International Journal of Analysis and Applications

Volume 16, Number 3 (2018), 317-327

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-317

LACUNARY I2-INVARIANT CONVERGENCE AND SOME PROPERTIES

UǦUR ULUSU, ERDİNÇ DÜNDAR∗ AND FATİH NURAY

Department of Mathematics, Faculty of Science and Literature, Afyon Kocatepe University, 03200,

Afyonkarahisar, Turkey

∗Corresponding author: edundar@aku.edu.tr

Abstract. In this paper, the concept of lacunary invariant uniform density of any subset A of the set

N×N is defined. Associate with this, the concept of lacunary I2-invariant convergence for double sequences

is given. Also, we examine relationships between this new type convergence concept and the concepts of

lacunary invariant convergence and p-strongly lacunary invariant convergence of double sequences. Finally,

introducing lacunary I∗2 -invariant convergence concept and lacunary I2-invariant Cauchy concepts, we give

the relationships among these concepts and relationships with lacunary I2-invariant convergence concept.

1. Introduction

Several authors have studied invariant convergent sequences (see, [8–10, 13, 15–17, 19]).

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 following

conditions:

(1) φ(x) ≥ 0, when the sequence x = (xn) has xn ≥ 0 for all n,

(2) φ(e) = 1, where e = (1, 1, 1, ...) and

(3) φ(xσ(n)) = φ(xn) for all x ∈ `∞.

The mappings σ are assumed to be one-to-one and such that σm(n) 6= n for all positive integers n and

m, where σm(n) denotes the m th iterate of the mapping σ at n. Thus, φ extends the limit functional on c,

2010 Mathematics Subject Classification. 40A05, 40A35.

Key words and phrases. double sequence; I-convergence; lacunary sequence; invariant convergence; I-Cauchy sequence.
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.

317

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-317


Int. J. Anal. Appl. 16 (3) (2018) 318

the space of convergent sequences, in the sense that φ(x) = lim x for all x ∈ c.

In the case σ is translation mappings σ(n) = n + 1, the σ-mean is often called a Banach limit.

By a lacunary sequence we mean an increasing integer sequence θ = {kr} such that k0 = 0 and hr =

kr −kr−1 →∞ as r →∞. The intervals determined by θ is denoted by Ir = (kr−1,kr] (see, [4]).

The concept of lacunary strongly σ-convergence was introduced by Savaş [17] as below:

Lθ =

{
x = (xk) : lim

r→∞

1

hr

∑
k∈Ir

|xσk(m) −L| = 0, uniformly in m

}
.

Pancaroǧlu and Nuray [13] defined the concept of lacunary invariant summability and the space [Vσθ]q as

follows:

A sequence x = (xk) is said to be lacunary invariant summable to L if

lim
r→∞

1

hr

∑
m∈Ir

xσm(n) = L,

uniformly in n.

A sequence x = (xk) is said to be strongly lacunary q-invariant convergent (0 < q < ∞) to L if

lim
r→∞

1

hr

∑
m∈Ir

|xσm(n) −L|q = 0,

uniformly in n and it is denoted by xk → L
(
[Vσθ]q

)
.

The idea of I-convergence was introduced by Kostyrko et al. [5] as a generalization of statistical conver-

gence which is based on the structure of the ideal I of subset of the set of natural numbers N.

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.

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.

For any ideal there is a filter F(I) corresponding with I, given by

F(I) =
{
M ⊂ N : (∃A ∈I)(M = N\A)

}
.

Recently, the concepts of lacunary σ-uniform density of the set A ⊆ N, lacunary Iσ-convergence, lacunary

I∗σ-convergence, lacunary Iσ-Cauchy and I∗σ-Cauchy sequences of real numbers were defined by Ulusu and



Int. J. Anal. Appl. 16 (3) (2018) 319

Nuray [20] and similar concepts can be seen in [12].

Let θ = {kr} be a lacunary sequence, A ⊆ N and

sr := min
n

{∣∣A∩{σm(n) : m ∈ Ir} ∣∣}
and

Sr := max
n

{∣∣A∩{σm(n) : m ∈ Ir}∣∣}.
If the following limits exist

Vθ(A) := lim
r→∞

sr
hr
, Vθ(A) := lim

r→∞

Sr
hr

then they are called a lower lacunary σ-uniform (lower σθ-uniform) density and an upper lacunary σ-uniform

(upper σθ-uniform) density of the set A, respectively. If Vθ(A) = Vθ(A), then

Vθ(A) = Vθ(A) = Vθ(A) is called the lacunary σ-uniform density or σθ-uniform density of A.

Denote by Iσθ the class of all A ⊆ N with Vθ(A) = 0.

Let Iσθ ⊂ 2N be an admissible ideal. A sequence (xk) is said to be lacunary Iσ-convergent or Iσθ-

convergent to the number L if for every ε > 0

Aε :=
{
k : |xk −L| ≥ ε

}
belongs to Iσθ; i.e., Vθ(Aε) = 0. In this case we write Iσθ − lim xk = L.

The set of all Iσθ-convergent sequences will be denoted by Iσθ.

Let Iσθ ⊂ 2N be an admissible ideal. A sequence x = (xk) is said to be I∗σθ-convergent to the number

L if there exists a set M = {m1 < m2 < ...} ∈ F(Iσθ) such that lim
k→∞

xmk = L. In this case we write

I∗σθ − lim xk = L.

A sequence (xk) is said to be lacunary Iσ-Cauchy sequence or Iσθ-Cauchy sequence if for every ε > 0,

there exists a number N = N(ε) ∈ N such that

A(ε) =
{
k : |xk −xN| ≥ ε

}
belongs to Iσθ; i.e., Vθ

(
A(ε)

)
= 0.

A sequence x = (xk) is said to be I∗σθ-Cauchy sequences if there exists a set M = {m1 < m2 < ... < mk <

...}∈F(Iσθ) such that

lim
k,p→∞

|xmk −xmp| = 0.



Int. J. Anal. Appl. 16 (3) (2018) 320

Convergence and I-convergence of double sequences in a metric space and some properties of this conver-

gence, and similar concepts which are noted following can be seen in [1, 2, 6, 7, 14, 18].

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 lim
k,j→∞

xkj = L.

A double sequence x = (xkj) is said to be bounded if supk,j xkj < ∞. The set of all bounded double

sequences of sets will be denoted by `2∞.

A nontrivial ideal I2 of N × N is called strongly admissible ideal if {i}× N and N ×{i} belong to I2 for

each i ∈ N.

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

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

I02 =
{
A ⊂ N × N : (∃m(A) ∈ N)(i,j ≥ m(A) ⇒ (i,j) 6∈ A)

}
. Then I02 is a strongly admissible ideal and

clearly an ideal I2 is strongly admissible if and only if I02 ⊂I2.

An admissible ideal I2 ⊂ 2N×N satisfies the property (AP2) if for every countable family of mutually

disjoint sets {E1,E2, ...} belonging to I2, there exists a countable family of sets {F1,F2, ...} such that

Ej∆Fj ∈ I02 , i.e., Ej∆Fj is included in the finite union of rows and columns in N × N for each j ∈ N and

F =
⋃∞
j=1 Fj ∈I2 (hence Fj ∈I2 for each j ∈ N).

Let (X,ρ) be a metric space. A sequence x = (xmn) in X is said to be I2-convergent to L ∈ X, if for any

ε > 0

A(ε) =
{

(m,n) ∈ N×N : ρ(xmn,L) ≥ ε
}
∈I2.

In this case, we write I2 − lim
m,n→∞

xmn = L.

The double sequence θ = {(kr,ju)} is called double lacunary sequence if there exist two increasing sequence

of integers such that

k0 = 0, hr = kr −kr−1 →∞ and j0 = 0, h̄u = ju − ju−1 →∞ as r,u →∞.

We use the following notations in the sequel:

kru = krju, hru = hrh̄u, Iru = {(k,j) : kr−1 < k ≤ kr and ju−1 < j ≤ ju},

qr =
kr
kr−1

and qu =
ju
ju−1

.



Int. J. Anal. Appl. 16 (3) (2018) 321

Also, the idea of I2-invariant convergence concepts and I2-invariant Cauchy concepts of double sequences

were defined by Dündar and Ulusu (see [3]).

2. Lacunary I2-Invariant Convergence

In this section, firstly, the concepts of lacunary invariant convergence of double sequence and lacunary

invariant uniform density of any subset A of the set N×N are defined. Associate with this uniform density, the

concept of lacunary I2-invariant convergence for double sequences is given. Also, we examine relationships

between this new type convergence concept and the concepts of lacunary invariant convergence, p-strongly

lacunary invariant convergence for double sequences.

Definition 2.1. A double sequence x = (xkj) is said to be lacunary invariant convergent to L if

lim
r,u→∞

1

hru

∑
k,j∈Iru

xσk(m),σj(n) = L,

uniformly in m,n and it is denoted by xkj → L
(
V σθ2

)
.

Definition 2.2. Let θ = {(kr,ju)} be a double lacunary sequence, A ⊆ N×N and

sru := min
m,n

∣∣∣A∩{(σk(m),σj(n)) : (k,j) ∈ Iru}∣∣∣
and

Sru := max
m,n

∣∣∣A∩{(σk(m),σj(n)) : (k,j) ∈ Iru}∣∣∣.
If the following limits exist

V θ2 (A) := lim
r,u→∞

sru
hru

, V θ2 (A) := lim
r,u→∞

Sru
hru

,

then they are called a lower lacunary σ-uniform density and an upper lacunary σ-uniform density of the set

A, respectively. If V θ2 (A) = V
θ
2 (A), then V

θ
2 (A) = V

θ
2 (A) = V

θ
2 (A) is called the lacunary σ-uniform density

of A.

Denote by Iσθ2 the class of all A ⊆ N×N with V θ2 (A) = 0.

Throughout the paper we take Iσθ2 as a strongly admissible ideal in N×N.

Definition 2.3. A double sequence x = (xkj) is said to be lacunary I2-invariant convergent or

Iσθ2 -convergent to the L if for every ε > 0, the set

Aε :=
{

(k,j) ∈ Iru : |xkj −L| ≥ ε
}

belongs to Iσθ2 ; i.e., V θ2 (Aε) = 0. In this case, we write

Iσθ2 − lim xkj = L or xkj → L
(
Iσθ2
)
.



Int. J. Anal. Appl. 16 (3) (2018) 322

The set of all Iσθ2 -convergent sequences will be denoted by Iσθ2 .

Theorem 2.1. If Iσθ2 − lim xkj = L1 and Iσθ2 − lim ykj = L2, then

(i) Iσθ2 − lim(xkj + ykj) = L1 + L2

(ii) Iσθ2 − lim αxkj = αL1 (α is a constant).

Proof. The proof is clear so we omit it. �

Theorem 2.2. Suppose that x = (xkj) is a bounded double sequence. If (xkj) is lacunary I2-invariant

convergent to L, then (xkj) is lacunary invariant convergent to L.

Proof. Let θ = {(kr,ju)} be a double lacunary sequence, m,n ∈ N be an arbitrary and ε > 0. Now, we

calculate

t(k,j,r,u) :=

∣∣∣∣∣∣ 1hru
∑

k,j∈Iru

xσk(m),σj(n) −L

∣∣∣∣∣∣ .
We have

t(k,j,r,u) ≤ t(1)(k,j,r,u) + t(2)(k,j,r,u),

where

t(1)(k,j,r,u) :=
1

hru

∑
k,j∈Iru

|x
σk(m),σj(n)

−L|≥ε

|xσk(m),σj(n) −L|

and

t(2)(k,j,r,u) :=
1

hru

∑
k,j∈Iru

|x
σk(m),σj(n)

−L|<ε

|xσk(m),σj(n) −L|.

We get t(2)(k,j,r,u) < ε, for every m,n = 1, 2, . . . . The boundedness of x = (xkj) implies that there exists

a K > 0 such that

|xσk(m),σj(n) −L| ≤ K, ((k,j) ∈ Iru; m,n = 1, 2, ...).

Then, this implies that

t(1)(k,j,r,u) ≤
K

hru

∣∣∣{(k,j) ∈ Iru : |xσk(m),σj(n) −L| ≥ ε}∣∣∣

≤ K
max
m,n

∣∣∣{(k,j) ∈ Iru : |xσk(m),σj(n) −L| ≥ ε}∣∣∣
hru

= K
Sru
hru

,

hence (xkj) is lacunary invariant summable to L. �



Int. J. Anal. Appl. 16 (3) (2018) 323

The converse of Theorem 2.2 does not hold. For example, x = (xkj) is the double sequence defined by

following;

xkj :=




1 , if
kr−1 < k < kr−1 + [

√
hr],

jr−1 < j < jr−1 + [
√
h̄u],

and k + j is an even integer.

0 , if
kr−1 < k < kr−1 + [

√
hr],

jr−1 < j < jr−1 + [
√
h̄u],

and k + j is an odd integer.

When σ(m) = m + 1 and σ(n) = n + 1, this sequence is lacunary invariant convergent to
1

2
but it is not

lacunary I2-invariant convergent.

In [20], Ulusu and Nuray gave some inclusion relations between [Vσθ]q-convergence and lacunary I-

invariant convergence, and showed that these are equivalent for bounded sequences. Now, we shall give

analogous theorems which states inclusion relations between [V σθ2 ]p-convergence and lacunary I2-invariant

convergence, and show that these are equivalent for bounded double sequences.

Definition 2.4. A double sequence x = (xkj) is said to be strongly lacunary invariant convergent to L if

lim
r,u→∞

1

hru

∑
k,j∈Iru

|xσk(m),σj(n) −L|,

uniformly in m,n and it is denoted by xkj → L
(
[V σθ2 ]

)
.

Definition 2.5. A double sequence x = (xkj) is said to be p-strongly lacunary invariant convergent (0 <

p < ∞) to L if

lim
r,u→∞

1

hru

∑
k,j∈Iru

|xσk(m),σj(n) −L|
p = 0,

uniformly in m,n and it is denoted by xkj → L
(
[V σθ2 ]p

)
.

Theorem 2.3. If a double sequence x = (xkj) is p-strongly lacunary invariant convergent to L, then (xkj)

is lacunary I2-invariant convergent to L.



Int. J. Anal. Appl. 16 (3) (2018) 324

Proof. Assume that xkj → L
(
[V σθ2 ]p

)
and given ε > 0. Then, for every double lacunary sequence θ =

{(kr,ju)} and for every m,n ∈ N, we have∑
k,j∈Iru

∣∣xσk(m),σj(n) −L∣∣p ≥ ∑
(k,j)∈Iru

|x
σk(m),σj(n)

−L|≥ε

|xσk(m),σj(n) −L|
p

≥ εp
∣∣{(k,j) ∈ Iru : |xσk(m),σj(n) −L| ≥ ε}∣∣

≥ εp max
m,n

∣∣{(k,j) ∈ Iru : |xσk(m),σj(n) −L| ≥ ε}∣∣
and

1

hru

∑
k,j∈Iru

∣∣xσk(m),σj(n) −L∣∣p ≥ εp maxm,n
∣∣{(k,j) ∈ Iru : |xσk(m),σj(n) −L| ≥ ε}∣∣

hru

= εp
Sru
hru

.

This implies

lim
r,u→∞

Sru
hru

= 0

and so (xkj) is Iσθ2 -convergent to L. �

Theorem 2.4. If a double sequence x = (xkj) ∈ `2∞ and (xkj) is lacunary I2-invariant convergent to L,

then (xkj) is p-strongly lacunary invariant convergent to L (0 < p < ∞).

Proof. Suppose that x = (xkj) ∈ `2∞ and xkj → L
(
Iσθ2
)
. Let 0 < p < ∞ and ε > 0. By assumption we have

V θ2
(
Aε
)

= 0. The boundedness of (xkj) implies that there exists K > 0 such that

|xσk(m),σj(n) −L| ≤ K, ((k,j) ∈ Ir,u; m,n = 1, 2, . . . ).

Observe that, for every m,n ∈ N we have

1

hru

∑
k,j∈Iru

∣∣xσk(m),σj(n) −L∣∣p = 1
hru

∑
k,j∈Iru

|x
σk(m),σj(n)

−L|≥ε

|xσk(m),σj(n) −L|
p

+
1

hru

∑
k,j∈Iru

|x
σk(m),σj(n)

−L|<ε

|xσk(m),σj(n) −L|
p

≤ K
max
m,n

∣∣{(k,j) ∈ Iru : |xσk(m),σj(n) −L| ≥ ε}∣∣
hru

+ εp

≤ K
Sru
hru

+ εp.



Int. J. Anal. Appl. 16 (3) (2018) 325

Hence, we obtain

lim
r,u→∞

1

hru

∑
k,j∈Iru

∣∣xσk(m),σj(n) −L∣∣p = 0,
uniformly in m,n. �

Theorem 2.5. A double sequence x = (xkj) ∈ `2∞ and (xkj) is lacunary I2-invariant convergent to L if and

only if (xkj) is p-strongly lacunary invariant convergent to L (0 < p < ∞.)

Proof. This is an immediate consequence of Theorem 2.3 and Theorem 2.4. �

Now, introducing lacunary I∗2 -invariant convergence concept, lacunary Iσ2 -Cauchy double sequence and

Iσθ2∗ -Cauchy double sequence concepts, we give the relationships among these concepts and relationships with

lacunary I2-invariant convergence concept.

Definition 2.6. A double sequence x = (xkj) is lacunary I∗2 -invariant convergent or Iσθ2∗ -convergent to L if

and only if there exists a set M2 ∈F(Iσθ2 ) (N×N\M2 = H ∈Iσθ2 ) such that

lim
k,j→∞

(k,j)∈M2

xkj = L. (2.1)

In this case, we write Iσθ2∗ − lim xkj = L or xkj → L
(
Iσθ2∗
)
.

Theorem 2.6. If a double sequence x = (xkj) is lacunary I∗2 -invariant convergent to L, then this sequence

is lacunary I2-invariant convergent to L.

Proof. Since Iσθ2∗ − lim
k,j→∞

xkj = L, there exists a set M2 ∈F(Iσθ2 ) (N×N\M2 = H ∈Iσθ2 ) such that

lim
k,j→∞

(k,j)∈M2

xkj = L.

Given ε > 0. By (2.1), there exists k0,j0 ∈ N such that |xkj −L| < ε, for all (k,j) ∈ M2 and k ≥ k0,j ≥ j0.

Hence, for every ε > 0, we have

T(ε) =
{

(k,j) ∈ N×N : |xkj −L| ≥ ε
}

⊂ H ∪
(
M2 ∩

(
({1, 2, ..., (k0 − 1)}×N) ∪ (N×{1, 2, ..., (k0 − 1)})

))
.

Since Iσθ2 ⊂ 2N×N is a strongly admissible ideal,

H ∪
(
M2 ∩

(
({1, 2, ..., (k0 − 1)}×N) ∪ (N×{1, 2, ..., (k0 − 1)})

))
∈Iσθ2 ,

so we have T(ε) ∈Iσθ2 that is V θ2
(
T(ε)

)
= 0. Hence, Iσθ2 − lim

k,j→∞
xkj = L. �

The converse of Theorem 2.6, which it’s proof is similar to the proof of Theorems in [1–3], holds if Iσθ2
has property (AP2).



Int. J. Anal. Appl. 16 (3) (2018) 326

Theorem 2.7. Let Iσθ2 has property (AP2). If a double sequence x = (xkj) is lacunary I2-invariant

convergent to L, then this sequence is lacunary I∗2 -invariant convergent to L.

Finally, we define the concepts of lacunary I2-invariant Cauchy and lacunary I∗2 -invariant Cauchy double

sequences.

Definition 2.7. A double sequence (xkj) is said to be lacunary I2-invariant Cauchy sequence or Iσθ2 -Cauchy

sequence, if for every ε > 0, there exist numbers s = s(ε), t = t(ε) ∈ N such that

A(ε) =
{

(k,j), (s,t) ∈ Iru : |xkj −xst| ≥ ε
}
∈Iσθ2 ,

that is, V θ2
(
A(ε)

)
= 0.

Definition 2.8. A double sequence (xkj) is lacunary I∗2 -invariant Cauchy sequence or Iσθ2∗ -Cauchy sequence

if there exists a set M2 ∈ F(Iσθ2 ) (i.e., N × N\M2 = H ∈ Iσθ2 ) such that for every

(k,j), (s,t) ∈ M2

lim
k,j,s,t→∞

|xkj −xst| = 0.

The proof of the following theorems are similar to the proof of Theorems in [2, 3, 11], so we omit them.

Theorem 2.8. If a double sequence x = (xkj) is Iσθ2 -convergent, then (xkj) is an Iσθ2 -Cauchy sequence.

Theorem 2.9. If a double sequence x = (xkj) is Iσθ2∗ -Cauchy sequence, then (xkj) is Iσθ2 -Cauchy sequence.

Theorem 2.10. Let Iσθ2 has property (AP2). If a double sequence x = (xkj) is Iσθ2 -Cauchy sequence then,

(xkj) is Iσθ2∗ -Cauchy sequence.

Acknowledgements: This study supported by Afyon Kocatepe University Scientific Research Coordination

Unit with the project number 17.Kariyer.21.

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	1. Introduction
	2. Lacunary I2-Invariant Convergence
	References