Some Lacunary Sequence Spaces of Invariant Means Defined by Musielak-Orlicz Functions on 2-Norm Space ()
1. Introduction
Let
be the set of all sequences of real numbers
and
be respectively the Banach spaces of bounded, convergent and null sequences
with
or
the usual norm
, where
, the positive integers.
The idea of difference sequence spaces was first introduced by Kizmaz [1] and then the concept was generalized by Et and Çolak [2] . Later on Et and Esi [3] extended the difference sequence spaces to the sequence spaces:
![](https://www.scirp.org/html/htmlimages\16-7402121x\345ad56d-a577-451b-be13-e8720bea2ada.png)
for
and
, where
be any fixed sequence of non zero complex numbers and
![](https://www.scirp.org/html/htmlimages\16-7402121x\7047ef0d-98b2-4a44-98aa-d4f747811a03.png)
The generalized difference operator has the following binomial representation,
![](https://www.scirp.org/html/htmlimages\16-7402121x\77c39894-d879-4069-afeb-9932722c185c.png)
The sequence spaces
,
and
are Banach spaces normed by
![](https://www.scirp.org/html/htmlimages\16-7402121x\cd5b8cde-2cf3-4e56-a192-0da892bbd275.png)
The concept of 2-normed space was initially introduces by Gahler [4] as an interesting linear generalization of normed linear space which was subsequently studied by many others [5] [6] ). Recently a lot of activities have started to study summability, sequence spaces and related topics in these linear spaces [7] [8] ).
Let
be a real vector space of dimension
, where
. A 2-norme on
is a function
which satisfies:
1)
if and only if
and
are linearly dependent2)
3)
4)
.
The pair
is called a 2-normed space. As an example of a 2-normed space we may take
being equiped with the 2-norm
= the area os paralelogram spaned by the vectors
and
, which may be given explicitly by the formula
.
Then clearly
is 2-normed space. Recall that
is a 2-Banach space if every cauchy sequence in
is convergent to some
.
Let
be a mapping of the positive integers into itself. A continuous linear functional
on
is said to be an invariant mean or
-mean if and only if 1)
, when the sequence
has,
for all
2)
3)
for all ![](https://www.scirp.org/html/htmlimages\16-7402121x\73f06d82-6683-419d-a48f-c82673d89558.png)
If
, where
. It can be shown that
![](https://www.scirp.org/html/htmlimages\16-7402121x\a19ef506-4d77-4fbc-b657-1a7f812f29d1.png)
where
[9] .
In the case
is the translation mapping
,
-mean is often called a Banach limit and
the set of bounded sequences of all whose invariant means are equal is the set of almost convergent sequence [10] .
By Lacunary sequence
where
we mean an increasing sequence of non negative integers
. The intervals determined by
are denoted by ![](https://www.scirp.org/html/htmlimages\16-7402121x\30808650-810f-4ed4-a9f4-c73a57551af7.png)
and the ratio
will be denoted by
. The space of lacunary strongly convergent sequence
was defined by Freedman et al. [11] as follows:
![](https://www.scirp.org/html/htmlimages\16-7402121x\65a3b8e5-5f80-48c9-9545-eec8712b20ff.png)
An Orlicz function is a function
which is continuous, non-decreasing and convex with
for
and
as ![](https://www.scirp.org/html/htmlimages\16-7402121x\3b585fc4-3b21-437a-8128-c337a0d739f5.png)
It is well known that if
is convex function and
then
, for all
with ![](https://www.scirp.org/html/htmlimages\16-7402121x\0efd82ef-5479-4c97-946b-2be5a2ac010f.png)
Lindenstrauss and Tzafriri [12] used the idea of Orlicz function and defined the sequence space which was called an Orlicz sequence space
such as
![](https://www.scirp.org/html/htmlimages\16-7402121x\5362c58a-a0c2-41df-9e67-63a81eeb41aa.png)
which was a Banach space with the norm
![](https://www.scirp.org/html/htmlimages\16-7402121x\eaf74d12-5dc1-49c0-82bb-610540d2861e.png)
which was called an Orlicz sequence space. The
was closely related to the space
which was an Orlicz sequence space with
for
. Later the Orlicz sequence spaces were investigated by Prashar and Choudhry [13] , Maddox [14] , Tripathy et al. [15] -[17] and many others.
A sequence of function
of Orlicz function is called a Musielak-Orlicz function [18] [19] . Also a Musielak-Orlicz function
is called complementary function of a Musielak-Orlicz function
if
![](https://www.scirp.org/html/htmlimages\16-7402121x\774c1b42-bf56-4c33-b58c-ce6c7ca5fa92.png)
For a given Musielak-Orlicz function
, the Musielak-Orlicz sequence space
and its subspaces
are defined as follow:
![](https://www.scirp.org/html/htmlimages\16-7402121x\98f14ddb-4844-48b3-b42d-df130f43ef59.png)
![](https://www.scirp.org/html/htmlimages\16-7402121x\689738a8-aa85-4051-8c4d-11316409b48e.png)
where
is a convex modular defined by
![](https://www.scirp.org/html/htmlimages\16-7402121x\3035a93e-6a76-4f81-b879-17fb76060bef.png)
We consider
equipped with the Luxemburg norm
![](https://www.scirp.org/html/htmlimages\16-7402121x\e96fea68-5f4e-4be9-8cff-4fd2458127e4.png)
or equipped with the Orlicz norm
![](https://www.scirp.org/html/htmlimages\16-7402121x\6f7e064a-c267-4126-88ec-8bf63c017dad.png)
The main purpose of this paper is to introduce the following sequence spaces and examine some properties of the resulting sequence spaces. Let
be a Musielak-Orlicz function,
is called a 2-normed space. Let
be any sequences of positive real numbers, for all
and
such that
. Let
be any real number such that
. By
we denote the space of all sequences defined over
. Then we define the following sequence spaces:
![](https://www.scirp.org/html/htmlimages\16-7402121x\d2cfb49e-6d1d-469e-aaad-155e7775b585.png)
![](https://www.scirp.org/html/htmlimages\16-7402121x\383374aa-9ffe-4469-983d-bffa1ad7aed2.png)
![](https://www.scirp.org/html/htmlimages\16-7402121x\53073dac-53ef-4ad2-8e67-303833f2357a.png)
Definition 1. A sequence space
is said to be solid or normal if
whenever
and for all sequences of scalar
with
[20] .
Definition 2. A sequence space
is said to be monotone if it contains the canonical pre-images of all its steps spaces, [20] .
Definition 3. If
is a Banach space normed by
, then
is also Banach space normed by
![](https://www.scirp.org/html/htmlimages\16-7402121x\efeb860a-0917-4c86-a748-4e8f40554a67.png)
Remark 1. The following inequality will be used throughout the paper. Let
be a positive sequence of real numbers with
,
. Then for all
for all
. We have
(1)
2. Main Results
Theorem 1. Let
be a Musielak-Orlicz function,
be a bounded sequence of positive real number and
be a lacunary sequence. Then
![](https://www.scirp.org/html/htmlimages\16-7402121x\ccb9d952-0955-43eb-a43c-2371a0a893bb.png)
and
are linear spaces over the field of complex numbers.
Proof 1. Let
and
. In order to prove the result we need to find some
such that,
![](https://www.scirp.org/html/htmlimages\16-7402121x\a8aed64b-9410-4555-8a1d-9f2eb48535cc.png)
Since
, there exist positive
such that
![](https://www.scirp.org/html/htmlimages\16-7402121x\e73cb110-2254-4d6a-9b81-f2ba6e2b09d4.png)
and
![](https://www.scirp.org/html/htmlimages\16-7402121x\7a1a10bd-82e1-4fe7-919b-433caea8dc9c.png)
Define
Since
is non decreasing and convex
![](https://www.scirp.org/html/htmlimages\16-7402121x\d4527bc9-40ed-4b54-8584-5814e7591e07.png)
So that
This completes the proof. Similarly, we can prove that
and
are linear spaces.
Theorem 2. Let
be a Musielak-Orlicz function,
be a bounded sequence of positive real number and
be a lacunary sequence. Then
is a topological linear space totalparanormed by
![](https://www.scirp.org/html/htmlimages\16-7402121x\908be0a9-8d98-4a1b-9363-5d2c29b61439.png)
Proof 2. Clearly
. Since
, for all
. we get
, for
Let
,
and let us choose
and
such that
![](https://www.scirp.org/html/htmlimages\16-7402121x\3c4372fe-aab8-4adf-999f-43cfcd61fdec.png)
and
![](https://www.scirp.org/html/htmlimages\16-7402121x\0b500494-0da2-4d85-bda9-5954c773e7d3.png)
Let
, then we have
![](https://www.scirp.org/html/htmlimages\16-7402121x\a418b350-95c4-442a-8e1c-65b3c47c97bc.png)
Since
, we have
![](https://www.scirp.org/html/htmlimages\16-7402121x\9e9f773e-b897-42d8-a2b1-a7069f67f97c.png)
![](https://www.scirp.org/html/htmlimages\16-7402121x\38e901d4-a8a4-4203-9fdd-3947e541575e.png)
Finally, we prove that the scalar multiplication is continuous. Let
be a given non zero scalar in
. Then the continuity of the product follows from the following expression.
![](https://www.scirp.org/html/htmlimages\16-7402121x\e553dcf5-b395-45da-bef7-432cb818637b.png)
where
Since
,
![](https://www.scirp.org/html/htmlimages\16-7402121x\b72e3189-c754-48f8-8404-373c388b247c.png)
This completes the proof of this theorem.
Theorem 3. Let
be a Musielak-Orlicz function,
be a bounded sequence of positive real number and
be a lacunary sequence. Then
![](https://www.scirp.org/html/htmlimages\16-7402121x\6eede263-4f47-4468-af3c-281cb188fb31.png)
Proof 3. The inclusion
is obvious. Let
. Then there exists some positive number
such that
![](https://www.scirp.org/html/htmlimages\16-7402121x\72236f8c-f53b-4646-b03e-e1e3784e2940.png)
as
, uniformly in
. Define
. Since
is non decreasing and convex for all
, we have
![](https://www.scirp.org/html/htmlimages\16-7402121x\bf181007-5b6b-4560-b258-8ca3142a02a2.png)
where
,
by (1).
Thus
.
Theorem 4. Let
be a Musielak-Orlicz functions. If
for all
, then
![](https://www.scirp.org/html/htmlimages\16-7402121x\a35753c9-9640-4735-819b-427d0e430c7c.png)
Proof 4. Let
by using (1), we have
![](https://www.scirp.org/html/htmlimages\16-7402121x\76a7e682-f9b1-4127-af10-bf0120dc6310.png)
Since
, we can take the
. Hence we can get
.
This complete the proof.
Theorem 5. Let
be fixed integer. Then the following statements are equivalent:
1)
2)
3) ![](https://www.scirp.org/html/htmlimages\16-7402121x\e199d765-4740-4132-91ac-a3d395d1ea6e.png)
Proof 5. Let
Then there exist
such that
![](https://www.scirp.org/html/htmlimages\16-7402121x\d006ef14-9124-46b6-908f-02389843b128.png)
Since
is non decreasing and convex, we have
![](https://www.scirp.org/html/htmlimages\16-7402121x\7ec694ab-843c-49cc-abcb-476a56b5f791.png)
Taking
, we have
![](https://www.scirp.org/html/htmlimages\16-7402121x\56e3759a-08e3-4cf2-a2b5-084152f39598.png)
i.e.
The rest of these cases can be proved in similar way.
Theorem 6. Let
and
be two Musielak-Orlicz functions. Then we have 1) ![](https://www.scirp.org/html/htmlimages\16-7402121x\854dcab5-f982-4381-9487-81e51ad71827.png)
2) ![](https://www.scirp.org/html/htmlimages\16-7402121x\69dc95d0-4da9-47c7-90aa-0e954d25ad82.png)
3) ![](https://www.scirp.org/html/htmlimages\16-7402121x\450e761f-dd00-43dc-aeb0-ef2f00f901cb.png)
Proof 6. Let
Then
![](https://www.scirp.org/html/htmlimages\16-7402121x\1190a9dc-7949-4503-8463-b8b41847b005.png)
and
![](https://www.scirp.org/html/htmlimages\16-7402121x\aa361f02-f300-47d4-902c-fb481a571353.png)
uniformly in n. We have
![](https://www.scirp.org/html/htmlimages\16-7402121x\88e6db79-6b1f-44d4-a512-45908037310e.png)
by (1). Applying
and multiplying by
and
both side of this inequality, we get
![](https://www.scirp.org/html/htmlimages\16-7402121x\4e3ac98f-20a6-4b4e-9fba-6cd9d9fa27d6.png)
uniformly in n. This completes the proof 2) and 3) can be proved similar to 1).
Theorem 7. 1) The sequence spaces
and
are solid and hence they are monotone.
2) The space
is not monotone and neither solid nor perfect.
Proof 7. We give the proof for
. Let
and
be a sequence of scalars such that
for all
. Then we have
![](https://www.scirp.org/html/htmlimages\16-7402121x\a2f4833a-1134-46da-a48c-d807d6abeb16.png)
, uniformly in n. Hence
for all sequence of scalars
with
for all
, whenever
. The spaces are monotone follows from the remark (1).