Filomat 30:5 (2016), 1273–1281
DOI 10.2298/FIL1605273D
Published by Faculty of Sciences and Mathematics,
University of Niš, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
I2 –Uniform Convergence of Double Sequences of Functions
Erdinç Dündara , Bilal Altayb
a Afyon
Kocatepe Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Afyonkarahisar-03200/Türkiye
b İnönü Üniversitesi, Eğitim Fakültesi, Malatya-44280/Türkiye
Abstract. In this work, we discuss various kinds of I2 -uniform convergence for double sequences of
functions and introduce the concepts of I2 and I∗2 -uniform convergence, I2 -uniformly Cauchy sequences
for double sequences of functions. Then, we show the relation between them.
1. Background and Introduction
The concept of convergence of a sequence of real numbers has been extended to statistical convergence
independently by Fast [11] and Schoenberg [28]. This concept was extended to the double sequences by
Mursaleen and Edely [21]. A lot of development have been made in this area after the works of Šalát
[27] and Fridy [13, 14]. Furthermore, Gökhan et al. [16] introduced the notion of pointwise and uniform
statistical convergent of double sequences of real-valued functions. In general, statistically convergent
sequences satisfy many of the properties of ordinary convergent sequences in metric spaces [11, 13, 14, 25].
Çakan and Altay [4] presented multidimensional analogues of the results presented by Fridy and Orhan
[12].
Throughout the paper N denotes the set of all positive integers and R the set of all real numbers. The
idea of I-convergence was introduced by Kostyrko et al. [18] as a generalization of statistical convergence
which is based on the structure of the ideal I of subset of the set of natural numbers. Nuray and Ruckle
[23] indepedently introduced the same with another name generalized statistical convergence. Das et
al. [5] introduced the concept of I-convergence of double sequences in a metric space and studied some
properties of this convergence. Balcerzak et al. [3] discussed various kinds of statistical convergence
and I-convergence for sequences of functions with values in R or in a metric space. Gezer and Karakuş
[15] investigated I-pointwise and I-uniform convergence and I∗ -pointwise and I∗ -uniform convergence
of function sequences and examined the relation between them. Dündar and Altay [8] investigated the
relation between I2 -convergence and I∗2 -convergence of double sequences of functions defined between
linear metric spaces. Some results on I-convergence may be found in [2, 6, 19, 20, 22, 29].
In this work, we discuss various kinds of uniformly ideal convergence for double sequences of functions
with values in R or in a metric space. We introduce the concepts of I2 , I∗2 -uniform convergence, I2 uniformly Cauchy sequences for double sequences of functions and show the relation between them.
2010 Mathematics Subject Classification. Primary 40A30; Secondary 40A35
Keywords. Ideal, Uniform Convergence, I-Convergence, Double Sequences of Functions.
Received: 04 April 2014; Accepted: 27 April 2014
Communicated by Eberhard Malkowsky
Email addresses: [email protected], [email protected] (Erdinç Dündar), [email protected] (Bilal Altay)
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2. Definitions and Notations
Now, we recall that the definitions of concepts of ideal convergence, ideal Cauchy sequences and basic
concepts. (See [1, 5, 9, 11, 16, 18, 21, 24, 26]).
A double sequence x = (xmn )m,n∈N of real numbers is said to be convergent to L ∈ R if for any ε > 0 ,
there exists Nε ∈ N such that
|xmn − L| < ε,
whenever m, n > Nε . In this case we write
lim xmn = L.
m,n→∞
A double sequence x = (xmn )m,n∈N of real numbers is said to be bounded if there exists a positive real
number M such that |xmn | < M, for all m, n ∈ N. That is
kxk∞ = sup |xmn | < ∞.
m,n
Let K ⊂ N × N. Let Kmn be the number of (j, k) ∈ K such that j ≤ m, k ≤ n. If the sequence
limit in Pringsheim’s sense then we say that K has double natural density and is denoted by
d2 (K) = lim
m,n→∞
n
Kmn
m.n
o
has a
Kmn
.
m.n
A double sequence x = (xmn )m,n∈N of real numbers is said to be statistically convergent to L ∈ R, if for
any ε > 0 we have d2 (A(ε)) = 0, where A(ε) = {(m, n) ∈ N × N : |xmn − L| ≥ ε}.
A double sequence of functions { fmn } is said to be pointwise convergent to f on a set S ⊂ R, if for each
point x ∈ S and for each ε > 0, there exists a positive integer N = N(x, ε) such that
| fmn (x) − f (x)| < ε,
for all m, n > N. In this case we write
lim fmn (x) = f (x) or fmn → f
m,n→∞
on S.
A double sequence of functions { fmn } is said to be uniformly convergent to f on a set S ⊂ R, if for each
ε > 0, there exists a positive integer N = N(ε) such that m, n > N implies
| fmn (x) − f (x)| < ε, for all x ∈ S.
In this case we write
fmn ⇒ f
on S.
A double sequence of functions { fmn } is said to be pointwise statistically convergent to f on a set S ⊂ R,
if for every ε > 0,
1 {(i, j), i ≤ m and j ≤ n : | fi j (x) − f (x)| ≥ ε} = 0,
lim
m,n→∞ mn
for each (fixed) x ∈ S, i.e., for each (fixed) x ∈ S,
| fi j (x) − f (x)| < ε, a.a. (i, j).
In this case we write
st − lim fmn (x) = f (x) or fmn →st f
m,n→∞
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on S.
A double sequence of functions { fmn } is said to be uniformly statistically convergent to f on a set S ⊂ R,
if for every ε > 0,
1 {(i, j), i ≤ m and j ≤ n : | fi j (x) − f (x)| ≥ ε} = 0, for all x ∈ S
m,n→∞ mn
lim
i.e., for all x ∈ S,
In this case we write
| fi j (x) − f (x)| < ε, a.a. (i, j).
st − lim fmn (x) = f (x) uniformly on S or fmn ⇒st f
m,n→∞
on S.
Let X , ∅. A class I of subsets of X is said to be an ideal in X provided:
i) ∅ ∈ I, ii) A, B ∈ I implies A ∪ B ∈ I, iii) A ∈ I, B ⊂ A implies B ∈ I.
I is called a nontrivial ideal if X < I.
Let X , ∅. A non empty class F of subsets of X is said to be a filter in X provided:
i) ∅ < F , ii) A, B ∈ F implies A ∩ B ∈ F , iii) A ∈ F , A ⊂ B implies B ∈ F .
Lemma 2.1. [18] If I is a nontrivial ideal in X, X , ∅, then the class
F (I) = {M ⊂ X : (∃A ∈ I)(M = X\A)}
is a filter on X, called the filter associated with I.
A nontrivial ideal I in X is called admissible if {x} ∈ I for each x ∈ X.
Throughout the paper we take I2 as a nontrivial admissible ideal in N × N.
A nontrivial ideal I2 of N × N is called strongly admissible if {i} × N and N × {i} belong to I2 for each
i ∈ N.
It is evident that a strongly
admissible
ideal is admissible also.
0
Let I2 = {A ⊂ N × N : ∃m(A) ∈ N i, j ≥ m(A) ⇒ (i, j) < A }. Then I02 is a nontrivial strongly admissible
ideal and clearly I2 is strongly admissible if and only if I02 ⊂ I2 .
Let (X, ρ) be a linear metric space and I2 ⊂ 2N×N be a strongly admissible ideal. A double sequence
x = (xmn ) of elements of X is said to be I2 -convergent to L ∈ X, if for any ε > 0 we have
A(ε) = {(m, n) ∈ N × N : ρ(xmn , L) ≥ ε} ∈ I2 .
In this case we say that x is I2 -convergent to L ∈ X and we write
I2 − lim xmn = L.
m,n→∞
If I2 is a strongly admissible ideal on N × N, then usual convergence implies I2 -convergence.
Let (X, ρ) be a linear metric space and I2 ⊂ 2N×N be a strongly admissible ideal. A double sequence
x = (xmn ) of elements of X is said to be I∗2 -convergent to L ∈ X, if and only if there exists a set M ∈ F (I2 )
(i.e., N × N\M ∈ I2 ) such that
lim xmn = L,
m,n→∞
for (m, n) ∈ M and we write
I∗2 − lim xmn = L.
m,n→∞
Let (X, ρ) be a linear metric space and I2 ⊂ 2N×N be a strongly admissible ideal. A double sequence x = (xmn )
of elements of X is said to be I2 -Cauchy if for every ε > 0, there exist s = s(ε), t = t(ε) ∈ N such that
A(ε) = {(m, n) ∈ N × N : ρ(xmn , xst ) ≥ ε} ∈ I2 .
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We say that an admissible ideal I2 ⊂ 2N×N satisfies 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
0
A j ∆B
Sj ∞∈ I2 , i.e., A j ∆B j is included in the finite union of rows and columns in N × N for each j ∈ N and
B = j=1 B j ∈ I2 (hence B j ∈ I2 for each j ∈ N).
Now we begin with quoting the lemmas due to Dündar and Altay [8, 9] which are needed throughout
the paper.
Lemma 2.2. [9] Let I2 ⊂ 2N×N be a strongly admissible ideal, { fmn } is a double sequence of functions and f is a
function on S ⊂ R. Then
I∗2 − lim fmn (x) = f (x) implies I2 − lim fmn (x) = f (x), (pointwise)
m,n→∞
m,n→∞
for each x ∈ S.
Lemma 2.3. [9] Let I2 ⊂ 2N×N be a strongly admissible ideal. { fmn } is a double sequence of functions is pointwise
I2 -convergent to f on S ⊂ R if and only if it is pointwise I2 -Cauchy sequences.
Lemma 2.4. [8] Let I2 ⊂ 2N×N be a strongly admissible ideal having the property (AP2), (X, dx ) and (Y, d y ) two
linear metric spaces, fmn : X → Y a double sequence of functions and f : X → Y. If { fmn } double sequence of functions
is I2 -convergent then it is I∗2 -convergent.
3. Main Results
First we prove the following theorem with an another way that it is given in [16].
Theorem 3.1. Let f and fmn , m, n = 1, 2, ..., be continuous functions on D = [a, b] ⊂ R. Then fmn ⇒ f on D = [a, b]
if and only if
lim cmn = 0,
m,n→∞
where cmn = maxx∈D | fmn (x) − f (x)|.
Proof. Suppose that fmn ⇒ f on D = [a, b]. Since f and fmn are continuous functions on D = [a, b] so
| fmn − f |
is continuous on D = [a, b], for each m, n ∈ N. Since limm,n→∞ fmn (x) = f (x) uniformly on D = [a, b] then, for
each ε > 0, there is a positive integer k0 = k0 (ε) ∈ N such that m, n > k0 implies
ε
| fmn (x) − f (x)| < ,
2
for all x ∈ D. Thus, when m, n > k0 we have
ε
cmn = max | fmn (x) − f (x)| ≤ < ε.
x∈D
2
This implies
lim cmn = 0.
m,n→∞
Now, suppose that limm,n cmn = 0. Then for each ε > 0, there is a positive integer k0 = k0 (ε) ∈ N such that
0 ≤ cmn = max | fmn (x) − f (x)| < ε,
x∈D
for m, n > k0 . This implies that
for all x ∈ D and m, n > k0 . Hence, we have
lim fmn (x) = f (x),
m,n→∞
for all x ∈ D.
| fmn (x) − f (x)| < ε,
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Definition 3.2. Let I2 ⊂ 2N×N be a strongly admissible ideal. A double sequence of functions { fmn } is said to be
I2 -uniformly convergent to f on a set S ⊂ R, if for every ε > 0
{(m, n) ∈ N × N : | fmn (x) − f (x)| ≥ ε} ∈ I2 , for each x ∈ S.
This can be written by the formula
(∀ε > 0) (∃H ∈ I2 ) (∀(m, n) < H) (∀x ∈ S) | fmn (x) − f (x)| < ε.
This convergence can be showed by
fmn ⇒I2 f.
Theorem 3.3. Let I2 ⊂ 2N×N be a strongly admissible ideal, f and fmn , m, n = 1, 2, ..., be continuous functions on
D = [a, b] ⊂ R. Then
fmn ⇒I2 f
on D = [a, b] if and only if
I2 − lim cmn = 0,
m,n
where cmn = maxx∈D | fmn (x) − f (x)|.
Proof. Suppose that fmn ⇒I2 f on D = [a, b]. Since f and fmn be continuous functions on D = [a, b], so
| fmn − f |
is continuous on D = [a, b] for each m, n ∈ N. By I2 -uniform convergence for ε > 0
ε
∈ I2 , for each x ∈ D.
(m, n) ∈ N × N : | fmn (x) − f (x)| ≥
2
Hence, for ε > 0 it is clear that
cmn = max | fmn (x) − f (x)| ≥ | fmn (x) − f (x)| ≥
x∈D
ε
, for each x ∈ D.
2
Thus, we have
I2 − lim cmn = 0.
m,n→∞
Now, suppose that I2 − limm,n cmn = 0. Then, for ε > 0
A(ε) = (m, n) ∈ N × N : max | fmn (x) − f (x)| ≥ ε ∈ I2 .
x∈D
Since, for ε > 0
max | fmn (x) − f (x)| ≥ | fmn (x) − f (x)| ≥ ε
x∈D
we have
(m, n) ∈ N × N : | fmn (x) − f (x)| ≥ ε ⊂ A(ε), for each x ∈ D.
This proves the theorem.
Definition 3.4. Let I2 ⊂ 2N×N be a strongly admissible ideal. A double sequence of functions { fmn } is said to be
I∗2 -uniformly convergent to f on a set S ⊂ R, if there exists a set M ∈ F (I2 ) (i.e., N × N\M ∈ I2 ) such that for
every ε > 0
lim fmn (x) = f (x), for each x ∈ S
m,n→∞
(m,n)∈M
and we write
fmn ⇒I∗2 f.
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Theorem 3.5. Let I2 ⊂ 2N×N be a strongly admissible ideal, { fmn } be a double sequence of continuous functions and
f be a function on S. If
fmn ⇒I∗2 f
then, f is continuous on S.
Proof. Assume fmn ⇒I∗2 f on S. Then for every ε > 0, there exists a set M ∈ F (I2 ) (i.e., H = N × N\M ∈ I2 )
and k0 = k0 (ε), l0 = l0 (ε) ∈ N such that
| fmn (x) − f (x)| <
ε
, (m, n) ∈ M
3
for each x ∈ S and for all m > k0 , n > l0 . Now, let x0 ∈ S is arbitrary. Since { fk0 l0 } is continuous at x0 ∈ S, there
is a δ > 0 such that |x − x0 | < δ implies
| fk0 l0 (x) − fk0 l0 (x0 )| <
ε
.
3
Then, for all x ∈ S for which |x − x0 | < δ, we have
| f (x) − fk0 l0 (x)| + | fk0 l0 (x) − fk0 l0 (x0 )| + | fk0 l0 (x0 ) − f (x0 )|
ε ε ε
+ + = ε.
<
3 3 3
Since x0 ∈ S is arbitrary, f is continuous on S.
| f (x) − f (x0 )|
≤
Theorem 3.6. Let I2 ⊂ 2N×N be a strongly admissible ideal with the property (AP2), S be a compact subset of R
and { fmn } be a double sequence of continuous functions on S. Assume that { fmn } be monotonic decreasing on S i.e.,
f(m+1),(n+1) (x) ≤ fmn (x), (m, n = 1, 2, ...)
for every x ∈ S, f is continuous and
I2 − lim fmn (x) = f (x)
m,n→∞
on S. Then
fmn ⇒I2 f
on S.
Proof. Let
1mn = fmn − f
(1)
a sequence of functions on S. Since { fmn } is continuous and monotonic decreasing and f is continuous on S,
then {1mn } is continuous and monotonic decreasing on S. Since
I2 − lim fmn (x) = f (x),
m,n→∞
then by (1)
I2 − lim 1mn (x) = 0
m,n→∞
on S and since I2 satisfy the condition (AP2) then we have
I∗2 − lim 1mn (x) = 0
m,n→∞
on S. Hence, for every ε > 0 and each x ∈ S there exists Kx ∈ F (I2 ) such that
ε 0 ≤ 1mn (x) < , (m, n), m(x) = m(x, ε), n(x) = n(x, ε) ∈ Kx
2
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1279
for m ≥ m(x) and n ≥ n(x). Since {1mn } is continuous at x ∈ S, for every ε > 0 there is an open set A(x) which
contains x such that
|1m(x)n(x) (t) − 1m(x)n(x) (x)| <
ε
,
2
for all t ∈ A(x). Then for ε > 0 by monotonicity, we have
0 ≤ 1mn (t) ≤ 1m(x)n(x) (t)
= 1m(x)n(x) (t) − 1m(x)n(x) (x) + 1m(x)n(x) (x)
≤
|1m(x)n(x) (t) − 1m(x)n(x) (x)| + 1m(x)n(x) (x), ((m, n) ∈ Kx )
for every t ∈ A(x) and for all m ≥ m(x), n ≥ n(x) and for each x ∈ S. Since S ⊂
set, by the Heine-Borel theorem S has a finite open covering such that
S
x∈S
A(x) and S is a compact
S ⊂ A(x1 ) ∪ A(x2 ) ∪ A(x3 ) ∪ ... ∪ A(xi ).
Now, let
K = Kx1 ∩ Kx2 ∩ Kx3 ∩ ... ∩ Kxi
and define
n
o
M = maks m(x1 ), m(x2 ), m(x3 ), ..., m(xi ) ,
n
o
N = maks n(x1 ), n(x2 ), n(x3 ), ..., n(xi ) .
Since for every Kxi belong to F (I2 ), we have K ∈ F (I2 ). Then, when all (m, n) ≥ (M, N)
0 ≤ 1mn (t) < ε, (m, n) ∈ K,
for every t ∈ A(x). So
1mn ⇒I∗2 0
on S. Since I2 is a strongly admissible ideal,
1mn ⇒I2 0
on S and by (1) we have
fmn ⇒I2 f
on S.
Theorem 3.7. Let I2 ⊂ 2N×N be a strongly admissible ideal, (X, dx ) and (Y, d y ) be two metric spaces, fmn : X → Y,
(m, n ∈ N), are equi-continuous and f : X → Y. Assume that
fmn →I2 f
on X. Then, f is continuous on X. Also, if X is compact then we have
fmn ⇒I2 f
on X.
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1280
Proof. First we will prove that f is continuous on X. Let x0 ∈ X and ε > 0. By the equi-continuity of fmn ’s,
there exists δ > 0 such that
ε
d y fmn (x), fmn (x0 ) < ,
3
for every m, n ∈ N and x ∈ Bδ (x0 ) (Bδ (x0 ) stands for an open ball in X with center x0 and radius δ). Let
x ∈ Bδ (x0 ) be fixed. Since fmn →I2 f , the set
ε
ε
∪ (m, n) ∈ N × N : d y ( fmn (x), f (x)) ≥
(m, n) ∈ N × N : d y ( fmn (x0 ), f (x0 )) ≥
3
3
is in I2 and is different from N × N. Hence, there exists (m, n) ∈ N × N such that
ε
ε
d y fmn (x0 ), f (x0 ) < and d y fmn (x), f (x) < .
3
3
Thus, we have
d y f (x0 ), f (x) ≤ d y f (x0 ), fmn (x0 ) + d y fmn (x0 ), fmn (x) + d y fmn (x), f (x)
ε ε ε
+ + =ε
<
3 3 3
so f is continuous on X.
Now, assume that X is compact. Let ε > 0. Since X is compact, it follows that f is uniformly continuous
and fmn ’s are equi-uniformly continuous on X. So, pick δ > 0 such that for any x, x0 ∈ X with
dx (x, x0 ) < δ,
then, by equi-uniformly and uniformly continuous we have
ε
ε
d y fmn (x), fmn (x0 ) < ve d y f (x), f (x0 ) < .
3
3
By the compactness of X, we can choose a finite subcover
Bx1 (δ), Bx2 (δ), ..., Bxk (δ)
from the cover {Bx (δ)}x∈X of X. Using fmn →I2 f pick a set M ∈ I2 such that
ε
d y fmn (xi ), f (xi ) < , i ∈ {1, 2, ..., k},
3
for all (m, n) < M. Let (m, n) < M and x ∈ X. Thus, x ∈ Bxi (δ) for some i ∈ {1, 2, ..., k}. Hence, we have
d y fmn (x), f (x) ≤ d y fmn (x), fmn (xi ) + d y fmn (xi ), f (xi ) + d y f (xi ), f (x)
ε ε ε
+ + = ε,
<
3 3 3
and so
fmn ⇒I2 f
on X.
Definition 3.8. Let I2 ⊂ 2N×N be a strongly admissible ideal and { fmn } be a double sequence of functions on S ⊂ R.
{ fmn } is said to be I2 -uniformly Cauchy if for every ε > 0 there exist s = s(ε), t = t(ε) ∈ N such that
{(m, n) ∈ N × N : | fmn (x) − fst (x)| ≥ ε} ∈ I2 , for each x ∈ S.
(2)
Now, we give I2 -Cauchy criteria for I2 -uniform convergence.
Theorem 3.9. Let I2 ⊂ 2N×N be a strongly admissible ideal with the property (AP2) and let { fmn } be a sequence of
bounded functions on S ⊂ R. Then { fmn } is I2 -uniformly convergent if and only if it is I2 -uniformly Cauchy on S.
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Proof. Necessity of Theorem is similar to that of Lemma 2.3.
Conversely, assume that { fmn } is I2 -uniformly Cauchy on S. Let x ∈ S be fixed. By (2), for every ε > 0
there exist s = s(ε) and t = t(ε) ∈ N such that
(m, n) ∈ N × N : | fmn (x) − fst (x)| < ε < I2 .
Hence, { fmn } is I2 -Cauchy, so by Lemma 2.3 we have that { fmn } is I2 -convergent to f (x). Then, I2 −
limm,n→∞ fmn (x) = f (x) on S. Note that since I2 satisfy the property (AP2), by (2) there is a M < I2 such that
| fmn (x) − fst (x)| < ε, (m, n), (s, t) ∈ M
(3)
for all m, n, s, t ≥ N and N = N(ε) ∈ N and for each x ∈ S. By (3), for s, t → ∞ we have
| fmn (x) − f (x)| < ε, (m, n) ∈ M ,
for all m, n > N and for each x ∈ S. This shows that
fmn ⇒I∗2 f
on S. Since I2 ⊂ 2N×N is a strongly admissible ideal we have
fmn ⇒I2 f.
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