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The Generalized Triple Difference Lacunary Statistical on and#915;<sup>3</sup> Over P-Metric Spaces Defined by Musielak Orlicz Function | OMICS International
ISSN: 2090-0902
Journal of Physical Mathematics
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The Generalized Triple Difference Lacunary Statistical on Γ3 Over P-Metric Spaces Defined by Musielak Orlicz Function

Mishra LN1*, Deepmala2 and Subramanian N3

1Department of Mathematics, National Institute of Technology, Silchar – 788 010, India

2SQC & OR Unit, Indian Statistical Institute, Kolkata-700 108, India

3Department of Mathematics, Sastra University, Thanjavur-613 401, India

*Corresponding Author:
Mishra LN
Department of Mathematics
National Institute of Technology
India
Tel: +91 9838375431
E-mail: [email protected]

Received date: May 30, 2016; Accepted date: July 01, 2016; Published date: July 04, 2016

Citation: Mishra LN, Deepmala, Subramanian N (2016) The Generalized Triple Difference Lacunary Statistical on Γ3 Over P-Metric Spaces Defined by Musielak Orlicz Function. J Phys Math 7:183. doi:10.4172/2090-0902.1000183

Copyright: © 2016 Mishra LN, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

We introduce the generalized triple sequence spaces of entire difference lacunary statistical convergence and discuss general topological properties also inclusion theorems are with respect to a sequence of Musielak-Orlicz function.

Keywords

Analytic sequence; Triple sequences; Difference sequence; Γ3 space; Musielak-Orlicz function; Lacunary sequence; Statistical convergence

Introduction

A triple sequence (real or complex) can be defined as a function image where image and image denote the set of natural numbers, real numbers and complex numbers respectively. The different types of notions of triple sequence was introduced and investigated at the initial by Sahiner [1,2], Esi [3-5], Datta [6], Subramanian [7], Debnath [8] and many others.

A triple sequence x=(xmk) is said to be triple analytic if

image

The space of all triple analytic sequences are usually denoted by Λ3. A triple sequence x=(xmk) is called triple entire sequence if

image

The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz [9] as follows

image

For Z=c,c0 and image where image for all image

The difference triple sequence space was introduced by Debnath et al. (see [8]) and is defined as

image

Def´╗┐initions and Preliminaries

Throughout the article w33(Δ), Λ3(Δ) denote the spaces of all, triple entire difference sequence spaces and triple analytic difference sequence spaces respectively.

Subramanian introduced by a triple entire sequence spaces, triple analytic sequences spaces and triple gai sequence spaces [7]. The triple sequence spaces of Γ3(Δ), Λ3(Δ) are defined as follows:

image image

1.Definition

An Orlicz function is a function M:[0,∞)→ [0,∞) which is continuous, non-decreasing and convex with M(0)=M(x)>0 forM(x)>0  and M(x)→∞ as x→∞[10]. If convexity of Orlicz function M is replaced by M(x+y)≤ M(x)+ M(y) then this function is called modulus function. M:[0,∞)→ [0,∞)

Lindenstrauss and Tzafriri [11] used the idea of Orlicz function to construct Orlicz sequence space. A sequence g=(gmn) defined by

image

 is called the complementary function of a Musielak-Orlicz functionf. For a given Musielak-Orlicz function fthe Musielak-Orlicz sequence space tf is defined as follows [12]

image

Where If is a convex modular defined by

image

We consider tf equipped with the Luxemburg metric

image

is an exteneded real number.

2.Definition

Let image X be a real vector space of dimension w where nm. A real valued function image on X satisfying the following four conditions:

(i) image if and and only if image are linearly dependent,

(ii) image is invariant under permutation,

(iii) image

(iv) image

(v) image,

For image is called the p product metric of the Cartesian product of n metric spaces [13].

3.Definition

Let X be a linear metric space. A function image is called paranorm, if

(1) ρ(x)≥0 for all xX;

(2) ρ(-x)= ρ(x) for all xX;

(3) image, for all x,y,z∈X,

(4) If image is a sequence of scalars with image as image and x=(xmk) is a sequence of vectors with

image as image then image as image

4.Definition

The triple sequence image is called triple lacunary if there exist three increasing sequences of integers such that

image

Main Results

The notion of λ-triple entire and triple analytic sequences as follows: Let image be a strictly increasing sequences of positive real numbers tending to infinity, that is

image

and said that a sequence image

is λ-convergent to 0, called a the λ-limit of x if image as image where

image

The sequence image is λ-triple difference analytic if image. If image in the ordinary sense of convergence, then

image

This implies that

image

which yields that image and hence image is λ-convergent to 0. Let image- be an admissible ideal of image be a triple difference lacunary sequence, image be a Musielak-Orlicz function and image be a p-metric space, q=(qmnk) be triple difference analytic sequence of strictly positive real numbers. By w3(p-X) we denote the space of all sequences defined over

image

In the present paper we define the following sequence spaces:

image

In the present paper we plan to study some topological properties and inclusion relation between the above defined sequence spaces. image and image which we shall discuss in this paper.

1.Theorem

Let image be a Musielak-Orlicz function, q=(qmnk) be a triple analytic difference sequence of strictly positive real numbers, the sequence spaces

image

Proof: It is routine verification. Therefore the proof is omitted.

2.Theorem

Let image be a Musielak-Orlicz function, q=(qmnk) be a triple analytic difference sequence of strictly positive real numbers, the sequence space image is a paranormed space with respect to the paranorm defined by

image

Proof: Clearly g(x)≥0 for

image

Since fmnk(0)=0 we get g(0)=0

Conversely, suppose that g(x) then

image

Suppose that image. for each image Then

image. It follows that

image

which is a contradiction. Therefore image. Let

image

and

image

Then by using Minkowski’s inequality, we have

image

g (x + y) ≤ g (x) + g ( y).

Finally, to prove that the scalar multiplication is continuous. Let λ be any complex number. By definition,

image

Then

image

where image. Since image, we have

image

This completes the proof.

3.Theorem

(i) If the Musielak Orlicz function (fmnk) satisfies Δ2- condition, then

image

(ii) If the Musielak Orlicz function (gmnk) satisfies Δ2- condition, then

image

Proof: Let the Musielak Orlicz function (fmk) satisfies Δ2-condition, we get

image

To prove the inclusion

image

Since the Musielak Orlicz function (fmk) satisfies condition, then

image , we get

image

we are granted with (1) and (2)

image

(ii)Similarly,onecanprovethat

image

if the Musielak Orlicz function (gmk) satisfies Δ2-condition.

1.Proposition

The sequence space

image is not solid Proof: The result follows from the following example.

Example: Consider

image

2.Proposition

The sequence space

image

is not monotone.

Proof: The proof follows from Proposition 3.4.

3.Proposition

The sequence space

image is not solid.

4.Proposition

The sequence space

image

is not monotone.

Conclusion

Through this paper we studied some topological properties and inclusion relation with respect to a sequence of Musielak-Orlicz function.

References

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