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^{1}SQC and OR Unit, Indian Statistical Institute, Kolkata, West Bengal, India

^{2}Department of Mathematics, SASTRA University, Thanjavur, India

^{3}Department of Mathematics, National Institute of Technology, Silchar, Cachar, Assam, India

- *Corresponding Author:
- Lakshmi Narayan M

Department of Mathematics

National Institute of Technology

Silchar, District - Cachar, Assam, India

**Tel:**03842 242 273

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

**Received Date:** May 23, 2016; **Accepted Date:** September 01, 2016; **Published Date:** September 05, 2016

**Citation: **Deepmala, Subramanian N, Mishra LN (2016) The ∫Γ^{3λI} Statistical Convergence of pre-Cauchy over the p- Metric Space Defined by Musielak Orlicz Function. Adv Robot Autom 5:155. doi: 10.4172/2168-9695.1000155

**Copyright:** © 2016 Deepmala, 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** Advances in Robotics & Automation

In this paper we are concerned with ∫Γ3λ I statistical convergence of pre-cauchy triple sequences. ∫Γ3λ I statistical convergence implies ∫Γ3λ I statistical pre-Cauchy condition and examine some properties of these concepts. We examine some properties of these concepts, if the triple entire sequence spaces is statistically convergent then statistically pre-Cauchy and also triple sequence of ideal (I3)- is statistically pre-Cauchy

Analytic sequence; Double sequences; Γ^{3} space; Musielak - Orlicz function; p- metric space; Ideal; Filter; ∫Γ3λ I statistical convergence; ∫Γ3λ I statistical pre-Cauchy

We introduce ∫Γ3λ I sequence space and also discuss ∫Γ3λ I is statistically convergent is pre-Cauchy and the ideal space is pre- Cauchy. Throughout *w*, *x* 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 (*x _{mnk}*), where

Some initial work on double series and interesting results are found in Apostol [1] and double sequence spaces is found in Hardy [2], Subramanian et al. [3], Deepmala et al. [4-7], Mishra et al. [8,9], Mishra and Mishra [10], Mishra [11] and many others. Later on investigated by some initial work on triple sequence spaces is found in sahiner et al. [6], Esi et al. [12-15], Savas et al. [16] , Subramanian et al. [17], Prakash et al. [18,19] 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 give one space is said to be convergent if and only if the triple sequence (

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 space Λ^{3} and Γ^{3} is a metric space with the metric

(1)

Forall *x* = { *x _{mnk}* } and

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

with 1 in the *(i, j, q) ^{th}* position and zero otherwise.

A modulus function was introduced by Nakano [20]. We recall that a modulus *f* is a function from [0,∞) →[0,∞), such that

(1) *f (x) = 0* if and only if *x = 0*

(2)* f (x+y) ≤ f (x)+ f (y,)* for all *x ≥ 0, y ≥ 0*,

(3) *f* is increasing,

(4) *f* is continuous from the right at 0. Since | *f (x) - f (y)* |≤* f (x-y)*, it follows from here that *f* is continuous on [0,∞).

Let *M* and *?* are mutually complementary **Orlicz** functions. Then, we have:

(i) For all *u,y* ≥ *0*,

*uy* ≤ *M(u) + ?(y), (Young’s inequality)* [3] (2)

(ii) For all *u* ≥ *0*, and *0* < *λ* < *1*,

*M (λu)* ≤ *λM (u)* (3)

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

becomes a Banach space which is called an Orlicz sequence space. For *M*(*t*)=*t ^{p}* (1≤

A sequence* f = (f _{mnk})* of Orlicz function 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}*.

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

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

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

Let *w*^{3} denote the set of all complex double sequences and *M* :[0,∞) →[0,∞), be an Orlicz function. Given a triple sequence, *x* ∈ *w*^{3}. Define the sets:

Let n∈ and X be a real vector space of dimension *w*, where *n* ≤ *m*. A real valued function

*d _{p} (x_{1},…,x_{n})* =||

(i) || *(d _{1}(x_{1},0),…,d_{n}(x_{n},0))* ||

(ii) || *(d _{1}(x_{1},0),…,d_{n}(x_{n},0))* ||

for *x _{1},x_{2},…x_{n}* ∈

A trivial example of *p* product metric of n metric space is the *p* norm space is *X* = equipped with the following Euclidean metric in the product space is the *p* norm:

Where for each *i = 1,2,…n i =1, 2,…n.*

If every Cauchy sequence in *X* converges to some *L* ∈ *X*, then *X* is said to be complete with respect to the *p*- metric. Any complete *p ^{-}* metric space is said to be

**Definition**

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

(1) *ρ (x)*≥ *0*, for all *x* ∈ *X*;

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

(3) *ρ (x+y)* ≤ *ρ (x)* + *ρ (y)*, for all *x,y* ∈ *X*;

(4) If (*σ _{mn}*) is a sequence of scalars with

A paranorm *w* for which *ρ (x)* = *0* implies *x* = *0* is called total paranorm and the pair (*X,w*) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm.

B) A family *I⊂ 2 ^{Y×YY}* of subsets of a non empty set

(1) *φ* ∈ *I*

(3) *A, B* ∈ *I, C* ⊂ *A* imply *C* ∈ *I*.

while an admissible ideal I of Y further satisfies {*x*}∈*I* for each *x*∈*Y*. Given *I* ⊂ be a non trivial ideal in A sequence in *X* is said to be *I*- convergent to 0 ∈ *X*, if for each ε > 0 the set

C) A non-empty family of sets *F* ⊂ *2 ^{X×XX}* is a

(1) *φ* ∈ *F*

(3) each *A*, *B* ∈ *F* and each *B* ⊂ *C*, we have *C* ∈ *F*

D) An ideal I is called non-trivial ideal if *I* ≠ *φ* and *X* ∉ *I*. Clearly *I* ⊂ *2 ^{X×XX}* is a non-trivial ideal if and only if

E) A non-trivial ideal *I* ⊂ *2 ^{X×XX}* is called (i) admissible if and only if

If we take *I = I _{f} = {A ⊆ : A is a finite subset}*. Then If is a non-trivial admissible ideal of and the corresponding convergence coincides with the usual convergence. If we take

F) A sequence space *E* is said to be monotone if *E* contains the canonical pre-images of all its step spaces.

**Remark**

Let *μ* = (λ_{rsu}) be a non-decreasing sequence of positive real numbers tending to **infinity** and λ_{111} = *1* and _{r+1,s+1,u+1} ≤ λ_{rsu} + *1*, for all r,s,u ∈ .

The generalized de la Vallee-Poussin means is defined by

Where *I*_{rsu} = [*r,s,u − λ _{rsu} + 1, rsu*]. A sequence

The main aim of this article to introduce the following sequence spaces and examine **topological** and **algebraic** properties of the resulting sequence spaces. Let *p* = (*p _{mnk}*) be a sequence of positive real numbers for all be a sequence of Musielak-Orlicz function, be a

**Definition**

Let f is a Musielak Orlicz function and a triple sequence is said to *I*− statistically convergent if, for any ε > 0 and δ > 0,

**Theorem**

Let *f* is a Musielak Orlicz function and if

for all (*m,n,k*) ∈ *A ^{c}*, where c stands for the complement of the set

we observe that *m,n,k,a,b,c* ∈ *B*

Therefore

which implies

Hence

for all *(m,n,k)* ∈ *A ^{c}*. Let δ

**Theorem**

Let f is a Musielak Orlicz function and a triple sequence *x* = (*x _{mnk}*) is

Therefore for any δ > 0,

Hence *x* is *I _{3}* statistically pre-Cauchy.

Conversely assume that *x* is *I _{3}*, where

This completes the proof.

We examine some properties of these concepts, if the triple entire sequence spaces is statistically convergent then statistically pre-Cauchy and also triple sequence of ideal is statistically pre-Cauchy. In future developed by rough statistical convergence on triple sequence and also rough sets in **statistical** convergence of fractional order of triple sequence of Γ.

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

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 second author NS wish to thank the Department of Science and Technology, Government of India for the financial sanction towards this work under FIST program SR/FST/MSI-107/2015. The research of the first author Deepmala is supported by the Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India under SERB National Post-Doctoral fellowship scheme File Number: PDF/2015/000799. The authors are thankful to the editor(s) and reviewers of esteemed journal i.e., Advances in Robotics and Automation.

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