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**Vishnu Narayan Mishra ^{1*}, Deepmala^{2}, Subramanian N^{3} and Lakshmi Narayan Mishra^{4}**

^{1}Department of Mathematics, Sardar Vallabhbhai National Institute of Technology, Surat, Gujarat, India

^{2}SQC and OR Unit, Indian Statistical Institute, 203 B. T. Road, 700 108, Kolkata, West Bengal, India

^{3}Department of Mathematics, SASTRA University, 613 401, Thanjavur, India

^{4}Department of Mathematics, National Institute of Technology, 788 010, Silchar, District Cachar, Assam, India

- *Corresponding Author:
- Vishnu Narayan Mishra

Department of Mathematics, Sardar Vallabhbhai National Institute of Technology

Surat, Gujarat, India

**Tel:**+91 9913387604

**E-mail:**[email protected]

**Received date:** June 08, 2016; **Accepted date:** July 18, 2016; **Published date:** July 22, 2016

**Citation: **Vishnu Narayan Mishra, Deepmala, Subramanian N, Lakshmi Narayan Mishra (2016) The Generalized Semi Normed Difference of χ^{3} Sequence Spaces Defined by Orlicz Function. J Appl Computat Math 5:316. doi:10.4172/2168-9679.1000316

**Copyright:** © 2016 Mishra, 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 Applied & Computational Mathematics

In this paper we introduced generalized semi normed difference of triple gai sequence spaces defined by an Orlicz function. We study their different properties and obtain some inclusion relations involving these semi normed difference triple gai sequence spaces.

Gai sequence; Analytic sequence; Triple sequence; Difference sequence

Throughout the paper *w*,χ and Λ denote the classes of all, gai and analytic scalar valued single sequences, respectively. We write *w ^{3}* for the set of all complex triple sequences (

Some initial work on double series is found in Apostol [1] and double sequence spaces is found in Hardy [2], Subramanian et al. [3-9], and many others. Later on, some work on triple sequence spaces can also be found in Sahiner et al. [10] , Esi et al. [11-15], Subramanian et al. [16-19], Prakash et al. [20-24] and many others.

Let (*x _{mnk}*) be a triple sequence of real or complex numbers. Then the series is called a triple series. The triple series is said to be

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

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

The vector space of all triple entire sequences are usually denoted by Γ^{3}. The spaces Λ^{3} and Γ^{3} are metric spaces with the metric

, (1)

for all *x=(x _{mnk})* and

Consider a triple sequence *x=(x _{mnk})*. The (

Let *M* and Φ be mutually complementary Orlicz functions. Then, we have:

(i) For all *u,y* ≥ 0,

*uy* ≤ *M(u)*+Φ(*y*), (*Young’s inequality*) [see [25]] (2)

(ii) For all *u* ≥ 0,

*uη(u)*=*M(u)*+Φ(*η(u)*). (3)

(iii) For all *u* ≥ 0, and 0<λ<1,

*M(λu)* ≤ *λM(u)* (4)

Lindenstrauss and Tzafriri [26] used the idea of Orlicz function to construct Orlicz sequence space

The space with the norm

becomes a Banach space which is called an Orlicz sequence space. For , the spaces coincide with the classical sequence space .

A sequence *f=(f _{mnk})* of Orlicz functions is called a Musielak-Orlicz function. A sequence

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

where *M _{f}* is a convex modular defined by

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

is an extended real number.

If *X* is a sequence space, we give the following definitions:

(i) *X*= the continuous *dual of X*;

*X, X, X* are called *α*-(or Kothe-Toeplitz) dual of *X, β*-(or **generalized**- Kothe-Toeplitz) dual of *X, γ*-dual of *X, δ*-dual of *X* respectively.

The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz [27] as follows

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

Later on the notion was further investigated by many others. We now introduce the following difference double sequence spaces defined by

where and for all .

Let be denote the spaces of all, triple gai **difference sequence** space and triple analytic difference sequence space respectively and is defined as

A sequence *x*=(*x _{mnk}*) is said to be triple analytic if . The vector space of all triple analytic sequences is usually denoted by Λ

(5)

for all *x*={*x _{mnk}*} and

Throughout the article denote the spaces of all, triple gai difference sequence spaces and triple analytic difference sequence spaces respectively [28].

For a triple sequence , we define the sets

The space Λ^{3}(Δ) is a metric space with the metric

for all *x*=(*x _{mnk}*) and

The space χ^{3}(Δ) is a metric space with the metric

for all *x*=(*x _{mnk}*) and

Let *p*=(*p _{mnk}*) be a sequence of positive real numbers. We have the following well known inequality, which will be used throughout this paper:

(6)

where *a _{mnk}* and

Spaces of strongly summable sequences were studied at the initial stage by Kuttner, Maddox and others. The class of sequences those are strongly Cesaro summable with respect to a modulus was introduced by Maddox as an extension of the definition of strongly Cesaro summable sequences. Jeff Connor further extended this definition to a definition of strongly A-summability with respect to a modulus when A is nonnegative regular matrix.

Let *η*=(*λabc*) be a non-decreasing sequence of positive real numbers tending to infinity and *λ*_{111}=1 and *λ _{a+b+c+3}≤λ_{a+b+c+3+1}, *for all .

The generalized de la Vall `e e-Poussin means are defined by , where . A sequence *x*=(*x _{mnk}*) is said to (

Throughout the article *E* will represent a semi normed space by a semi norm *q*. We define *w ^{3}(E)* to be the vector space of all

We define the following sets of sequences in this article:

uniformly in *m,n,k*.

uniformly in *m,n,k*.

For *γ*=1 these spaces are denoted by , for and Λ^{3}respectively. We define

Similarly and can be defined.

For , the set of complex numbers, for all , *r*=0, *γ*=0, the spaces , for and Λ^{3} represent the spaces [*V,λ*]_{z}, for and Λ^{3}. These spaces are called as *λ*-strongly gai to zero, *λ*-strongly entire to zero and *λ*-strongly **analytic** by the de la Valle-Poussin method. In the special case, where *λ _{pqr}*=

In this chapter we introduced generalized semi normed difference of triple gai sequence spaces defined by an Orlicz function. We study their different properties and obtain some inclusion relations involving these semi normed difference triple gai sequence spaces.

**Theorem**

Let the sequence *p*=(*p _{mnk}*) be analytic. Then the sequence space , are linear spaces over the complex field for and Λ

**Proof:** It is easy. Therefore the proof is omitted.

**Theorem**

Let *f* be an Orlicz function, then

**Proof:** Let will represent a semi normed space by a semi norm q. Here there exists a positive integer *M*_{1} such that *q*≤*M*_{1}. Then we have

Thus . Since . This completes the proof.

**Theorem**

Let , then is a paranormed space with

where

**Proof:** From Theorem 3.2, for each , *g*(*x*) exists. Clearly *g*(*-x*)=*g*(*x*). It is trivial that for Hence, we get . By Minkowski inequality, we have *g(x+y)≤g(x)+g(y)*. Now we show that the scalar multiplication is continuous. Let α be any fixed complex number. By definition of *f*, we have *x*→θ implies, *g(ax)*→0. Similarly we have for fixed *X* and α→0 implies *g(αx)*→0. Finally *x*→θ and α→0 implies *g(αx)*→0. This completes the proof.

**Theorem**

If *r*≥1 then the inclusion is strict. In general for *j*=0,1,2,…*r*-1 and the inclusions are strict.

**Proof:** The result follows from the following inequality

proceeding inductively, we have for *j*=0,1,2,…*r*-1. The inclusion is strict and it follows from the following example.

**Example:** Let *E=C*, *q(x)=|x|*; *λ _{pqr}*=1 for all for all . Let

**Theorem**

Let *f* be an Orlicz function, then

(a) Let , for all and be analytic, then

(b) If for all then

(c) If then

**Theorem**

Let *f* be an Orlicz function and *s* be a positive integer. Then,

**Proof:** Let *ε*>0 be given and choose *δ* with 0<*δ*<1 such that for . Write and consider

Since *f* is continuous, we have

(7)

and for , we use the fact that, and so, by the definition of *f*, we have for ,

Hence

(8)

From (7) and (8) we obtain . This completes the proof.

The authors are extremely grateful to the anonymous learned referee(s) for their keen reading, valuable suggestion and constructive comments for the improvement of the manuscript. The authors are thankful to the editor(s) and reviewers of Communications Faculty of Sciences University of Ankara Series A1.

The authors declare that there is no conflict of interests regarding the publication of this research paper.

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