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1. *Corresponding Author: Ayhan Esi, Email: aesi23@hotmail.com
RESEARCH ARTICLE
Available Online at www.ajms.in
Asian Journal of Mathematical Sciences 2017; 1(1):19-25
The triple sequence spaces of on rough statistical convergence defined by Musielak-Orlicz
function of metric
1
Ayhan Esi* , 2
N. Subramanian
1
Adiyaman University, Department of Mathematics,02040, Adiyaman, Turkey
2
Department of Mathematics, SASTRA University, Thanjavur-613 401, India.
Received on: 14/02/2017, Revised on: 24/02/2017, Accepted on: 28/02/2017
ABSTRACT
We introduce the triple sequence spaces notions of point wise rough statistical convergence and rough
statistically Cauchy sequences of real valued function and study their inclusion.
2010 Mathematics Subject Classification. 40F05, 40J05, 40G05.
Keywords: Point wise rough statistical convergence, rough convergence, rough statistically Cauchy
criterion, triple sequences, chi sequence
INTRODUCTION
The idea of rough convergence was introduced by Phu [11], who also introduced the concepts of rough
limit points and roughness degree. The idea of rough convergence occurs very naturally in numerical
analysis and has interesting applications. Aytar [1] extended the idea of rough convergence into rough
statistical convergence using the notion of natural density just as usual convergence was extended to
statistical convergence. Pal et al. [10] extended the notion of rough convergence using the concept of
ideals which automatically extends the earlier notions of rough convergence and rough 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 et al. [12,13], Esi et
al. [2-4], Dutta et al. [5],Subramanian et al. [14], Debnath et al. [6], Esi et al.[16] and many others.
A triple sequence is said to be triple analytic if
The space of all triple analytic sequences are usually denoted by . A triple sequence is
called triple gai sequence if
as
The space of all triple gai sequences are usually denoted by .
DEFINITIONS AND PRELIMINARIES
1 Definition
An Orlicz function ([see [7]) is a function which is continuous, non-decreasing and
convex with for and as If convexity of Orlicz function
is replaced by then this function is called modulus function.

2. Ayhan Esi et al./ The triple sequence spaces of on rough statistical convergence defined by Musielak-Orlicz function of
metric
20
© 2017, AJMS. All Rights Reserved
Lindenstrauss and Tzafriri ([8]) used the idea of Orlicz function to construct Orlicz sequence space.
A sequence defined by
is called the complementary function of a Musielak-Orlicz function . For a given Musielak-Orlicz
function [see [9] ] the Musielak-Orlicz sequence space is defined as follows
where is a convex modular defined by
We consider equipped with the Luxemburg metric
is an exteneded real number.
2 Definition
Let be a real vector space of dimension where A real valued function
on satisfying the following four conditions:
(i) if and only if are linearly dependent,
(ii) is invariant under permutation,
(iii)
(iv) (or)
(v)
for is called the product metric of the Cartesian product of metric
spaces (see [15]) .
3 Definition
A triple sequence spaces of is said to be rough convergent ( convergent) to if for every
there exists a positive integer such that for all
where is a non negative real number called the convergence degree of three.
4 Definition
A triple sequence spaces of is said to be rough convergent to if for each
: ,0|> + ∈ .
Here is called the rough limit of the sequence and we write
5 Definition
A triple sequence spaces of is said to be rough convergent to if for each ,
Here is called the rough limit of the triple sequence spaces of and we write
6 Definition
Let be a subset of the set of positive integers Let be a set defined as follows,
Then the natural density of is defined as
where denotes the number of elements in . Clearly finite set has natural
density zero.
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3. Ayhan Esi et al./ The triple sequence spaces of on rough statistical convergence defined by Musielak-Orlicz function of
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7 Definition
A real triple sequence spaces of is said to be point wise statistically
convergent on a set if for every ,
Rough statistical convergent triple sequence spaces of of real valued functions
1 Definition
A triple sequence spaces of of real valued functions on a set is said
to be point wise rough statistically convergent to function on a set if for every there exists a
real number such that
for every
That is
(3.1)
We write on . That is for every there exists an integer such
that
for every and , for all .
2 Remark
Let be a Musielak-Orlicz function of a triple sequence spaces of real valued functions which is not
point wise statistically convergent but point wise rough statistically convergent.
3 Example
Define a triple sequence spaces of real valued functions
on
The triple sequence spaces of is pointwise rough statistically convergent to zero
with roughness degree 3. Therefore this triple sequence spaces is not point wise statistically convergent.
4 Definition
Let be a triple sequence spaces of real valued functions define on a set
. Let be a real number. The triple sequence
spaces of real valued functions is said to be rough statistically Cauchy sequence if for
every there exists a natural number such that
a.a.k. . That is
forevery
. Now define the following sequence spaces:
,
where and is a real valued functions.
Main Results
In this section we introduce the notion of different types of convergent double sequences. This
generalizes and unifies different notions of convergence for We shall denote the ideal of by
Let be a non empty set. A non-void class (power set, of ) is called an ideal if is
additive (i.e ) and hereditary (i.e and ). A non-empty family
of sets is said to be a filter on if and
For each ideal there is a filter given by A non-trivial ideal
is called admissible if and only if
Theorem-1
Let and
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4. Ayhan Esi et al./ The triple sequence spaces of on rough statistical convergence defined by Musielak-Orlicz function of
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be two triple sequence spaces of real valued
functions defined on a set . If
and
on then
,
for all
Proof: It is obvious for . Let and . Let be a real number. Let be
given. Then
3 + 3 , 1, 2,⋯, −1 ≥ + for every
, ∈ 3 ≤ , ≤ , ≤ : 3 − , 1, 2,⋯, −1 ≥ + , ∈ 3. Since
3 + 3 , 1, 2,⋯, −1 ≥ 3 − , 1, 2,⋯, −1 + 3 − ,
1, 2,⋯, −1 , hence we obtain
on .
Theorem-2
Let be a triple sequence spaces of real valued
functions. Then the following statements are equivalent.
(i) is a point wise rough statistically convergent.
(ii) is a rough statistically Cauchy sequence.
(iii)If
then
a.a.k. for every
Proof: (i) (ii) Let be a real number.
Let
Then for every
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5. Ayhan Esi et al./ The triple sequence spaces of on rough statistical convergence defined by Musielak-Orlicz function of
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. a.a.k. Let be a
natural number so choosen that
. Then we have
.
Hence is a rough statistically Cauchy sequence.
(ii) (iii) Assume that (ii) is true. Choose a natural number
such that contains
a.a.k for every .By (ii) we get
contains
a.a.k for every Hence
a.a.k for every Now
for every
for every
for every . Hence
for every
+ →∞1 ≤ , ≤ , ≤ : 3 , 1, 2,⋯, −1 ∉ ′
for every Therefore , which contains
a.a.k for every
Proceeding in this way contains
a.a.k any by above argument contains
a.a.k for every and . By induction
principle, In general we can choosen positive integer triple sequence spaces such that
(4.1)
for every if Now construct a triple sub sequence of
containing of all terms such that and
when then
for every .
Define a triple sequence spaces of real valued function
by
if is a term of
for every Then
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6. Ayhan Esi et al./ The triple sequence spaces of on rough statistical convergence defined by Musielak-Orlicz function of
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If and then either
is a term of
on
on and
for every Now if
then
. So by equation (4.1)
for every
By taking we get the limit is zero. Hence
a.a.k for every Therefore (ii) implies (iii).
(iii) (i) Let us assume that (iii) holds. That is
a.a.k. for every and
on
Let be real number and . Then for every such that
3⊆ ≤ , ≤ , ≤ : 3 , 1, 2,⋯, −1 ∈ 3≠ ≤ , ≤ , ≤ : 3 , 1,
2,⋯, −1 ∈ 3 ≤ , ≤ , ≤ : 3 − , 1, 2,⋯, −1 ≥ + ∈ 3.
Since
on We have a finite number of integers, (say)
Therefore since
a.a.k for every we get
+ ∈ 3≤lim →∞1 ≤ , ≤ , ≤ : 3 ≠ , 1, 2,⋯, −1 ≥ +
∈ 3+ →∞ =0. Hence
So (i) holds.
Competing Interests
The authors declare that there is not any conflict of interests regarding the publication of this manuscript.
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