Applied Mathematics
Vol.5 No.16(2014), Article ID:49594,10 pages DOI:10.4236/am.2014.516248

Some Lacunary Sequence Spaces of Invariant Means Defined by Musielak-Orlicz Functions on 2-Norm Space

Mohammad Aiyub

Department of Mathematics, University of Bahrain, Sakhir, Bahrain


Copyright © 2014 by author and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

Received 11 July 2014; revised 15 August 2014; accepted 25 August 2014


The purpose of this paper is to introduce and study some sequence spaces which are defined by combining the concepts of sequences of Musielak-Orlicz functions, invariant means and lacunary convergence on 2-norm space. We establish some inclusion relations between these spaces under some conditions.

Keywords:Invariant Means, Musielak-Orlicz Functions, 2-Norm Space, Lacunary Sequence

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:

for and, where be any fixed sequence of non zero complex numbers and

The generalized difference operator has the following binomial representation,

The sequence spaces, and are Banach spaces normed by

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 all2)3) for all

If, where. It can be shown that


[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

and the ratio will be denoted by. The space of lacunary strongly convergent sequence was defined by Freedman et al. [11] as follows:

An Orlicz function is a function which is continuous, non-decreasing and convex with for and as

It is well known that if is convex function and then, for all with

Lindenstrauss and Tzafriri [12] used the idea of Orlicz function and defined the sequence space which was called an Orlicz sequence space such as

which was a Banach space with the norm

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

For a given Musielak-Orlicz function, the Musielak-Orlicz sequence space and its subspaces are defined as follow:

where is a convex modular defined by

We consider equipped with the Luxemburg norm

or equipped with the Orlicz norm

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:

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

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


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

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,

Since, there exist positive such that


Define Since is non decreasing and convex

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

Proof 2. Clearly. Since, for all. we get, for Let

, and let us choose and such that


Let, then we have

Since, we have

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.

where Since,

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

Proof 3. The inclusion is obvious. Let

. Then there exists some positive number such that

as, uniformly in. Define. Since is non decreasing and convex for all, we have

where, by (1).


Theorem 4. Let be a Musielak-Orlicz functions. If for all, then

Proof 4. Let by using (1), we have

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:


Proof 5. Let Then there exist such that

Since is non decreasing and convex, we have

Taking, we have

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)



Proof 6. Let Then


uniformly in n. We have

by (1). Applying and multiplying by and both side of this inequality, we get

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

, uniformly in n. Hence for all sequence of scalars with for all, whenever. The spaces are monotone follows from the remark (1).


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