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**Mishra LN ^{1*}, Deepmala^{2} and Subramanian N^{3}**

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

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

^{3}Department 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 Γ^{3} 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.

**Visit for more related articles at** Journal of Physical Mathematics

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.

Analytic sequence; Triple sequences; Difference sequence; Γ^{3} space; Musielak-Orlicz function; Lacunary sequence; Statistical convergence

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 [6], Subramanian [7], Debnath [8] and many others.

A triple sequence *x*=(*x _{mk}*) is said to be triple analytic if

The space of all triple analytic sequences are usually denoted by Λ^{3}. A triple sequence *x*=(*x _{mk}*) is called triple entire sequence if

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

For *Z*=*c*,*c*_{0} and where for all

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

Throughout the article *w*^{3},Γ^{3}(Δ), Λ^{3}(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 Γ^{3}(Δ), Λ^{3}(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 for*M*(*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*=(*g _{mn}*) defined by

is called the complementary function of a Musielak-Orlicz function*f*. For a given Musielak-Orlicz function *f*the Musielak-Orlicz sequence space *t _{f}* is defined as follows [12]

Where *I _{f}* is a convex modular defined by

We consider *t _{f}* equipped with the Luxemburg metric

is an exteneded real number.

**2.Definition**

Let *X *be a real vector space of dimension *w* where *n**m*. 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 [13].

**3.Definition**

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

(1) *ρ*(*x*)≥0 for all *x*∈*X*;

(2) *ρ*(*-x*)= *ρ*(*x*) for all *x*∈*X*;

(3) , for all *x,y,z∈X*,

(4) If is a sequence of scalars with as and *x*=(*x _{mk}*) 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

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*=(*q _{mnk}*) be triple difference analytic sequence of strictly positive real numbers. By

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*=(*q _{mnk}*) 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*=(*q _{mnk}*) 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 *f _{mnk}*(0)=0 we get

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 (*f _{mnk}*) satisfies Δ

(ii) If the Musielak Orlicz function (*g _{mnk}*) satisfies Δ

**Proof:** Let the Musielak Orlicz function (*f _{mk}*) satisfies Δ

To prove the inclusion

Since the Musielak Orlicz function (*f _{mk}*) satisfies condition, then

, we get

we are granted with (1) and (2)

(ii)Similarly,onecanprovethat

if the Musielak Orlicz function (*g _{mk}*) satisfies Δ

**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.

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

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