In this article we introduce the zweier double sequence spaces
using the Orlicz function
M
. We study the algebraic properties and inclusion relations on these spaces.
AMS Mathematics Subject Classification : 65H05, 65F10.
1. Introduction
Let
be the sets of all natural, real and complex numbers respectively. We write
the space of all double sequences real or complex.
Let
ℓ_{∞}
,
c
and
c
_{0}
denote the Banach spaces of bounded, convergent and null sequences respectively normed by
At the initial stage the notion of Iconvergence was introduced by Kostyrko,Šalát and Wilczyński
[1]
. Later on it was studied by Šalát, Tripathy and Ziman
[2]
, Demirci
[3]
and many others. Iconvergence is a generalization of Statistical Convergence.
Now we have a list of some basic definitions used in the paper .
Definition 1.1
(
[4
,
5]
). Let X be a non empty set. Then a family of sets I⊆2
^{X}
(2
^{X}
denoting the power set of X) is said to be an ideal in X if

(i)

(ii) I is additive i.e A,B∈I ⇒ A ∪ B∈I.

(iii) I is hereditary i.e A∈I, B⊆A⇒B∈I.
For more details see
[6
,
7
,
8
,
9
,
10]
. An Ideal I⊆ 2
^{X}
is called nontrivial if I≠ 2
^{X}
. A nontrivial ideal I⊆ 2
^{X}
is called admissible if {{
x
} :
x
∈
X
} ⊆I.
A nontrivial ideal I is maximal if there cannot exist any nontrivial ideal J≠I containing I as a subset.
For each ideal I, there is a filter
£
(I) corresponding to I. i.e
Definition 1.2.
A double sequence of complex numbers is defined as a function
x
: ℕ × ℕ → ℂ. We denote a double sequence as (
x_{ij}
), where the two subscripts run through the sequence of natural numbers independent of each other. A number
a
∈ ℂ is called a double limit of a double sequence (
x_{ij}
) if for every
ϵ
> 0 there exists some
N
=
N
(
ϵ
) ∈
N
such that
Definition 1.3
(
[12]
). A double sequence (
x_{ij}
) ∈
ω
is said to be Iconvergent to a number L if for every
ϵ
> 0,
In this case we write
I
− lim
x_{ij}
=
L
.
Definition 1.4
(
[12]
). A double sequence (
x_{ij}
) ∈
ω
is said to be Inull if L = 0. In this case we write
Definition 1.5
(
[12]
). A double sequence (
x_{ij}
) ∈
ω
is said to be ICauchy if for every
ϵ
> 0 there exist numbers m = m(
ϵ
), n= n(
ϵ
) such that
Definition 1.6
(
[12]
). A double sequence (
x_{ij}
) ∈
ω
is said to be Ibounded if there exists
M
> 0 such that
Definition 1.7
(
[12]
). A double sequence space E is said to be solid or normal if (
x_{ij}
) ∈ E implies (
α_{ij}x_{ij}
) ∈ E for all sequence of scalars (
α_{ij}
) with 
α_{ij}
 < 1 for all i,j ∈ ℕ.
Definition 1.8
(
[12]
). A double sequence space
E
is said to be monotone if it contains the canonical preimages of its stepspaces.
Definition 1.9
(
[12]
). A double sequence space
E
is said to be convergence free if (
y_{ij}
) ∈
E
whenever (
x_{ij}
) ∈
E
and
x_{ij}
= 0 implies
y_{ij}
= 0.
Definition 1.10
(
[12]
). A double sequence space
E
is said to be a sequence algebra if (
x_{ij}
.
y_{ij}
) ∈
E
whenever (
x_{ij}
), (
y_{ij}
) ∈
E
.
Definition 1.11
(
[12]
). A double sequence space
E
is said to be symmetric if (
x_{ij}
) ∈
E
implies (
x
_{π(ij)}
) ∈
E
, where
π
is a permutation on
Any linear subspace of
ω
, is called a sequence space.
A sequence space
λ
with linear topology is called a Kspace provided each of maps
p_{i}
→
¢
defined by
p_{i}
(
x
) =
x_{i}
is continuous for all
i
∈
A Kspace
λ
is called an FKspace provided
λ
is a complete linear metric space.
An FKspace whose topology is normable is called a BKspace.
Let
λ
and
μ
be two sequence spaces and
A
= (
a_{nk}
) be an infinite matrix of real or complex numbers
a_{nk}
, where
n, k
∈
Then we say that
A
defines a matrix mapping from
λ
to
μ
, and we denote it by writing
A
:
λ
→
μ
.
If for every sequence
x
= (
x_{k}
) ∈
λ
the sequence
Ax
= {(
Ax
)
_{n}
}, the
A
transform of
x
is in
μ
, where
By (
λ
:
μ
), we denote the class of matrices
A
such that
A
:
λ
→
μ
.
Thus,
A
∈ (
λ
:
μ
) if and only if the series on the right side of (1) converges for each
n
∈
and every
x
∈
λ
. (see
[14]
).
The approach of constructing the new sequence spaces by means of the matrix domain of a particular limitation method have been recently studied by Başar and Altay
[15]
, Malkowsky
[16]
, Ng and Lee
[17]
and Wang
[18]
, Başar, Altay and Mursaleen
[19]
.
Şengönül
[20]
defined the sequence
y
= (
y_{i}
) which is frequently used as the
Z^{p}
transform of the sequence
x
= (
x_{i}
) i.e,
where
x
_{−1}
= 0,
p
≠ 1, 1 <
p
< ∞ and
Z^{p}
denotes the matrix
Z^{p}
= (
z_{ik}
) defined by
Following Basar and Altay
[15]
, Şengönül
[20]
introduced the Zweier sequence spaces
Z
and
Z
_{0}
as follows
An Orlicz function is a function
M
: [0,∞) → ]undefined[0,∞), which is continuous, nondecreasing and convex with
M
(0) = 0,
M
(
x
) > 0 for
x
> 0 and
M
(
x
) → ∞ as
x
→ ∞.(see]undefined
[21
,
22]
).
Lindenstrauss and Tzafriri
[22]
used the idea of Orlicz functions to construct the sequence space
The space
ℓ_{M}
is a Banach space with the norm
The space
ℓ_{M}
is closely related to the space
ℓ_{p}
which is an Orlicz sequence space with
M
(
x
) =
x^{p}
for 1 ≤
p
< ∞ (c.f
[23]
,
[24]
,
[25]
).
The following Lemmas will be used for establishing some results of this article.
Lemma 1.12
(
[24]
).
A sequence space E is solid implies that E is monotone.
Lemma 1.13
(
[26
,
27
,
28]
).
Let K
∈
£
(
I
)
and M
⊆
N. If M
∉
I, thenM
⋂
K
∉ I.
Lemma 1.14
(
[26
,
27
,
28]
).
If I
⊂ 2
^{N}
and M
⊆
N. If M
∉
I, then M
∩
K
∉
I.
Recently Vakeel.A.Khan et. al.
[29]
introduced and studied the following classes of sequence spaces.
We also denote by
and
2. Main results
In this article we introduce the following classes of zweier IConvergent double sequence spaces defined by the Orlicz function.
Also we denote by
and
Throughout the article, for the sake of convenience, we will denote by
Z^{p}
(
x_{k}
) =
x′
,
Z^{p}
(
y_{k}
) =
y′
,
Z^{p}
(
z_{k}
) =
z′
for
x, y, z
∈
ω
.
Theorem 2.1.
For any Orlicz function M, the classes of sequences
_{2}
Z^{I}
(
M
),
are linear spaces.
Proof
. We shall prove the result for the space
_{2}
Z^{I}
(
M
). The proof for the other spaces will follow similarly. Let (
x_{ij}
), (
y_{ij}
) ∈
_{2}
Z^{I}
(
M
) and let
α, β
be scalars. Then there exists positive numbers
ρ
_{1}
and
ρ
_{2}
such that
That is for a given
ϵ
> 0, we have
Let
ρ
_{3}
= max{2
α

ρ
_{1}
, 2
β

ρ
_{2}
}. Since
M
is nondecreasing and convex function, we have
Now, by (1) and (2),
Therefore (
αx_{ij}
+
βy_{ij}
) ∈
_{2}
Z^{I}
(
M
). Hence
_{2}
Z^{I}
(
M
) is a linear space.
Theorem 2.2.
The spaces
are Banach spaces normed by
Proof
. Proof of this result is easy in view of the existing techniques and therefore is omitted.
Theorem 2.3.
Let
M
_{1}
and
M
_{2}
be Orlicz functions that satisfy the
△
_{2}

condition. Then
(a)
X
(
M
_{2}
) ⊆
X
(
M
_{1}
.
M
_{2}
);
(b)
X
(
M
_{1}
) ∩
X
(
M
_{2}
) ⊆
X
(
M
_{1}
+
M
_{2}
) For
Proof
. (a) Let
Then there exists
ρ
> 0 such that
Let
ϵ
> 0 and choose
δ
with 0 <
δ
< 1 such that
M
_{1}
(
t
) <
ϵ
for 0 ≤
t
≤
δ
.
Write
and consider for all
we have
We have
For (
y_{ij}
) >
δ
, we have
Since
M
_{1}
is nondecreasing and convex, it follows that
Since
M
_{1}
satisfies the △
_{2}
condition, we have
Hence
From (3), (4) and (5), we have
Thus
The other cases can be proved similarly.
(b) Let
Then there exists
ρ
> 0 such that
The rest of the proof follows from the following equality
Theorem 2.4.
The spaces
are solid and monotone.
Proof
. We shall prove the result for
the result can be proved similarly. Let
Then there exists
ρ
> 0 such that
Let (
α_{ij}
) be a sequence of scalars with 
α_{ij}
 ≤ 1 for all
Then the result follows from (6) and the following inequality for all
By Lemma 1.12, a sequence space E is solid implies that E is monotone.
We have the space
is monotone.
Theorem 2.5.
The spaces
are neither solid nor monotone in general.
Proof
. Here we give a counter example. Let
I
=
I_{δ}
and
M
(
x
) =
x
^{2}
for all
x
∈ [0,∞). Consider the Kstep space
X_{K}
(
M
) of
X
(
M
) defined as follows,
let (
x_{ij}
) ∈
X
(
M
) and let (
y_{ij}
) ∈
X_{K}
(
M
) be such that
Consider the sequence (
x_{ij}
) defined by
x_{ij}
= 1 for all
Then (
x_{ij}
) ∈
_{2}
Z^{I}
(
M
) but its Kstepspace preimage does not belong to
_{2}
Z^{I}
(
M
).
Thus
_{2}
Z^{I}
(
M
) is not monotone. Hence
_{2}
Z^{I}
(
M
) is not solid.
Theorem 2.6.
The spaces
and
_{2}
Z^{I}
(
M
)
are not convergence free in general.
Proof
. Here we give a counter example. Let
I
=
I_{f}
and
M
(
x
) =
x
^{3}
for all
x
∈ [0,∞). Consider the sequence (
x_{ij}
) and (
y_{ij}
) defined by
Then (
x_{ij}
) ∈
_{2}
Z^{I}
(
M
) and
but (
y_{ij}
) ∉
_{2}
Z^{I}
(
M
) and
Hence the spaces
_{2}
Z^{I}
(
M
) and
are not convergence free.
Theorem 2.7.
The spaces
and
_{2}
Z^{I}
(
M
)
are sequence algebras.
Proof
. We prove that
is a sequence algebra. For the space
_{2}
Z^{I}
(
M
), the result can be proved similarly. Let
Then
Let
ρ
=
ρ
_{1}
.
ρ
_{2}
> 0. Then we can show that
Thus
Hence
is a sequence algebra.
Theorem 2.8.
If I is not maximal and I
≠
I_{f}
,
then the spaces
_{2}
Z^{I}
(
M
)
and
are not symmetric.
Proof
. Let
A
∈
I
be infinite and
M
(
x
) =
x
for all
x
= (
x_{ij}
). If
Then
by lemma 1.14. Let
be such that
K
∉
I
and
Let
ϕ
:
K
→
A
and
be bijections, then the map
defined by
is a permutation on
but (
x
_{π(i)π(j)}
) ∉
_{2}
Z^{I}
(
M
) and
Hence
and
_{2}
Z^{I}
(
M
) are not symmetric.
Acknowledgements
The authors would like to record their gratitude to the reviewer for his careful reading and making some useful corrections which improved the presentation of the paper.
BIO
Vakeel A. Khan received M.Sc., M.Phil and Ph.D at Aligarh Muslim University. He is currently an Associate Professor at Aligarh Muslim University. He has published a number of research articles and some books to his name. His research interests include Functional Analysis, sequence spaces, Iconvergence, invariant means, zweier sequences and so on.
Department of Mathematics, Aligarh Muslim University, Aligarh, 200002, India.
email:vakhanmaths@gmail.com
Nazneen Khan received M.Sc. and M.Phil. from Aligarh Muslim University, and is currently a Ph.D. scholar at Aligarh Muslim University. Her research interests are Functional Analysis, sequence spaces and double sequences.
Department of Mathematics, Aligarh Muslim University, Aligarh, 200002, India.
email:nazneen4maths@gmail.com
Kostyrko P.
,
Šalát T.
,
Wilczyński W.
(2000)
Iconvergence.
Real Anal. Exch.
26
(2)
669 
686
Šalát T.
,
Tripathy B.C.
,
Ziman M.
(2004)
On some properties of Iconvergence.
Tatra Mt. Math. Publ.
(28)
279 
286
Demirci K.
(2001)
Ilimit superior and limit inferior
Math. Commun.
6
165 
172
Das P.
,
Kostyrko P.
,
Malik P.
,
Wilczyński W.
(2008)
I and I*Convergence of Double Sequences.
Math. Slovaca
58
605 
620
DOI : 10.2478/s121750080096x
Güurdal M.
,
Huban M.B.
2013
On IConvergence of Double Sequences in the Topology induced by Random 2Norms.
Matematicki Vesnik
65
(3)
113 
Esi Ayhan
,
Hazarika Bipan
(2012)
Lacunary summable sequence spaces of fuzzy numbers defined by ideal convergence and an Orlicz function
Afrika Matematika November
1 
7
Esi Ayhan
,
Kemal zdemir M.
(2013)
0strongly summable sequence spaces in nnormed spaces defined by ideal convergence and an Orlicz function
Mathematica Slovaca
63
(4)
829 
838
DOI : 10.2478/s121750130137y
Esi Ayhan
,
Hazarika Bipan
(2013)
λideal convergence in intuitionistic fuzzy 2normed linear space
Journal of Intelligent Fuzzy Systems: Applications in Engineering and Technology
24
(4)
725 
732
Esi Ayhan
,
Sharma S.K
(2013)
Some Iconvergent sequence spaces defined by using sequence of moduli and nnormed space
Journal of the Egyptian Mathematical Society
21
(2)
103 
107
DOI : 10.1016/j.joems.2013.01.005
Esi Ayhan
,
Dutta Hemen
,
Khalaf Alias B
2013
Some Orlicz extended Iconvergent Asummable classes of sequences of fuzzy numbers
Journal of Inequalities and Applications
Gürdal M.
,
Ahmet S.
(2008)
Extremal ILimit Points of Double sequences.
Applied Mathematics ENotes
8
131 
137
Khan V.A.
,
Khan N.
(2013)
On a new Iconvergent double sequence spaces.
Int J of Anal
Hindawi Publishing Corporation
Article ID126163
2013
1 
7
Khan V.A.
,
Sabiha T.
(2011)
On Some New double sequence spaces of Invariant Means defined by Orlicz function.
Commun. Fac. Sci.
60
11 
21
Khan V.A.
,
Ebadullah K.
On Zweier Iconvergent sequence spaces defined by Orlicz functions. Submitted.
Başar F.
,
Altay B.
(2003)
On the spaces of sequences of pbounded variation and related matrix mappings.
Ukr. Math.J.
55
Malkowsky E.
(1997)
Recent results in the theory of matrix transformation in sequence spaces.
Math.Vesnik.
(49)
187 
196
Ng P.N.
,
Lee P.Y.
(1978)
Cesaro sequence spaces of nonabsolute type.
Comment.Math. Prace.Mat.
20
(2)
429 
433
Wang C.S.
(1978)
On Nörlund sequence spaces.
Tamkang J.Math.
(9)
269 
274
Altay B.
,
Başar F.
,
Mursaleen M.
(2006)
On the Euler sequence space which include the spaces lpand l∞.
I,Inform.Sci.
176
(10)
1450 
1462
DOI : 10.1016/j.ins.2005.05.008
Şengönül M.
(2007)
On The Zweier Sequence Space.
Demonstratio Math.
XL
(1)
181 
196
Bhardwaj V.K.
,
Singh N.
(2000)
Some sequence spaces defined by Orlicz functions.
Demons. Math.
33
(3)
571 
582
Kamthan P.K.
,
Gupta M.
(1980)
Sequence spaces and series.
Marcel Dekker Inc
New York
Tripathy B.C.
,
Hazarika B.
(2011)
Some IConvergent sequence spaces defined by Orlicz function.
Acta Math. Appl. Sin.,Engl.Ser.
27
(1)
149 
154
DOI : 10.1007/s102550110048z
Khan V.A.
,
Khan N.
(2013)
On some IConvergent double sequence spaces defined by a sequence of modulii
Ilirias J. Math
4
(2)
1 
8
Khan V.A.
,
Khan N.
(2013)
On some IConvergent double sequence spaces defined by a modulus function
Eng. Sci. Res.
5
35 
40
Khan V.A.
,
Khan N.
(2013)
IPreCauchy Double Sequences and Orlicz Functions
Eng.Sci. Res.
5
52 
56
Khan V.A.
,
Ebadullah K.
,
Esi Ayhan
,
Khan N.
,
Shafiq M.
(2013)
On Paranorm Zweier IConvergent Sequence Spaces
Journal of Mathematics
Hindawi Publishing Corporation
Article ID 613501
2013
1 
6