Volume 16, Number 2 (2018), 222-231
URL:https://doi.org/10.28924/2291-8639 DOI:10.28924/2291-8639-16-2018-222
QUASI-ALMOST LACUNARY STATISTICAL CONVERGENCE OF SEQUENCES OF SETS
ESRA G ¨ULLE∗, U ˘GUR ULUSU
Department of Mathematics, Faculty of Science and Literature, Afyon Kocatepe University, 03200,
Afyonkarahisar, Turkey
∗Corresponding author: egulle@aku.edu.tr
Abstract. In this study, we defined concepts of Wijsman quasi-almost lacunary convergence, Wijsman quasi-strongly almost lacunary convergence and Wijsman quasiq-strongly almost lacunary con-vergence. Also we give the concept of Wijsman quasi-almost lacunary statistically concon-vergence. Then, we study relationships among these concepts. Furthermore, we investigate relationship between these concepts and some convergences types given earlier for sequences of sets, too.
1. INTRODUCTION AND BACKGROUNDS
The concept of statistical convergence was first introduced by Fast [10]. Also this concept was studied by
Fridy [12], ˇSal´at [17] and many others.
A sequencex= (xk) isstatistically convergent to the number Lif for everyε >0,
lim n→∞
1 n
k≤n:|xk−L| ≥ε
= 0
where the vertical bars indicate the number of elements in the enclosed set.
Freedman et al. [1] established the connection between the strongly Ces`aro summable sequences space
|σ1| and the strongly lacunary summable sequences spaceNθ.
Received 2017-10-26; accepted 2017-12-20; published 2018-03-07. 2010Mathematics Subject Classification. 40A05, 40A35.
Key words and phrases. almost convergence; quasi-almost convergence; quasi-almost statistical convergence; lacunary se-quence; sequences of sets; Wijsman convergence.
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.
By a lacunary sequence we mean an increasing integer sequence θ ={kr} such that k0 = 0 and hr =
kr−kr−1 → ∞ as r → ∞. Throughout this study the intervals determined by θ will be denoted by
Ir= (kr−1, kr] and ratio kkr
r−1 will be abbreviated byqr.
The concept of lacunary statistical convergence was introduced by Fridy and Orhan [13].
Letθ={kr} be a lacunary sequence. A sequencex= (xk) islacunary statistically convergent to Lif for
everyε >0,
lim r→∞
1 hr
k∈Ir:|xk−L| ≥ε
= 0.
The idea of almost convergence was introduced by Lorentz [9]. Maddox [11] and (independently) Freedman
[1] gave the concept of strong almost convergence. Similar concepts can be seen in [2].
LetX be any non-empty set andNbe the set of natural numbers. The function
f :N→P(X)
is defined by f(k) = Ak ∈ P(X) for each k ∈ N, where P(X) is power set of X. The sequence {Ak} =
(A1, A2, . . .), which is the range’s elements of f, is said to besequences of sets.
Let (X, ρ) be a metric space. For any point x∈ X and any non-empty subset A of X, we define the
distance fromxtoA by
d(x, A) = inf
a∈Aρ(x, a).
Throughout the paper we take (X, ρ) as a metric space andA, Ak as any non-empty closed subsets ofX.
There are different convergence notions for sequence of sets. One of them handled in this paper is the
concept of Wijsman convergence (see, [6–8,14–16]).
A sequence{Ak} is said to beWijsman convergent to Aif for eachx∈X
lim
k→∞d(x, Ak) =d(x, A)
and denoted by Ak W
→A or W−limAk =A.
A sequence{Ak} is said to bebounded if for each x∈X
sup k
d(x, Ak) <∞.
The set of all bounded sequences of sets is denoted byL∞.
The concepts of Wijsman statistical convergence was introduced by Nuray and Rhoades [6].
A sequence{Ak} isWijsman statistically convergent toAif for each x∈X and everyε >0
lim n→∞
1 n
k≤n:|d(x, Ak)−d(x, A)| ≥ε
= 0
and it is denoted byst−limWAk=A.
The concepts of Wijsman lacunary summability (W Nθ),[W N]θ,[W N]pθand concept of Wijsman
lacu-nary statistical convergence (W Sθ)
Letθ={kr}be a lacunary sequence. A sequence {Ak} is Wijsman lacunary summable toA if for each
x∈X,
lim r→∞
1 hr
X
k∈Ir
d(x, Ak) =d(x, A)
and it is denoted byAk
(W Nθ)
−→ A.
Let θ = {kr} be a lacunary sequence. A sequence {Ak} is Wijsman strongly lacunary summable to A
if for eachx∈X,
lim r→∞
1 hr
X
k∈Ir
|d(x, Ak)−d(x, A)|= 0
and it is denoted byAk
[W N]θ
−→ A.
Letθ={kr}be a lacunary sequence. A sequence{Ak}is Wijsmanp-strongly lacunary summable toAif
for eachx∈X and 0< p <∞,
lim r→∞
1 hr
X
k∈Ir
|d(x, Ak)−d(x, A)| p
= 0
and it is denoted byAk
[W N]pθ −→ A.
A sequence{Ak} isWijsman lacunary statistically convergent to Aif for everyε >0 and eachx∈X,
lim r→∞
1 hr
k∈Ir:|d(x, Ak)−d(x, A)| ≥ε
= 0
and it is denoted byAk
(W Sθ)
−→ A.
Also the concepts of Wijsman almost lacunary convergence and Wijsman almost lacunary statistical
convergence were introduced by Ulusu [18,19], too.
Let θ = {kr} be a lacunary sequence. A sequence {Ak} is Wijsman almost lacunary convergent to A
if for eachx∈X,
lim r→∞
1 hr
X
k∈Ir
d(x, Ak+i) =d(x, A)
uniformly ini= 0,1,2, . . ..
Letθ={kr} be a lacunary sequence. A sequence{Ak} is Wijsman strongly almost lacunary convergent
toAif for eachx∈X,
lim r→∞
1 hr
X
k∈Ir
|d(x, Ak+i)−d(x, A)|= 0
uniformly ini.
Letθ={kr}be a lacunary sequence. A sequence{Ak}isWijsman strongly p-almost lacunary convergent
toAif for eachx∈X and 0< p <∞,
lim r→∞
1 hr
X
k∈Ir
|d(x, Ak+i)−d(x, A)|p= 0
A sequence {Ak} is Wijsman almost lacunary statistically convergent to A if for every ε > 0 and each
x∈X,
lim r→∞
1 hr
k∈Ir:|d(x, Ak+i)−d(x, A)| ≥ε
= 0
uniformly ini.
The idea of quasi-almost convergence in a normed space was introduced by Hajdukovi´c [3]. Then, Nuray [5]
studied concepts of quasi-invariant convergence and quasi-invariant statistical convergence in a normed space.
The concepts of Wijsman quasi-strongly almost convergence and Wijsman quasi-almost statistically
con-vergence were studied by G¨ulle and Ulusu [4].
A sequence{Ak} ∈L∞ isWijsman quasi-strongly almost convergent toAif for eachx∈X,
1 p
np+p−1
X
k=np
dx(Ak)−dx(A)
→0 (as p→ ∞)
uniformly innand it is denoted byAk
[W QF] −→ A.
A sequence{Ak}isWijsman quasi-almost statistically convergent toAif for eachx∈X and everyε >0
lim p→∞
1 p
k≤p:|dx(Ak+np)−dx(A)| ≥ε
= 0,
uniformly innand it is denoted byAk W QS
−→ A.
The set of all Wijsman quasi-almost statistically convergence sequences will be denoted by
W QS .
2. MAIN RESULTS
In this section, we defined concepts of Wijsman almost lacunary convergence, Wijsman
quasi-strongly almost lacunary convergence and Wijsman quasi q-strongly almost lacunary convergence. Also
we give the concept of Wijsman quasi-almost lacunary statistically convergence. Then, we study
relation-ships among these concepts. Furthermore, we investigate relationship between these concepts and some
convergences types given earlier for sequences of sets, too.
Definition 2.1. Let θ ={kr} be a lacunary sequence. A sequence {Ak} ∈ L∞ is Wijsman quasi-almost
lacunary convergent toA if for eachx∈X,
1 hr
X
k∈Ir
dx(Ak+nr)−dx(A)
−→0 (as r→ ∞), (2.1)
uniformly inn= 0,1,2, . . . wheredx(Ak+nr) =d(x, Ak+nr)anddx(A) =d(x, A). In this case, we will write
(W QF)θ−limAk=A orAk
(W QF)θ
Example 2.1. Let we define a sequence{Ak} as follows:
Ak:=
{x∈R: 2≤x≤kr−kr−1} , if k≥2 andk is square integer,
{1} , otherwise.
This sequence is not Wijsman lacunary summable. But, since for eachx∈X
lim r→∞
1 hr
X
k∈Ir
dx(Ak+nr)−dx({1})
= 0
uniformlyn, this sequence is Wijsman quasi-almost lacunary convergent to the setA={1}.
Theorem 2.1. If a sequence{Ak} ∈L∞is Wijsman almost lacunary convergent toA, then{Ak}is Wijsman
quasi-almost lacunary convergent toA.
Proof. Suppose that the sequence{Ak}is Wijsman almost lacunary convergent toA. Then, for eachx∈X
and everyε >0 there exists an integer r0>0 such that for allr > r0
1 hr
X
k∈Ir
dx(Ak+i)−dx(A)
< ε,
uniformly ini. Ifi is taken asi=nr, then we have
1 hr
X
k∈Ir
dx(Ak+nr)−dx(A)
< ε,
uniformly inn. Sinceε >0 is an arbitrary, the limit is taken forr→ ∞we can write
1 hr
X
k∈Ir
dx(Ak+nr)−dx(A)
−→0
uniformly inn. That is, the sequence{Ak}is Wijsman quasi-almost lacunary convergent toA.
Theorem 2.2. If a sequence {Ak} ∈L∞ is Wijsman quasi-almost lacunary convergent toA, then{Ak} is
Wijsman lacunary summable toA.
Proof. Assume that the sequence {Ak} ∈ L∞ is Wijsman quasi-almost lacunary convergent to A. Then,
Equation (2.1) is true which forn= 0 implies for everyε >0 and eachx∈X,
1 hr
X
k∈Ir
dx(Ak)−dx(A)
−→0 (as r→ ∞);
so,{Ak}is Wijsman lacunary summable toA.
Definition 2.2. Let θ={kr} be a lacunary sequence. A sequence {Ak} is Wijsman quasi-almost lacunary
statistically convergent to Aif for eachx∈X and everyε >0
lim r→∞
1 hr
k∈Ir:|dx(Ak+nr)−dx(A)| ≥ε
uniformly in n. In this case, we will write (W QS)θ−limAk =AorAk
(W QS)θ
−→ A.
The set of all Wijsman quasi-almost lacunary statistically convergence sequences will be denoted by
W QSθ :
W QSθ =
{Ak}: lim r→∞ 1 hr
k∈Ir:|dx(Ak+nr)−dx(A)| ≥ε
= 0
.
Theorem 2.3. If a sequence {Ak} is Wijsman almost lacunary statistically convergent toA, then{Ak} is
Wijsman quasi-almost lacunary statistically convergent toA.
Proof. Suppose that the sequence{Ak}is Wijsman almost lacunary statistically convergent toA. Then, for
everyε, δ >0 and for eachx∈X there exists an integerr0>0 such that for allr > r0
1 hr
k∈Ir:|dx(Ak+i)−dx(A)| ≥ε
< δ,
uniformly ini. Ifi is taken asi=nr, then we have
1 hr
k∈Ir:|dx(Ak+nr)−dx(A)| ≥ε
< δ,
uniformly inn. Sinceδ >0 is an arbitrary, we have
lim r→∞ 1 hr
k∈Ir:|dx(Ak+nr)−dx(A)| ≥ε
= 0,
uniformly innwhich means thatAk
(W QS)θ
−→ A.
Theorem 2.4. For any lacunary sequence θ={kr}; if lim infrqr>1, then
W QS ⊂
W QSθ .
Proof. Suppose that lim infrqr >1. Then for each r ≥1, there is a number δ ≥0 such that qr ≥1 +δ.
Sinceqr≥1 +δ and hr=kr−kr−1,we have
hr
kr
≥ δ
1 +δ.
Assume thatAk W QS
−→ A. For eachx∈X, we can write
1 kr
k≤kr:|dx(Ak+nkr)−dx(A)| ≥ε ≥ 1 kr
k∈Ir:|dx(Ak+nkr)−dx(A)| ≥ε
= hr kr 1 hr
k∈Ir:|dx(Ak+nkr)−dx(A)| ≥ε
≥ δ
1 +δ
1 hr
k∈Ir:|dx(Ak+nkr)−dx(A)| ≥ε that is, 1 kr
k≤kr:|dx(Ak+nkr)−dx(A)| ≥ε
≥
δ 1 +δ
1 hr
uniformly inn. If the limit is taken for the above inequality; sinceAk W QS
−→ A, we have
0≥ δ
1 +δ ·rlim→∞
1 hr
k∈Ir:|dx(Ak+nkr)−dx(A)| ≥ε .
By the definition of lacunary sequence, we can writer instead ofkr. Hence, for eachx∈X we have
lim r→∞ 1 hr
k∈Ir:|dx(Ak+nr)−dx(A)| ≥ε
= 0,
uniformly inn, that isAk
(W QS)θ
−→ A.
Definition 2.3. Let θ ={kr} be a lacunary sequence. A sequence {Ak} ∈L∞ is Wijsman quasi-strongly
almost lacunary convergent toAif for eachx∈X,
1 hr
X
k∈Ir
|dx(Ak+nr)−dx(A)| −→0 (as r→ ∞);
uniformly in n. In this case, we will write [W QF]θ−limAk=Aor Ak
[W QF]θ
−→ A.
Theorem 2.5. For any lacunary sequence θ={kr}; if lim infrqr>1, then
Ak
[W QF]
−→ A⇒Ak
[W QF]θ
−→ A.
Proof. Let lim infrqr > 1. Then for each r ≥ 1, there is a number δ ≥ 0 such that qr ≥ 1 +δ. Since
qr≥1 +δ and hr=kr−kr−1, we have
kr
hr
≤ 1 +δ δ and
kr−1 hr
≤ 1
δ · (2.2)
Assume thatAk
[W QF]
−→ A. For eachx∈X, we can write
1 hr
X
k∈Ir
|dx(Ak+nr)−dx(A)|= 1 hr
kr X
i=1
|dx(Ai+nr)−dx(A)| − 1 hr
kr−1
X
i=1
|dx(Ai+nr)−dx(A)|
= kr hr 1 kr kr X i=1
|dx(Ai+nr)−dx(A)|
−kr−1 hr
1
kr−1
kr−1
X
i=1
|dx(Ai+nr)−dx(A)|
.
Hence, for eachx∈X we have
lim r→∞
1 hr
X
k∈Ir
|dx(Ak+nr)−dx(A)|= lim r→∞ kr hr 1 kr kr X i=1
|dx(Ai+nr)−dx(A)|
− lim r→∞
kr−1 hr
1
kr−1
kr−1
X
i=1
|dx(Ai+nr)−dx(A)|
uniformly inn. SinceAk
[W QF]
−→ A, for eachx∈X we have
1 kr
kr X
i=1
|dx(Ai+nr)−dx(A)| →0 and 1 kr−1
kr−1
X
i=1
uniformly inn. By using the Inequalities (2.2) and the Status (2.3), we handle
lim r→∞
1 hr
X
k∈Ir
|dx(Ak+nr)−dx(A)|= 0.
The proof of theorem is completed.
Definition 2.4. Letθ={kr} be a lacunary sequence. A sequence{Ak} ∈L∞ is Wijsman quasi q-strongly
almost lacunary convergent toAif for eachx∈X and0< q <∞,
1 hr
X
k∈Ir
|dx(Ak+nr)−dx(A)|q −→0 (as r→ ∞); (2.4)
uniformly in n. In this case, we will write [W QF]qθ−limAk=Aor Ak
[W QF]qθ −→ A.
Theorem 2.6. Let 0< q <∞. Then, we have following assertions:
i. If a sequence{Ak} is Wijsman quasiq-strongly almost lacunary convergent toA, then the sequence
{Ak} is Wijsman quasi-almost lacunary statistically convergent toA.
ii. If a sequence{Ak} ∈L∞ and Wijsman quasi-almost lacunary statistically convergent toA, then the
sequence {Ak} is Wijsman quasiq-strongly almost lacunary convergent toA.
Proof. (i) Letε >0 be given. Then, for each x∈X following inequality is proved
X
k∈Ir
|dx(Ak+nr)−dx(A)|q ≥εq
k∈Ir:|dx(Ak+nr)−dx(A)| ≥ε
, (2.5)
uniformly in n. Since the sequence {Ak} is Wijsman quasi q-strongly almost lacunary convergent to A;
if the both side of Inequality (2.5) are multipled by 1 hr
and after that the limit is taken forr→ ∞, then we
have
lim r→∞
1 hr
X
k∈Ir
|dx(Ak+nr)−dx(A)|q ≥εq lim r→∞
1 hr
k∈Ir:|dx(Ak+nr)−dx(A)| ≥ε
0≥εq lim r→∞
1 hr
k∈Ir:|dx(Ak+nr)−dx(A)| ≥ε
.
Hence, we handle
lim r→∞
1 hr
k∈Ir:|dx(Ak+nr)−dx(A)| ≥ε
= 0,
uniformly inn. That isAk
(W QS)θ
−→ A.
(ii) Since{Ak}is bounded, we can write
sup k
dx(Ak) +dx(A) =M, (0< M <∞),
If {Ak} is Wijsman quasi-almost lacunary statistically convergent to A, then for
a givenε >0 a number Nε∈Ncan be chosen such that for allr > Nε and eachx∈X
1 hr
k∈Ir:|dx(Ak+nr)−dx(A)| ≥
ε
2
1/q
< ε 2Mq
uniformly inn. Let take the set
Lp=
k≤p:|dx(Ak+nr)−dx(A)| ≥
ε
2
1/q
.
Thus, for eachx∈X we have
1 hr
X
k∈Ir
dx(Ak+nr)−dx(A)
q = 1
hr
X
k∈Ir
k∈Lp
|dx(Ak+nr)−dx(A)|q+
X
k∈Ir
k /∈Lp
|dx(Ak+nr)−dx(A)|q
!
< 1 hr
hr
ε 2MqM
q+ 1
hr
hr
ε 2
= ε 2+
ε 2 =ε
uniformly inn. So, the proof is completed.
Theorem 2.7. If the sequence {Ak} is Wijsman quasi q-strongly almost lacunary convergence to A, then
{Ak} is Wijsmanq-strongly lacunary summable to A.
Proof. Suppose that the sequence{Ak} ∈L∞is Wijsman quasiq-strongly almost lacunary convergent toA.
Then, Equation (2.4) is true which forn= 0 implies for everyε >0 and eachx∈X,
1 hr
X
k∈Ir
|dx(Ak)−dx(A)|q −→0 (as r→ ∞);
so,{Ak}is Wijsmanq-strongly lacunary summable toA.
Theorem 2.8. If a sequence{Ak}is Wijsman quasiq-strongly almost lacunary convergence to A, then the
sequence {Ak} is Wijsman lacunary statistically convergent toA.
Proof. Assume that the sequence{Ak}is Wijsman quasiq-strongly almost lacunary convergence toA. Then,
by Theorem2.7, the sequence {Ak} is Wijsman q-strongly lacunary summable toA. For each x∈X and
everyε >0, we can write
X
k∈Ir
|dx(Ak)−dx(A)|q ≥εq
k∈Ir:|dx(Ak)−dx(A)| ≥ε
. (2.6)
Since the sequence{Ak}is Wijsmanq-strongly lacunary summable toA; if the both side of Inequality (2.6)
are multipled by 1 hr
and after that the limit is taken forr→ ∞, left side of the Inequality (2.6) is equal to
0. Hence, we handle
lim r→∞
1 hr
k∈Ir:|dx(Ak)−dx(A)| ≥ε
The proof of theorem is completed.
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