A$-cluster points via ideals |
| title_sort | $a$-cluster points via ideals |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1699 |
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