On the lacunary $(A, ϕ)$ -statistical convergence of double sequences
We extend some results known from the literature for ordinary (single) sequences to multiple sequences of real numbers. Further, we introduce a concept of double lacunary strong $(A, ϕ)$-convergence with respect to a modulus function. In addition, we also study some relationships between double lacu...
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| Дата: | 2018 |
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2018
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507408835543040 |
|---|---|
| author | Savaş, E. Саваш, Є. |
| author_facet | Savaş, E. Саваш, Є. |
| author_sort | Savaş, E. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:19:59Z |
| description | We extend some results known from the literature for ordinary (single) sequences to multiple sequences of real numbers.
Further, we introduce a concept of double lacunary strong $(A, ϕ)$-convergence with respect to a modulus function. In
addition, we also study some relationships between double lacunary strong $(A, ϕ)$-convergence with respect to a modulus
and double lacunary statistical convergence. |
| first_indexed | 2026-03-24T02:08:51Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 519.21, 517.5
E. Savaş (Uşak Univ., Turkey)
ON THE LACUNARY (\bfitA ,\bfitvarphi )-STATISTICAL CONVERGENCE
OF DOUBLE SEQUENCES
ПРО ЛАКУНАРНУ (\bfitA ,\bfitvarphi )-СТАТИСТИЧНУ ЗБIЖНIСТЬ
ПОДВIЙНИХ ПОСЛIДОВНОСТЕЙ
We extend some results known from the literature for ordinary (single) sequences to multiple sequences of real numbers.
Further, we introduce a concept of double lacunary strong (A,\varphi )-convergence with respect to a modulus function. In
addition, we also study some relationships between double lacunary strong (A,\varphi )-convergence with respect to a modulus
and double lacunary statistical convergence.
Деякi вiдомi результати для звичайних (одинарних) послiдовностей поширено на багатократнi послiдовностi дiйсних
чисел. Крiм того, введено поняття подвiйної лакунарної сильної (A,\varphi )-збiжностi вiдносно функцiї модуля, а також
вивчено деякi спiввiдношення мiж подвiйною лакунарною сильною (A,\varphi )-збiжнiстю вiдносно модуля та подвiйною
лакунарною статистичною збiжнiстю.
1. Introduction. A notion of a modulus function was introduced by Nakano [10]. We recall that a
modulus f is a function from [0,\infty ) to [0,\infty ) such that
(i) f(x) = 0 if and only if x = 0,
(ii) f(x+ y) \leq f(x) + f(y) for all x, y \geq 0,
(iii) f is increasing,
(iv) f is continuous from the right at zero.
A modulus may be bounded or unbounded. For example, f(x) = xp, for 0 < p \leq 1 is un-
bounded, but f(x) =
x
1 + x
is bounded (see [13]).
The class of sequences which are strongly Cesàro summable with respect to a modulus was in-
troduced by Maddox [6] as an extension of the definition of strongly Cesàro summable sequences.
Connor [1] further extended this definition by replacing the Cesàro matrix with an abritrary non-
negative regular matrix summability A and established some elementary connections between strong
A-summability with respect to a modulus and A-statistical convergence. Recently E. Savas [14]
generalized the concept of strong almost convergence by using a modulus function and examined
some properties of the corresponding new sequence spaces. Malkowsky and Savas [8] introduced
and studied some sequence spaces which arise from the notation of generalized de la Vallée Poussin
means and the concept of a modulus function. Furthermore, the four dimensional matrix transfor-
mation (Ax)m,n =
\sum \infty
k,l=1
am,n,k,lxk,l was studied extensively by Robison [12] and Hamilton [5],
respectively. In their work and throughout this paper, the four dimensional matrices and double
sequences have real-valued entries unless specified otherwise.
In [11] the notion of convergence for double sequences was presented by A. Pringsheim.
Before continuing with this paper we present a few definitions and preliminaries.
c\bigcirc E. SAVAŞ, 2018
848 ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 6
ON THE LACUNARY (A,\varphi )-STATISTICAL CONVERGENCE OF DOUBLE SEQUENCES 849
A lacunary sequence \theta = (kr), r = 0, 1, 2, . . . , where k0 = 0, is an increasing sequence of
nonnegative integers such that hr = kr - kr - 1 \rightarrow \infty as r \rightarrow \infty . The intervals determined by \theta will
be denoted by Ir = (kr - 1, kr] and qr =
kr
kr - 1
.
The space of lacunary strongly convergent sequences N\theta was defined by Freedman et al. [3] as
follows:
N\theta =
\left\{ x = (xk) : \mathrm{l}\mathrm{i}\mathrm{m}
r
1
hr
\sum
k\in Ir
| xk - L| ) = 0 for some L
\right\} .
The double sequence \theta r,s = \{ (kr, ls)\} is called \bfd \bfo \bfu \bfb \bfl \bfe \bfl \bfa \bfc \bfu \bfn \bfa \bfr \bfy if there exist two increasing
sequences of integers such that
k0 = 0, hr = kr - kr - 1 \rightarrow \infty as r \rightarrow \infty
and
l0 = 0, \=hs = ls - ls - 1 \rightarrow \infty as s\rightarrow \infty .
Let us denote kr,s = krls, hr,s = hr\=hs and \theta r,s is determine by Ir,s = \{ (k, l) : kr - 1 < k \leq
\leq kr and ls - 1 < l \leq ls\} , qr =
kr
kr - 1
, \=qs =
ls
ls - 1
, and qr,s = qr\=qs.
For more recent developments on double sequences one can consult the papers (see [17 – 24]),
where more references can be found.
By a \varphi -function we understand a continuous nondecreasing function \varphi (u) defined for u \geq 0 and
such that \varphi (0) = 0, \varphi (u) > 0 for u > 0 and \varphi (u) \rightarrow \infty as u\rightarrow \infty (see [25]).
A \varphi -function \varphi is called non weaker than a \varphi -function \psi if there are constants c, b, k, l > 0
such that c\psi (lu) \leq b\varphi (ku) (for all large u) and we write \psi \prec \varphi . A \varphi -functions \varphi and \psi are
called equivalent if there are positive constants b1, b2, c, k1, k2, l such that b1\varphi (k1u) \leq c\psi (lu) \leq
\leq b2\varphi (k2u) (for all large u) and we write \varphi \sim \psi .
In the present paper, we introduce and study an idea of double lacunary strong (A,\varphi )-convergence
with respect to a modulus function. We also investigate the relationship between double lacunary
strong (A,\varphi )-convergence with respect to a modulus and double lacunary (A,\varphi )-statistical conver-
gence.
2. Main results. Throughout this paper we shall examine our sequence spaces using the
following type of transformation:
Definition 2.1. Let A = (am,n,k,l) denote a four dimensional summability method that maps the
real double sequences x into the double sequence Ax where the (mn)th term to Ax is as follows:
(Ax)m,n =
\infty \sum
k,l=1
am,n,k,lxk,l.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 6
850 E. SAVAŞ
Such transformation is said to be nonnegative if all am,n,k,l is nonnegative.
By the convergence of a double sequence we mean the convergence in the Pringsheim sense that
is, a double sequence x = (xk,l) has Pringsheim limit L (denoted by \mathrm{P} - \mathrm{l}\mathrm{i}\mathrm{m}x = L) provided that
given \varepsilon > 0 there exists an N \in \BbbN such that | xk,l - L| < \varepsilon whenever k, l > \BbbN [11]. We shall
describe such an x more briefly as “P-convergent”.
Let \varphi and f be given \varphi -function and modulus function, respectively. Moreover, let A =
= (am,n,k,l) be a nonnegative four dimensional matrix of real entries and double lacunary sequence
\theta be given. Then we define the following:
N2
\theta (A,\varphi , f) =
=
\left\{ x = (xk,l) : \mathrm{P} - \mathrm{l}\mathrm{i}\mathrm{m}
r,s
1
hr,s
\sum
(m,n)\in Ir,s
f
\left( \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l - L| )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right) = 0 for some L
\right\}
and
N2
\theta (A,\varphi , f)0 =
\left\{ x = (xk,l) : \mathrm{P} - \mathrm{l}\mathrm{i}\mathrm{m}
r,s
1
hr,s
\sum
(m,n)\in Ir,s
f
\left( \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l| )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right) = 0
\right\} .
If x \in N2
\theta (A,\varphi , f)0, the sequence x is said to be double lacunary strong (A,\varphi )-convergent to
zero with respect to a modulus f.
If f(x) = x, we write
N2
\theta (A,\varphi ) =
\left\{ x = (xk,l) : \mathrm{P} - \mathrm{l}\mathrm{i}\mathrm{m}
r,s
1
hr,s
\sum
(m,n)\in Ir,s
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l - L| )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| = 0 for some L
\right\} .
If we take A = I and \varphi (x) = x respectively, then we have
N2
\theta (f) =
\left\{ x = (xk,l) : \mathrm{P} - \mathrm{l}\mathrm{i}\mathrm{m}
r,s
1
hr,s
\sum
(k,l)\in Ir,s
f (| xk,l - L| ) = 0 for some L
\right\} .
If we take A = I, \varphi (x) = x and f(x) = x respectively, then we obtain
N2
\theta =
\left\{ x = (xk,l) : \mathrm{P} - \mathrm{l}\mathrm{i}\mathrm{m}
r,s
1
hr,s
\sum
(k,l)\in Ir,s
| xk,l - L| = 0 for some L
\right\} ,
which was defined and studied in [18].
In the next theorem we establish inclusion relations between w2(A,\varphi , f)0 and N2
\theta (A,\varphi , f)0. We
now have the following theorem.
Theorem 2.1. Let f be any modulus function, \varphi -function \varphi , and let A = (am,n,k,l) be a
nonnegative four dimensional matrix of real entries and double lacunary sequence \theta be given. If
w2(A,\varphi , f)0 =
\left\{ x = (xk,l) : \mathrm{P} - \mathrm{l}\mathrm{i}\mathrm{m}
i,j
1
ij
ij\sum
m,n=1
f
\left( \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k,l=1
am,n,k,l\varphi (| xkl| )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right) = 0
\right\} ,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 6
ON THE LACUNARY (A,\varphi )-STATISTICAL CONVERGENCE OF DOUBLE SEQUENCES 851
then the following relation is true:
if \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} qr > 1 and \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} \=qs > 1, then we have w2(A,\varphi , f)0 \subseteq N2
\theta (A,\varphi , f)0.
Proof. Let us suppose that x \in w2(A,\varphi , f)0. There exists \delta > 0 such that qr > 1 + \delta for
sufficiently large r and \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f} \=qs > 1 + \delta for sufficiently large s we get hr/kr \geq \delta /(1 + \delta ) for
sufficiently large r and
\=hs
ls
\geq \delta
1 + \delta
for sufficiently large s. Then
1
kr,s
kr,s\sum
n,m=1
f
\left( \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k,l=1
am,n,k,l\varphi (| xkl| )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right) \geq
\geq 1
kr,s
\sum
(m,n)\in Ir,s
f
\left( \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k,l=1
am,n,k,l\varphi (| xkl| )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right) =
=
hr,s
kr,s
1
hr,s
\sum
(m,n)\in Ir,s
f
\left( \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k,l=1
am,n,k,l\varphi (| xkl| )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right) \geq
\geq
\biggl(
\delta
1 + \delta
\biggr) 2 1
hr,s
\sum
(m,n)\in Ir,s
f
\left( \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k,l=1
am,n,k,l\varphi (| xkl| )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right) .
Hence, x \in N2
\theta (A,\varphi , f)0.
Theorem 2.1 is proved.
We now have the following theorem.
Theorem 2.2. N2
\theta (A,\varphi ) \subset N2
\theta (A,\varphi , f).
Proof. Let x \in N2
\theta (A,\varphi ). For a given \varepsilon > 0 we choose 0 < \delta < 1 such that f(x) < \varepsilon for every
x \in [0, \delta ]. We have
1
hr,s
\sum
(m,n)\in Ir,s
f
\left( \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l - L| )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right) = S11 + S22,
where
S11 =
1
hr,s
\sum
(m,n)\in Ir,s
f
\left( \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l - L| )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right)
and this sum is taken over
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l - L| ) \leq \delta ,
and
S22 =
1
hr,s
\sum
(m,n)\in Ir,s
f
\left( \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l - L| )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 6
852 E. SAVAŞ
and this sum is taken over
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l - L| ) > \delta .
By definition of the modulus f we obtain
S11 =
1
hr,s
\sum
(m,n)\in Ir,s
f(\delta ) = f(\delta ) < \varepsilon
and further
S22 = f(1)
1
\delta
1
hr,s
\sum
(m,n)\in Ir,s
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l - L| ).
Finally, we have x \in N2
\theta (A,\varphi , f).
Theorem 2.2 is proved.
3. Double A-statistical convergence. The concept of statistical convergence was introduced
by Fast [2] in 1951. A real number sequence x is said to be statistically convergent to the number L
if for every \varepsilon > 0
\mathrm{l}\mathrm{i}\mathrm{m}
n
1
n
| \{ k \leq n : | xk - L| \geq \varepsilon \} | = 0,
where by k \leq n we mean that k = 0, 1, 2, . . . , n and the vertical bars indicate the number of
elements in the enclosed set. In this case we write st1 - \mathrm{l}\mathrm{i}\mathrm{m}x = L or xk \rightarrow L(st1).
We first recall the definition of lacunary statistical convergence of a sequence of real numbers
which is defined by Friday and Orhan [4] as follows. Let \theta be a lacunary sequence; the number
sequence x is S\theta -convergent to L provided that for every \varepsilon > 0
\mathrm{l}\mathrm{i}\mathrm{m}
r
1
hr
| \{ k \in Ir : | xk - L| \geq \varepsilon \} | = 0.
In this case we write S\theta - \mathrm{l}\mathrm{i}\mathrm{m}x = L or xk \rightarrow L(S\theta ).
Let K \subseteq \BbbN \times \BbbN be a two dimensional set of natural numbers and let 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
\mathrm{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
m,n=1
has a limit then we say that K has a natural density
and is defined
\mathrm{P} - \mathrm{l}\mathrm{i}\mathrm{m}
m,n
Km,n
mn
= \delta 2(K).
For example, let K = \{ (i2, j2) : i, j \in \BbbN \} . Then
\delta 2(K) = \mathrm{P} - \mathrm{l}\mathrm{i}\mathrm{m}
m,n
Km,n
mn
\leq \mathrm{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., 2018, т. 70, № 6
ON THE LACUNARY (A,\varphi )-STATISTICAL CONVERGENCE OF DOUBLE SEQUENCES 853
Recently, Mursaleen and Edely [9], defined the statistical analogue for double sequences x =
= (xk,l) as follows: a real double sequences x = (xk,l) is said to be P -statistically convergent to L
provided that for each \varepsilon > 0
\mathrm{P} - \mathrm{l}\mathrm{i}\mathrm{m}
k,l
1
kl
\{ number of (m,n) : k \leq m and l \leq n, | xk,l - L| \geq \varepsilon \} = 0.
In this case we write st2 - \mathrm{l}\mathrm{i}\mathrm{m}kl xk,l = L and we denote the set of all statistical convergent
double sequences by st2.
Furthermore, Savas and Patterson [15] studied double lacunary sequence spaces as follows:
Definition 3.1. Let \theta r,s be a double lacunary sequence; the double sequence x is S\theta r,s -convergent
to L provided that for every \varepsilon > 0,
\mathrm{P} - \mathrm{l}\mathrm{i}\mathrm{m}
r,s
1
hr,s
| \{ (k, l) \in Ir,s : | xk,l - L| \geq \varepsilon \} | = 0.
In this case we write S2
\theta - \mathrm{l}\mathrm{i}\mathrm{m}k,l xk,l = L and we denote the set of all statistical convergent
double sequences by S2
\theta .
We now define the following: Let \theta be a double lacunary sequence, and let the nonnegative
matrix A = (am,n,k,l), the sequence x = (xkl), the \varphi - function \varphi (x) and a positive number \varepsilon > 0
be given. We write
K2
\theta (A,\varphi , \varepsilon ) =
\left\{ (n,m) \in Ir,s :
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l - L| ) \geq \varepsilon
\right\} .
The sequence x is said to be double lacunary (A,\varphi )-statistically convergent to a number zero if for
every \varepsilon > 0
\mathrm{P} - \mathrm{l}\mathrm{i}\mathrm{m}
r,s
1
hr,s
\mu (K2
\theta (A,\varphi , \varepsilon )) = 0,
where \mu (K2
\theta (A,\varphi , \varepsilon )) denotes the number of elements belonging to K2
\theta (A,\varphi , \varepsilon ). We denote by
S2
\theta (A,\varphi ), the set of sequences x = (xk,l) which are double lacunary (A,\varphi )-statistical convergent to
zero. We write
S2
\theta (A,\varphi ) =
\biggl\{
x = (xk,l) : \mathrm{P} - \mathrm{l}\mathrm{i}\mathrm{m}
r,s
1
hr,s
\mu (K2
\theta (A,\varphi , \varepsilon )) = 0
\biggr\}
.
If we take A = I and \varphi (x) = x respectively, then S2
\theta (A,\varphi ) reduce to S2
\theta (see [16]).
In the next theorem we prove the following inclusion.
Theorem 3.1. If \psi \prec \varphi , then S2
\theta (A,\psi ) \subset S2
\theta (A,\varphi ).
Proof. By assumption we have \psi (| xk,l - L| ) \leq b\varphi (c| xk,l - L| ) and
\infty \sum
k,l=0
am,n,k,l\psi (| xk,l - L| ) \leq b
\infty \sum
k,l=0
am,n,k,l\varphi (c| xk,l - L| ) \leq M
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l - L| )
for b, c > 0, where the constant M is connected with properties of \varphi . Thus, the condition
\infty \sum
k,l=0
am,n,k,l\psi (| xk,l - L| ) \geq \varepsilon
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 6
854 E. SAVAŞ
implies the condition
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l - L| ) \geq \varepsilon
and in consequence we get
\mu (K2
\theta (A,\varphi , \varepsilon )) \subset \mu (K2
\theta (A,\psi , \varepsilon ))
and
\mathrm{l}\mathrm{i}\mathrm{m}
r,s
1
hr,s
\mu (K2
\theta (A,\varphi , \varepsilon )) \leq \mathrm{l}\mathrm{i}\mathrm{m}
r,s
1
hr,s
\mu (K2
\theta (A,\psi , \varepsilon )).
Theorem 3.1 is proved.
We establish a relation between the sets N2
\theta (A,\varphi , f) and S2
\theta (A,\varphi ) as follows:
Theorem 3.2. (i) If the nonnegative double matrix A, the lacunary sequence \theta and functions
f and \varphi are given, then
N2
\theta (A,\varphi , f) \subset S2
\theta (A,\varphi ).
(ii) If the \varphi -function \varphi (u) and the nonnegative double matrix A are given, and the modulus
function f is bounded, then
S2
\theta (A,\varphi ) \subset N2
\theta (A,\varphi , f).
(iii) If the \varphi -function \varphi (u) and the nonnegative double matrix A are given, and the modulus
function f is bounded, then
S2
\theta (A,\varphi ) = N2
\theta (A,\varphi , f).
Proof. (i) Let f be a modulus function and \varepsilon > 0. We can write the inequalities
1
hr,s
\sum
(m,n)\in Ir,s
f
\left( \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l - L| )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right) \geq
\geq 1
hr,s
\sum
(m,n)\in I1r,s
f
\left( \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l - L| )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right) \geq
\geq 1
hr,s
f(\varepsilon )\mu (K\theta (A,\varphi , \varepsilon )),
where
I1r,s =
\left\{ (m,n) \in Ir,s :
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l - L| )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \geq \varepsilon
\right\} .
Finally, if x \in N2
\theta (A,\varphi , f), then x \in S2
\theta (A,\varphi ).
(ii) Let us suppose that x \in S2
\theta (A,\varphi ). If the modulus function f is a bounded function, then
there exists an integer M such that f(x) < M for all x \geq 0. Let us write
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 6
ON THE LACUNARY (A,\varphi )-STATISTICAL CONVERGENCE OF DOUBLE SEQUENCES 855
I2r,s =
\left\{ (m,n) \in Ir,s :
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l - L| )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| < \varepsilon
\right\} .
Thus we write
1
hr,s
\sum
(m,n)\in Ir,s
f
\left( \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l - L| )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right) \leq
\leq 1
hr,s
\sum
(m,n)\in I1r,s
f
\left( \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l - L| )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right) +
+
1
hr,s
\sum
(m,n)\in I2r,s
f
\left( \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \sum
k,l=0
am,n,k,l\varphi (| xk,l - L| )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\right) \leq
\leq 1
hr,s
M\mu (K\theta (A,\varphi , \varepsilon )) + f(\varepsilon ).
Taking the limit as \varepsilon \rightarrow 0, we obtain that x \in N2
\theta (A,\varphi ).
The proof of (iii) follows from (i) and (ii).
Theorem 3.2 is proved.
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856 E. SAVAŞ
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Received 27.12.16
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 6
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| id | umjimathkievua-article-1599 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:08:51Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f8/067e20357a1231fc0c9206cbfdef8bf8.pdf |
| spelling | umjimathkievua-article-15992019-12-05T09:19:59Z On the lacunary $(A, ϕ)$ -statistical convergence of double sequences Про лакунарну $(A, ϕ)$-статистичну збiжнiсть подвiйних послiдовностей Savaş, E. Саваш, Є. We extend some results known from the literature for ordinary (single) sequences to multiple sequences of real numbers. Further, we introduce a concept of double lacunary strong $(A, ϕ)$-convergence with respect to a modulus function. In addition, we also study some relationships between double lacunary strong $(A, ϕ)$-convergence with respect to a modulus and double lacunary statistical convergence. Деякi вiдомi результати для звичайних (одинарних) послiдовностей поширено на багатократнi послiдовностi дiйсних чисел. Крiм того, введено поняття подвiйної лакунарної сильної $(A, ϕ)$-збiжностi вiдносно функцiї модуля, а також вивчено деякi спiввiдношення мiж подвiйною лакунарною сильною $(A, ϕ)$-збiжнiстю вiдносно модуля та подвiйною лакунарною статистичною збiжнiстю. Institute of Mathematics, NAS of Ukraine 2018-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1599 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 6 (2018); 848-858 Український математичний журнал; Том 70 № 6 (2018); 848-858 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1599/581 Copyright (c) 2018 Savaş E. |
| spellingShingle | Savaş, E. Саваш, Є. On the lacunary $(A, ϕ)$ -statistical convergence of double sequences |
| title | On the lacunary $(A, ϕ)$ -statistical convergence of double sequences
|
| title_alt | Про лакунарну $(A, ϕ)$-статистичну збiжнiсть
подвiйних послiдовностей |
| title_full | On the lacunary $(A, ϕ)$ -statistical convergence of double sequences
|
| title_fullStr | On the lacunary $(A, ϕ)$ -statistical convergence of double sequences
|
| title_full_unstemmed | On the lacunary $(A, ϕ)$ -statistical convergence of double sequences
|
| title_short | On the lacunary $(A, ϕ)$ -statistical convergence of double sequences
|
| title_sort | on the lacunary $(a, ϕ)$ -statistical convergence of double sequences |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1599 |
| work_keys_str_mv | AT savase onthelacunaryaphstatisticalconvergenceofdoublesequences AT savašê onthelacunaryaphstatisticalconvergenceofdoublesequences AT savase prolakunarnuaphstatističnuzbižnistʹpodvijnihposlidovnostej AT savašê prolakunarnuaphstatističnuzbižnistʹpodvijnihposlidovnostej |