International Journal of Analysis and Applications

Volume 16, Number 5 (2018), 643-653

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-643

WIJSMAN ROUGH LACUNARY STATISTICAL CONVERGENCE ON I CESÀRO

TRIPLE SEQUENCES

N. SUBRAMANIAN1 AND A. ESI2,∗

1Department of Mathematics, SASTRA University, Thanjavur-613 401, India

2Department of Mathematics, Adiyaman University, 02040, Adiyaman, Turkey

∗Corresponding author: aesi23@hotmail.com

Abstract. In this paper, we defined concept of Wijsman I-Cesàro summability for sequences of sets and

investigate the relationships between the concepts of Wijsman strongly I-Cesàro summability and Wijsman

statistical I− Cesàro summability by using the concept of a triple sequence spaces.

1. Introduction

The idea of statistical convergence was introduced by Steinhaus and also independently by Fast for real

or complex sequences. Statistical convergence is a generalization of the usual notion of convergence, which

parallels the theory of ordinary convergence.

Let K be a subset of the set of positive integers N×N×N, and let us denote the set

{(m,n,k) ∈ K : m ≤ u,n ≤ v,k ≤ w} by Kuvw. Then the natural density of K is given by

δ (K) = limuvw→∞
|Kuvw|
uvw

, where |Kuvw| denotes the number of elements in Kuvw. Clearly, a finite subset

has natural density zero, and we have δ (Kc) = 1 − δ (K) where Kc = N\K is the complement of K. If

K1 ⊆ K2, then δ (K1) ≤ δ (K2) .

Throughout the paper, R denotes the real of three dimensional space with metric (X,d) . Consider a triple

sequence x = (xmnk) such that xmnk ∈ R,m,n,k ∈ N.

Received 2017-10-29; accepted 2018-01-13; published 2018-09-05.

2010 Mathematics Subject Classification. 40F05, 40J05.

Key words and phrases. Wijsman rough statistical convergence; natural density; triple sequences; I− Cesàro.
c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

643

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-643


Int. J. Anal. Appl. 16 (5) (2018) 644

A triple sequence x = (xmnk) is said to be statistically convergent to 0 ∈ R, written as st − lim x = 0,

provided that the set {
(m,n,k) ∈ N3 : |xmnk, 0| ≥ �

}
has natural density zero for any � > 0. In this case, 0 is called the statistical limit of the triple sequence

x.

If a triple sequence is statistically convergent, then for every � > 0, infinitely many terms of the sequence

may remain outside the �− neighbourhood of the statistical limit, provided that the natural density of the

set consisting of the indices of these terms is zero. This is an important property that distinguishes statistical

convergence from ordinary convergence. Because the natural density of a finite set is zero, we can say that

every ordinary convergent sequence is statistically convergent.

If a triple sequence x = (xmnk) satisfies some property P for all m,n,k except a set of natural density

zero, then we say that the triple sequence x satisfies P for almost all (m,n,k) and we abbreviate this by a.a.

(m,n,k).

Let
(
xminjk`

)
be a sub sequence of x = (xmnk). If the natural density of the set

K =
{

(mi,nj,k`) ∈ N3 : (i,j,`) ∈ N3
}

is different from zero, then
(
xminjk`

)
is called a non thin sub sequence

of a triple sequence x.

c ∈ R is called a statistical cluster point of a triple sequence x = (xmnk) provided that the natural density

of the set {
(m,n,k) ∈ N3 : |xmnk − c| < �

}
is different from zero for every � > 0. We denote the set of all statistical cluster points of the sequence x

by Γx.

A triple sequence x = (xmnk) is said to be statistically analytic if there exists a positive number M such

that

δ
({

(m,n,k) ∈ N3 : |xmnk|
1/m+n+k ≥ M

})
= 0

The theory of statistical convergence has been discussed in trigonometric series, summability theory,

measure theory, turnpike theory, approximation theory, fuzzy set theory and so on.

The idea of rough convergence was introduced by [8], 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 [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 [7]

extended the notion of rough convergence using the concept of ideals which automatically extends the earlier

notions of rough convergence and rough statistical convergence.

Let (X,ρ) be a metric space. For any non empty closed subsets A,Amnk ⊂ X (m,n,k ∈ N) , we say that



Int. J. Anal. Appl. 16 (5) (2018) 645

the triple sequence (Amnk) is Wijsman statistical convergent to A is the triple sequence (d (x,Amnk)) is

statistically convergent to d (x,A) , i.e., for � > 0 and for each x ∈ X

limrst
1
rst
|{m ≤ r,n ≤ s,k ≤ t : |d (x,Amnk) −d (x,A)| ≥ �}| = 0.

In this case, we write St−limmnkAmnk = A or Amnk −→ A (WS) . The triple sequence (Amnk) is bounded

if supmnkd (x,Amnk) < ∞ for each x ∈ X.

In this paper, we introduce the notion of Wijsman rough statistical convergence of triple sequences. Defining

the set of Wijsman rough statistical limit points of a triple sequence, we obtain to Wijsman statistical

convergence criteria associated with this set. Later, we prove that this set of Wijsman statistical cluster

points and the set of Wijsman rough statistical limit points of a triple sequence.

A triple sequence (real or complex) can be defined as a function x : N×N×N → R (C) , where N,R and

C 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 ( [9], [10]), ( [2], [3], [4]), [5],

[11], [6], [12] and many others.

Throughout the paper let r be a nonnegative real number.

2. Definitions and Preliminaries

Definition 2.1. A triple sequence x = (xmnk) of real numbers is said to be statistically convergent to l ∈ R3

if for any � > 0 we have d (A (�)) = 0, where

A (�) =
{

(m,n,k) ∈ N3 : |xmnk − l| ≥ �
}
.

Definition 2.2. A triple sequence x = (xmnk) is said to be statistically convergent to l ∈ R3, written as

st− limx = l, provided that the set {
(m,n,k) ∈ N3 : |xmnk − l| ≥ �

}
,

has natural density zero for every � > 0.

In this case, l is called the statistical limit of the sequence x.

Definition 2.3. Let x = (xmnk)m,n,k∈N×N×N be a triple sequence in a metric space (X, |., .|) and r be a

non-negative real number. A triple sequence x = (xmnk) is said to be r−convergent to l ∈ X, denoted by

x →r l, if for any � > 0 there exists N� ∈ N×N×N such that for all m,n,k ≥ N� we have

|xmnk − l| < r + �

In this case l is called an r− limit of x.

Remark 2.1. We consider r− limit set x which is denoted by LIMrx and is defined by

LIMrx = {l ∈ X : x →r l} .



Int. J. Anal. Appl. 16 (5) (2018) 646

Definition 2.4. A triple sequence x = (xmnk) is said to be r− convergent if LIMrx 6= φ and r is called a

rough convergence degree of x. If r = 0 then it is ordinary convergence of triple sequence.

Definition 2.5. Let x = (xmnk) be a triple sequence in a metric space (X, |., .|) and r be a non-negative

real number is said to be r− statistically convergent to l, denoted by x →r−st3 l, if for any � > 0 we have

d (A (�)) = 0, where

A (�) = {(m,n,k) ∈ N×N×N : |xmnk − l| ≥ r + �} .

In this case l is called r− statistical limit of x. If r = 0 then it is ordinary statistical convergent of triple

sequence.

Definition 2.6. A class I of subsets of a nonempty set X is said to be an ideal in X provided

(i) φ ∈ I

(ii) A,B ∈ I implies A
⋃
B ∈ I.

(iii) A ∈ I,B ⊂ A implies B ∈ I.

I is called a nontrivial ideal if X /∈ I.

Definition 2.7. A nonempty class F of subsets of a nonempty set X is said to be a filter in X. Provided

(i) φ ∈ F.

(ii) A,B ∈ F implies A
⋂
B ∈ F.

(iii) A ∈ F,A ⊂ B implies B ∈ F.

Definition 2.8. I is a non trivial ideal in X, X 6= φ, then the class

F (I) = {M ⊂ X : M = X\A for some A ∈ I}

is a filter on X, called the filter associated with I.

Definition 2.9. A non trivial ideal I in X is called admissible if {x}∈ I for each x ∈ X.

Case 2.1. If I is an admissible ideal, then usual convergence in X implies I convergence in X.

Remark 2.2. If I is an admissible ideal, then usual rough convergence implies rough I− convergence.

Definition 2.10. Let x = (xmnk) be a triple sequence in a metric space (X, |., .|) and r be a non-negative

real number is said to be rough ideal convergent or rI− convergent to l, denoted by x →rI l, if for any � > 0

we have

{(m,n,k) ∈ N×N×N : |xmnk − l| ≥ r + �}∈ I.

In this case l is called rI− limit of x and a triple sequence x = (xmnk) is called rough I− convergent to l

with r as roughness of degree. If r = 0 then it is ordinary I− convergent.



Int. J. Anal. Appl. 16 (5) (2018) 647

Case 2.2. Generally, a triple sequence y = (ymnk) is not I− convergent in usual sense and |xmnk −ymnk| ≤

r for all (m,n,k) ∈ N×N×N or

{(m,n,k) ∈ N×N×N : |xmnk −ymnk| ≥ r}∈ I.

for some r > 0. Then the triple sequence x = (xmnk) is rI− convergent.

Case 2.3. It is clear that rI− limit of x is not necessarily unique.

Definition 2.11. Consider rI− limit set of x, which is denoted by

I −LIMrx =
{
L ∈ X : x →rI l

}
,

then the triple sequence x = (xmnk) is said to be rI− convergent if I −LIMrx 6= φ and r is called a rough

I− convergence degree of x.

Definition 2.12. A triple sequence x = (xmnk) ∈ X is said to be I− analytic if there exists a positive real

number M such that {
(m,n,k) ∈ N×N×N : |xmnk|

1/m+n+k ≥ M
}
∈ I.

Definition 2.13. A point L ∈ X is said to be an I− accumulation point of a triple sequence x = (xmnk) in

a metric space (X,d) if and only if for each � > 0 the set{
(m,n,k) ∈ N3 : d (xmnk, l) = |xmnk − l| < �

}
/∈ I.

We denote the set of all I− accumulation points of x by I (Γx) .

Definition 2.14. A triple sequence x = (xmnk) is said to be Wijsman r− convergent to A denoted by

Amnk →r A, provided that

∀� > 0 ∃(m�,n�,k�) ∈ N3 : m ≥ m�,n ≥ n�,k ≥ k� =⇒

limrst
1
rst
|{m ≤ r,n ≤ s,k ≤ t : |d (x,Amnk) −d (x,A)| < r + �}| = 0

The set

LIMrA =
{
L ∈ R3 : Amnk →r A

}
is called the Wijsman r− limit set of the triple sequences.

Definition 2.15. A triple sequence x = (xmnk) is said to be Wijsman r− convergent if LIMrA 6= φ. In

this case, r is called the Wijsman convergence degree of the triple sequence x = (xmnk). For r = 0, we get

the ordinary convergence.

Definition 2.16. A triple sequence (xmnk) is said to be Wijsman r− statistically convergent to A, denoted

by Amnk →rst A, provided that the set

limrst
1
rst

∣∣{(m,n,k) ∈ N3 : |d (x,Amnk) −d (x,A)| ≥ r + �}∣∣ = 0
has natural density zero for every � > 0, or equivalently, if the condition



Int. J. Anal. Appl. 16 (5) (2018) 648

st− lim sup |d (x,Amnk) −d (x,A)| ≤ r

is satisfied.

In addition, we can write Amnk →rst A if and only if the inequality

limrst
1
rst
|{m ≤ r,n ≤ s,k ≤ t : |d (x,Amnk) −d (x,A)| < r + �}| = 0

holds for every � > 0 and almost all (m,n,k) . Here r is called the wijsman roughness of degree. If we

take r = 0, then we obtain the ordinary Wijsman statistical convergence of triple sequence.

Definition 2.17. A triple sequence (xmnk) is said to be Wijsman Cesáro convergent to A, denoted by

Amnk →ces A, provided that the set

limrst
1
rst

∑r
m=1

∑s
n=1

∑t
k=1 d (x,Amnk) = d (x,A) .

Definition 2.18. A triple sequence (xmnk) is said to be Wijsman strongly Cesáro convergent to A, denoted

by Amnk →stces A, provided that the set

limrst
1
rst

∑r
m=1

∑s
n=1

∑t
k=1 |d (x,Amnk) −d (x,A)| = 0.

Definition 2.19. A triple sequence (xmnk) is said to be Wijsman p− Cesáro convergent to A, denoted by

Amnk →stces A, if for each p positive real number and for each x ∈ X,

limrst
1
rst

∑r
m=1

∑s
n=1

∑t
k=1 |d (x,Amnk) −d (x,A)|

p
= 0.

Definition 2.20. The triple sequence θi,`,j = {(mi,n`,kj)} is called triple lacunary if there exist three

increasing sequences of integers such that

m0 = 0,hi = mi −mi−1 →∞ as i →∞ and

n0 = 0,h` = n` −n`−1 →∞ as ` →∞.

k0 = 0,hj = kj −kj−1 →∞ as j →∞.

Let mi,`,j = min`kj,hi,`,j = hih`hj, and θi,`,j is determine by

Ii,`,j = {(m,n,k) : mi−1 < m < mi andn`−1 < n ≤ n` andkj−1 < k ≤ kj} ,qk = mkmk−1 ,q` =
n`
n`−1

,qj =
kj
kj−1

.

Let θi,`,j be a lacunary sequence. A triple sequence (xmnk) is said to be Wijsman strongly lacunary convergent

to A, denoted by Amnk →stlac A, if

limrst
1

hrst

∑
(m,n,k)∈Irst |d (x,Amnk) −d (x,A)| = 0.

In a similar fashion to the idea of classic Wijsman rough convergence, the idea of Wijsman rough statistical

convergence of a triple sequence spaces can be interpreted as follows:

Assume that a triple sequence y = (ymnk) is Wijsman statistically convergent and cannot be measured or

calculated exactly; one has to do with an approximated (or Wijsman statistically approximated) triple sequence

x = (xmnk) satisfying |d (x−y,Amnk) −d (x−y,A)| ≤ r for all m,n,k (or for almost all (m,n,k) , i.e.,

δ
(
limrst

1
rst
|{m ≤ r,n ≤ s,k ≤ t : |d (x−y,Amnk) −d (x−y,A)| > r}|

)
= 0.



Int. J. Anal. Appl. 16 (5) (2018) 649

Then the triple sequence x is not statistically convergent any more, but as the inclusion

limrst
1

rst
{|d (y,Amnk) −d (y,A)| ≥ �}⊇ limrst

1

rst
{|d (x,Amnk) −d (x,A)| ≥ r + �} (2.1)

holds and we have

δ
(
limrst

1
rst

∣∣{(m,n,k) ∈ N3 : |ymnk − l| ≥ �}∣∣) = 0,
i.e., we get

δ
(
limrst

1
rst
|{m ≤ r,n ≤ s,k ≤ t : |d (x,Amnk) −d (x,A)| ≥ r + �}|

)
= 0,

i.e., the triple sequence spaces x is Wijsman r− statistically convergent in the sense of definition (2.21)

In general, the Wijsman rough statistical limit of a triple sequence may not unique for the Wijsman

roughness degree r > 0. So we have to consider the so called Wijsman r− statistical limit set of a triple

sequence x = (xmnk) , which is defined by

st−LIMrAmnk = {L ∈ R : Amnk →rst A} .

The triple sequence x is said to be Wijsman r− statistically convergent provided that st−LIMrAmnk 6= φ.

It is clear that if st−LIMrAmnk 6= φ for a triple sequence x = (xmnk) of real numbers, then we have

st−LIMrAmnk = [st− lim sup Amnk −r,st− lim inf Amnk + r] (2.2)

We know that LIMr = φ for an unbounded triple sequence x = (xmnk) . But such a triple sequence might be

Wijsman rough statistically convergent. For instance, define

d (x,Amnk) =




(−1)mnk , if (m,n,k) 6= (i,j,`)2 (i,j,` ∈ N) ,

(mnk) , otherwise


 .

in R. Because the set {1, 64, 739, · · ·} has natural density zero, we have

st−LIMrAmnk =




φ, if r < 1,

[1 −r,r − 1] , otherwise




and LIMrAmnk = φ for all r ≥ 0.

As can be seen by the example above, the fact that st−LIMrAmnk 6= φ does not imply LIMrAmnk 6= φ.

Because a finite set of natural numbers has natural density zero, LIMrAmnk 6= φ implies st−LIMrAmnk 6=

φ. Therefore, we get LIMrAmnk ⊆ st−LIMrAmnk. This obvious fact means {r ≥ 0 : LIMrAmnk 6= φ} ⊆

{r ≥ 0 : st−LIMrAmnk 6= φ} in this language of sets and yields immediately

inf {r ≥ 0 : LIMrAmnk 6= φ}≥ inf {r ≥ 0 : st−LIMrAmnk 6= φ} .

Moreover, it also yields directly diam (LIMrAmnk) ≤ diam (st−LIMrAmnk) .

Throughout the paper, we let (X; ρ) be a separable metric space, I ⊆ 2N
3

be an admissible ideal and A; Amnk

be any non-empty closed subsets of X.



Int. J. Anal. Appl. 16 (5) (2018) 650

Definition 2.21. A triple sequence (xmnk) is said to be Wijsman r − I convergent to A, if for every � > 0

and for each x ∈ X,

A (x,�) =
{

(m,n,k) ∈ N3 : |d (x,Amnk) −d (x,A)| ≥ r + �
}
∈ I

Definition 2.22. A triple sequence (xmnk) is said to be Wijsman r − I statistical convergent to A, if for

every � > 0,δ > 0 and for each x ∈ X,{
(r,s,t) ∈ N3 : 1

rst
|{(r,s,t) ≤ (m,n,k) : |d (x,Amnk) −d (x,A)| ≥ r + �}|≥ δ

}
∈ I.

In this case, we write Amnk →s(IW ) A.

Definition 2.23. Let θ be a lacunary sequence. A triple sequence (xmnk) is said to be Wijsman strongly

r − I convergent to A, if for every � > 0 and for each x ∈ X,{
(r,s,t) ∈ N3 : 1

hrst

∑
(m,n,k)∈Irst |d (x,Amnk) −d (x,A)| ≥ r + �

}
∈ I.

In this case, we write Amnk →Nθ(IW ) A.

Definition 2.24. A triple sequence (xmnk) is said to be Wijsman r−I Cesáro convergent to A, if for every

� > 0 and for each x ∈ X,{
(r,s,t) ∈ N3 :

∣∣∣ 1rst ∑rm=1 ∑sn=1 ∑tk=1 d (x,Amnk) −d (x,A)∣∣∣ ≥ r + �} ∈ I.
In this case, we write Amnk →C(IW ) A.

Definition 2.25. A triple sequence (xmnk) is said to be Wijsman strongly r−I Cesáro convergent to A, if

for every � > 0 and for each x ∈ X,{
(r,s,t) ∈ N3 : 1

rst

∑r
m=1

∑s
n=1

∑t
k=1 |d (x,Amnk) −d (x,A)| ≥ r + �

}
∈ I.

In this case, we write Amnk →C(IW ) A.

Definition 2.26. A triple sequence (xmnk) is said to be Wijsman p strongly r−I Cesáro convergent to A,

if for each p positive real number, if for every � > 0 and for each x ∈ X,{
(r,s,t) ∈ N3 : 1

rst

∑r
m=1

∑s
n=1

∑t
k=1 |d (x,Amnk) −d (x,A)|

p ≥ r + �
}
∈ I.

In this case, we write Amnk →Cp(IW ) A.

3. Main Results

Theorem 3.1. Let the triple sequence (Amnk) ∈ Λ3. If (Amnk) is Wijsman r − I statistical convergent to

A, then (Amnk) is Wijsman p strongly r − I Cesáro convergent to A.

Proof: Suppose that (Amnk) is triple analytic and Amnk →S(IW ) A. Then, there is an M > 0 such that

|d (x,Amnk) −d (x,A)|
1/m+n+k ≤ M,



Int. J. Anal. Appl. 16 (5) (2018) 651

for each x ∈ X and for all m,n,k. Given � > 0, we have
1
rst

∑r
m=1

∑s
n=1

∑t
k=1 |d (x,Amnk) −d (x,A)|

p/m+n+k
=

1
rst

∑r
m=1

∑s
n=1

∑t
k=1,|d(x,Amnk)−d(x,A)|≥r+� |d (x,Amnk) −d (x,A)|

p/m+n+k
+

1
rst

∑r
m=1

∑s
n=1

∑t
k=1,|d(x,Amnk)−d(x,A)|<r+� |d (x,Amnk) −d (x,A)|

p/m+n+k

≤ 1
rst
Mp/m+n+k

∣∣∣{(m,n,k) ≤ (r,s,t) : |d (x,Amnk) −d (x,A)|1/m+n+k ≥ r + �}∣∣∣ +
1
rst
�p/m+n+k

∣∣∣{(m,n,k) ≤ (r,s,t) : |d (x,Amnk) −d (x,A)|1/m+n+k < r + �}∣∣∣
≤ M

p/m+n+k

rst

∣∣∣{(m,n,k) ≤ (r,s,t) : |d (x,Amnk) −d (x,A)|1/m+n+k ≥ r + �}∣∣∣ + �p/m+n+k. Then for any δ >
0{

(r,s,t) ∈ N3 : 1
rst

∑r
m=1

∑s
n=1

∑t
k=1 |d (x,Amnk) −d (x,A)|

p/m+n+k ≥ δ
}
⊆{

(r,s,t) ∈ N3 : 1
rst
|{(m,n,k) ≤ (r,s,t) |d (x,Amnk) −d (x,A)| ≥ r + �}|≥ δ

p/m+n+k

Mp/m+n+k

}
∈ I, for each x ∈ X.

Hence Amnk →Cp(IW ) A.

Theorem 3.2. Let the triple sequence (Amnk) is Wijsman p strongly r − I Cesáro convergent to A then

(Amnk) is Wijsman r − I statistical convergent to A.

Proof: Let Amnk →Cp(IW ) A and given � > 0. Then, we have∑r
m=1

∑s
n=1

∑t
k=1 |d (x,Amnk) −d (x,A)|

p ≥∑r
m=1

∑s
n=1

∑t
k=1,|d(x,Amnk)−d(x,A)|≥r+� |d (x,Amnk) −d (x,A)|

p

�p |{(m,n,k) ≤ (r,s,t) : |d (x,Amnk) −d (x,A)| ≥ r + �}|

for each x ∈ X and so
1

�p(rst)

∑r
m=1

∑s
n=1

∑t
k=1 |d (x,Amnk) −d (x,A)|

p ≥
1
rst
|{(m,n,k) ≤ (r,s,t) : |d (x,Amnk) −d (x,A)| ≥ r + �}| .

Hence for given δ > 0{
(r,s,t) ∈ N3 : 1

rst
|{(m,n,k) ≤ (r,s,t) |d (x,Amnk) −d (x,A)| ≥ r + �}|≥ δ

}
⊆{

(r,s,t) ∈ N3 : 1
rst

∑r
m=1

∑s
n=1

∑t
k=1 |d (x,Amnk) −d (x,A)|

p ≥ (r + �)p δ
}
∈ I, for each x ∈ X. Hence

Amnk →S(IW ) A.

Theorem 3.3. Let θ be a triple lacunary sequence (Amnk) . If liminfuvwquvw > 1 then, Amnk →C1(IW )

A ⇒ Amnk →Nθ(IW ) A.

Proof: If liminfuvwquvw > 1, then there exists δ > 0 such that quvw ≥ 1 + δ for all u,v,w ≥ 1. Since

huvw = (munvkw) − (mu−1nv−1kw−1) , we have

munvkw
hu−1,v−1,w−1

≤ 1+δ
δ

and
mu−1nv−1kw−1
hu−1,v−1,w−1

≤ 1
δ
.

Let � > 0 and we define the set

S =
{

(munvkw) ∈ N3 : 1munvkw
∑mu
m=1

∑nv
n=1

∑kw
k=1 |d (x,Amnk) −d (x,A)| < r + �

}
, for each x ∈ X and al-

so S ∈ F (I) , which is a filter of the ideal I, we have
1

huvw

∑
mnk∈Iuvw |d (x,Amnk) −d (x,A)| =

1
huvw

∑mu
m=1

∑nv
n=1

∑kw
k=1 |d (x,Amnk) −d (x,A)|−

1
huvw

∑mu−1
m=1

∑nv−1
n=1

∑kw−1
k=1 |d (x,Amnk) −d (x,A)|



Int. J. Anal. Appl. 16 (5) (2018) 652

= mrnvkw
huvw

1
mrnvkw

∑mu
m=1

∑nv
n=1

∑kw
k=1 |d (x,Amnk) −d (x,A)|−

mr−1nv−1kw−1
huvw

1
mr−1nv−1kw−1

∑mu−1
m=1

∑nv−1
n=1

∑kw−1
k=1 |d (x,Amnk) −d (x,A)|

≤
(
1+δ
δ

)(
r + �

′
)
− 1

δ

(
r + �

′
)

for each x ∈ X and (munvkw) ∈ S. Choose η =
(
1+δ
δ

)(
r + �

′
)

+ 1
δ

(
r + �

′
)
. Therefore, for each x ∈ X{

(u,v,w) ∈ N3 : 1
huvw

∑
mnk∈Iuvw |d (x,Amnk) −d (x,A)| < η

}
∈ F (I) .

Theorem 3.4. Let θ be a triple lacunary sequence (Amnk) . If limsupuvwquvw < ∞ then, Amnk →Nθ(IW )

A ⇒ Amnk →C1(IW ) A.

Proof: If limsupuvwquvw < ∞ then there exists M > 0 such that quvw < M, for all u,v,w ≥ 1. Let

Amnk →Nθ(IW ) A and we define the sets T and R such that

T =
{

(u,v,w) ∈ N3 : 1
huvw

∑
mnk∈Iuvw |d (x,Amnk) −d (x,A)| < r + �1

}
and

R =
{

(a,b,c) ∈ N3 : 1
rst

∑a
m=1

∑b
n=1

∑c
k=1 |d (x,Amnk) −d (x,A)| < r + �2

}
,

for every �1,�2 > 0 for each x ∈ X. Let

aj =
1
hj

∑
mnk∈Ij |d (x,Amnk) −d (x,A)| < r + �1

for each x ∈ X and for all j ∈ T. It is obvious that T ∈ F (I) . Choose (a,b,c) in any integer with

(mu−1nv−1kw−1) < (a,b,c) < (munvkw) , where (u,v,w) ∈ T. Then, we have
1

huvw

∑
mnk∈Iuvw |d (x,Amnk) −d (x,A)| ≤

1
mu−1nv−1kw−1

∑mu
m=1

∑nv
n=1

∑kw
k=1 |d (x,Amnk) −d (x,A)|

= 1
mu−1nv−1kw−1

 ∑(m,n,k)∈I111 |d (x,Amnk) −d (x,A)| + ∑(m,n,k)∈I222 |d (x,Amnk) −d (x,A)|+
· · · +

∑
(m,n,k)∈Iuvw |d (x,Amnk) −d (x,A)|




= m1n1k1
mu−1nv−1kw−1

(
1

h111

∑
(m,n,k)∈I111 |d (x,Amnk) −d (x,A)|

)
+

(m2n2k2)−(m1n1k1)
mu−1nv−1kw−1

(
1

h222

∑
(m,n,k)∈I222 |d (x,Amnk) −d (x,A)|

)
+ · · ·

(munvkw)−(mu−1nv−1kw−1)
mu−1nv−1kw−1

(
1

huvw

∑
(m,n,k)∈Iuvw |d (x,Amnk) −d (x,A)|

)
= m1n1k1

mu−1nv−1kw−1
a111 +

(m2n2k2)−(m1n1k1)
mu−1nv−1kw−1

a222 + · · · +
(munvkw)−(mu−1nv−1kw−1)

mu−1nv−1kw−1
auvw

≤ (supj∈Taj) · m1n1k1mu−1nv−1kw−1 < r + �1 ·M for each x ∈ X. Choose r + �2 =
r+�1
M

and the fact that⋃
{(a,b,c) : mu−1 < a < mu,nv−1 < b < nv,kw−1 < c < kw, (u,v,w) ∈ T}⊂ R,

where T ∈ F (I) . It follows from our assumption on θ that the set R ∈ F (I) .

Competing Interests: The authors declare that there is not any conflict of interests regarding the

publication of this manuscript.

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Int. J. Anal. Appl. 16 (5) (2018) 653

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	1. Introduction
	2. Definitions and Preliminaries
	3. Main Results
	References