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Some New Double Sequence Spaces Defined by Orlicz Function in
-Normed Space
Journal of Inequalities and Applications volume 2011, Article number: 592840 (2011)
Abstract
The aim of this paper is to introduce and study some new double sequence spaces with respect to an Orlicz function, and also some properties of the resulting sequence spaces were examined.
1. Introduction
We recall that the concept of a 2-normed space was first given in the works of Gähler ([1, 2]) as an interesting nonlinear generalization of a normed linear space which was subsequently studied by many authors (see, [3, 4]). Recently, a lot of activities have started to study summability, sequence spaces, and related topics in these nonlinear spaces (see, e.g., [5–9]). In particular, Savaş [10] combined Orlicz function and ideal convergence to define some sequence spaces using 2-norm.
In this paper, we introduce and study some new double-sequence spaces, whose elements are form -normed spaces, using an Orlicz function, which may be considered as an extension of various sequence spaces to
-normed spaces. We begin with recalling some notations and backgrounds.
Recall in [11] that an Orlicz function is continuous, convex, and nondecreasing function such that
and
for
, and
as
.
Subsequently, Orlicz function was used to define sequence spaces by Parashar and Choudhary [12] and others. An Orlicz function can always be represented in the following integral form:
, where
is the known kernel of
, right differential for
,
,
for
,
is nondecreasing, and
as
.
If convexity of Orlicz function is replaced by
, then this function is called Modulus function, which was presented and discussed by Ruckle [13] and Maddox [14].
Remark 1.1.
If is a convex function and
, then
for all
with
.
Let and
be real vector space of dimension
, where
. An
-norm on
is a function
which satisfies the following four conditions:
(i) if and only if
are linearly dependent,
(ii) are invariant under permutation,
(iii),
,
(iv).
The pair is then called an
-normed space [3].
Let be equipped with the
-norm, then
the volume of the
-dimensional parallelepiped spanned by the vectors,
which may be given explicitly by the formula
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ1_HTML.gif)
where denotes inner product. Let
be an
-normed space of dimension
and
a linearly independent set in
. Then, the function
on
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ2_HTML.gif)
is defines an norm on
with respect to
(see, [15]).
Definition 1.2 (see [7]).
A sequence in
-normed space
is aid to be convergent to an
in
(in the
-norm) if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ3_HTML.gif)
for every .
Definition 1.3 (see [16]).
Let be a linear space. Then, a map
is called a paranorm (on
) if it is satisfies the following conditions for all
and
scalar:
(i)
,
(ii),
(iii),
(iv) and
imply
.
2. Main Results
Let be any
-normed space, and let
denote
-valued sequence spaces. Clearly
is a linear space under addition and scalar multiplication.
Definition 2.1.
Let be an Orlicz function and
any
-normed space. Further, let
be a bounded sequence of positive real numbers. Now, we define the following new double sequence space as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ4_HTML.gif)
for each .
The following inequalities will be used throughout the paper. Let be a double sequence of positive real numbers with
, and let
. Then, for the factorable sequences
and
in the complex plane, we have as in Maddox [16]
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ5_HTML.gif)
Theorem 2.2.
sequences space is a linear space.
Proof.
Now, assume that and
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ6_HTML.gif)
Since is a
-norm on
, and
is an Orlicz function, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ7_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ8_HTML.gif)
and this completes the proof.
Theorem 2.3.
space is a paranormed space with the paranorm defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ9_HTML.gif)
where ,
.
Proof.
-
(i)
Clearly,
and (ii)
. (iii) Let
, then there exists
such that
(2.7)
So, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ11_HTML.gif)
and thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ12_HTML.gif)
-
(iv)
Now, let
and
. Since
(2.10)
This gives us .
Theorem 2.4.
If for each
and
, then
.
Proof.
If , then there exists some
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ14_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ15_HTML.gif)
for sufficiently large values of and
. Since
is nondecreasing, we are granted
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ16_HTML.gif)
Thus, . This completes the proof.
The following result is a consequence of the above theorem.
Corollary 2.5.
-
(i)
If
for each
and
, then
(2.14)
-
(ii)
If
for each
and
, then
(2.15)
Theorem 2.6.
, where
is the double space of bounded sequences and
.
Proof.
. Then, there exists an
such that
for each
,
. We want to show
. But
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ19_HTML.gif)
and this completes the proof.
Theorem 2.7.
Let and
be Orlicz function. Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ20_HTML.gif)
Proof.
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ21_HTML.gif)
Let ; when adding the above inequality from
to
we get
and this completes the proof.
Definition 2.8 (see [10]).
Let be a sequence space. Then,
is called solid if
whenever
for all sequences
of scalars with
for all
.
Definition 2.9.
Let be a sequence space. Then,
is called monotone if it contains the canonical preimages of all its step spaces (see, [17]).
Theorem 2.10.
The sequence space is solid.
Proof.
Let ; that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ22_HTML.gif)
Let () be double sequence of scalars such that
for all
. Then, the result follows from the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ23_HTML.gif)
and this completes the proof.
We have the following result in view of Remark 1.1 and Theorem 2.10.
Corollary 2.11.
The sequence space is monotone.
Definition 2.12 (see [18]).
Let denote a four-dimensional summability method that maps the complex double sequences
into the double-sequence
, where the
th term to
is as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ24_HTML.gif)
Such transformation is said to be nonnegative if is nonnegative for all
, and
.
Definition 2.13.
Let be a nonnegative matrix. Let
be an Orlicz function and
a factorable double sequence of strictly positive real numbers. Then, we define the following sequence spaces:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ25_HTML.gif)
for each . If
, then we say
is
summable to
, where
.
If we take and
for all
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ26_HTML.gif)
Theorem 2.14.
is linear spaces.
Proof.
This can be proved by using the techniques similar to those used in Theorem 2.2.
Theorem 2.15.
-
(1)
If
, then
(2.24)
-
(2)
If
, then
(2.25)
Proof.
-
(1)
Let
; since
, we have
(2.26)
and hence .
-
(2)
Let
for each
and
. Let
.
Then, for each , there exists a positive integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ30_HTML.gif)
for all . This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F592840/MediaObjects/13660_2011_Article_2353_Equ31_HTML.gif)
Thus, , and this completes the proof.
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The author wishes to thank the referees for their careful reading of the paper and for their helpful suggestions.
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Savaş, E. Some New Double Sequence Spaces Defined by Orlicz Function in -Normed Space.
J Inequal Appl 2011, 592840 (2011). https://doi.org/10.1155/2011/592840
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DOI: https://doi.org/10.1155/2011/592840