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Journal of
Classical
Analysis
Volume 17, Number 2 (2021), 119–128 doi:10.7153/jca-2021-17-08
WIJSMAN LACUNARY INVARIANT STATISTICAL CONVERGENCE
FOR TRIPLE SEQUENCES VIA ORLICZ FUNCTION
MUALLA BIRG ¨
UL HUBAN∗AND MEHMET G¨
URDAL
Abstract. In this paper, we generalized the Wijsman lacunary invariant statistical convergence
of closed sets in metric space by introducing the Wijsman lacunary invariant statistical
φ
-
convergence for the sets of triple sequences. We introduce the concepts of Wijsman invari-
ant
φ
-convergence, Wijsman invariant statistical
φ
-convergence, Wijsman lacunary invariant
φ
-convergence, Wijsman lacunary invariant statistical
φ
-convergence for the sets of triple se-
quences. In addition, we investigate existence of some relations among these new notations for
the sets of triple sequences.
1. Introduction and background
The idea of statistical convergence was first introduced by Fast [6] and Steinhaus
[28] independently in the same year 1951 and since then several generalizations and
applications of this concept have been investigated by various authors, namely Fridy
[7], G¨urdal and Huban [10], G¨urdal and Pehlivan [11,12], Nabiev et al. [17], and many
others (see [4,8,24]).
Statistical convergence depends on the natural density of subsets of the set Nof
positive integers. The natural density
δ
(A)of a subset Aof Nis defined by
δ
(A)= lim
n→∞n−1|{kn:k∈A}|
where |{kn:k∈A}| denotes the number of elements of Anot exceeding n.Ase-
quence (xk)⊂Ris said to be statistically convergent to ∈Rif, for each
ε
>0,the
set {k∈N:|xk−|
ε
}has the zero natural density.
The concept of convergence of sequences of points has been extended by sev-
eral authors [19,20,29,30] to convergence of sequences of sets. One of such ex-
tensions considered in this paper is the concept of Wijsman convergence. Nuray and
Rhoades [18] extended the notion of Wijsman convergence of sequences of sets to that
of Wijsman statistical convergence and introduced the notion of Wijsman strong Cesaro
summability of sequences of sets and discussed its relations with Wijsman statistical
convergence.
In this study, we introduce the concepts of Wijsman invariant
φ
-convergence, Wi-
jsman invariant statistical
φ
-convergence,Wijsman lacunary invariant
φ
-convergence,
Mathematics subject classification (2010): 40A05, 40C05, 40D25.
Keywords and phrases: Triple sequence, Orlicz function, lacunary sequence, triple statistical conver-
gence, Wijsman convergence.
∗Corresponding author.
c
,Zagreb
Paper JCA-17-08 119
120 M. BIRG ¨
UL HUBAN AND M. G ¨
URDAL
Wijsman lacunary invariant statistical
φ
-convergence for the sets of triple sequences.
Also, we investigate existence of some relations among these new
φ
-convergence con-
cepts for the sets of triple sequences.
We now recall the following basic concepts from [2,15,19,26,27] which will be
needed throughout the paper.
Let
σ
be a mapping of the positive integers into themselves. A continuous linear
functional
ϕ
on ∞,the space of real bounded sequences, is said to be an invariant
mean or a
σ
-mean if it satisfies the following conditions:
(i)
ϕ
(x)0, for the sequence x=(xn)with xn0foralln∈N,
(ii)
ϕ
(e)=1, where e=(1,1,1,...)and
(iii)
ϕ
x
σ
(n)=
ϕ
(xn)for all x∈∞.
The mapping
σ
is assumed to be one-to-one and such that
σ
m(n)=nfor all
n,m∈Z+,where
σ
m(n)denotes the mth iterate of the mapping
σ
at n.Thus,
ϕ
extends the limit functional on c, the space of convergent sequences, in the sense that
ϕ
(xn)=lim xnfor all x∈c. In the case
σ
is translation mappings
σ
(n)=n+1, the
σ
-mean is often called a Banach limit. The space V
σ
of the bounded sequences whose
invariant means are equal may be defined, as follows;
V
σ
=x∈∞: lim
m→∞
1
m
m
∑
k=1
x
σ
k(n)=L,uniformly in m.
In [27], Schaefer proved that a bounded sequence x=(xn)of real numbers is
σ
-
convergent to Lif and only if
lim
p→∞
1
p
p
∑
k=1
x
σ
k(m)=L,
uniformly in m.
Let (X,
ρ
)be a metric space. For any point x∈Xand any non-empty subset Aof
X,the distance d(x,A)from xto Ais defined by
d(x,A)=inf
a∈A
ρ
(x,a).
DEFINITION 1. Let (X,
ρ
)be a metric space. For any non-empty closed subsets
A;Ak⊆X; we say that the sequence (Ak)is Wijsman convergent to Aif
lim
k→∞d(x,Ak)=d(x,A)
We now recall that the concept of statistical convergence for triple sequences was
presented by S¸ahiner, G¨urdal and D¨uden [23] as follows:
A function x:N×N×N→R(or C)is called a real (complex) triple sequence.
A triple sequence xjkl is said to be convergent to Lin Pringsheim’s sense if for every
ε
>0, there exists n0(
ε
)∈Nsuch that |xjkl −L|<
ε
whenever j,k,ln0.
WIJSMAN LACUNARY INVARIANT STATISTICALCONVERGENCE 121
DEFINITION 2. A subset Kof N×N×Nis said to have natural density
δ
3(K)if
δ
3(K)=P−lim
n,k,l→∞
|Knkl|
nkl
exists, where the vertical bars denotethe number of (n,k,l)in Ksuch that pn,qk,
rl. Then, a real triple sequence x=(xpqr)is said to be statistically convergent to L
in Pringsheim’s sense if for every
ε
>0,
δ
3(n,k,l)∈N×N×N:pn,qk,rl,xpqr −L
ε
=0.
Recently, Mursaleen and Edely [16] presented the idea of statistical convergence
for multiple sequences, and there are several papers dealing with double and triple
statistical and ideal convergence (see literature [1,5,9,13,21]). Also, the readers
should refer to the monographs [3]and[14] for the background on the sequence spaces
and related topics.
In several literary works, statistical convergence of any real sequence is identified
relatively to absolute value. While we have known that the absolute value of real num-
bers is special of an Orlicz function [22], that is, a function
φ
:R→Rin such a way
that it is even, non-decreasing on R+, continuous on R, and satisfying
φ
(x)=0 if and only if x=0and
φ
(x)→∞as x→∞.
Further, an Orlicz function
φ
:R→Ris said to satisfy the 2condition, if there exists
an positive real number Msuch that
φ
(2x)M.
φ
(x)for every x∈R+.
DEFINITION 3. ([25]) Let
φ
:R→Rbe an Orlicz function. A sequence x=(xn)
is said to be statistically
φ
-convergent to Lif for each
ε
>0,
lim
n
1
nkn:
φ
(xk−L)
ε
=0.
Furthermore, a new type of sequence called triple lacunary sequence was intro-
duced in Esi and Savas¸[5]. The triple sequence
θ
3=
θ
p,q,r=(jp,kq,lr)is called
triple lacunary sequence if there exist three increasing sequences of integers such that
j0=0,hp=jp−jp−1→∞as p→∞,
k0=0,hq=kq−kq−1→∞as q→∞,
and
l0=0,hr=lr−lr−1→∞as r→∞.
Let kp,q,r=jpkqlr,hp,q,r=hphqhrand
θ
p,q,ris determined by
Ip,q,r=(j,k,l):jp−1<jjp,kq−1<kkqand lr−1<llr,
sp=jp
jp−1
,sq=kq
kq−1
,sr=lr
lr−1
and sp,q,r=spsqsr.
Let D⊂N×N×N.The number
δθ
3
3(D)=lim
p,q,r
1
hp,q,r(j,k,l)∈Ip,q,r:(j,k,l)∈D
is said to be the
θ
p,q,r-density of D,provided the limit exists.
122 M. BIRG ¨
UL HUBAN AND M. G ¨
URDAL
2. Main results
Following the above definitions and results, we aim in this section to introduce
some new notions of Wijsman invariant statistical convergence with the use of Orlicz
function, lacunary and triple sequences and obtain some analogous results from the new
definitions point of views.
DEFINITION 4. A triple sequence x=xjklof real numbers is said to be
σ
3-
convergence to Lif
lim
p,q,r→∞
1
pqr
p
∑
j=1
q
∑
k=1
r
∑
l=1
x
σ
j(m),
σ
k(n),
σ
l(o)=L,
uniformly in m,nand o.
DEFINITION 5. A triple sequence Ajklis Wijsman invariant convergent to Aif
for each x∈X
lim
p,q,r→∞
1
pqr
p,q,r
∑
j,k,l=1,1,1
dx,A
σ
j(m),
σ
k(n),
σ
l(o)=d(x,A),
uniformly in m,nand o, and is written Ajkl →A(W3V
σ
).
DEFINITION 6. Let
φ
:R→Rbe an Orlicz function. A triple sequences Ajkl
is Wijsman strongly invariant
φ
-convergent to Aif for each x∈X
lim
p,q,r→∞
1
pqr
p,q,r
∑
j,k,l=1,1,1
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)=0,
uniformly in m,nand o, and is written Ajkl →AW
φ
3V
σ
.
If double sequence Ajkis taken instead of a triple sequence Ajkland
φ
(x)=
|x|,then the concept Wijsman strongly invariant
φ
-convergence is reduced to Wijsman
strongly invariant convergence.
DEFINITION 7. Let
φ
:R→Rbe an Orlicz function. A triple sequence Ajkl
is Wijsman invariant statistically
φ
-convergent to Aif for every
ε
>0 and for each
x∈X
lim
p,q,r→∞
1
pqr jp,kq,lr:
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
=0
or
δ
3(p,q,r)∈N×N×N:
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
=0
uniformly in m,nand o, and is written Ajkl →AW
φ
3S
σ
.
WIJSMAN LACUNARY INVARIANT STATISTICALCONVERGENCE 123
DEFINITION 8.
θ
3=
θ
r,s,tbe a lacunary triple sequence. A triple sequence Ajkl
is Wijsman lacunary invariant convergent to Aif for each x∈X
lim
p,q,r→∞
1
hp,q,r∑
j,k,l∈Ip,q,r
dx,A
σ
j(m),
σ
k(n),
σ
l(o)=d(x,A)
uniformly in m,nand o, and is written Ajkl →AW3V
θ
σ
.
DEFINITION 9. Let
φ
:R→Rbe an Orlicz function and
θ
3=
θ
r,s,tbe a lacunary
triple sequence. A triple sequence Ajklis Wijsman strongly lacunary invariant
φ
-
convergence to Aif for each x∈X
lim
p,q,r→∞
1
hp,q,r∑
j,k,l∈Ip,q,r
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)=0,
uniformly in m,nand o, and is written Ajkl →AW
φ
3V
θ
σ
.
DEFINITION 10. Let
φ
:R→Rbe an Orlicz function and
θ
3=
θ
r,s,tbe a lacu-
nary triple sequence. A triple sequence Ajklis Wijsman lacunary invariant statisti-
cally
φ
-convergent to Aif for every
ε
>0 and for each x∈X
lim
p,q,r→∞
1
hp,q,r(j,k,l)∈Ip,q,r:
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
=0
or
δθ
3(j,k,l)∈Ip,q,r:
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
=0
uniformly in m,nand o, and is written Ajkl →AW
φ
3S
θ
3
σ
.
DEFINITION 11. Let
φ
:R→Rbe an Orlicz function. A triple sequence Ajkl
is said to be bounded if there exists M>0 such that
φ
AjklMfor all j,k,l∈N.
We denote the space of all bounded triple sequences by 3
∞.
THEOREM 1. Let
φ
:R→Rbe an Orlicz function and
θ
3=
θ
r,s,t={(jr,ks,lt)}
be a lacunary triple sequence. Then, the following statements hold:
(i) If Ajklis Wijsman strongly lacunary invariant
φ
-convergent to A,then
Ajklis Wijsman lacunary invariant statistically
φ
-convergent to A.
(ii) If Ajkl ∈3
∞and Ajklis Wijsman lacunary invariant statistically
φ
-
convergent to A,then Ajkl is Wijsman strongly lacunary invariant
φ
-convergent
to A.
124 M. BIRG ¨
UL HUBAN AND M. G ¨
URDAL
Proof. (i) Ajkl →AW
φ
3V
θ
σ
.For every
ε
>0 and for each x∈X,then we have
∑
(j,k,l)∈Ip,q,r
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
∑
j,k,l∈Ip,q,r
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
.(j,k,l)∈Ip,q,r:
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
.
This shows that Ajkl →AW
φ
3S
θ
3
σ
.
(ii) Suppose that Ajklbelongs to the space 3
∞and Ajkl →AW
φ
3S
θ
3
σ
.Then
we can assume that
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)M
for each x∈Xand all j,kand l.Given every
ε
>0 and for each x∈X,we have
1
hp,q,r∑
(j,k,l)∈Ip,q,r
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
=1
hp,q,r∑
(j,k,l)∈Ip,q,r
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
+1
hp,q,r∑
(j,k,l)∈Ip,q,r
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)<
ε
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
M
hp,q,r(j,k,l)∈Ip,q,r:
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
+
ε
.
This shows that Ajkl →AW
φ
3V
θ
σ
.
THEOREM 2. Let
φ
:R→Ris an Orlicz function. Suppose for given
δ
>0and
every
ε
>0,there exists
1
pqr 0jp−1,0kq−1,0lr−1:
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
<
δ
for all p p0,qq0,rr0,mm0,nn0,oo0,then Ajklis Wijsman
invariant statistically
φ
-convergent to A.
WIJSMAN LACUNARY INVARIANT STATISTICALCONVERGENCE 125
Proof. Let
δ
>0.For every
ε
>0,we choose p
0,q
0,r
0,m0,n0and o0such that
for all x∈X,
1
pqr 0jp−1,0kq−1,0lr−1:
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
<
δ
2(1)
for all pp
0,qq
0,rr
0,mm0,nn0,oo0.It is enough to prove that
there exists p
0,q
0,r
0such that for each x∈X,
1
pqr 0jp−1,0kq−1,0lr−1:
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
<
δ
(2)
for pp
0,q>q
0,r>r
0,0mm0,0nn0and 0 oo0.
Since taking p0=max p
0,p
0,q0=maxq
0,q
0and r0=maxr
0,r
0,(2)
holds for each x∈X,pp0,qq0,rr0and for all m,nand o,which gives the
result. Once m0,n0and o0have been chosen 0 mm0,0nn0,0oo0,
m0,n0and o0fixed. So, suppose that x
SL
θ
3(
φ
)
∼y,and let
F=0jm0−1,0kn0−1,0lo0−1:
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
.
Now taking 0 mm0,0nn0,0oo0and pm0,qn0,ro0,by (1)
for each x∈X,we get
1
pqr 0jp−1,0kq−1,0lr−1:
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
1
pqr 0jm0−1,0kn0−1,0lo0−1:
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
+1
pqr m0jp−1,n0kq−1,o0lr−1:
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
F
pqr +
δ
2
and taking p,q,rsufficiently large, we can write
F
pqr +
δ
2<
δ
126 M. BIRG ¨
UL HUBAN AND M. G ¨
URDAL
which gives (2)and thus, the result follows.
THEOREM 3. Let
φ
:R→Rbe an Orlicz function and
θ
3=
θ
r,s,t={(jr,ks,lt)}
be a lacunary triple sequence. Then Ajklis Wijsman lacunary invariant statistically
φ
-convergent to A iff Ajklis Wijsman invariant statistically
φ
-convergent to A.
Proof. Let Ajkl →AW
φ
3S
θ
3
σ
.Then, for given
δ
>0 there exists p0,q0,r0such
that for all
ε
>0 and for each x∈X,
1
hp,q,r0jhp−1,0khq−1,0lhr−1:
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
<
δ
for pp0,qq0,rr0and m=jp−1+1+v,v0,n=kq−1+1+w,w0,
o=lr−1+1+z,z0.Let shp,thqand uhr.Write s=
α
hp+e,t=
β
hq+f
and u=
γ
hr+gwhere 0 ehp,0fhqand 0 ghr,
α
,
β
and
γ
are integers.
Since shp,thqand uhr,we can write
1
st u 0js−1,0kt−1,0lu−1:
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
1
st u 0j(
α
+1)hp−1,0k(
β
+1)hq−1,0l(
γ
+1)hr−1:
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
=1
stu
α
∑
f=0
β
∑
g=0
γ
∑
h=0ehpj(e+1)hp−1,fh
qk(f+1)hq−1,
ghrl(g+1)hr−1:
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
1
stu (
α
+1)(
β
+1)(
γ
+1)hphqhr
δ
and since for 1
m
α
hp1,1
m
β
hq1and 1
m
γ
hr1,we have
1
st u 0js−1,0kt−1,0lu−1:
φ
dx,A
σ
j(m),
σ
k(n),
σ
l(o)−d(x,A)
ε
8.
δ
.
Thus , by Theorem 2, W
φ
3S
θ
3
σ
⊂W
φ
3S
σ
.It is easy to see that W
φ
3S
σ
⊂W
φ
3S
θ
3
σ
.This
completes the proof.
From Theorem 3, we have
WIJSMAN LACUNARY INVARIANT STATISTICALCONVERGENCE 127
THEOREM 4. Let
φ
:R→Rbe an Orlicz function and
θ
3=
θ
r,s,t={(jr,ks,lt)}
be a lacunary triple sequence. Then Ajklis Wijsman lacunary invariant convergent
to A iff Ajklis Wijsman strongly invariant
φ
-convergent to A.
When (
σ
(m),
σ
(n),
σ
(o)) = (m+1,n+1,o+1),from Definitions 5 - 10 , we
have the definitions of almost Wijsman, almost convergence, Wijsman strongly almost
φ
-convergence, Wijsman almost statistical
φ
-convergence, Wijsman lacunary almost
convergence, Wijsman strongly lacunary almost
φ
-convergence, Wijsman lacunary al-
most statistical
φ
-convergence for the sets of triple sequences.
So, similar inclusions to Theorems 1 - 4 hold between Wijsman strongly lacu-
nary almost
φ
-convergent triple set sequences, Wijsman lacunary almost statistical
φ
-
convergent triple set sequences, Wijsman almost statistical
φ
-convergent triple set se-
quences, Wijsman lacunary almost convergent triple set sequences and Wijsman strongly
almost
φ
-convergent triple set sequences.
Acknowledgement. The authors thank to the referee for valuable comments and
fruitful suggestions which enhanced the readability of the paper.
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(Received February 6, 2021) Mualla Birg¨ul Huban
Isparta University of Applied Sciences
Isparta, Turkey
e-mail: muallahuban@isparta.edu.tr
Mehmet G¨urdal
Department of Mathematics
Suleyman Demirel University
32260, Isparta, Turkey
e-mail: gurdalmehmet@sdu.edu.tr
Journal of Classical Analysis
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