Available at: http://www.pmf.ni.ac.rs/filomat
On I− Lacunary Statistical Convergence of Order
α of Sequences of Sets
Hacer S¸eng ¨ula, Mikail Etb
aDepartment of Mathematics; Siirt University 56100; Siirt ; TURKEY bDepartment of Mathematics; Fırat University 23119; Elazıg ; TURKEY
Abstract.The idea of I−convergence of real sequences was introduced by Kostyrko et al. [ Kostyrko, P. ; ˇSal´at, T. and Wilczy ´nski, W. I−convergence, Real Anal. Exchange 26(2) (2000/2001), 669-686 ] and also independently by Nuray and Ruckle [ Nuray, F. and Ruckle, W. H. Generalized statistical convergence and convergence free spaces, J. Math. Anal. Appl. 245(2) (2000), 513–527 ]. In this paper we introduce the concepts of Wijsman I−lacunary statistical convergence of orderα and Wijsman strongly I−lacunary statistical convergence of order α, and investigated between their relationship.
1. Introduction
The concept of statistical convergence was introduced by Steinhaus [36] and Fast [15]. Schoenberg [34] established some basic properties of statistical convergence and studied the concept as a summability method. Later on it was further investigated from the sequence space point of view and linked with summability theory by Altın et al. [1], Bas¸arır and Konca [2], Caserta et al. [3], Connor [4], C¸ akallı [5], C¸ olak ([8],[9]), Et et al. ([11],[12],[20],[21]), Fridy [17], Gadjiev and Orhan [19], Kolk [22], Mursaleen et al. ([25],[26]), Salat [29], Savas¸ et al. ([10],[32],[33]) and many others. Nuray and Rhoades [28] extended the notion to statistical convergence of sequences of sets and gave some basic theorems. Ulusu and Nuray [38] defined the Wijsman lacunary statistical convergence of sequence of sets, and considered its relation with Wijsman statistical convergence.
Let X be a non-empty set. Then a family of sets I ⊆ 2X(power sets of X) is said to be an ideal if I is
additive i.e. A, B ∈ I implies A ∪ B ∈ I and hereditary, i.e. A ∈ I, B ⊂ A implies B ∈ I. A non-empty family of sets F ⊆ 2Xis said to be a filter of X if and only if
(i)φ < F, (ii) A, B ∈ F implies A ∩ B ∈ F and (iii) A ∈ F, A ⊂ B implies B ∈ F. An ideal I ⊆ 2Xis called non-trivial if I , 2X.
A non-trivial ideal I is said to be admissible if I ⊃ {{x} : x ∈ X}. If I is a non-trivial ideal in XX , φthen the family of sets
F(I)= {M ⊂ X : (∃A ∈ I) (M = X \ A)} is a filter of X, called the filter associated with I.
Let (X, d) be a metric space. For any non-empty closed subset Akof X, we say that the sequence {Ak} is
bounded if supkd(x, Ak)< ∞ for each x ∈ X. In this case we write {Ak} ∈L∞.
Throughout the paper I will stand for a non-trivial admissible ideal of N.
2010 Mathematics Subject Classification. 40A05, 40C05, 46A45 Keywords. I−convergence, Wijsman convergence, lacunary sequence Received: 04 December 2015; Accepted: 25 April 2016
Communicated by Ljubiˇsa D.R. Koˇcinac
The idea of I−convergence of real sequences was introduced by Kostyrko et al. [23] and also indepen-dently by Nuray and Ruckle [27] (who called it generalized statistical convergence) as a generalization of statistical convergence. Later on I−convergence was studied in ([6],[7],[14],[24],[30], [31],[32],[33],[37],[39]).
2. Main Results
In this section, we will extend the results of Et and S¸eng ¨ul ([13], [35]) to statistical convergence of set sequences, namely; the relationship between the concepts of Wijsman I−lacunary statistical convergence of orderα and Wijsman strongly I−lacunary statistical convergence of order α are given
Definition 2.1. Let (X, d) be a metric space, θ be a lacunary sequence, α ∈ (0, 1] and I ⊆ 2Nbe an admissible ideal
of subsets of N. For any non-empty closed subsets A, Ak⊂X, we say that the sequence {Ak}is Wijsman I−lacunary
statistical convergent to A of orderα ( or Sαθ(Iw) −convergent to A ) if for eachε > 0, δ > 0 and x ∈ X,
(
r ∈ N : 1 hαr
|{k ∈ Ir: |d (x, Ak) − d (x, A)| ≥ ε}| ≥ δ )
belongs to I. In this case, we write Ak−→A
Sαθ(Iw) . For θ = (2r), we shall write Sα(Iw) instead of Sαθ(Iw) and in
the special caseα = 1 and θ = (2r) we shall write S (I
w) instead of Sαθ(Iw).
As an example, consider the following sequence: Ak= ( {3x}, kr−1 < k < kr−1+ √ hr {0}, otherwise.
Let(R, d) be a metric space such that for x, y ∈ X, d x, y= x−y , A = {1} , x > 1 and α = 1. Since 1 hαr |{k ∈ Ir: |d (x, Ak) − d (x, 1)| ≥ ε}| ≥ δ
belongs to I, the sequences {Ak} is Wijsman I−lacunary statistical convergent to {1} of order α ; that is Ak −→
{1}Sαθ(Iw) .
Definition 2.2. Let (X, d) be a metric space, θ be a lacunary sequence, α ∈ (0, 1] and I ⊆ 2Nbe an admissible ideal
of subsets of N. For any non-empty closed subsets A, Ak⊂X, we say that the sequence {Ak}is said to be Wijsman
strongly I−lacunary statistical convergent to A of orderα ( or Nθα[Iw] −convergent to A ) if for eachε > 0 and x ∈ X,
r ∈ N : 1 hαr X k∈Ir |d(x, Ak) − d (x, A)| ≥ ε belongs to I. In this case, we write Ak−→A
Nαθ[Iw] . For θ = (2r), we shall write Nα[Iw] instead of Nαθ[Iw] and in
the special caseα = 1 and θ = (2r) we shall write N [Iw] instead of Nαθ[Iw].
As an example, consider the following sequence:
Ak= ( nxk 2o , kr−1< k < kr−1+ √ hr {0}, otherwise.
Let(R, d) be a metric space such that for x, y ∈ X, d x, y= x−y , A = {1} , x > 1 and α = 1. Since 1 hαr X k∈Ir |d(x, Ak) − d (x, 1)| ≥ ε,
the sequences {Ak}is Wijsman I−lacunary statistical convergent to {1} of orderα ; that is Ak−→ {1}
Theorem 2.3. Sαθ(Iw) ∩ L∞is a closed subset of L∞for 0< α ≤ 1.
Proof. Omitted.
Theorem 2.4. Let (X, d) be a metric space, θ = (kr) be a lacunary sequence and A, Ak(for all k ∈ N) be non-empty
closed subsets of X, then (i) Ak→A Nθα[Iw] ⇒Ak→ASα θ(Iw)
and Nθα[Iw] is a proper subset of Sαθ(Iw),
(ii) {Ak} ∈L∞and Ak→A Sαθ(Iw) ⇒Ak→ANα θ[Iw] , (iii) Sαθ(Iw) ∩ L∞= Nαθ[Iw] ∩ L∞.
Proof. (i) The inclusion part of proof is easy. In order to show that the inclusion Nαθ[Iw] ⊆ Sαθ(Iw) is proper,
letθ be given and we define a sequence {Ak} as follows
Ak= ( n x2o , k = 1, 2, 3, ..., h√h r i {0}, otherwise .
Let (R, d) be a metric space such that for x, y ∈ X, d x, y = x−y
. We have for every ε > 0, x > 0 and
1 2 < α ≤ 1, 1 hαr |{k ∈ Ir: |d (x, Ak) − d (x, {0})| ≥ ε}| ≤ h√ hr i hαr ,
and for anyδ > 0 we get ( r ∈ N : 1 hαr |{k ∈ Ir: |d (x, Ak) − d (x, {0})| ≥ ε}| ≥ δ ) ⊆ r ∈ N : h√ hr i hαr ≥δ .
Since the set on the right-hand side is a finite set and so belongs to I, it follows that for 1
2 < α ≤ 1,
Ak→ {0}
Sαθ(Iw) .
On the other hand, for1
2 < α ≤ 1 and x > 0, 1 hαr X k∈Ir |d(x, Ak) − d (x, {0})| = x2− 2x h√h r i hαr → 0 and for 0< α < 12 x2− 2x h √ hr i hαr → ∞. Hence we have r ∈ N : 1 hαr X k∈Ir |d(x, Ak) − d (x, {0})| ≥ 0 = r ∈ N : x2− 2x h√h r i hαr ≥ 0 = {a, a + 1, a + 2, ...}
for some a ∈ N which belongs to F (I) , since I is admissible. So Ak9 {0}
Nαθ[Iw] .
ii) Suppose that {Ak} ∈L∞and Ak→A
Sαθ(Iw) . Then we can assume that
for each x ∈ X and all k ∈ N. Given ε > 0, we get 1 hαr X k∈Ir |d(x, Ak) − d (x, A)| = 1 hαr X k∈Ir |d(x,Ak)−d(x,A)|≥ε |d(x, Ak) − d (x, A)| +1 hαr X k∈Ir |d(x,Ak)−d(x,A)|<ε |d(x, Ak) − d (x, A)| ≤ M hαr |{k ∈ Ir: |d (x, Ak) − d (x, A)| ≥ ε}| + ε. Hence we have r ∈ N : 1 hαr X k∈Ir |d(x, Ak) − d (x, A)| ≥ Mδ + ε ⊂ ( r ∈ N : 1 hαr |{k ∈ Ir: |d (x, Ak) − d (x, A)| ≥ ε}| ≥ δ ) ∈I. Therefore Ak→A Nαθ[Iw] .
iii) Follows from (i) and (ii).
Theorem 2.5. Letθ = (kr) be a lacunary sequence andα be a fixed real number such that 0 < α ≤ 1. If lim infrqr> 1,
then Sα(Iw) ⊂ Sαθ(Iw).
Proof. Suppose first that lim infrqr> 1; then there exists a λ > 0 such that qr≥ 1+ λ for sufficiently large r,
which implies that hr kr ≥ λ 1+ λ=⇒ hr kr !α ≥ λ 1+ λ α =⇒ 1 kαr ≥ λ α (1+ λ)α 1 hαr .
If Ak→A(Sα(Iw)), then for every ε > 0, for each x ∈ X, and for sufficiently large r, we have
1 kαr |{k ≤ kr: |d (x, Ak) − d (x, A)| ≥ ε}| ≥ 1 kαr |{k ∈ Ir: |d (x, Ak) − d (x, A)| ≥ ε}| ≥ λ α (1+ λ)α 1 hαr |{k ∈ Ir: |d (x, Ak) − d (x, A)| ≥ ε}| . Forδ > 0, we have ( r ∈ N : 1 hαr |{k ∈ Ir: |d (x, Ak) − d (x, A)| ≥ ε}| ≥ δ ) ⊆ ( r ∈ N : 1 kαr |{k ≤ kr: |d (x, Ak) − d (x, A)| ≥ ε}| ≥ δλ α (1+ λ)α ) ∈I. This completes the proof.
Theorem 2.6. Letθ = (kr) be a lacunary sequence and the parametersα and β be fixed real numbers such that
0< α ≤ β ≤ 1, then Nαθ[Iw] ⊆ Nβθ[Iw] and the inclusion is strict.
Proof. The inclusion part of proof is easy. To show that the inclusion is strict define {Ak} such that for (R, d),
x> 1 and A = {0} , Ak= ( {3x+ 5} , kr−1< k < kr−1+ √ hr {0}, otherwise .
Theorem 2.7. Letθ = (kr) be a lacunary sequence and the parametersα and β be fixed real numbers such that
0< α ≤ β ≤ 1, then Sαθ(Iw) ⊆ Sβθ(Iw) and the inclusion is strict.
Proof. The inclusion part of proof is easy. To show that the inclusion is strict define {Ak} such that for X= R2
Ak=
(
x, y ∈ R2, x2+ y − 12= k2, if k is square
{(0, 0)} , otherwise .
Then {Ak} ∈Sβθ(Iw) for 12 < β ≤ 1 but {Ak} < Sαθ(Iw) for 0< α ≤ 12.
Theorem 2.8. Let the parametersα and β be fixed real numbers such that 0 < α ≤ β ≤ 1, then Sβ(Iw) ⊆ Nα[Iw].
Proof. For any sequence {Ak} andε > 0, we have
1 nα n X k=1 |d(x, Ak) − d (x, A)| ≥ 1 nα|{k ≤ n : |d (x, Ak) − d (x, A)| ≥ ε}| ε ≥ 1 nβ|{k ≤ n : |d (x, Ak) − d (x, A)| ≥ ε}| ε and so n ∈ N : 1 nα n X k=1 |d(x, Ak) − d (x, A)| ≥ δ ⊆ n ∈ N : 1 nβ |{k ≤ n : |d (x, Ak) − d (x, A)| ≥ ε}| ≥ δ ε ∈I.
This gives that Sβ(Iw) ⊆ Nα[Iw].
Theorem 2.9. Letθ = (kr) be a lacunary sequence andα be a fixed real number such that 0 < α ≤ 1. If limr→∞infh α r kr >
0 then S (Iw) ⊆ Sαθ(Iw).
Proof. Let (X, d) be a metric space, θ = (kr) be a lacunary sequence and A, Ak(for all k ∈ N) be non-empty
closed subsets of X. If limr→∞infh α r
kr > 0, then we can write
{k ≤ kr: |d (x, Ak) − d (x, A)| ≥ ε} ⊃ {k ∈ Ir: |d (x, Ak) − d (x, A)| ≥ ε} 1 kr |{k ≤ kr: |d (x, Ak) − d (x, A)| ≥ ε}| ≥ 1 kr |{k ∈ Ir: |d (x, Ak) − d (x, A)| ≥ ε}| = hαr kr 1 hαr |{k ∈ Ir: |d (x, Ak) − d (x, A)| ≥ ε}| . So ( r ∈ N : 1 hαr |{k ∈ Ir: |d (x, Ak) − d (x, A)| ≥ ε}| ≥ δ ) ⊆ ( r ∈ N : 1 kr |{k ≤ kr: |d (x, Ak) − d (x, A)| ≥ ε}| ≥ δh α r kr )
which implies that S (Iw) ⊆ Sαθ(Iw).
Theorem 2.10. Let (X, d) be a metric space and A, Ak(for all k ∈ N) be non-empty closed subsets of X. If θ = (kr) is
a lacunary sequence with lim sup(kj−kj−1)
α kαr−1 < ∞ (j = 1, 2, ..., r), then Ak→A Sαθ(Iw) implies Ak→A(Sα(Iw)).
Proof. If lim sup(kj−kj−1)
α
kαr−1 < ∞, then without any loss of generality, we can assume that there exists a
0 < Bj < ∞ such that ( kj−kj−1)α
kr−1α < Bj, (j = 1, 2, ..., r) for all r ≥ 1. Suppose that Ak → A
Sαθ(Iw)
and for ε, δ, δ1> 0 define the sets
C= ( r ∈ N : 1 hαr |{k ∈ Ir: |d (x, Ak) − d (x, A)| ≥ ε}| < δ ) and T= r ∈ N : 1 nα|{k ≤ n : |d (x, Ak) − d (x, A)| ≥ ε}| < δ1 .
It is obvious from our assumption that C ∈ F(I), the filter associated with the ideal I. Further observe that
Ai=
1
hαi |{k ∈ Ii: |d (x, Ak) − d (x, A)| ≥ ε}| < δ
for all i ∈ C. Let n ∈ N be such that kr−1 < n < krfor some r ∈ C. Now
1 nα|{k ≤ n : |d (x, Ak) − d (x, A)| ≥ ε}| ≤ 1 kαr−1 |{k ≤ kr: |d (x, Ak) − d (x, A)| ≥ ε}| = 1 kαr−1 |{k ∈ I1: |d (x, Ak) − d (x, A)| ≥ ε}| + ... + 1 kαr−1 |{k ∈ Ir: |d (x, Ak) − d (x, A)| ≥ ε}| = kα1 kαr−1 1 hα1 |{k ∈ I1: |d (x, Ak) − d (x, A)| ≥ ε}| +(k2−k1)α kαr−1 1 hα2 |{k ∈ I2: |d (x, Ak) − d (x, A)| ≥ ε}| +... +(kr−kr−1)α kαr−1 1 hαr |{k ∈ Ir: |d (x, Ak) − d (x, A)| ≥ ε}| ≤ sup i∈C Ai. kα1 + (k2−k1)α+ ... + (kr−kr−1)α kαr−1 ≤ sup i∈C Ai(B1+ B2+ ... + Br)< δ r X j=1 Bj. Choosingδ1= rδ P j=1Bj
and in view of the fact that ∪n : kr−1< n < kr,r ∈ C ⊂ T where C ∈ F(I). This completes
the proof of the theorem.
In [16], it is defined that the lacunary sequenceθ0 = (sr) is called a lacunary refinement of the lacunary
sequenceθ = (kr) if (kr) ⊆ (sr). In [18], the inclusion relationship between Sθand Sθ0 is studied.
Theorem 2.11. Supposeθ0= (sr) is a lacunary refinement of the lacunary sequenceθ = (kr). Let Ir= (kr−1, kr] and
Jr= (sr−1, sr], r = 1, 2, 3, .... If there exists > 0 such that for 0 < α ≤ β ≤ 1,
Jj β |Ii|α ≥ for every Jj⊆Ii. Then Ak→A Sαθ(Iw) implies Ak→A Sβθ0(Iw) , i.e., Sα θ(Iw) ⊆ Sβθ0(Iw).
Proof. For anyε > 0, and every Jj, we can find Iisuch that Jj⊆Ii; then we have 1 Jj β n k ∈ Jj: |d (x, Ak) − d (x, A)| ≥ ε o = |Ii|α Jj β 1 |Ii|α ! n k ∈ Jj: |d (x, Ak) − d (x, A)| ≥ ε o ≤ |Ii|α Jj β 1 |Ii|α ! |{k ∈ Ii: |d (x, Ak) − d (x, A)| ≥ ε}| ≤ 1 1 |Ii|α ! |{k ∈ Ii: |d (x, Ak) − d (x, A)| ≥ ε}| , and so r ∈ N : 1 Jj β n k ∈ Jj: |d (x, Ak) − d (x, A)| ≥ ε o ≥δ ⊆ ( r ∈ N : |1 Ii|α ! |{k ∈ Ii: |d (x, Ak) − d (x, A)| ≥ ε}| ≥ δ ) ∈I.
The proof completes immediately.
Theorem 2.12. Suppose θ = (kr) and θ0 = (sr) are two lacunary sequences. Let Ir = (kr−1, kr], Jr = (sr−1, sr],
r= 1, 2, 3, ..., and Ii j= Ii∩Jj, i, j = 1, 2, 3, .... If there exists > 0 such that for 0 < α ≤ β ≤ 1,
Ii j β
|Ii|α ≥ for every i, j = 1, 2, 3, ..., provided Ii j, ∅. Then Ak→A Sαθ(Iw) implies Ak→A Sβθ0(Iw) , i.e., Sα θ(Iw) ⊆ Sβθ0(Iw).
Proof. Letθ00 = θ0∪θ. Then θ00
is a lacunary refinement of the lacunary sequenceθ0, also θ. Then interval
sequence ofθ00 isnIi j= Ii∩Jj: Ii j, ∅o . From Theorem 2.11, the condition in Theorem 2.12:
|Ii j|β |Ii|α
≥ , for every i, j = 1, 2, 3, ...,provided Ii j, ∅ yields that Ak→A
Sαθ(Iw) implies Ak→A Sβθ00 (Iw) . Since θ 00 is also a lacunary refinement of the lacunary sequenceθ0, we have that Ak→A
Sαθ00(Iw) implies Ak→A Sβθ0(Iw) .
The proof follows immediately.
Letθ = (kr) andθ0 = (sr) be two lacunary sequences such that Ir ⊂ Jrfor all r ∈ N and let α and β be
positive real numbers such that 0< α ≤ β ≤ 1. Now we shall give some general inclusion relations between the sets of Sαθ(Iw) −convergent sequences and Nαθ[Iw] −summable sequences for different α’s andθ’s which
also include Theorem 2.4, Theorem 2.6, Theorem 2.7 and Theorem 2.8 as a special case.
Theorem 2.13. Letθ = (kr) andθ0= (sr) be two lacunary sequences such that Ir⊂Jrfor all r ∈ N and let α and β
be such that 0< α ≤ β ≤ 1, (i) If lim r→∞inf hαr `rβ > 0 (1) then Sβθ0(Iw) ⊆ Sα θ(Iw), (ii) If
lim r→∞ `r hβr = 1 (2) then Sαθ(Iw) ⊆ Sβθ0(Iw).
Proof. (i) Let (X, d) be a metric space, θ = (kr) be a lacunary sequence and A, Ak(for all k ∈ N) be non-empty
closed subsets of X. Suppose that Ir⊂Jrfor all r ∈ N and let (1) be satisfied. For given ε > 0 we have
{k ∈ Jr: |d (x, Ak) − d (x, A)| ≥ ε} ⊇ {k ∈ Ir: |d (x, Ak) − d (x, A)| ≥ ε} , and so 1 `rβ |{k ∈ Jr: |d (x, Ak) − d (x, A)| ≥ ε}| ≥ h α r `βr 1 hαr |{k ∈ Ir: |d (x, Ak) − d (x, A)| ≥ ε}| . Hence ( r ∈ N : 1 hαr |{k ∈ Ir: |d (x, Ak) − d (x, A)| ≥ ε}| ≥ δ ) ⊆ r ∈ N : 1 `βr |{k ∈ Jr: |d (x, Ak) − d (x, A)| ≥ ε}| ≥ δh α r `βr ∈I
for all r ∈ N, where Ir= (kr−1, kr], Jr = (sr−1, sr], hr= kr−kr−1, `r = sr−sr−1. Now taking the limit as r → ∞
in the last inequality and using (1) we get Sβθ0(Iw) ⊆ Sα θ(Iw).
(ii) Omitted.
Theorem 2.14. Letθ = (kr) andθ0= (sr) be two lacunary sequences such that Ir⊆Jrfor all r ∈ N, α and β be fixed
real numbers such that 0< α ≤ β ≤ 1. Then we have (i) If (1) holds then Nθβ0[Iw] ⊂ Nα
θ[Iw],
(ii) If (2) holds and {Ak} ∈L∞then Nαθ[Iw] ⊂ Nβθ0[Iw].
Proof. Omitted.
Theorem 2.15. Letθ = (kr) andθ0= (sr) be two lacunary sequences such that Ir⊆Jrfor all r ∈ N, α and β be fixed
real numbers such that 0< α ≤ β ≤ 1. Then
(i) Let (1) holds, if a sequence is strongly Nβθ0[Iw] −summable to A, then it is Sα
θ(Iw) −statistically convergent to
A,
(ii) Let (2) holds and {Ak}be a bounded sequence, if a sequence is Sαθ(Iw) −statistically convergent to A then it is
strongly Nβθ0[Iw] −summable to A.
(ii) Suppose that Sαθ(Iw) − lim Ak = A and {Ak} ∈ L∞. Then there exists some M > 0 such that
|d(x, Ak) − d (x, A)| ≤ M for all k, then for every ε > 0 we may write 1 `rβ X k∈Jr |d(x, Ak) − d (x, A)| = 1 `βr X k∈Jr−Ir |d(x, Ak) − d (x, A)| + 1 `βr X k∈Ir |d(x, Ak) − d (x, A)| ≤ `r−hr `βr M+ 1 `βr X k∈Ir |d(x, Ak) − d (x, A)| ≤ `r−hβr `rβ M+ 1 `βr X k∈Ir |d(x, Ak) − d (x, A)| ≤ `r hβr − 1 M+ 1 hβr X k∈Ir |d(x,Ak)−d(x,A)|≥ε |d(x, Ak) − d (x, A)| +1 hβr X k∈Ir |d(x,Ak)−d(x,A)|<ε |d(x, Ak) − d (x, A)| ≤ `r hβr − 1 M+ M hαr |{k ∈ Ir: |d (x, Ak) − d (x, A)| ≥ ε}| + `r hβr ε and so r ∈ N : 1 `βr X k∈Jr |d(x, Ak) − d (x, A)| ≥ δ ⊆ ( r ∈ N : 1 hαr |{k ∈ Ir: |d (x, Ak) − d (x, A)| ≥ ε}| ≥ δ M ) ∈I, for all r ∈ N. Using (2) we obtain that Nβθ0[Iw] − lim Ak= A, whenever Sα
θ(Iw) − lim Ak= A.
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