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The Generalized Semi Normed Difference of andchi;<sup>3</sup> Sequence Spaces Defined by Orlicz Function | OMICS International
ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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The Generalized Semi Normed Difference of χ3 Sequence Spaces Defined by Orlicz Function

Vishnu Narayan Mishra1*, Deepmala2, Subramanian N3 and Lakshmi Narayan Mishra4

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

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

3Department of Mathematics, SASTRA University, 613 401, Thanjavur, India

4Department 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 χ3 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.

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Abstract

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.

Keywords

Gai sequence; Analytic sequence; Triple sequence; Difference sequence

Introduction

Throughout the paper w,χ and Λ denote the classes of all, gai and analytic scalar valued single sequences, respectively. We write w3 for the set of all complex triple sequences (xmnk), where image, the set of positive integers. Then, w3 is a linear space under the coordinatewise addition and scalar multiplication.

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 (xmnk) be a triple sequence of real or complex numbers. Then the series image is called a triple series. The triple series image is said to be convergent if and only if the triple sequence (Smnk) is convergent, where

image

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

image

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

image

The vector space of all triple entire sequences are usually denoted by Γ3. The spaces Λ3 and Γ3 are metric spaces with the metric

image, (1)

 

for all x=(xmnk) and y={ymnk} in Γ3. Let φ be the set of finite sequences.

Consider a triple sequence x=(xmnk). The (m,n,k)th section x[m,n,k] of the sequence is defined by image for all image, where image is a three dimensional matrix with 1 in the (i,j,k)th position and zero otherwise.

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

(i) For all u,y ≥ 0,

uyM(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

image

The space image with the norm

image

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

A sequence f=(fmnk) of Orlicz functions is called a Musielak-Orlicz function. A sequence g=(gmnk) defined by

image

is called the complementary function of a Musielak-Orlicz function f. For a given Musielak-Orlicz function f, the Musielak-Orlicz sequence space tf is defined as follows

image

where Mf is a convex modular defined by

image

We consider tf equipped with the Luxemburg metric

image

is an extended real number.

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

(i) X= the continuous dual of X;

image

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

image

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

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

image

where image and imageimage for all image.

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

image

Definitions and Preliminaries

A sequence x=(xmnk) is said to be triple analytic if image. The vector space of all triple analytic sequences is usually denoted by Λ3. A sequence x=(xmnk) is called triple entire sequence if image as image. The vector space of triple entire sequences is usually denoted by Γ3. A sequence x=(xmnk) is called triple gai sequence if image as image. The vector space of triple gai sequences is usually denoted by χ3. The space χ3 is a metric space with the metric

image (5)

for all x={xmnk} and y={ymnk} in χ3.

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

For a triple sequence image, we define the sets

image

The space Λ3(Δ) is a metric space with the metric

image

for all x=(xmnk) and y=(ymnk) in Λ3(Δ).

The space χ3(Δ) is a metric space with the metric

image

for all x=(xmnk) and y=(ymnk) in χ3(Δ).

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

image (6)

where amnk and bmnk are complex numbers, D=max{1,2H-1} and image.

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 λ111=1 and λa+b+c+3≤λa+b+c+3+1, for all image.

The generalized de la Vall `e e-Poussin means are defined by image, where image. A sequence x=(xmnk) is said to (V,λ)-summable to a number L if image

Throughout the article E will represent a semi normed space by a semi norm q. We define w3(E) to be the vector space of all E-valued sequences. Let f be an Orlicz function and p=(pmnk) be any sequence of positive real numbers. Let image be four dimensional infinite regular matrix of non-negative complex numbers such that image.

We define the following sets of sequences in this article:

image

uniformly in m,n,k.

image

uniformly in m,n,k.

image

For γ=1 these spaces are denoted by image, for image and Λ3respectively. We define

image

Similarly image and image can be defined.

For image, the set of complex numbers, image for all image, r=0, γ=0, the spaces image, for image and Λ3 represent the spaces [V,λ]z, for image and Λ3. 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 λpqr=pqr, for all p,q,r=1,2,3…. the sets [V,λ]χ3, [V,λ] Γ3 and [V,λ]Λ3 reduce to the sets image.

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.

Main Results

Theorem

Let the sequence p=(pmnk) be analytic. Then the sequence space image, are linear spaces over the complex field image for image and Λ3.

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

Theorem

Let f be an Orlicz function, then image

Proof: Let imagewill represent a semi normed space by a semi norm q. Here there exists a positive integer M1 such that qM1. Then we have

image

Thus image. Since image. This completes the proof.

Theorem

Let image, then image is a paranormed space with image

where image

Proof: From Theorem 3.2, for each image, g(x) exists. Clearly g(-x)=g(x). It is trivial that image for image Hence, we get image. 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 image is strict. In general image for j=0,1,2,…r-1 and the inclusions are strict.

Proof: The result follows from the following inequality

image

proceeding inductively, we have image 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|; λpqr=1 for all imagefor all image. Let f(x)=x, for all image for image. Then consider the sequence x=(xmnk) defined by image for all image. Hence image but image

Theorem

Let f be an Orlicz function, then

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

(b) If image for all image then image

(c) If image then image

Theorem

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

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

Since f is continuous, we have

image (7)

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

image

Hence

image (8)

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

Acknowledgement

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.

Competing Interests

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

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