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On asymptotically double lacunary statistically equivalent sequences
Applied Mathematics Letters 22 (2009) 1781–1785
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
On asymptotically double lacunary statistically equivalent sequences Ayhan Esi Adiyaman University, Science and Art Faculty, Department of Mathematics, 02040 Adiyaman, Turkey
article
info
Article history: Received 21 May 2009 Received in revised form 29 June 2009 Accepted 29 June 2009 Keywords: Pringsheim limit point P-convergent Double lacunary sequence Double statistical convergence
abstract This work presents the following definition which is a natural combination of the definition for asymptotically equivalent, double statistically limited and double lacunary sequences. Let θr ,s = {(kr , ls )} be a double lacunary sequence; the two nonnegative sequences x = xk,l and y = yk,l are said to be asymptotically double lacunary statistically equivalent of multiple L provided that for every ε > 0 P − lim r ,s
(k, l) ∈ Ir ,s
1 hr ,s
xk,l − L ≥ ε = 0 : y k,l
SθL r ,s
(denoted by x v y) and simply asymptotically double lacunary statistically equivalent if L = 1. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction In 1993, Marouf [1] presented definitions for asymptotically equivalent sequences and asymptotically regular matrices. In 2003, Patterson [2] extended these concepts by presenting an asymptotically statistically equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices. Later these definitions were extended to lacunary sequences by Patterson and Savaş in [3]. This work extends the definitions presented in [3] to double lacunary sequences. In addition to these definitions, natural inclusion theorems will also be presented. 2. Definitions and notation Definition 2.1 (Marouf [1]). Two nonnegative sequences x = (xk ) and y = (yk ) are said to be asymptotically equivalent if lim k
xk yk
=1
(denoted by x v y). Definition 2.2 (Fridy [4]). The sequence x = (xk ) has statistical limit L provided that for every ε > 0, lim n
1 n
{the number of k ≤ n : |xk − L| ≥ ε} = 0.
The next definition is a natural combination of Definitions 2.1 and 2.2.
E-mail addresses: [email protected], [email protected]. URL: http://www.adiyaman.edu.tr. 0893-9659/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2009.06.018
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A. Esi / Applied Mathematics Letters 22 (2009) 1781–1785
Definition 2.3 (Patterson [2]). Two nonnegative sequences x = (xk ) and y = (yk ) are said to be asymptotically statistically equivalent of multiple L provided that for every ε > 0,
xk lim the number of k ≤ n : − L ≥ ε = 0 n n yk 1
S
(denoted by x vL y) and simply asymptotically statistically equivalent if L = 1. By a lacunary sequence θ = (kr ); r = 0, 1, 2, . . ., where ko = 0, we shall mean an increasing sequence of nonnegative integers with kr − kr −1 → ∞ as r → ∞. The intervals determined by θ will be denoted by Ir = (kr −1 , kr ] and hr = kr − kr −1 . The ratio k kr will be denoted by qr . r −1
The next definition is a natural combination of Definitions 2.1 and 2.3. Definition 2.4 (Patterson and Savaş [3]). Let θ = (kr ) be a lacunary sequence; the two sequences x = (xk ) and y = (yk ) are said to be asymptotically lacunary statistically equivalent of multiple L provided that for every ε > 0, lim r
k ∈ Ir hr 1
xk : − L ≥ ε = 0 y k
SL θ
(denoted by x v y) and simply asymptotically lacunary statistically equivalent if L = 1, where the vertical bars indicate the numbers of elements in the enclosed set. Definition 2.5 (Patterson and Savaş [3]). Let θ = (kr ) be a lacunary sequence; the two sequences x = (xk ) and y = (yk ) are strongly asymptotically lacunary equivalent of multiple L provided that lim r
− L = 0, y
1 X xk hr k∈I r
k
NθL
(denoted by x v y) and simply strongly asymptotically lacunary equivalent if L = 1. In 1900 Pringsheim presented the following definition for the convergence of double sequences.
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 Definition 2.6 (Pringsheim [5]). A double sequence x = xk,l has Pringsheim limit L (denoted by P- lim x = L) provided that
as ‘‘P-convergent’’. We shall denote the space of all P-convergent sequences by c ıı . 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 forms by kxk(∞,2) = supk,l xk,l < ∞. We shall also denote the set of all bounded double sequences by lıı∞ . We also note that, in contrast to the case for a single sequence, a P-convergent double sequence need not be bounded. Definition 2.7 (Savaş and Patterson [6]). The double sequence θr ,s = {(kr , ls )} is called a double lacunary sequence if there exist two increasing of integers such that ko = 0,
hr = kr − kr −1 → ∞ as r → ∞
lo = 0,
hs = ls − ls−1 → ∞ as s → ∞.
and −
−
Notation. kr ,s = kr ls , hr ,s = hr hs , θr ,s is determined by Ir ,s = {(k, l) : kr −1 < k ≤ kr and ls−1 < l ≤ ls } , qr =
kr kr −1
−
,
qs =
ls ls−1
−
and
qr ,s = qr q s .
Definition 2.8 (Mursaleen and Edely [7]). 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 mn
(k, l) : k ≤ m and l ≤ n, xk,l − L ≥ ε = 0.
In this case we write S L − lim x = L or xk,l → L S L .
A. Esi / Applied Mathematics Letters 22 (2009) 1781–1785
1783
Now we give some new definitions which are natural combinations of Definitions 2.7 and 2.8. Definition 2.9. Let θr ,s = {(kr , ls )} be a double lacunary sequence; the two nonnegative double sequences x = xk,l and
y = yk,l are said to be asymptotically double lacunary statistically equivalent of multiple L provided that for every ε > 0,
P − lim r ,s
(k, l) ∈ Ir ,s
1 h r ,s
xk,l − L ≥ ε = 0 : y k,l
SθL r ,s
(denoted by x v y) and simply asymptotically double lacunary statistically equivalent if L = 1. Furthermore, let SθLr ,s denote SθL r ,s
the set of all sequences x = xk,l and y = yk,l such that x v y.
Definition 2.10. Let θr ,s = {(kr , ls )} be a double lacunary sequence; the two double sequences x = xk,l and y = yk,l are said to be strongly asymptotically double lacunary equivalent of multiple L provided that
P − lim r ,s
X xk,l y − L = 0,
1
hr ,s (k,l)∈I r ,s
k,l
NθL
r ,s
(denoted by x v y) and simply strongly asymptotically double lacunary equivalent if L = 1. In addition, let NθLr ,s denote the NθL r ,s
set of all sequences x = xk,l and y = yk,l such that x v y.
3. Main results Theorem 3.1. Let θr ,s = {(kr , ls )} be a double lacunary sequence. Then: NθL r ,s
SθL r ,s
(i) (a) If x v y then x v y. (b) NθLr ,s is a proper subset of SθLr ,s . SθL r ,s
NθL
r ,s
(ii) If x = xk,l ∈ lıı∞ and x v y then x v y. (iii) SθLr ,s ∩ lıı∞ = NθLr ,s ∩ lıı∞ . Proof.
NθL r ,s
(i) (a) If ε > 0 and x v y then X X xk,l y − L ≥ x k,l k,l (k,l)∈I r ,s
(k,l)∈Ir ,s & y
k,l
−L ≥ε
≥ ε (k, l) ∈ Ir ,s
xk,l − L y k,l
xk,l : − L ≥ ε . y k,l
SθL r ,s
Thereforex v y. p p (b) x = xk,l is defined as follows: xk,l is 1, 2, . . . , hr ,s for the first hr ,s integers in Ir ,s and zero otherwise. SθL r ,s
NθL r ,s
y = yk,l = 1 for all k, l = 1. These two satisfy the following: x v y, but the following fails: x v y.
SθL r ,s
(ii) Suppose that x = xk,l and y = yk,l are in lıı∞ and x v y. Then we can assume that
xk,l < H, − L y k,l
for all k and l.
Given ε > 0 1 h r ,s
X xk,l 1 y − L = h k,l r ,s (k,l)∈Ir ,s
X x (k,l)∈Ir ,s & yk,l −L ≥ε
r ,s
r ,s
Therefore x v y. (iii) It follows from (i) and (ii).
k,l
k,l
H (k, l) ∈ Ir ,s ≤ h NθL
xk,l 1 y − L + h
r ,s
X x (k,l)∈Ir ,s & yk,l −L <ε k,l
xk,l : − L ≥ ε + ε. y k,l
xk,l y − L k,l
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A. Esi / Applied Mathematics Letters 22 (2009) 1781–1785 SL
Theorem 3.2. Let θr ,s = {(kr , ls )} be a double lacunary sequence with lim infr qr > 1 and lim infs qs > 1; then x v y implies SθL r ,s
x v y. Proof. Suppose that lim infr qr > 1 and lim infs qs > 1; then there exists δ > 0 such that qr > 1 + δ and qs > 1 + δ . This hr kr
implies that kr ls h r ,s
≤
≥
δ
1+δ
1+δ
hs ls
and
δ
1+δ
. Since hr ,s = kr ls − kr −1 ls−1 , we are granted the following:
kr −1 ls−1
and
δ
≥
h r ,s
≤
1
δ
.
SL
Then for x v y, we can write, for every ε > 0 and for sufficiently large r and s, that we have
(k, l) ∈ Ir ,s : k ≤ kr and l ≤ ls , xk,l − L ≥ ε kr ls yk,l 1 (k, l) ∈ Ir ,s : xk,l − L ≥ ε ≥ kr ls yk,l xk,l 1 δ . − L ≥ ε . ≥ (k, l) ∈ Ir ,s : 1 + δ h r ,s yk,l 1
This completes the proof.
SθL r ,s
Theorem 3.3. Let θr ,s = {(kr , ls )} be a double lacunary sequence with lim supr qr < ∞ and lim sups qs < ∞; then x v y SL
implies x v y. Proof. Since lim supr qr < ∞ and lim sups qs < ∞ there exists H > 0 such that qr < H and qs < H for all r and s. Let SθL r ,s
x v y and ε > 0. Then there exist ro > 0 and so > 0 such that for every i ≥ ro and j ≥ so
(k, l) ∈ Ii,j : xk,l − L ≥ ε < ε. 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 Bi,j =
1
the following:
(k, l) ∈ Ii,j : k ≤ m and l ≤ n, xk,l − L ≥ ε mn yk,l xk,l 1 ≥ε k , l ∈ I : k ≤ k and l ≤ l , ≤ − L ( ) i , j r s 1
kr −1 ls−1
≤
≤
1
yk,l
ro ,so X
kr −1 ls−1 M
t ,u ro ,so X
kr −1 ls−1
t ,u
≤
Mkro lso ro so
≤
Mkro lso ro so
kr −1 ls−1
kr −1 ls−1
≤
Mkro lso ro so
≤
Mkro lso ro so
kr −1 ls−1
kr −1 ls−1
ht ,u Bt ,u +
h t ,u +
1
kr −1 ls−1 (r
X
kr −1 ls−1 (r
1
+
X
X
kr −1 ls−1 (r
+
sup t ≥ro ∪u≥so
+
ε
1
ht ,u Bt ,u
ht ,u Bt ,u
ht ,u Bt ,u
X
kr −1 ls−1 (r
X
kr −1 ls−1 (r
h t ,u
h t ,u
+ εH 2 .
Since kr and ls both approach infinity as both m and n approach infinity, it follows that
(k, l) ∈ Ii,j : k ≤ m and l ≤ n, xk,l − L ≥ ε → 0. mn y 1
k,l
This completes the proof.
A. Esi / Applied Mathematics Letters 22 (2009) 1781–1785
1785
The following theorem is an immediate consequence of Theorems 3.2 and 3.3. SθL r ,s
Theorem 3.4. Let θr ,s = {(kr , ls )} be a double lacunary sequence with 1 < lim infr ,s qrs ≤ lim supr ,s qrs < ∞; then x v y SL
= x v y. References [1] [2] [3] [4] [5] [6] [7]
M. Marouf, Asymptotic equivalence and summability, Internat. J. Math. Math. Sci. 16 (4) (1993) 755–762. R.F. Patterson, On asymptotically statistically equivalent sequences, Demonstratio Math. 36 (1) (2003) 149–153. R.F. Patterson, E. Savaş, On asymptotically lacunary statistically equivalent sequences, Thai J. Math. 4 (2) (2006) 267–272. J.A. Fridy, On statistical convergence, Analysis 5 (1985) 301–313. A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Annal. 53 (1900) 289–321. E. Savaş, R.F. Patterson, Some double lacunary sequence spaces defined by Orlicz functions (preprint). M. Mursaleen, O.H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (1) (2003) 223–231.
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
On asymptotically double lacunary statistically equivalent sequences Ayhan Esi Adiyaman University, Science and Art Faculty, Department of Mathematics, 02040 Adiyaman, Turkey
article
info
Article history: Received 21 May 2009 Received in revised form 29 June 2009 Accepted 29 June 2009 Keywords: Pringsheim limit point P-convergent Double lacunary sequence Double statistical convergence
abstract This work presents the following definition which is a natural combination of the definition for asymptotically equivalent, double statistically limited and double lacunary sequences. Let θr ,s = {(kr , ls )} be a double lacunary sequence; the two nonnegative sequences x = xk,l and y = yk,l are said to be asymptotically double lacunary statistically equivalent of multiple L provided that for every ε > 0 P − lim r ,s
(k, l) ∈ Ir ,s
1 hr ,s
xk,l − L ≥ ε = 0 : y k,l
SθL r ,s
(denoted by x v y) and simply asymptotically double lacunary statistically equivalent if L = 1. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction In 1993, Marouf [1] presented definitions for asymptotically equivalent sequences and asymptotically regular matrices. In 2003, Patterson [2] extended these concepts by presenting an asymptotically statistically equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices. Later these definitions were extended to lacunary sequences by Patterson and Savaş in [3]. This work extends the definitions presented in [3] to double lacunary sequences. In addition to these definitions, natural inclusion theorems will also be presented. 2. Definitions and notation Definition 2.1 (Marouf [1]). Two nonnegative sequences x = (xk ) and y = (yk ) are said to be asymptotically equivalent if lim k
xk yk
=1
(denoted by x v y). Definition 2.2 (Fridy [4]). The sequence x = (xk ) has statistical limit L provided that for every ε > 0, lim n
1 n
{the number of k ≤ n : |xk − L| ≥ ε} = 0.
The next definition is a natural combination of Definitions 2.1 and 2.2.
E-mail addresses: [email protected], [email protected]. URL: http://www.adiyaman.edu.tr. 0893-9659/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2009.06.018
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A. Esi / Applied Mathematics Letters 22 (2009) 1781–1785
Definition 2.3 (Patterson [2]). Two nonnegative sequences x = (xk ) and y = (yk ) are said to be asymptotically statistically equivalent of multiple L provided that for every ε > 0,
xk lim the number of k ≤ n : − L ≥ ε = 0 n n yk 1
S
(denoted by x vL y) and simply asymptotically statistically equivalent if L = 1. By a lacunary sequence θ = (kr ); r = 0, 1, 2, . . ., where ko = 0, we shall mean an increasing sequence of nonnegative integers with kr − kr −1 → ∞ as r → ∞. The intervals determined by θ will be denoted by Ir = (kr −1 , kr ] and hr = kr − kr −1 . The ratio k kr will be denoted by qr . r −1
The next definition is a natural combination of Definitions 2.1 and 2.3. Definition 2.4 (Patterson and Savaş [3]). Let θ = (kr ) be a lacunary sequence; the two sequences x = (xk ) and y = (yk ) are said to be asymptotically lacunary statistically equivalent of multiple L provided that for every ε > 0, lim r
k ∈ Ir hr 1
xk : − L ≥ ε = 0 y k
SL θ
(denoted by x v y) and simply asymptotically lacunary statistically equivalent if L = 1, where the vertical bars indicate the numbers of elements in the enclosed set. Definition 2.5 (Patterson and Savaş [3]). Let θ = (kr ) be a lacunary sequence; the two sequences x = (xk ) and y = (yk ) are strongly asymptotically lacunary equivalent of multiple L provided that lim r
− L = 0, y
1 X xk hr k∈I r
k
NθL
(denoted by x v y) and simply strongly asymptotically lacunary equivalent if L = 1. In 1900 Pringsheim presented the following definition for the convergence of double sequences.
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 Definition 2.6 (Pringsheim [5]). A double sequence x = xk,l has Pringsheim limit L (denoted by P- lim x = L) provided that
as ‘‘P-convergent’’. We shall denote the space of all P-convergent sequences by c ıı . 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 forms by kxk(∞,2) = supk,l xk,l < ∞. We shall also denote the set of all bounded double sequences by lıı∞ . We also note that, in contrast to the case for a single sequence, a P-convergent double sequence need not be bounded. Definition 2.7 (Savaş and Patterson [6]). The double sequence θr ,s = {(kr , ls )} is called a double lacunary sequence if there exist two increasing of integers such that ko = 0,
hr = kr − kr −1 → ∞ as r → ∞
lo = 0,
hs = ls − ls−1 → ∞ as s → ∞.
and −
−
Notation. kr ,s = kr ls , hr ,s = hr hs , θr ,s is determined by Ir ,s = {(k, l) : kr −1 < k ≤ kr and ls−1 < l ≤ ls } , qr =
kr kr −1
−
,
qs =
ls ls−1
−
and
qr ,s = qr q s .
Definition 2.8 (Mursaleen and Edely [7]). 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 mn
(k, l) : k ≤ m and l ≤ n, xk,l − L ≥ ε = 0.
In this case we write S L − lim x = L or xk,l → L S L .
A. Esi / Applied Mathematics Letters 22 (2009) 1781–1785
1783
Now we give some new definitions which are natural combinations of Definitions 2.7 and 2.8. Definition 2.9. Let θr ,s = {(kr , ls )} be a double lacunary sequence; the two nonnegative double sequences x = xk,l and
y = yk,l are said to be asymptotically double lacunary statistically equivalent of multiple L provided that for every ε > 0,
P − lim r ,s
(k, l) ∈ Ir ,s
1 h r ,s
xk,l − L ≥ ε = 0 : y k,l
SθL r ,s
(denoted by x v y) and simply asymptotically double lacunary statistically equivalent if L = 1. Furthermore, let SθLr ,s denote SθL r ,s
the set of all sequences x = xk,l and y = yk,l such that x v y.
Definition 2.10. Let θr ,s = {(kr , ls )} be a double lacunary sequence; the two double sequences x = xk,l and y = yk,l are said to be strongly asymptotically double lacunary equivalent of multiple L provided that
P − lim r ,s
X xk,l y − L = 0,
1
hr ,s (k,l)∈I r ,s
k,l
NθL
r ,s
(denoted by x v y) and simply strongly asymptotically double lacunary equivalent if L = 1. In addition, let NθLr ,s denote the NθL r ,s
set of all sequences x = xk,l and y = yk,l such that x v y.
3. Main results Theorem 3.1. Let θr ,s = {(kr , ls )} be a double lacunary sequence. Then: NθL r ,s
SθL r ,s
(i) (a) If x v y then x v y. (b) NθLr ,s is a proper subset of SθLr ,s . SθL r ,s
NθL
r ,s
(ii) If x = xk,l ∈ lıı∞ and x v y then x v y. (iii) SθLr ,s ∩ lıı∞ = NθLr ,s ∩ lıı∞ . Proof.
NθL r ,s
(i) (a) If ε > 0 and x v y then X X xk,l y − L ≥ x k,l k,l (k,l)∈I r ,s
(k,l)∈Ir ,s & y
k,l
−L ≥ε
≥ ε (k, l) ∈ Ir ,s
xk,l − L y k,l
xk,l : − L ≥ ε . y k,l
SθL r ,s
Thereforex v y. p p (b) x = xk,l is defined as follows: xk,l is 1, 2, . . . , hr ,s for the first hr ,s integers in Ir ,s and zero otherwise. SθL r ,s
NθL r ,s
y = yk,l = 1 for all k, l = 1. These two satisfy the following: x v y, but the following fails: x v y.
SθL r ,s
(ii) Suppose that x = xk,l and y = yk,l are in lıı∞ and x v y. Then we can assume that
xk,l < H, − L y k,l
for all k and l.
Given ε > 0 1 h r ,s
X xk,l 1 y − L = h k,l r ,s (k,l)∈Ir ,s
X x (k,l)∈Ir ,s & yk,l −L ≥ε
r ,s
r ,s
Therefore x v y. (iii) It follows from (i) and (ii).
k,l
k,l
H (k, l) ∈ Ir ,s ≤ h NθL
xk,l 1 y − L + h
r ,s
X x (k,l)∈Ir ,s & yk,l −L <ε k,l
xk,l : − L ≥ ε + ε. y k,l
xk,l y − L k,l
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A. Esi / Applied Mathematics Letters 22 (2009) 1781–1785 SL
Theorem 3.2. Let θr ,s = {(kr , ls )} be a double lacunary sequence with lim infr qr > 1 and lim infs qs > 1; then x v y implies SθL r ,s
x v y. Proof. Suppose that lim infr qr > 1 and lim infs qs > 1; then there exists δ > 0 such that qr > 1 + δ and qs > 1 + δ . This hr kr
implies that kr ls h r ,s
≤
≥
δ
1+δ
1+δ
hs ls
and
δ
1+δ
. Since hr ,s = kr ls − kr −1 ls−1 , we are granted the following:
kr −1 ls−1
and
δ
≥
h r ,s
≤
1
δ
.
SL
Then for x v y, we can write, for every ε > 0 and for sufficiently large r and s, that we have
(k, l) ∈ Ir ,s : k ≤ kr and l ≤ ls , xk,l − L ≥ ε kr ls yk,l 1 (k, l) ∈ Ir ,s : xk,l − L ≥ ε ≥ kr ls yk,l xk,l 1 δ . − L ≥ ε . ≥ (k, l) ∈ Ir ,s : 1 + δ h r ,s yk,l 1
This completes the proof.
SθL r ,s
Theorem 3.3. Let θr ,s = {(kr , ls )} be a double lacunary sequence with lim supr qr < ∞ and lim sups qs < ∞; then x v y SL
implies x v y. Proof. Since lim supr qr < ∞ and lim sups qs < ∞ there exists H > 0 such that qr < H and qs < H for all r and s. Let SθL r ,s
x v y and ε > 0. Then there exist ro > 0 and so > 0 such that for every i ≥ ro and j ≥ so
(k, l) ∈ Ii,j : xk,l − L ≥ ε < ε. 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 Bi,j =
1
the following:
(k, l) ∈ Ii,j : k ≤ m and l ≤ n, xk,l − L ≥ ε mn yk,l xk,l 1 ≥ε k , l ∈ I : k ≤ k and l ≤ l , ≤ − L ( ) i , j r s 1
kr −1 ls−1
≤
≤
1
yk,l
ro ,so X
kr −1 ls−1 M
t ,u ro ,so X
kr −1 ls−1
t ,u
≤
Mkro lso ro so
≤
Mkro lso ro so
kr −1 ls−1
kr −1 ls−1
≤
Mkro lso ro so
≤
Mkro lso ro so
kr −1 ls−1
kr −1 ls−1
ht ,u Bt ,u +
h t ,u +
1
kr −1 ls−1 (r
X
kr −1 ls−1 (r
1
+
X
X
kr −1 ls−1 (r
+
sup t ≥ro ∪u≥so
+
ε
1
ht ,u Bt ,u
ht ,u Bt ,u
ht ,u Bt ,u
X
kr −1 ls−1 (r
X
kr −1 ls−1 (r
h t ,u
h t ,u
+ εH 2 .
Since kr and ls both approach infinity as both m and n approach infinity, it follows that
(k, l) ∈ Ii,j : k ≤ m and l ≤ n, xk,l − L ≥ ε → 0. mn y 1
k,l
This completes the proof.
A. Esi / Applied Mathematics Letters 22 (2009) 1781–1785
1785
The following theorem is an immediate consequence of Theorems 3.2 and 3.3. SθL r ,s
Theorem 3.4. Let θr ,s = {(kr , ls )} be a double lacunary sequence with 1 < lim infr ,s qrs ≤ lim supr ,s qrs < ∞; then x v y SL
= x v y. References [1] [2] [3] [4] [5] [6] [7]
M. Marouf, Asymptotic equivalence and summability, Internat. J. Math. Math. Sci. 16 (4) (1993) 755–762. R.F. Patterson, On asymptotically statistically equivalent sequences, Demonstratio Math. 36 (1) (2003) 149–153. R.F. Patterson, E. Savaş, On asymptotically lacunary statistically equivalent sequences, Thai J. Math. 4 (2) (2006) 267–272. J.A. Fridy, On statistical convergence, Analysis 5 (1985) 301–313. A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Annal. 53 (1900) 289–321. E. Savaş, R.F. Patterson, Some double lacunary sequence spaces defined by Orlicz functions (preprint). M. Mursaleen, O.H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (1) (2003) 223–231.