Asymptotically I_2-lacunary statistical equivalence of double sequences of sets

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Journal of Inequalities and Special Functions ISSN: 2217-4303, URL: http://ilirias.com/jiasf Volume 7 Issue 2(2016), Pages 44-56.

ASYMPTOTICALLY I2-LACUNARY STATISTICAL

EQUIVALENCE OF DOUBLE SEQUENCES OF SETS

U ˇGUR ULUSU, ERD˙INC¸ D ¨UNDAR

Abstract. In this paper, we introduce the concepts of Wijsman asymptoti-cally I2-statistical equivalence, Wijsman strongly asymptotically I2-lacunary

equivalence and Wijsman asymptotically I2-lacunary statistical equivalence of

double sequences of sets and investigate the relationship between them.

1. introduction, definitions and notations

Throughout the paper N denotes the set of all positive integers and R the set of all real numbers. The concept of convergence of a sequence of real numbers has been extended to statistical convergence independently by Fast [11] and Schoenberg [30]. This concept was extended to the double sequences by Mursaleen and Edely [19]. C¸ akan and Altay [6] presented multidimensional analogues of the results presented by Fridy and Orhan [13].

The idea of I-convergence was introduced by Kostyrko et al. [17] as a general-ization of statistical convergence which is based on the structure of the idea I of subset of the set of natural numbers. Recently, Das et al. [7] introduced new no-tions, namely I-statistical convergence and I-lacunary statistical convergence by using ideal. Das, Kostyrko, Wilczy´nski and Malik [8] introduced the concept of I-convergence of double sequences in a metric space and studied some properties of this convergence.

The concept of convergence of sequences of numbers has been extended by several authors to convergence of sequences of sets (see, [3–5, 34, 35]). Nuray and Rhoades [20] extended the notion of convergence of set sequences to statistical convergence and gave some basic theorems. Ulusu and Nuray [32] defined the Wijsman lacunary statistical convergence of sequence of sets and considered its relation with Wijsman statistical convergence, which was defined by Nuray and Rhoades. Ki¸si and Nuray [15] introduced a new convergence notion, for sequences of sets, which is called Wijsman I-convergence by using ideal. Recently, Ulusu and D¨undar [31] studied the concepts of Wijsman I-statistical convergence, Wijsman I-lacunary statistical convergence and Wijsman strongly I-lacunary convergence of sequences of sets.

2010 Mathematics Subject Classification. 34C41, 40A05, 40A35.

Key words and phrases. Statistical convergence, lacunary sequence, I2-convergence,

asymp-totically equivalence, double sequence of sets, Wijsman convergence. c

2016 Ilirias Publications, Prishtin¨e, Kosov¨e. Submitted January 5, 2016. Published June 27, 2016.

This study supported by Afyon Kocatepe University Scientific Research Coordination Unit with the project number 15.HIZ.DES.46.

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Nuray et al. [22] studied Wijsman statistical convergence, Hausdorff statisti-cal convergence and Wijsman statististatisti-cal Cauchy double sequences of sets and in-vestigate the relationship between them. Nuray et al. [21] studied the concepts of Wijsman I2, I2∗-convergence and Wijsman I2, I2∗-Cauchy double sequences of sets. D¨undar et al. [10] introduced the concepts of the Wijsman I2-statistical convergence, Wijsman I2-lacunary statistical convergence and Wijsman strongly I2-lacunary convergence of double sequences of sets.

Marouf [18] peresented definitions for asymptotically equivalent and asymptotic regular matrices. Patterson [26] extend these concepts by presenting an asymptoti-cally statistical equivalent analog of these definitions. Patterson and Sava¸s [27] ex-tend the definitions presented in [26] to lacunary sequences. In addition to these def-initions, natural inclusion theorems were presented. Recently, Sava¸s [28] presented the concept of I-asymptotically lacunary statistically equivalence which is a nat-ural combination of the definitions for asymptotically equivalence and I-lacunary statistical convergence.

The concept of asymptotically equivalence of sequences of real numbers which is defined by Marouf [18] has been extended by Ulusu and Nuray [33] to concept of Wijsman asymptotically equivalence of set sequences. In addition to these def-initions, natural inclusion theorems are presented. Ki¸si et al. [16] introduced the concept of Wijsman I-asymptotically equivalence of sequences of sets.

Now, we recall the basic definitions and concepts (See [1–3, 8–10, 12–14, 17, 18, 21–25, 29]).

Two nonnegative sequences x = (xk) and y = (yk) are said to be asymptotically equivalent if lim k xk yk = 1. It is denoted by x ∼ y.

Let (X, ρ) be a metric space. For any point x ∈ X and any non-empty subset A of X, we define the distance from x to A by

d(x, A) = inf

a∈Aρ(x, a).

Let (X, ρ) be a metric space and A, Ak be any non-empty closed subsets of X. The sequence {Ak} is Wijsman convergent to A if for each x ∈ X,

lim

k→∞d(x, Ak) = d(x, A). A family of sets I ⊆ 2N_{is called an ideal if and only if}

(i) ∅ ∈ I, (ii) For each A, B ∈ I we have A ∪ B ∈ I, (iii) For each A ∈ I and each B ⊆ A we have B ∈ I.

An ideal is called non-trivial if N /∈ I and non-trivial ideal is called admissible if {n} ∈ I for each n ∈ N.

Throughout the paper we take I2 as an admissible ideal in N × N.

A non-trivial ideal I2of N × N is called strongly admissible if {i} × N and N × {i} belongs to I2 for each i ∈ N .

It is evident that a strongly admissible ideal is admissible also. I0

2 = {A ⊂ N × N : (∃m(A) ∈ N)(i, j ≥ m(A) ⇒ (i, j) 6∈ A)}. Then I20 is a strongly admissible ideal and clearly an ideal I2 is strongly admissible if and only if I0

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The class of all A ⊂ N × N which has natural density zero denoted by I2f. Then I₂f is strongly admissible ideal.

A family of sets F ⊆ 2N_{is called a filter if and only if}

(i) ∅ /∈ F, (ii) For each A, B ∈ F we have A ∩ B ∈ F, (iii) For each A ∈ F and each B ⊇ A we have B ∈ F.

I is a non-trivial ideal in N if and only if F (I) = {M ⊂ N : (∃A ∈ I)(M = N\A)} is a filter in N.

An admissible ideal I2⊂ 2N×Nsatisfies the property (AP2) if for every countable family of mutually disjoint sets {A1, A2, ...} belonging to I2, there exists a countable family of sets {B1, B2, ...} such that Aj∆Bj ∈ I20, i.e., Aj∆Bj is included in the finite union of rows and columns in N × N for each j ∈ N and B =S∞

j=1Bj ∈ I2 (hence Bj∈ I2 for each j ∈ N).

A double sequence x = (xkj)k,j∈N of real numbers is said to be convergent to L ∈ R in Pringsheim’s sense if for any ε > 0, there exists Nε ∈ N such that |xkj− L| < ε, whenever k, j > Nε. In this case, we write

P − lim

k,j→∞xkj= L or k,j→∞lim xkj= L.

Throughout the paper, we let (X, ρ) be a separable metric space, I2⊆ 2N×N be a strongly admissible ideal and A, Akj be any non-empty closed subsets of X.

The double sequence {Akj} is Wijsman convergent to A if for each x ∈ X, P − lim

k,j→∞d(x, Akj) = d(x, A) or k,j→∞lim d(x, Akj) = d(x, A).

The double sequence {Akj} is Wijsman statistically convergent to A if for every ε > 0 and for each x ∈ X,

lim m,n→∞

1

mn|{k ≤ m, j ≤ n : |d(x, Akj) − d(x, A)| ≥ ε}| = 0, that is, |d(x, Akj) − d(x, A)| < ε for almost every (k, j).

By a lacunary sequence we mean an increasing integer sequence θ = {kr} such that k0= 0 and hr= kr− kr−1→ ∞ as r → ∞.

The double sequence θ = {(kr, js)} is called double lacunary sequence if there exist two increasing sequence of integers such that

k0= 0, hr= kr− kr−1→ ∞ as r → ∞ and

j0= 0, ¯hu= ju− ju−1→ ∞ as u → ∞. We use following notations in the sequel:

kru= krju, hru= hr¯hu, Iru= {(k, j) : kr−1< k ≤ kr and ju−1< j ≤ ju}, qr= kr kr−1 and qu= ju ju−1 .

Let θ be a double lacunary sequence. The double sequence {Akj} is Wijsman strongly lacunary convergent to A if for each x ∈ X,

lim r,u→∞ 1 hr¯hu kr X k=kr−1+1 ju X j=ju−1+1 |d(x, Akj) − d(x, A)| = 0.

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Let θ be a double lacunary sequence. The double sequence {Akj} is Wijsman lacunary statistically convergent to A, if for every ε > 0 and for each x ∈ X,

lim r,u→∞

1 hr¯hu

|{(k, j) ∈ Iru: |d(x, Akj) − d(x, A)| ≥ ε}| = 0.

The double sequence of sets {Akj} is Wijsman I2-convergent to A, if for every ε > 0 and for each x ∈ X,

{(k, j) ∈ N × N : |d(x, Akj) − d(x, A)| ≥ ε} ∈ I2. In this case we write IW2− lim

k,j→∞Akj= A. We define d(x; Akj, Bkj) as follows: d(x; Akj, Bkj) =        d(x, Akj) d(x, Bkj) , x 6∈ Akj∪ Bkj L , x ∈ Akj∪ Bkj.

The double sequences {Akj} and {Bkj} are Wijsman asymptotically equivalent of multiple L if for each x ∈ X, lim

k,j→∞d(x; Akj, Bkj) = L.

The double sequences {Akj} and {Bkj} are Wijsman asymptotically statistical equivalent of multiple L if for every ε > 0 and for each x ∈ X,

lim m,n→∞ 1 mn k ≤ m, j ≤ n : |d(x; Akj, Bkj) − L| ≥ ε = 0.

Let θ be a double lacunary sequence. The double sequences {Akj} and {Bkj} are Wijsman strongly asymptotically lacunary equivalent of multiple L if for each x ∈ X, lim r,u→∞ 1 hrhu X k,j∈Iru |d(x; Akj, Bkj) − L| = 0.

Let θ be a double lacunary sequence. The double sequences {Akj} and {Bkj} are Wijsman asymptotically lacunary statistical equivalent of multiple L if for every ε > 0 and each x ∈ X, lim r,u→∞ 1 hrhu (k, j) ∈ Iru: |d(x; Akj, Bkj) − L| ≥ ε = 0.

The sequence {Akj} is Wijsman I2-statistical convergent to A or S (IW2)-convergent

to A if for every ε > 0, δ > 0 and for each x ∈ X,

(m, n) ∈ N × N :_mn1 |{k ≤ m, j ≤ n : |d(x, Akj) − d(x, A)| ≥ ε}| ≥ δ

∈ I2. In this case, we write Akj→ A (S (IW2)) .

Let θ be a double lacunary sequence. The sequence {Akj} is said to be Wijsman strongly I2-lacunary convergent to A or Nθ[IW2]-convergent to A if for every ε > 0

and for each x ∈ X,    (r, u) ∈ N × N : 1 hrhu X (k,j)∈Iru |d(x, Akj) − d(x, A)| ≥ ε    ∈ I2. In this case, we write Akj→ A (Nθ[IW2]) .

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Let θ be a double lacunary sequence. The sequence {Akj} is Wijsman I2 -lacunary statistical convergent to A or Sθ(IW2)-convergent to A if for every ε > 0,

δ > 0 and for each x ∈ X, (r, u) ∈ N × N : 1 hrhu | {(k, j) ∈ Iru: |d(x, Akj) − d(x, A)| ≥ ε} | ≥ δ ∈ I2. In this case, we write Akj→ A (Sθ(IW2)) .

X ⊂ R, f, g : X → R functions and a point a ∈ X0are given. If f (x) = α(x)g(x) for ∀x ∈

o

Uδ(a) ∩ X, then for x ∈ X we write f = O(g) as x → a, where for any δ > 0, α : X → R is bounded function on

o

Uδ(a) ∩ X. In this case, if there exists a c ≥ 0 such that |f (x)| ≤ c|g(x)| for ∀x ∈

o

Uδ(a) ∩ X, then for x ∈ X, f = O(g) as x → a.

2. main results

In this section, we define the concepts of Wijsman asymptotically I2-statistical equivalence, Wijsman strongly asymptotically I2-lacunary equivalence and Wijs-man asymptotically I2-lacunary statistical equivalence of double sequences of sets and investigate the relationship between them.

Definition 2.1. The double sequences {Akj} and {Bkj} are Wijsman asymptoti-cally I2-equivalent of multiple L if for every ε > 0 and each x ∈ X

{(k, j) ∈ N × N : |d(x; Akj, Bkj) − L| ≥ ε} ∈ I2. In this case, we write Akj

IL W2

∼ Bkjand simply Wijsman asymptotically I2-equivalent if L = 1.

Definition 2.2. The double sequences {Akj} and {Bkj} are Wijsman asymptoti-cally I2-statistical equivalent of multiple L if for every ε > 0, δ > 0 and for each x ∈ X,

(m, n) ∈ N × N : _mn1 |{k ≤ m, j ≤ n : |d(x; Akj, Bkj) − L| ≥ ε}| ≥ δ

∈ I2.

In this case, we write Akj S(IL

W2)

∼ Bkj and simply Wijsman asymptotically I2 -statistical equivalent if L = 1. The set of Wijsman asymptotically I2-statistical equivalent double sequences will be denoted byS I_WL₂ .

For I2 = I f

2, Wijsman asymptotically I2-statistical equivalent of multiple L coincides with Wijsman asymptotically statistical equivalent of multiple L which is defined in [23].

As an example, consider the following double sequences; Akj=

{(x, y) ∈ R2_{: x}2_{+ y}2_{+ kjy = 0}} _, _{if k and j are a square integer,}

{(1, 1)} , otherwise.

and Bkj=

{(x, y) ∈ R2_{: x}2_{+ y}2_{− kjy = 0}} _, _{if k and j are a square integer,}

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If we take I2= I2f, since

(m, n) ∈ N × N : _mn1 |{k ≤ m, j ≤ n : |d(x; Akj, Bkj) − L| ≥ ε}| ≥ δ

∈ I₂f, then the double sequences {Akj} and {Bkj} are Wijsman asymptotically I2-statistical equivalent.

Definition 2.3. Let θ be a double lacunary sequence. The double sequences {Akj} and {Bkj} are Wijsman asymptotically I2-lacunary equivalent of multiple L if for every ε > 0 and for each x ∈ X,

   (r, u) ∈ N × N : 1 hrhu X (k,j)∈Iru d(x; Akj, Bkj) − L ≥ ε    ∈ I2.

In this case, we write Akj

Nθ(I_W2L )

∼ Bkj and simply Wijsman asymptotically I2 -lacunary equivalent if L = 1.

Definition 2.4. Let θ be a double lacunary sequence. The double sequences {Akj} and {Bkj} are said to be Wijsman strongly asymptotically I2-lacunary equivalent of multiple L if for every ε > 0 and for each x ∈ X,

   (r, u) ∈ N × N : 1 hrhu X (k,j)∈Iru |d(x; Akj, Bkj) − L| ≥ ε    ∈ I2.

Nθ[IL_W2]

∼ Bkjand simply Wijsman strongly asymptotically I2-lacunary equivalent if L = 1. The set of Wijsman strongly asymptotically I2 -lacunary equivalent double sequences will be denoted by NθIWL2 .

As an example, consider the following double sequences;

Akj:=    (x, y) ∈ R2_: (x − √ kj)2 kj + y2 2kj = 1 , if kr−1< k < kr−1+ [ √ hr], ju−1< j < ju−1+ [ p hu]. {(1, 1)} , otherwise. and Bkj:=    (x, y) ∈ R2: (x + √ kj)2 kj + y2 2kj = 1 , if kr−1< k < kr−1+ [ √ hr], ju−1< j < ju−1+ [ p hu]. {(1, 1)} , otherwise. If we take I2= I2f, since    (r, u) ∈ N × N : 1 hrhu X (k,j)∈Iru |d(x; Akj, Bkj) − L| ≥ ε    ∈ I₂f,

then the double sequences {Akj} and {Bkj} are Wijsman strongly asymptotically I2-lacunary equivalent.

Definition 2.5. Let θ be a double lacunary sequence. The double sequences {Akj} and {Bkj} are Wijsman asymptotically I2-lacunary statistical equivalent of multiple L if for every ε > 0, δ > 0 and for each x ∈ X,

(r, u) ∈ N × N : 1 hrhu | {(k, j) ∈ Iru: |d(x; Akj, Bkj) − L| ≥ ε} | ≥ δ ∈ I2.

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Sθ(I_W2L )

∼ Bkj and simply Wijsman asymptotically I2 -lacunary statistical equivalent if L = 1. The set of Wijsman asymptotically I2 -lacunary statistical equivalent double sequences will be denoted bySθ IWL2 .

For I2= I2f, Wijsman asymptotically I2-lacunary statistical equivalent of mul-tiple L coincides with Wijsman asymptotically lacunary statistical equivalent of multiple L which is defined in [23].

As an example, consider the following double sequences;

Akj:=        (x, y) ∈ R2_{: x}2_{+ (y − 1)}2₌ 1 kj , if kr−1 < k < kr−1+ [ √ hr], ju−1< j < ju−1+ [ p hu] and k is a square integer,

{(0, 0)} , otherwise. and Bkj:=        (x, y) ∈ R2_{: x}2_{+ (y + 1)}2₌ 1 kj , if kr−1< k < kr−1+ [ √ hr], ju−1< j < ju−1+ [ p hu] and k is a square integer,

{(0, 0)} , otherwise. If we take I2= I2f, since (r, u) ∈ N × N : 1 hrhu | {(k, j) ∈ Iru: |d(x; Akj, Bkj) − L| ≥ ε} | ≥ δ ∈ I₂f, then the sequences {Akj} and {Bkj} is Wijsman asymptotically I2-lacunary sta-tistical equivalent.

Theorem 2.6. Let θ be a double lacunary sequence. Then, Akj

Nθ[I_W2L ]

∼ Bkj⇒ Akj

Sθ(I_W2L )

∼ Bkj.

Proof. Suppose that {Akj} and {Bkj} is Wijsman strongly asymptotically I2 -lacunary equivalent of multiple L. Given ε > 0 and for each x ∈ X we can write

(r, u) ∈ N × N : 1 hrhu (k, j) ∈ Iru: |d(x; Akj, Bkj) − L| ≥ ε ≥ δ ⊆ ( (r, u) ∈ N × N : 1 hrhu P (k,j)∈Iru |d(x; Akj, Bkj) − L| ≥ ε · δ ) ∈ I2 and so Akj Sθ(IL_W2) ∼ Bkj.

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Theorem 2.7. Let θ be a double lacunary sequence and d(x, Akj)O d(x, Bkj). Then, Akj Sθ(IL_W2) ∼ Bkj⇒ Akj Nθ[I_W2L ] ∼ Bkj.

Proof. Suppose that {Akj} and {Bkj} is Wijsman asymptotically I2-lacunary sta-tistical equivalent of multiple L and d(x, Akj)O d(x, Bkj). Then, there exists an M > 0 such that

|d(x; Akj, Bkj) − L| ≤ M,

( (r, u) ∈ N × N : 1 hrhu P (k,j)∈Iru |d(x; Akj, Bkj) − L| ≥ ε ) ⊆ (r, u) ∈ N × N : 1 hrhu (k, j) ∈ Iru: |d(x; Akj, Bkj) − L| ≥ ε 2 ≥ ε 2M ∈ I2 and so Akj Nθ[I_W2L ] ∼ Bkj.

We have the following Theorem by Theorem 2.6 and Theorem 2.7.

Theorem 2.8. Let θ be a double lacunary sequence. If d(x, Akj)O d(x, Bkj), then Sθ IWL2 = NθI

L W2 .

Theorem 2.9. Let θ be a double lacunary sequence. If lim infrqr > 1 and lim infuqu> 1 then,

Akj S(IL W2) ∼ Bkj⇒ Akj Sθ(I_W2L ) ∼ Bkj.

Proof. Assume that lim infrqr > 1 and lim infuqu > 1, then there exist λ, µ > 0 such that

qr≥ 1 + λ and qu≥ 1 + µ for sufficiently large r, u which implies that

hrhu kru

≥ λµ

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If {Akj} and {Bkj} is Wijsman asymptotically I2-statistical equivalent of mul-tiple L, then for every ε > 0, for each x ∈ X and for sufficiently large r, u, we get 1 krju k ≤ kr, j ≤ ju: |d(x; Akj, Bkj) − L| ≥ ε ≥ 1 krju (k, j) ∈ Iru: |d(x; Akj, Bkj) − L| ≥ ε ≥ λµ (1 + λ)(1 + µ). ₁ hrhu (k, j) ∈ Iru: |d(x; Akj, Bkj) − L| ≥ ε . Hence, for each x ∈ X and for any δ > 0 we have

(r, u) : 1 hrhu (k, j) ∈ Iru: |d(x; Akj, Bkj) − L| ≥ ε ≥ δ ⊆ (r, u) : 1 krju k ≤ kr, j ≤ ju: |d(x; Akj, Bkj) − L| ≥ ε ≥ δλµ (1+λ)(1+µ) ∈ I2 and so Akj S_θL(I_W2) ∼ Bkj.

Theorem 2.10. Let θ be a double lacunary sequence. If lim suprqr < ∞ and lim sup_uqu< ∞, then

Akj Sθ(IL_W2) ∼ Bkj⇒ Akj S(IL W2) ∼ Bkj.

Proof. If lim suprqr < ∞ and lim supuqu < ∞, then there is an M, N > 0 such that qr< M and qu < N , for all r, u. Suppose that {Akj} and {Bkj} is Wijsman asymptotically I2-lacunary statistical equivalent of multiple L and let

Uru= U (r, u, x) := (k, j) ∈ Iru: |d(x; Akj, Bkj) − L| ≥ ε .

Since {Akj} and {Bkj} is Wijsman asymptotically I2-lacunary statistical equivalent of multiple L, it follows that for every ε > 0 and δ > 0, for each x ∈ X,

(r, u) ∈ N × N : 1 hrhu (k, j) ∈ Iru: |d(x; Akj, Bkj) − L| ≥ ε ≥ δ = (r, u) ∈ N × N : Uru hrhu ≥ δ ∈ I2. Hence, we can choose a positive integers r0, u0∈ N such that

Uru hrhu < δ, for all r > r0, u > u0. Now let K := maxUru: 1 ≤ r ≤ r0, 1 ≤ u ≤ u0

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Then, we have 1 tv k ≤ t, j ≤ v : |d(x; Akj, Bkj) − L| ≥ ε ≤ 1 kr−1ju−1 k ≤ kr, j ≤ ju: |d(x; Akj, Bkj) − L| ≥ ε = 1 kr−1ju−1 U11+ U12+ U21+ U22+ · · · + Ur0u0+ · · · + Uru ≤ K kr−1ju−1 · r0u0 + 1 kr−1ju−1 hr0hu0+1 Ur0,u0+1 hr0hu0+1 + hr0+1hu0 Ur0+1,u0 hr0+1hu0 + · · · + hrhu Uru hrhu ! ≤ r0u0· K kr−1ju−1 + 1 kr−1ju−1   sup r>r0 u>u0 Uru hrhu    hr0hu0+1+ hr0+1hu0+ · · · + hrhu ≤ r0u0· K kr−1ju−1 + ε ·(kr− kr0)(ju− ju0) kr−1ju−1 ≤ r0u0· K kr−1ju−1 + ε · qr· qu≤ r0u0· K kr−1ju−1 + ε · M · N. Since kr−1ju−1→ ∞ as t, v → ∞, it follows that

1 tv k ≤ t, j ≤ v : |d(x; Akj, Bkj) − L| ≥ ε → 0 and consequently, for any δ1> 0 the set

n (t, v) ∈ N × N : _tv1 k ≤ t, j ≤ v : |d(x; Akj, Bkj) − L| ≥ ε ≥ δ1 o ∈ I2. This shows that {Akj} and {Bkj} is Wijsman asymptotically I2-statistical

equiva-lent of multiple L.

We have the following Theorem by Theorem 2.9 and Theorem 2.10. Theorem 2.11. Let θ be a double lacunary sequence. If

1 < lim inf

r qr≤ lim sup_r qr< ∞ and 1 < lim infu qu≤ lim sup_u qu< ∞, then

Sθ(IWL2) = S(I L W2) .

Theorem 2.12. Let I2 ⊆ 2N×N be a strongly admissible ideal satisfying property (AP 2) and θ ∈ F (I2). If {Akj} , {Bkj} ∈S(IWL12) ∩ Sθ(I

L2

W2) , then L1= L2.

Proof. Assume that Akj

SL1(I_W2) ∼ Bkj, Akj S_θL2(I_W2) ∼ Bkj and L16= L2. Let 0 < ε < 1 2|L1− L2|.

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Since I2satisfies the property (AP 2), there exists M ∈ F (I2) (i.e., N × N\M ∈ I2) such that for each x ∈ X and for (m, n) ∈ M,

lim m,n→∞ 1 mn k ≤ m, j ≤ n : |d(x; Akj, Bkj) − L1| ≥ ε = 0. Let the sets

P = {k ≤ m, j ≤ n : |d(x; Akj, Bkj) − L1|)| ≥ ε|} and

R = {k ≤ m, j ≤ n : |d(x; Akj, Bkj) − L2| ≥ ε} . Then, mn = |P ∪ R| ≤ |P | + |R|. This implies that

1 ≤ |P | mn+ |R| mn. Since |R| mn ≤ 1 and m,n→∞lim |P | mn = 0, so we must have lim m,n→∞ |R| mn = 1.

Let M∗= M ∩ θ ∈ F (I2). Then, for (kl, jt) ∈ M∗the kljtth term of the statistical limit expression 1 mn k ≤ m, j ≤ n : |d(x; Akj, Bkj) − L2| ≥ ε is 1 kljt ( (k, j) ∈ l,t [ r,u=1,1 Iru: |d(x; Akj, Bkj) − L2| ≥ ε ) = 1 l,t P r,u=1,1 hrhu l,t X r,u=1,1 vruhrhu, (2.1) where vru= 1 hrhu (k, j) ∈ Iru: |d(x; Akj, Bkj) − L2| ≥ ε I2 → 0

because {Akj} and {Bkj} is Wijsman asymptotically I2-lacunary statistical equiva-lent of multiple L2. Since θ is a double lacunary sequence, (2.1) is a regular weighted mean transform of vru’s and therefore it is also I2-convergent to 0 as l, t → ∞, and so it has a subsequence which is convergent to 0 since I2 satisfies property (AP 2). But since this is a subsequence of

1 mn k ≤ m, j ≤ n : |d(x; Akj, Bkj) − L2| ≥ ε (m,n)∈M , we infer that ₁ mn k ≤ m, j ≤ n : |d(x; Akj, Bkj) − L2| ≥ ε (m,n)∈M

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Uˇgur Ulusu

department of mathematics, faculty of science and literature, afyon kocatepe univer-sity, afyonkarahisar, turkey

E-mail address: ulusu@aku.edu.tr Erdinc¸ D¨undar

department of mathematics, faculty of science and literature, afyon kocatepe univer-sity, afyonkarahisar, turkey