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

# 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 where and 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 , Subramanian , Debnath  and many others.

A triple sequence x=(xmk) is said to be triple analytic if The space of all triple analytic sequences are usually denoted by Λ3. A triple sequence x=(xmk) is called triple entire sequence if The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz  as follows For Z=c,c0 and where for all The difference triple sequence space was introduced by Debnath et al. (see ) and is defined as #### Def﻿initions 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 . The triple sequence spaces of Γ3(Δ), Λ3(Δ) are defined as follows:  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→∞. 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  used the idea of Orlicz function to construct Orlicz sequence space. A sequence g=(gmn) defined by 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 Where If is a convex modular defined by We consider tf equipped with the Luxemburg metric is an exteneded real number.

2.Definition

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

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

(ii) is invariant under permutation,

(iii) (iv) (v) ,

For is called the p product metric of the Cartesian product of n metric spaces .

3.Definition

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

(1) ρ(x)≥0 for all xX;

(2) ρ(-x)= ρ(x) for all xX;

(3) , for all x,y,z∈X,

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

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

The notion of λ-triple entire and triple analytic sequences as follows: Let be a strictly increasing sequences of positive real numbers tending to infinity, that is and said that a sequence is λ-convergent to 0, called a the λ-limit of x if as where The sequence is λ-triple difference analytic if . If in the ordinary sense of convergence, then This implies that which yields that and hence is λ-convergent to 0. Let - be an admissible ideal of be a triple difference lacunary sequence, be a Musielak-Orlicz function and 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 In the present paper we define the following sequence spaces: In the present paper we plan to study some topological properties and inclusion relation between the above defined sequence spaces. and which we shall discuss in this paper.

1.Theorem

Let be a Musielak-Orlicz function, q=(qmnk) be a triple analytic difference sequence of strictly positive real numbers, the sequence spaces Proof: It is routine verification. Therefore the proof is omitted.

2.Theorem

Let be a Musielak-Orlicz function, q=(qmnk) be a triple analytic difference sequence of strictly positive real numbers, the sequence space is a paranormed space with respect to the paranorm defined by Proof: Clearly g(x)≥0 for Since fmnk(0)=0 we get g(0)=0

Conversely, suppose that g(x) then Suppose that . for each Then . It follows that which is a contradiction. Therefore . Let and Then by using Minkowski’s inequality, we have g (x + y) ≤ g (x) + g ( y).

Finally, to prove that the scalar multiplication is continuous. Let λ be any complex number. By definition, Then where . Since , we have This completes the proof.

3.Theorem

(i) If the Musielak Orlicz function (fmnk) satisfies Δ2- condition, then (ii) If the Musielak Orlicz function (gmnk) satisfies Δ2- condition, then Proof: Let the Musielak Orlicz function (fmk) satisfies Δ2-condition, we get To prove the inclusion Since the Musielak Orlicz function (fmk) satisfies condition, then , we get we are granted with (1) and (2) (ii)Similarly,onecanprovethat if the Musielak Orlicz function (gmk) satisfies Δ2-condition.

1.Proposition

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

Example: Consider 2.Proposition

The sequence space is not monotone.

Proof: The proof follows from Proposition 3.4.

3.Proposition

The sequence space is not solid.

4.Proposition

The sequence space 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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