Available at
http://pvamu.edu/aam
Appl. Appl. Math.
ISSN: 1932-9466
Applications and Applied
Mathematics:
An International Journal
(AAM)
Vol. 10, Issue 2 (December 2015), pp. 1093 – 1103
Asymptotically Double Lacunary Equivalent Sequences
in Topological Groups
1
1
Ayhan Esi and 2 M. Kemal Ozdemir*
Adiyaman University, Science and Art Faculty, Department of Mathematics
2040, Adiyaman, Turkey
aesi23@hotmail.com
2
Inonu University, Science and Art Faculty, Department of Mathematics
44280, Malatya, Turkey
∗
Corresponding Author: kozdemir73@gmail.com
Received: August 8, 2014; Accepted: July 14, 2015
Abstract
In this paper we study the concept of asymptotically double lacunary statistical convergent
sequences in topological groups and prove some inclusion theorems.
Keywords: Pringsheim Limit Point; Asymptotically equivalence; P-convergent; Double Lacunary
Sequence; Double Statistical Convergence; Topological Groups
MSC 2010 No.: 46A45; 40J05; 22A05
1.
Introduction
The idea of statistical convergence of double sequences was first introduced by (Mursaleen and
Edely, 2003) while the idea of statistical convergence of single sequences was first studied by
(Fast, 1951) and the rapid developments were started after the papers of (Salat, 1980) and (Fridy,
1985). Nowadays it has become one of the most active areas of research in the field of summability
theory. (Schoenberg, 1959) gave some basic properties of statistical convergence and also studied
the concept as a summability method. Later on statistical convergence was studied by (Connor,
1093
1094
A. Esi & M. K. Ozdemir
1992), (Freedman, A.R., Sember, J.J. and Raphael, M., 1978), (Esi and Esi, 2010), (Esi and
Tripathy, 2007), (Tripathy, 2003), (Esi, 2012) and many others.
(Marouf, 1993) presented definitions for asymptotically equivalent sequences and asymptotically
regular matrices. In (Patterson, 2003), Patterson extended those concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices. In (Patterson and Savas, 2006), Patterson and Savas extended the
definitions presented in (Patterson, 2003) to lacunary sequences.
This paper extend the definitions presented in (Esi, 2009) to double lacunary sequences in
topological groups.
In this paper we study the concept of asymptotically double lacunar statistical sequences on
topological groups. Since the study of convergence in topological groups is important, we feel
that the concept of asymptotically double lacunary statistical convergent sequences in topological
groups would provide a more general framework for the subject.
Definition 1. (Fast, 1951; Fridy, 1985) A single sequence x = (xk ) is said to be statistically
convergent to the number L if for each ε > 0,
lim
n
1
|{ k ≤ n : |xk − L| ≥ ε}| = 0.
n
In this case we write st − lim x = L or xk → L (st).
Definition 2. (Freedman, A.R., Sember, J.J. and Raphael, M., 1978) By a lacunary sequence
θ = (kr ), r = 0, 1, 2, . . ., where k0 = 0, we mean an increasing sequence of non-negative
integers with hr = kr − kr−1 → ∞ as r → ∞. The intervals determined by θ are denoted
kr
by Ir = (kr−1 , kr ] and the ratio kr−1
will be denoted by qr . The space of lacunary strongly
convergent sequence Nθ was defined in (Freedman, A.R., Sember, J.J. and Raphael, M., 1978)
as follows:
(
)
1 X
Nθ = x = (xi ) : lim
|xi − L| = 0 for some L .
r→∞ hr
i∈I
r
The following definition is due to (Fridy and Orhan, 1993).
Definition 3. (Fridy and Orhan, 1993) Let θ = (kr ) be a lacunary sequence; the single sequence
x = (xk ) is stθ −convergent to L provided that for every ε > 0
1
|{ k ∈ Ir : |xk − L| ≥ ε}| = 0.
r hr
In this case we write stθ − lim x = L or xk → L stθ .
lim
In 1900 Pringsheim presented the following definition for the convergence of double sequences.
Definition 4. (Pringsheim, 1900) A double sequence x = (xk,l ) has Pringsheim limit L (denoted
by P − lim x = L) provided that given ε > 0 there exists N ∈ N such that |xk,l − L| < ε
whenever k, l > N . We shall describe such an x = (xk,l ) more briefly as“P −convergent”.
AAM: Intern. J., Vol. 10, Issue 2 (December 2015)
1095
We shall denote the space of all P-convergent sequences by c2 . By a bounded double sequence
we shall mean there exists a positive number K such that |xk,l | < K for all (k, l) and denote
such bounded sequences by kxk(∞,2) = supk,l |xk,l | < ∞. We shall also denote the set of all
2
. We also note that in contrast to the case for single sequence
bounded double sequences by l∞
a P −convergent double sequence need not be bounded.
Definition 5. (Mursaleen and Edely, 2003) A real double sequence x = (xk,l ) is said to be
statistically convergent to L, provided that for each ε > 0
P − lim
m,n
1
|{(k, l) : k ≤ m and l ≤ n, |xk,l − L| ≥ ε}| = 0.
mn
In this case we write st2 − lim x = L or xk,l → L (st2 ).
Definition 6. (Savas and Patterson, 2006) The double sequence θr,s = {(kr , ls )} is called double
lacunary sequence if there exist two increasing sequences of integers such that
ko = 0, hr = kr − kr−1 → ∞ as r → ∞
and
−
lo = 0, hs = ls − ls−1 → ∞ as s → ∞.
Notation 1: kr,s = kr ks , hr,s = hr hs , θr,s is determined by
Ir,s = {(k, l) : kr−1 < k ≤ kr and ls−1 < l ≤ ls } ,
qr =
ls
kr
, qs =
and qr,s = qr qs .
kr−1
ls−1
Definition 7. ((Savas and Patterson, 2006)) Let θr,s = {(kr , ls )} be a double lacunary sequence;
the double number sequence x = (xk,l ) is stθ2 − convergent to L provided that for every ε > 0,
1
|{ (k, l) ∈ Ir,s : |xk,l − L| ≥ ε}| = 0.
r,s hr,s
In this case, we write stθ2 − limk,l xk,l = L or xk,l → L stθ2 .
P − lim
2.
Definitions and Notations
By X, we will denote an abelian topological Hausdorff group, written additively, which satisfies
the first axiom of countability. In (Cakalli, 1996), a single sequence x = (xk ) in X is said to be
statistically convergent to an element L ∈ X if for each neighbourhood U of 0,
lim
n
1
|{k ≤ n : xk − L ∈
/ U }| = 0.
n
In this case we write st (X) − limk xk = L or xk → L (st (X)).
In (Cakalli, 1995), the concept of lacunary statistical convergence was defined by Cakalli as
follows: Let θ = (kr ) be a lacunary sequence; the single sequence x = (xk ) in X is said to be
1096
A. Esi & M. K. Ozdemir
stθ − convergent to L (or lacunary statistically convergent to L in X) if for each neighbourhood
U of 0,
1
/ U }| = 0.
lim |{ k ∈ Ir : xk − L ∈
r hr
In this case we write stθ (X) − limk xk = L or xk → L stθ (X) .
In 2010, Cakalli and Savas defined double statistical convergence in topological group X as
follows:
Definition 8. ((Cakalli and Savas, 2010)) A double sequence x = (xk,l ) is said to be statistically
convergent to a point L ∈ X if for each neighbourhood U of 0,
P − lim
n,m
1
|{(k, l) : k ≤ n, l ≤ m; xk,l − L ∈
/ U }| = 0.
nm
In this case we write st2 (X) − limk,l xk,l = L or xk,l → L (st2 (X)) and denote the set of all
statistically convergent double sequences in X by st2 (X).
Now we give the two new definitions related to double lacunary asymptotically equivalent
sequences in topological groups as follows:
Definition 9. (Patterson and Savas, 2006) Let θr,s = {(kr , ls )} be a double lacunary sequence.
A sequence x = (xk,l ) is said to be stθ2 -convergent to L in X (or double lacunary statistical
convergent to L in X) if for each neighbourhood U of 0
1
|{ (k, l) ∈ Ir,s : xk,l − L ∈
/ U }| = 0.
r,s hr,s
In this case, we write stθ2 (X) − limk,l xk,l = L or xk,l → L stθ2 (X) and
θ
θ
st2 (X) = x = (xk,l ) : f or some L, st2 − lim xk,l = L
P − lim
k,l
and in particular
stθ2
θ
(X)o = x = (xk,l ) : st2 − lim xk,l = 0 .
k,l
Definition 10. (Esi, 2009) Let θr,s = {(kr , ls )} be a double lacunary sequence. A double number
sequence x = (xk,l ) is Nθ (X) −convergent to L in X provided that
X
1
P − lim
|xk,l − L| = 0.
r,s hr,s
(k,l)∈Ir,s & xk,l −L∈U
/
Definition 11. ((Esi, 2009)) Let θr,s = {(kr , ls )} be a double lacunary sequence. The double nonnegative sequences x = (xk,l ) and 0 ∈
/ y = (yk,l ) are said to be asymptotically double lacunary
statistical equivalent to L in X
xk,l
1 P − lim
(k, l) ∈ Ir,s : − L ≥ ε = 0.
r,s hr,s
yk,l
AAM: Intern. J., Vol. 10, Issue 2 (December 2015)
1097
SθLr,s
In this case, we write x ∼ y and simply asymptotically double lacunary statistical equivalent
if L = 1.
Definition 12. The double non-negative sequences x = (xk,l ) and 0 ∈
/ y = (yk,l ) are said to be
asymptotically double equivalent to L in X if for each neighbourhood U of 0
1 xk,l
P − lim
(k, l) : k ≤ n, l ≤ m; :
−L∈
/ U = 0.
n,m mn
yk,l
S L (X)
In this case, we write x ∼ y and simply asymptotically double statistical equivalent if L = 1
in X. Furthermore, let S L (X) denote the set of all double sequences x = (xk,l ) and y = (yk,l )
S L (X)
such that x ∼ y in X.
Definition 13. Let θr,s = {(kr , ls )} be a double lacunary sequence. The double non-negative
sequences x = (xk,l ) and 0 ∈
/ y = (yk,l ) are said to be asymptotically double lacunary equivalent
to L in X if for each neighbourhood U of 0
xk,l
1 −L∈
/ U = 0.
(k, l) ∈ Ir,s :
P − lim
r,s hr,s
yk,l
SθLr,s (X)
In this case, we write x ∼ y and simply asymptotically double lacunary statistical equivalent
if L = 1 in X. Furthermore, let SθLr,s (X) denote the set of all double sequences x = (xk,l ) and
0∈
/ y = (yk,l ) such that x
SθLr,s (X)
∼
y in X.
Definition 14. Let θr,s = {(kr , ls )} be a double lacunary sequence. The double non-negative
sequences x = (xk,l ) and 0 ∈
/ y = (yk,l ) are said to be asymptotically double lacunary
Nθr,s (X)L −equivalent to L in X if for each neighbourhood U of 0
X
xk,l
1
P − lim
− L = 0.
r,s hr,s
yk,l
(k,l)∈Ir,s & xk,l −L∈U
/
In this case, we write x
equivalent if L = 1 in X.
3.
Nθr,s (X)L
∼
y and simply asymptotically double lacunary Nθr,s (X) −
Main Results
Theorem 1.
Let θr,s = {(kr , ls )} be a double lacunary sequence. Then,
(i). (a) If x
Nθr,s (X)L
∼
y, then x
SθLr,s (X)
∼
y.
SθLr,s (X)
(i). (b) Nθr,s (X)L is a proper subset of x ∼ y.
SθLr,s (X)
Nθr,s (X)L
xk,l
2
∼
y.
(ii). If yk,l ∈ l∞ and x ∼ y then x
L
Sθr,s (X)
xk,l
2
, then Nθr,s (X)L = x ∼ y.
(iii). If yk,l
∈ l∞
1098
A. Esi & M. K. Ozdemir
Proof:
(i). (a) If x
Nθr,s (X)L
∼
y then
(k, l) ∈ Ir,s : xk,l − L ∈
/ U ≤
yk,l
and
1
P − lim
r,s hr,s
This implies that
X
(k,l)∈Ir,s & xk,l −L∈U
/
X
(k,l)∈Ir,s & xk,l −L∈U
/
xk,l
yk,l − L
xk,l
yk,l − L = 0.
1 xk,l
= 0.
P − lim
(k,
l)
∈
I
:
−
L
∈
/
U
r,s
r,s hr,s yk,l
This completes the proof (a).
r−1
(i). (b) Let (kr ) = (2r−1 ) and (ls ) = (2s−1 ). Then
< m ≤ 2r
we have Ir,s = {(m, n) : 2
m,n
as follows: We divide the total double
and 2s−1 < n ≤ 2s }. Define the double sequence xym,n
i
p
h√
x2r−1 +i,2r−1
3
3
r−1
s−1
sequences into blocks of Ir,s type such that y r−1 r−1 = i for 1 ≤ i ≤
hr,s =
2 2
2
+i,2
p
xm,n
3
in each of the first
hr,s number of rows. At other places = 0. The Ir,s block will look as
follows:
p


3
1 2 3 ··· p
hr,s 0 0 0 · · ·
 1 2 3 · · · 3 hr,s 0 0 0 · · · 


 1 2 3 ··· p
3
hr,s 0 0 0 · · · 


 · · · ·

·
·
·
·
·


p


 1 2 3 · · · 3 hr,s 0 0 0 · · · 


 0 0 0 ···
0
0 0 0 ··· 


 0 0 0 ···
0
0 0 0 ··· 


 · · · ·
·
· · · · 
· · · ·
·
· · · ·
m,n xm,n We consider the sup-norm for the double sequences, i.e. xym,n
= supm,n ym,n
. We have
p
3
hr,s → ∞ as r, s → ∞ independent of each other. Thus for a given ε > 0, (very small) we
have
p
3
xm,n h
1 r,s
≥ ε = P − lim
P − lim
(m, n) ∈ Ir,s , = 0.
r,s→∞ hr,s r,s→∞
ym,n hr,s
This is equivalent to
p
3
h
1 xm,n
r,s
P − lim
(m, n) ∈ Ir,s ,
6∈ U = P − lim
= 0,
r,s→∞ hr,s r,s→∞
ym,n
hr,s
where U is an ε−neighborhood
of zero, ε being very small.p
Since we have the first
p
3
3
number of rows having first
hr,s elements as 1, 2, 3, . . . ,
hr,s , so we have
p
p p
3
X
hr,s 3 hr,s 3 hr,s + 1
1
p
P − lim
|xm,n | = P − lim
= .
3
r,s→∞
r,s→∞
2
2
hr,s
(m,n)∈I
r,s
p
3
hr,s
AAM: Intern. J., Vol. 10, Issue 2 (December 2015)
Therefore, we have x
SθLr,s (X)
∼
y but not x
1099
Nθr,s (X)L
∼
y.
xk,l
2
and x
in l∞
yk,l
(ii). Suppose we have the double sequence
that
xk,l
≤ H, for all k and l.
−
L
yk,l
SθLr,s (X)
∼
y. Then we can assume
Given ε > 0
1
hr,s
Therefore x
X xk,l
= 1
−
L
hr,s
yk,l
xk,l
−
L
yk,l
(k,l)∈Ir,s & xk,l −L∈U
/
X
xk,l
1
+
−
L
yk,l
hr,s
(k,l)∈Ir,s & xk,l −L∈U
xk,l
H (k, l) ∈ Ir,s :
≤
−L∈
/ U + ε.
hr,s
yk,l
(k,l)∈Ir,s
Nθr,s (X)L
∼
X
y.
(iii). It follows from (i) and (ii). Theorem 2.
Let θr,s = {(kr , ls )} be a double lacunary sequence with lim inf r qr > 1 and lim inf s q s > 1 then
S L (X)
x ∼ y⇒x
SθLr,s (X)
∼
y.
Proof:
Suppose that lim inf r qr > 1 and lim inf s q s > 1 then there exists δ > 0 such that qr > 1 + δ and
δ
δ
q s > 1+δ. This implies that hkrr ≥ 1+δ
and hlss ≥ 1+δ
. Since hr,s = kr ls −kr ls−1 −kr−1 ls −kr−1 ls−1 ,
we are granted the following
kr ls
(1 + δ)2
kr−1 ls−1
1
≤
and
≤ .
2
hr,s
δ
hr,s
δ
S L (X)
θr,s (X)
Let x ∼ y. We will prove x ∼ L y. Let us take any neighbourhood U of 0. Then for
sufficiently large r and s, we have
1 xk,l
(k, l) ∈ Ir,s : k ≤ kr and l ≤ ls ,
−L∈
/ U kr ls
y
k,l
xk,l
1 ≥
(k, l) ∈ Ir,s :
−L∈
/ U kr ls
yk,l
2
δ
1 xk,l
≥
.
(k,
l)
∈
I
:
−
L
∈
/
U
r,s
(1 + δ)2 hr,s yk,l
This completes the proof. 1100
A. Esi & M. K. Ozdemir
Theorem 3.
Let θr,s = {(kr , ls )} be a double lacunary sequence with lim supr qr < ∞ and lim sups q s < ∞
then x
SθLr,s (X)
∼
S L (X)
y ⇒ x ∼ y.
Proof:
Since lim supr qr < ∞ and lim sups q s < ∞ there exists H > 0 such that qr < H and q s < H
SθLr,s (X)
for all r and s. Let x ∼
every i ≥ ro and j ≥ so
y and ε > 0. Then there exists ro > 0 and so > 0 such that for
xk,l
1 < ε.
(k,
l)
∈
I
:
−
L
∈
/
U
Bi,j =
i,j
hi,j yk,l
Let M = max {Bi,j : 1 ≤ i ≤ ro and 1 ≤ j ≤ so } and m and n be such that kr−1 < m ≤ kr
and ls−1 < n ≤ ls . Thus we obtain the following
xk,l
1 (k, l) ∈ Ii,j : k ≤ m and l ≤ n,
−L∈
/ U mn
yk,l
1
xk,l
≤
(k, l) ∈ Ii,j : k ≤ kr and l ≤ ls ,
−L∈
/ U kr−1 ls−1
yk,l
ro ,so
X
X
1
1
ht,u Bt,u +
≤
ht,u Bt,u
kr−1 ls−1 t,u
kr−1 ls−1
(ro <t≤r)∪(so <u≤s)
≤
M
kr−1 ls−1
ro ,so
X
t,u
ht,u +
1
kr−1 ls−1
X
ht,u Bt,u
(ro <t≤r)∪(so <u≤s)
X
1
M kro lso ro so
+
ht,u Bt,u
kr−1 ls−1
kr−1 ls−1
(ro <t≤r)∪(so <u≤s)
X
M kro lso ro so
1
≤
+
sup Bt,u
kr−1 ls−1
kr−1 ls−1
t≥ro ∪u≥so
≤
ht,u
(ro <t≤r)∪(so <u≤s)
≤
M kro lso ro so
ε
+
kr−1 ls−1
kr−1 ls−1
X
ht,u
(ro <t≤r)∪(so <u≤s)
M kro lso ro so
+ εH 2 .
kr−1 ls−1
Since kr and ls both approach infinity as both m and n approach infinity, it follows that
1 xk,l
(k, l) ∈ Ii,j : k ≤ m and l ≤ n,
−L∈
/ U → 0.
mn
yk,l
This completes the proof. ≤
The following theorem is an immediate consequence of Theorem 2 and Theorem 3.
Theorem 4.
Let θr,s = {(kr , ls )} be a double lacunary sequence with 1 < lim inf r,s qrs ≤ lim supr,s qrs < ∞.
S L (X)
Then x ∼ y ⇔ x
SθLr,s (X)
∼
y.
AAM: Intern. J., Vol. 10, Issue 2 (December 2015)
1101
Theorem 5.
If the two double sequences x = (xk,l ) and y = (yk,l ) both satisfy x
then L1 = L2 .
S L1 (X)
∼
y and x
SθLr,s (X)
∼
y,
Proof:
SθLr,s (X)
S L1 (X)
Let x ∼ y and x ∼ y. Suppose that L1 6= L2 . Since X is a Hausdorff space, then
there exists a symmetric neighbourhood U of 0 such that L1 − L2 ∈
/ U . Then we may choose a
symmetric neighbourhood W of 0 such that W + W ⊂ U . We obtain the following inequality:
1
|{(k, l) : k ≤ km , l ≤ ln ; zk,l ∈
/ U }|
km ln
1 xk,l
≤
(k,
l)
:
k
≤
k
,
l
≤
l
;
−
L
∈
/
W
m
n
1
km ln yk,l
xk,l
1 ,
∈
/
W
(k,
l)
:
k
≤
k
,
l
≤
l
;
L
−
+
m
n
2
km ln yk,l
where zk,l = L1 − L2 for all k, l ∈ N. It follows from this inequality that
xk,l
1 −
L
∈
/
W
1≤
(k,
l)
:
k
≤
k
,
l
≤
l
;
1
m
n
km ln yk,l
1 xk,l
.
+
(k,
l)
:
k
≤
k
,
l
≤
l
;
L
∈
/
W
m
n
2
km ln yk,l
The second term on the right side of this inequality tends to 0 as m, n → ∞ in Pringsheim sense.
To see this, we write
x
k,l
(k, l) : k ≤ km , l ≤ ln ; L2 −
∈
/
W
yk,l
xk,l
(k, l) ∈ ∪m,n
I
:
L
−
∈
/
W
r,s
2
r,s=1,1
yk,l
m,n
1 X xk,l
=
(k, l) ∈ Ir,s : L2 −
∈
/ W km ln r,s=1,1
yk,l
!−1
!
m,n
m,n
X
X
=
hr,s
hr,s tr,s ,
1
km ln
1
=
km ln
r,s=1,1
where tr,s =
0. So,
1
hr,s
n
(k, l) ∈ Ir,s : L2 −
r,s=1,1
xk,l
yk,l
o
SθLr,s (X)
∈
/ W . Since x ∼ y, we know that P −limr,s tr,s =
1 xk,l
P − lim
(k, l) : k ≤ km , l ≤ ln ; L2 −
∈
/ W = 0.
m,n km ln
yk,l
(1)
1102
A. Esi & M. K. Ozdemir
On the other hand, since x
S L1 (X)
∼
y,
1 xk,l
P − lim
(k, l) : k ≤ km , l ≤ ln ;
− L1 ∈
/ W = 0.
m,n km ln
yk,l
(2)
By (1) and (2) it follows that
1
|{(k, l) : k ≤ km , l ≤ ln ; zk,l ∈
/ U }| = 0
km ln
which is a contradiction. This completes the proof. P − lim
m,n
4.
Conclusion
The definition of asymptotically equivalent sequences was introduced by Marouf in 1993. Later
on it was further investigated from the sequence space point of view and linked with summability
theory by several authors. The results obtained in this study are much more general than those
obtained earlier.
REFERENCES
Cakalli, H. (1995). Lacunary Statistical Convergence in Topological Groups. Indian J. Pure
Appl. Math., 26(2):113–119.
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