I2-STATISTICALLY AND I2-LACUNARY STATISTICALLY CONVERGENT DOUBLE SET SEQUENCES OF ORDER η U ˇGUR ULUSU, ESRA G ¨ULLE Abstract

ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 13 Issue 1 (2021), Pages 1-15.

I2-STATISTICALLY AND I2-LACUNARY STATISTICALLY CONVERGENT DOUBLE SET SEQUENCES OF ORDER η

U ˇGUR ULUSU, ESRA G ¨ULLE

Abstract. In this study, for double set sequences, as a new approach to the notion of statistical convergence of orderη, the notions of Wijsman I2- statistically convergence of orderη, Wijsman strongI₂-Ces`aro summability of orderη, WijsmanI₂-lacunary statistically convergence of orderηand Wijsman strongI₂-lacunary summability of orderη are introduced, where 0< η≤1.

Also, some properties of these notions are investigated, some investigations about these are made and the existence of some relationships between them are examined.

1. Introduction

The notion of statistical convergence, introduced in the 1950’s, was extended to double sequences by Mursaleen and Edely [17], which generalizes the notion of convergence for double sequences introduced by Pringshiem [25]. Then, using double lacunary sequence notion, Patterson and Sava¸s [24] studied the notion of lacunary statistical convergence for double sequences. Moreover, Das et. al [7]

presented the concept of I-convergence for double sequences via ideals. Recently, for double sequences, C¸ olak and Altın [6] defined the concept of statistically con- vergence of order α and investigated some properties of this notion. From past to present, many authors have studied and have developed these notions in their papers. More information on the notions of convergence for real sequences can be found in [1, 5, 8, 9, 12, 13, 16, 26, 27, 29]. The readers should refer to the monographs [2] and [18] for the background on the sequence spaces and related topics.

For years, many authors have examined on the notions of various convergence for set sequences. One of these convergence notions, handled in this study, is the notion of Wijsman convergence (see, [3, 4, 35]). Using the notions of statistical conver- gence, lacunary sequence, ideal and invariant mean, many authors have extended the notion of Wijsman convergence to the new convergence notions in Wijsman sense for set sequences (for examples, see [10, 15, 19, 23, 31]).

2010Mathematics Subject Classification. 40 A 05, 40 A 35 .

Key words and phrases. Orderη; sequences of sets; Wijsman convergence; double sequence;

I-convergence; statistical convergence; lacunary sequence.

c

2021 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e.

Submitted December 2, 2020. Published January 11, 2021.

Communicated by F. Basar.

1

The basic notions of Wijsman convergence for double sequences of sets were intro- duced by Nuray et. al [21, 22] and D¨undar et. al [11], some of which are Wijsman statistically, Wijsman lacunary statistically, WijsmanI2-statistically and Wijsman I2-lacunary statistically convergence. Recently, the new concepts about Wijsman convergence of orderαfor double sequences of sets were studied by G¨ulle and Ulusu [14, 34]. More information on the notions of convergence for set sequences can be found in [20, 28, 30, 33].

2. Preliminaries

First of all, we give the basic notations necessary for a better understanding of our study (see, [3, 4, 7, 11, 16, 22, 21, 25, 24, 33]).

A double sequence (amn) is called convergent toL(in Pringsheim sense) if every ξ >0, there existsNξ ∈N, the set of natural numbers, such that|amn−L| < ξ, when everm, n > Nξ.

A family of setsI ⊆2^N, the power set ofN, is said to be an ideal if and only if (i) ∅ ∈ I,

(ii) E∪F ∈ I for eachE, F ∈ I, (iii) F ∈ I for eachE∈ I andF ⊆E.

An idealI ⊆2^Nis said to be non-trivial ifN∈ I/ and a non-trivial idealI ⊆2^N is said to be admissible if{m} ∈ I for eachm∈N.

A non-trivial ideal I₂ ⊆ 2^N×N is said to be strong admissible if {m} ×N and N× {m} belongs to I₂ for each m ∈ N. Obviously, a strong admissible ideal is admissible.

Throughout the study,I2⊆2^N×Nwill taken a strong admissible ideal.

LetY be non-empty set. A functiong:N→2^Y is definedg(m) =Vm∈2^Y for eachm∈N. The sequence {Vm}= (V1, V2, . . .), which the range elements of g, is called sequences of sets.

Let (Y, d) be a metric space. For anyy ∈ Y and any non-empty V ⊆ Y, the distance fromyto V is defined

ρ(y, V) = inf

v∈Vd(y, v).

Throughout the study, (Y, d) will taken a metric space andV, V_mnwill taken any non-empty closed subsets ofY.

A double sequence{V_mn} is called Wijsman convergent toV if each y∈Y,

m,n→∞lim ρ(y, Vmn) =ρ(y, V).

A double sequence{Vmn}is called Wijsman statistically convergent toV if every ξ >0 and eachy∈Y,

i,j→∞lim 1 ij

(m, n) : m≤i, n≤j, |ρ(y, V_mn)−ρ(y, V)| ≥ξ = 0.

A double sequence{Vmn}is called Wijsman strong I2-Ces`aro summable toV if everyξ >0 and eachy∈Y,

(

(i, j)∈N×N: 1 ij

i,j

X

m,n=1,1

|ρ(y, Vmn)−ρ(y, V)| ≥ξ )

∈ I2.

A double sequence{Vmn} is called Wijsman I2-statistically convergent to V if everyξ >0,δ >0 and each y∈Y,

(i, j)∈N×N: 1 ij

(m, n) : m≤i, n≤j, |ρ(y, Vmn)−ρ(y, V)| ≥ξ ≥δ

∈ I2. The class of WijsmanI2-statistically convergent double sequences is denoted by S(IW₂).

A double sequence θ2 = {(js, kt)} is said to be a double lacunary sequence if there exists an increasing sequences (js) and (kt) of the integers such that

j0= 0, hs=js−js−1→ ∞ and k0= 0, h¯t=kt−kt−1→ ∞ as s, t→ ∞.

Throughout the study, regarding lacunary sequence θ₂={(j_s, k_t)}, we will use the following notations:

hst=hs¯ht, Ist={(j, k) :js−1< j≤js and kt−1< k≤kt} qs= j_s

j_s−1 and qt= k_t k_t−1.

Throughout the study,θ2={(js, kt)}will taken a double lacunary sequence.

A double sequence{Vmn}is called Wijsman lacunary statistically convergent to V if everyξ >0 and eachy ∈Y,

s,t→∞lim 1 hst

(m, n)∈I_st:|ρ(y, V_mn)−ρ(y, V)| ≥ξ = 0.

A double sequence{V_mn}is called Wijsman strongI₂-lacunary summable toV if everyξ >0 and eachy∈Y,

(

(s, t)∈N×N: 1 hst

X

(m,n)∈I_st

|ρ(y, V_mn)−ρ(y, V)| ≥ξ )

∈ I₂.

The class of Wijsman strongI2-lacunary summable double sequences is denoted byN_θ[IW2].

A double sequence{Vmn}is called WijsmanI2-lacunary statistically convergent toV if everyξ >0,δ >0 and eachy ∈Y,

(s, t)∈N×N: 1 h_st

(m, n)∈Ist:|ρ(y, Vmn)−ρ(y, V)| ≥ξ ≥δ

∈ I2. The class of Wijsman I2-lacunary statistically convergent double sequences is denoted bySθ(IW₂).

From now on, for short, we use ρy(V) and ρy(Vmn) instead of ρ(y, V) and ρ(y, Vmn), respectively.

3. New Concepts

In this section, for double set sequences, as a new approach to the notion of sta- tistical convergence of orderη, the notions of WijsmanI2-statistically convergence of orderη, Wijsman strongI2-Ces`aro summability of orderη, WijsmanI2-lacunary statistically convergence of order η and Wijsman strong I2-lacunary summability of orderη are introduced, where 0< η≤1.

Definition 3.1. Let0< η≤1. A double sequence{Vmn}is WijsmanI2-statistical convergent of order η toV or S(I_W^η

2)-convergent to V if every ξ >0, δ > 0 and each y∈Y,

(i, j)∈N×N: 1 (ij)^η

(m, n) : m≤i, n≤j, |ρy(Vmn)−ρy(V)| ≥ξ ≥δ

∈ I2.

Also, we writeVmn S(I_W^η

2)

−→ V orVmn−→V S(I_W^η

2) .

The class of Wijsman I₂-statistical convergent double sequences is denoted by S(I_W^η

2).

Example 3.1. Let Y =R² and double sequence{Vmn}be defined as follows:

Vmn:=







(a, b)∈R²: (a+m)²+ (b+n)²= 1 ; if mandnare square integers

{(1,1)} ; otherwise.

If we take I2 =I₂^δ, (I₂^δ is the class of E ⊂N×N with density of E equals to 0), then double sequence {Vmn} is WijsmanI2-statistical convergent of order η to V ={(1,1)}.

Remark. Forη= 1, the concept ofS(I_W^η

2)-convergence coincides with the notion of WijsmanI2-statistical convergence for double sequences of sets in [11].

Definition 3.2. Let 0 < η≤1. A double sequence{Vmn} is WijsmanI2-Ces`aro summable of order η to V or C1(I_W^η

2)-summable to V if every ξ > 0 and each y∈Y,

(

(i, j)∈N×N:

1 (ij)^η

i,j

X

m,n=1,1

ρ_y(V_mn)−ρ_y(V)

≥ξ )

∈ I2.

Also, we writeVmn C₁(I_W^η

2)

−→ V orVmn−→V C1(I_W^η

2) .

Definition 3.3. Let 0 < η ≤ 1. A double sequence {V_mn} is Wijsman strong I₂-Ces`aro summable of orderη toV orC₁[I_W^η

2]-summable toV if every ξ >0and each y∈Y,

(

(i, j)∈N×N: 1 (ij)^η

i,j

X

m,n=1,1

ρy(Vmn)−ρy(V) ≥ξ

)

∈ I2.

Also, we writeV_mn

C₁[I_W^η

2]

−→ V orV_mn−→V C₁[I_W^η

2] .

The class of Wijsman strong I2-Ces`aro summable double sequences is denoted byC1[I_W^η

2].

Example 3.2. Let Y =R² and double sequence{Vmn}be defined as follows:

V_mn:=







(a, b)∈R²: (a+ 1)²+b²=_mn¹ ; if mandnare square integers

{(0,1)} ; otherwise.

If I2 =I₂^f, (I₂^f is the class of finite subsets of N×N), then double sequence {Vmn} is Wijsman strongI2-Ces`aro summable of order η toV ={(0,1)}.

Remark. Forη= 1, the concepts ofC1(I_W^η

2)-summability andC1[I_W^η

2]-summability coincide with the notions of WijsmanI2-Ces`aro summability, Wijsman strong I2- Ces`aro summability for double sequences of sets in[33], respectively.

Definition 3.4. Let 0 < η ≤ 1 and 0 < p < ∞. A double sequence {V_mn} is Wijsman strongp− I2-Ces`aro summable of orderη toV orC1[I_W^η

2]^p-summable to V if everyξ >0and each y∈Y,

(

(i, j)∈N×N: 1 (ij)^η

i,j

X

m,n=1,1

ρy(Vmn)−ρy(V)

p≥ξ )

∈ I2.

Also, we writeVmn C₁[I_W^η

2]^p

−→ V or Vmn−→V C1[I_W^η

2]^p .

The class of Wijsman strongp−I2-Ces`aro summable double sequences is denoted byC1[I_W^η

2]^p.

Definition 3.5. Let 0< η≤1 andθ2={(js, kt)} be a double lacunary sequence.

Double sequence {Vmn} is WijsmanI2-lacunary statistically convergent of order η toV or Sθ(I_W^η

2)-convergent toV if everyξ >0, δ >0 and each y∈Y,

(s, t)∈N×N: 1 (hst)^η

(m, n)∈Ist:|ρy(Vmn)−ρy(V)| ≥ξ ≥δ

∈ I2.

Also, we writeVmn S_θ(I_W^η

2)

−→ V orVmn−→V Sθ(I_W^η

2) .

The class of Wijsman I2-lacunary statistical convergent double sequences is denoted bySθ(I_W^η

2).

Example 3.3. Let Y =R² and double sequence{Vmn}be defined as follows:

Vmn:=







(a, b)∈R²: (a−m)²+ (b+n)²= 1 ; if (m, n)∈Ist, mandn are square integers {(−1,1)} ; otherwise.

If we take I2=I₂^δ, then double sequence{Vmn} is WijsmanI2-lacunary statis- tical convergent of orderη toV ={(−1,1)}.

Remark. Forη= 1, the concept ofSθ(I_W^η

2)-convergence coincides with the notion of WijsmanI₂-lacunary statistical convergence for double sequences of sets in[11].

Definition 3.6. Let 0 < η ≤ 1 and θ2 = {(js, kt)} be a double lacunary se- quence. Double sequence{Vmn}is WijsmanI2-lacunary summable of orderηtoV orNθ(I_W^η

2)-summable toV if everyξ >0 and each y∈Y,







(s, t)∈N×N:

1 (hst)^η

X

(m,n)∈Ist

ρy(Vmn)−ρy(V)

≥ξ







∈ I2.

Also, we writeVmn N_θ(I_W^η

2)

−→ V orVmn−→V Nθ(I_W^η

2) .

Definition 3.7. Let 0< η≤1 andθ2={(js, kt)} be a double lacunary sequence.

Double sequence{Vmn} is Wijsman strong I2-lacunary summable of order η toV orNθ[I_W^η

2]-summable toV if everyξ >0 and eachy∈Y,







(s, t)∈N×N: 1 (hst)^η

X

(m,n)∈Ist

ρy(Vmn)−ρy(V) ≥ξ







∈ I2.

Also, we writeVmn N_θ[I_W^η

2]

−→ V or Vmn−→V Nθ[I_W^η

2] .

The class of Wijsman strongI2-lacunary summable double sequences is denoted byN_θ[I_W^η

2].

Example 3.4. Let Y =R² and double sequence{V_mn}be defined as follows:

V_mn:=







(a, b)∈R²: a²+ (b−1)²= _mn¹ ; if (m, n)∈I_st, mandn are square integers

{(1,0)} ; otherwise.

IfI2=I₂^f, then double sequence{Vmn}is Wijsman strongI2-lacunary summa- ble of order η toV ={(1,0)}.

Remark. Forη= 1, the concepts ofNθ(I_W^η

2)-summability andNθ[I_W^η

2]-summability coincide with the notions of WijsmanI2-lacunary convergence, Wijsman strongI2- lacunary convergence for double sequences of sets in[11].

Definition 3.8. Let0< η≤1,0< p <∞andθ2={(js, kt)}be a double lacunary sequence. Double sequence{Vmn} is Wijsman strong p− I2-lacunary summable of orderη toV orNθ[I_W^η

2]^p-summable toV if everyξ >0and each y∈Y,







(s, t)∈N×N: 1 (h_st)^η

X

(m,n)∈Ist

ρy(Vmn)−ρy(V)

p≥ξ







∈ I2.

Also, we writeV_mn^Nθ[I

η W2]^p

−→ V orV_mn−→V N_θ[I_W^η

2]^p .

The class of Wijsman strong p− I2-lacunary summable double sequences is denoted byNθ[I_W^η

2]^p.

4. Inclusions Teorems

In this section, firstly, some properties of the new notions introduced in Section 3 are examined with some investigations and emphasized on the existence of some relationships between them.

Theorem 4.1. If 0< η≤µ≤1, then S(I_W^η

2)⊆S(I_W^µ

2).

Proof. Let 0< η ≤µ≤1. Also, we suppose thatVmn S(I^η_W

2)

−→ V. For every ξ >0 and eachy∈Y, we have

1 (ij)^µ

{(m, n) : m≤i, n≤j, |ρ_y(V_mn)−ρ_y(V)| ≥ξ}

≤ 1 (ij)^η

{(m, n) : m≤i, n≤j, |ρy(V_mn)−ρ_y(V)| ≥ξ}

and so for everyδ >0, n

(i, j)∈N×N: _(ij)¹µ

{(m, n) : m≤i, n≤j, |ρy(Vmn)−ρy(V)| ≥ξ}

≥δo

⊆n

(i, j)∈N×N: _(ij)¹_η

{(m, n) : m≤i, n≤j, |ρ_y(V_mn)−ρ_y(V)| ≥ξ}

≥δo . Hence, by our assumption, the set on right side belongs to the ideal I2, obviously the set on left side also belongs toI₂. Consequently,S(I_W^η

2)⊆S(I_W^µ

2).

Ifµ= 1 is taken in Theorem 4.1, then the following corollary is obtained.

Corollary 4.2. A double sequence {Vmn} is WijsmanI2-statistical convergent to V if the double sequence is Wijsman I2-statistical convergent of order η toV for some0< η≤1, i.e.,S(I_W^η

2)⊆S(IW₂).

Similarly, we can give the following theorem without proof.

Theorem 4.3. Let θ₂={(j_s, k_t)} be a double lacunary sequence. Then, i. If 0< η≤µ≤1, thenSθ(I_W^η

2)⊆Sθ(I_W^µ

2).

ii. Particularly, forµ= 1,Sθ(I_W^η

2)⊆Sθ(IW₂).

Theorem 4.4. If 0< η≤µ≤1 and0< p <∞, thenC1[I_W^η

2]^p⊆C1[I_W^µ

2]^p. Proof. Let 0< η≤µ≤1. Also, we suppose that Vmn

C₁[I_W^η

2]^p

−→ V. For each y∈Y, we have

1 (ij)^µ

i,j

X

m,n=1,1

ρy(Vmn)−ρy(V)

p ≤ 1 (ij)^η

i,j

X

m,n=1,1

ρy(Vmn)−ρy(V)

p

and so for everyξ >0, (

(i, j)∈N×N: 1 (ij)^µ

i,j

X

m,n=1,1

ρy(Vmn)−ρy(V)

p≥ξ )

⊆ (

(i, j)∈N×N: 1 (ij)^η

i,j

X

m,n=1,1

ρ_y(V_mn)−ρ_y(V)

p≥ξ )

.

Hence, by our assumption, the set on right side belongs to the ideal I2, obviously the set on left side also belongs toI2. Consequently,C1[I_W^η

2]^p⊆C1[I_W^µ

2]^p.

Ifµ= 1 is taken in Theorem 4.4, then the following corollary is obtained.

Corollary 4.5. A double sequence{V_mn} is Wijsman strong p− I₂-Ces`aro sum- mable to V if the double sequence is Wijsman strong p− I2-Ces`aro summable of orderη toV for some0< η≤1.

Now, we can state the theorem giving the relation betweenC1[I_W^η

2]^pandC1[I_W^η

2]^q, where 0< η≤1 and 0< p < q <∞.

Theorem 4.6. Let 0< η≤1 and0< p < q <∞. Then,C1[I_W^η

2]^q⊂C1[I_W^η

2]^p. Similarly, we can give the following theorem without proof.

Theorem 4.7. Let θ2={(js, kt)} be a double lacunary sequence. Then, i. If 0< η≤µ≤1, thenN_θ[I_W^η

2]⊆N_θ[I_W^µ

2].

ii. Particularly, forµ= 1,N_θ[I_W^η

2]⊆N_θ[I_W2].

iii. If 0< η≤1 and0< p < q <∞, thenN_θ[I_W^η₂]^q ⊂N_θ[I_W^η₂]^p.

Theorem 4.8. Let 0< η≤µ≤1 and0< p <∞. If a double sequence{Vmn} is Wijsman strongp− I2-Ces`aro summable of orderη toV, then the sequence{Vmn} is WijsmanI₂-statistical convergent of orderµtoV.

Proof. Let 0 < η ≤µ ≤1 and 0 < p < ∞. Also, we suppose that the sequence {Vmn}is Wijsman strongp− I2-Ces`aro summable of orderηtoV. For everyξ >0 and eachy∈Y, we have

i,j

X

m,n=1,1

|ρy(Vmn)−ρy(V)|^p =

i,j

X

m,n=1,1

|ρy(Vmn)−ρy(V)|≥ξ

|ρy(Vmn)−ρy(V)|^p

+

i,j

X

m,n=1,1

|ρy(Vmn)−ρy(V)|<ξ

|ρy(Vmn)−ρy(V)|^p

≥

i,j

X

m,n=1,1

|ρy(Vmn)−ρy(V)|≥ξ

|ρy(Vmn)−ρy(V)|^p

≥ ξ^p

(m, n) : m≤i, n≤j, |ρy(Vmn)−ρy(V)| ≥ξ and so

1 ξ^p(ij)^η

i,j

X

m,n=1,1

|ρy(Vmn)−ρy(V)|^p

≥ 1 (ij)^η

(m, n) : m≤i, n≤j, |ρy(Vmn)−ρy(V)| ≥ξ

≥ 1 (ij)^µ

(m, n) : m≤i, n≤j, |ρ_y(V_mn)−ρ_y(V)| ≥ξ . Thus, for everyδ >0

(i, j)∈N×N: 1 (ij)^µ

(m, n) : m≤i, n≤j, |ρy(Vmn)−ρy(V)| ≥ξ ≥δ

⊆ (

(i, j)∈N×N: 1 (ij)^η

i,j

X

m,n=1,1

|ρy(Vmn)−ρy(V)|^p≥ξ^pδ )

. Hence, by our assumption, the set on right side belongs to the ideal I₂, obviously the set on left side also belongs toI₂. Consequently, the sequence{V_mn}is Wijsman

I2-statistical convergent of orderµtoV.

Ifµ=η is taken in Theorem 4.8, then the following corollary is obtained.

Corollary 4.9. Let 0 < η ≤ 1 and 0 < p < ∞. A double sequence {Vmn} is WijsmanI2-statistical convergent of orderηtoV if the double sequence is Wijsman strongp− I2-Ces`aro summable of orderη toV.

Similarly, we can give the following theorem without proof.

Theorem 4.10. Let 0 < η ≤ µ ≤1 and 0 < p < ∞. If the sequence {Vmn} is Wijsman strongp−I2-lacunary summable of orderηtoV, then the double sequence is WijsmanI2-lacunary statistical convergent of orderµtoV.

Ifµ=η is taken in Theorem 4.10, then the following corollary is obtained.

Corollary 4.11. Let 0 < η ≤ 1 and 0 < p < ∞. A double sequence {Vmn} is Wijsman I2-lacunary statistical convergent of order η to V if the double sequence is Wijsman strong p− I2-lacunary summable of order η toV.

Theorem 4.12. Let 0< η≤1 andθ2={(js, kt)} be a double lacunary sequence.

If lim infsq_s^η>1 andlim inftq^η_t >1, thenVmn S(I_W^η

2)

−→ V impliesVmn S_θ(I_W^η

2)

−→ V. Proof. Let 0 < η ≤ 1 be given. Also, we assume that lim infsq_s^η > 1 and lim inftq^η_t >1. Then, there existα >0, β >0 such thatq_s^η≥1 +αandq_t^η≥1 +β for allsandt, which implies that

h^η_st

(jskt)^η ≥ αβ (1 +α)(1 +β). Suppose thatV_mn^S(I

η W2)

−→ V. For everyξ >0 and eachy∈Y, we have 1

(jskt)^η

(m, n) : m≤j_s, n≤k_t:|ρ_y(V_mn)−ρ_y(V)| ≥ξ

≥ 1 (jskt)^η

(m, n)∈I_st:|ρy(V_mn)−ρ_y(V)| ≥ξ

= h^η_st (jskt)^η

1 h^η_st

(m, n)∈Ist:|ρy(Vmn)−ρy(V)| ≥ξ

≥ αβ

(1 +α)(1 +β) 1 h^η_st

(m, n)∈Ist :|ρy(Vmn)−ρy(V)| ≥ξ . Thus, for anyδ >0

n(s, t)∈N×N: _(h¹

st)^η

(m, n)∈I_st:|ρy(V_mn)−ρ_y(V)| ≥ξ ≥δo

⊆n

(s, t)∈N×N: _(j ¹

sk_t)^η

(m, n) : m≤js, n≤kt:|ρy(Vmn)−ρy(V)| ≥ξ

≥_(1+α)(1+β)^{α β δ} o . Hence, by our assumption, the set on right side belongs to the ideal I₂, obviously the set on left side also belongs toI2. Consequently,V_mn^Sθ(I

η W2)

−→ V.

Theorem 4.13. Let 0< η≤1 andθ2={(js, kt)} be a double lacunary sequence.

If lim sup_sqs<∞andlim sup_tqt<∞, thenVmn S_θ(I_W^η

2)

−→ V impliesVmn S(I_W^η

2)

−→ V.

Proof. Let 0 < η ≤ 1 be given. Also, we assume that lim sup_sqs < ∞ and lim sup_tqt < ∞. Then, there exist M, N > 0 such that qs < M and qt < N for allsandt. Suppose thatV_mn^Sθ(I

η W2)

−→ V and let Tst:=

(m, n)∈Ist:|ρy(Vmn)−ρy(V)| ≥ξ .

SinceV_mn

S_θ(I_W^η

2)

−→ V, it holds for everyξ >0, δ >0 and eachy∈Y,

(s, t)∈N×N: 1 (hst)^η

(m, n)∈I_st:|ρy(V_mn)−ρ_y(V)| ≥ξ ≥δ

=

(s, t)∈N×N: Tst

(hst)^η ≥δ

∈ I2.

Hence, we can choose positive integerss0, t0∈Nsuch that Tst

(h_st)^η < δ for alls≥s0, t≥t0. Now let

H := max{T_st : 1≤s≤s₀,1≤t≤t₀}

and let iand j be integers satisfying j_s−1 < i≤j_s andk_t−1 < j≤k_t. Then, for everyξ >0 and eachy∈Y we have

1 (ij)^η

(m, n) : m≤i, n≤j, |ρy(Vmn)−ρy(V)| ≥ξ

≤ 1

(j_s−1k_t−1)^η

(m, n) : m≤js, n≤kt: |ρy(Vmn)−ρy(V)| ≥ξ

= 1

(j_s−1k_t−1)^η

T11+T12+T21+T22+· · ·+Ts₀t₀+· · ·+Tst

≤ s0t0

(j_s−1k_t−1)^η max

1≤m≤s0

1≤n≤t0

{Tmn}

!

+ 1

(j_s−1k_t−1)^η (

h^η_s

0(t0+1)

T_s0(t0+1) h^η_s

0(t₀+1)

+h^η_(s

0+1)t0

T_(s0+1)t0 h^η_(s

0+1)t0

+h^η_(s

0+1)(t0+1)

T_(s0+1)(t0+1) h^η_(s

0+1)(t0+1)

+· · ·+h^η_stT_st h^η_st

)

≤ s0t0H

(j_s−1k_t−1)^η + 1

(j_s−1k_t−1)^η sup

s>s0

t>t0

Tst

h^η_st

! _s,t X

m≥s0

n≥t0

h^η_mn

!

≤ s0t0H

(j_s−1k_t−1)^η + 1

(j_s−1k_t−1) sup

s>s₀ t>t₀

Tst

h^η_st

! _s,t X

m≥s₀ n≥t₀

hmn

!

≤ s₀t₀H

(js−1kt−1)^η +δ(j_s−j_s0)(k_t−k_t0) js−1kt−1

≤ s0t0H

(j_s−1k_t−1)^η +δ qsqt

≤ s0t0H

(j_s−1k_t−1)^η +δ M N.

Sincej_s−1, k_t−1→ ∞asi, j→ ∞, it follows that for eachy∈Y 1

(ij)^η

(m, n) : m≤i, n≤j, |ρy(Vmn)−ρy(V)| ≥ξ →0 and so for anyδ1>0,

(i, j)∈N×N: 1 (ij)^η

(m, n) : m≤i, n≤j, |ρy(V_mn)−ρ_y(V)| ≥ξ ≥δ₁

∈ I2.

Consequently,Vmn S(I^η_W

2)

−→ V.

Theorem 4.14. Let θ2={(js, kt)}be a double lacunary sequence. If 1<lim inf

s q_s^η≤lim sup

s

qs<∞ and 1<lim inf

t q_t^η≤lim sup

t

qt<∞,

thenVmn S_θ(I^η_W

2)

−→ V if and only if Vmn S(I_W^η

2)

−→ V.

Proof. This can be obtained from Theorem 4.12 and Theorem 4.13, immediately.

Theorem 4.15. Let 0< η≤1 andθ2={(js, kt)} be a double lacunary sequence.

If lim infsq_s^η>1 andlim inftq^η_t >1, thenC1[I_W^η

2]⊆Nθ[I_W^η

2].

Proof. Let 0 < η ≤ 1 be given. Also, we assume that lim inf_sq_s^η > 1 and lim inf_tq^η_t > 1. Then, there exist α > 0 and β > 0 such that q^η_s ≥ 1 +α and q^η_t ≥1 +β for allsandt, which implies that

(j_sk_t)^η

h^η_st ≤ (1 +α)(1 +β)

αβ and (j_s−1k_t−1)^η h^η_st ≤ 1

αβ.

Suppose thatV_mn^C1[I

η W2]

−→ V. For each y∈Y, we have 1

h^η_st X

(m,n)∈Ist

|ρy(Vmn)−ρy(V)| = 1 h^η_st

js,kt

X

r,u=1,1

|ρy(Vru)−ρy(V)|

− 1 h^η_st

js−1,kt−1

X

r,u=1,1

|ρy(Vru)−ρy(V)|

= (j_sk_t)^η h^η_st

1 (jskt)^η

j_s,k_t

X

r,u=1,1

|ρy(Vru)−ρy(V)|

!

−(j_s−1k_t−1)^η h^η_st

1 (j_s−1k_t−1)^η

js−1,kt−1

X

r,u=1,1

|ρy(Vru)−ρy(V)|

! .

SinceVmn C₁[I_W^η

2]

−→ V, then for eachy∈Y 1

(jskt)^η

j_s,k_t

X

r,u=1,1

|ρy(Vru)−ρy(V)|→^I2 0 and 1 (j_s−1k_t−1)^η

js−1,kt−1

X

r,u=1,1

|ρy(Vru)−ρy(V)|→^I2 0.

Thus, when the above equality is considered, for eachy∈Y we have 1

h^η_st X

(m,n)∈Ist

|ρy(Vmn)−ρy(V)|→^I2 0,

that is,Vmn N_θ[I_W^η

2]

−→ V. Consequently,C1[I_W^η

2]⊆Nθ[I_W^η

2].

Theorem 4.16. Let 0< η≤1 andθ2={(js, kt)} be a double lacunary sequence.

If lim sup_sqs<∞andlim sup_tqt<∞, then Nθ[I_W^η

2]⊆C1[I_W^η

2].

Proof. Let 0 < η ≤ 1 be given. Also, we assume that lim sup_sqs < ∞ and lim sup_tqt < ∞. Then, there exist M, N > 0 such that qs < M and qt < N for allsandt. Suppose thatVmn

Nθ[I^η_W

2]

−→ V. Then, for everyξ >0 and eachy∈Y we can findS, T >0 andH >0 such that

sup

m≥S n≥T

τmn< ξ and τmn< H for all m, n= 1,2, . . .

where

τst= 1 h^η_st

X

(m,n)∈Ist

|ρy(Vmn)−ρy(V)|.

Ifiandj are integers satisfyingjs−1< i≤js andkt−1< j≤ktwhere s > S and t > T, then for eachy∈Y we have

1 (ij)^η

i,j

X

m,n=1,1

|ρ_y(V_mn)−ρ_y(V)|

≤ 1

(j_s−1k_t−1)^η

j_s,k_t

X

m,n=1,1

|ρy(Vmn)−ρy(V)|

= 1

(js−1kt−1)^η X

I₁₁

|ρ_y(V_mn)−ρ_y(V)|+X

I₁₂

|ρ_y(V_mn)−ρ_y(V)|

+X

I₂₁

|ρ_y(V_mn)−ρ_y(V)|+X

I₂₂

|ρ_y(V_mn)−ρ_y(V)|

+· · ·+X

Ist

|ρy(Vmn)−ρy(V)|

!

= h^η₁₁

(j_s−1k_t−1)^η.τ11+ h^η₁₂

(j_s−1k_t−1)^η.τ12+ h^η₂₁

(j_s−1k_t−1)^η.τ21

+ h^η₂₂

(j_s−1k_t−1)^η.τ22+· · ·+ h^η_st

(j_s−1k_t−1)^η ·τst

≤

S,T

X

m,n=1,1

h_mn js−1kt−1

.τ_mn+

s,t

X

m=S+1 n=T+1

h_mn js−1kt−1

.τ_mn

≤ sup

m≥1 n≥1

τmn

! jSkT

j_s−1k_t−1 + sup

m≥S n≥T

τmn

!(js−jS)(kt−kT) j_s−1k_t−1

≤H jSkT

j_s−1k_t−1 +ξ M N.

Sincej_s−1, k_t−1→ ∞asi, j→ ∞, it follows that for eachy∈Y 1

(ij)^η

i,j

X

m,n=1,1

|ρ_y(V_mn)−ρ_y(V)|→^I2 0,

that is,Vmn C₁[I_W^η

2]

−→ V. Consequently,Nθ[I_W^η

2]⊆C1[I_W^η

2].

Theorem 4.17. Let 0< η≤1 andθ2={(js, kt)} be a double lacunary sequence.

If

1<lim inf

s q_s^η≤lim sup

s

qs<∞ and 1<lim inf

t q_t^η≤lim sup

t

qt<∞, thenNθ2[I_W^η

2] =C1[I_W^η

2].

Proof. This can be obtained from Theorem 4.15 and Theorem 4.16, immediately.

Acknowledgments. The authors would like to thank the anonymous referee for his/her comments that helped us improve this article.

References

[1] Y. Altın, R. C¸ olak, B. Torgut,I₂(u)-convergence of double sequences of order(α, β), Geor- gian Math. J.22 2(2015) 153-158.

[2] F. Ba¸sar,Summability Theory and its Applications, Bentham Science Publishers, ˙Istanbul, (2012).

[3] G. Beer,On convergence of closed sets in a metric space and distance functions, Bull. Aust.

Math. Soc.31(1985) 421–432.

[4] G. Beer,Wijsman convergence: A survey, Set-Valued Anal.2(1994) 77–94.

[5] R. C¸ olak,Statistical convergence of orderα, In: Modern Methods in Analysis and Its Appli- cations (pp. 121–129), Anamaya Publishers, New Delhi, (2010).

[6] R. C¸ olak, Y. Altın,Statistical convergence of double sequences of orderα, J. Funct. Spaces Appl.2013(Article ID 682823) (2013) 5 pages.

[7] P. Das, P. Kostyrko, W. Wilczy´nski, P. Malik,I and I^∗-convergence of double sequences, Math. Slovaca58 5(2008) 605–620.

[8] P. Das, E. Sava¸s, S.Kr. Ghosal, On generalizations of certain summability methods using ideals, Appl. Math. Lett.24 9(2011) 1509–1514.

[9] P. Das, E. Sava¸s,On I-statistical andI-lacunary statistical convergence of orderα, Bull.

Iranian Math. Soc.40 2(2014) 459–472.

[10] E. D¨undar, U. Ulusu, and N. Pancaroˇglu,StronglyI2-lacunary convergence andI2-lacunary Cauchy double sequences of sets, Aligarh Bull. Math.35 1-2(2016) 1-15.

[11] E. D¨undar, U. Ulusu, B. Aydın,I2-lacunary statistical convergence of double sequences of sets, Konuralp J. Math.5 1(2017) 1–10.

[12] M. Et, H. S¸eng¨ul,Some Ces`aro-type summability spaces of orderαand lacunary statistical convergence of orderα, Filomat28 8(2014) 1593–1602.

[13] A.D. Gadjiev, C. Orhan,Some approximation theorems via statistical convergence, Rocky Mountain J. Math.32 1(2001) 129–138.

[14] E. G¨ulle, U. Ulusu, Double Wijsman lacunary statistical convergence of order α, (under review).

[15] ¨O. Ki¸si, F. Nuray,New convergence definitions for sequences of sets, Abstr. Appl. Anal.2013 2013Article ID 852796 6 pages. doi:10.1155/2013/852796.

[16] P. Kostyrko, T. ˇSal´at, W. Wilczy´nski, I-convergence, Real Anal. Exchange 26 2 (2000) 669–686.

[17] M. Mursaleen, O.H.H. Edely, Statistical convergence of double sequences, J. Math. Anal.

Appl.288(2003) 223–231.

[18] M. Mursaleen, F. Ba¸sar, Sequence Spaces: Topics in Modern Summability Theory, CRC Press, Taylor and Francis Group, Series: Mathematics and Its Applications, Boca Raton- London-New York, (2020).

[19] F. Nuray, B.E. Rhoades,Statistical convergence of sequences of sets, Fasc. Math.49(2012) 87–99.

[20] F. Nuray, U. Ulusu, E. D¨undar,Ces`aro summability of double sequences of sets, Gen. Math.

Notes25 1(2014) 8–18.

[21] F. Nuray, U. Ulusu, E. D¨undar,Lacunary statistical convergence of double sequences of sets, Soft Computing20(2016) 2883–2888.

[22] F. Nuray, E. D¨undar, U. Ulusu,Wijsman statistical convergence of double sequences of sets, Iran. J. Math. Sci. Inform. (in press).

[23] N. Pancaroˇglu, F. Nuray,On invariant statistically convergence and lacunary invariant sta- tistical convergence of sequences of sets, Prog. Appl. Math.5 2(2013) 23-29.

[24] R.F. Patterson, E. Sava¸s,Lacunary statistical convergence of double sequences, Math. Com- mun.10(2005) 55–61.

[25] A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann.53(1900) 289–321.

[26] E. Sava¸s,Double almost statistical convergence of orderα, Adv. Difference Equ.2013 62 (2013) 9 pages.

[27] E. Sava¸s,Double almost lacunary statistical convergence of orderα, Adv. Difference Equ.

2013 254(2013) 10 pages.

[28] E. Sava¸s,OnI-lacunary statistical convergence of orderαfor sequences of sets, Filomat29 6(2015) 1223–1229.

[29] H. S¸eng¨ul, M. Et,On lacunary statistical convergence of order α, Acta Math. Sci. Ser. B (Engl. Ed.)34 2(2014) 473–482.

[30] H. S¸eng¨ul, M. Et, On I-lacunary statistical convergence of order α of sequences of sets, Filomat31 8(2017) 2403–2412.

[31] U. Ulusu, F. Nuray,Lacunary statistical convergence of sequences of sets, Prog. Appl. Math.

4 2(2012) 99–109.

[32] U. Ulusu, E. D¨undar,I-lacunary statistical convergence of sequences of sets, Filomat28 8 (2014) 1567–1574.

[33] U. Ulusu, E. D¨undar, E. G¨ulle,I₂-Ces`aro summability of double sequences of sets, Palest. J.

Math.9 1(2020) 561–568.

[34] U. Ulusu, E. G¨ulle,Some statistical convergence types of orderαfor double set sequences, Facta Univ. Ser. Math. Inform.35 3(2020) 595–603.

[35] R.A. Wijsman,Convergence of sequences of convex sets, cones and functions, Bull. Amer.

Math. Soc.70(1964) 186–188.

Uˇgur Ulusu

Sivas Cumhuriyet University, 58140 Sivas, Turkey E-mail address:[email protected]

Esra G¨ulle

Department of Mathematics, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey E-mail address:[email protected]