Hindawi Publishing Corporation
Chinese Journal of Mathematics
Volume 2014, Article ID 805857, 5 pages
http://dx.doi.org/10.1155/2014/805857
Research Article
Asymptotic I-Equivalence of Two Number Sequences and
Asymptotic I-Regular Matrices
Hafize Gumus,1 Jeff Connor,2 and Fatih Nuray3
1
Faculty of Eregli Education, Necmettin Erbakan University, Eregli, Konya, Turkey
Department of Mathematics, Ohio University, Athens, OH, USA
3
Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, Afyon, Turkey
2
Correspondence should be addressed to Fatih Nuray; [email protected]
Received 8 November 2013; Accepted 5 January 2014; Published 24 February 2014
Academic Editors: W. Xiao and J. Zhang
Copyright © 2014 Hafize Gumus et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study I-equivalence of the two nonnegative sequences 𝑥 = (𝑥𝑘 ) and 𝑦 = (𝑦𝑘 ). Also we define asymptotic I-regular matrices and
obtain conditions for a matrix 𝐴 = (𝑎𝑗𝑘 ) to be asymptotic I-regular.
Proposition 3. 𝐼 is a nontrivial ideal in N if and only if
1. Introduction
The notion of 𝐼-convergence was introduced by Kostyrko et
al. for real sequences (see [1]) and then extended to metric
spaces by Kostyrko et al. (see [2]). Fast [3] introduced statistical convergence and 𝐼-convergence, which is based on using
ideals of N to define sets of density 0; is a natural extension of
Fast’s definition.
Definition 1. A family of sets 𝐼 ⊆ 2N is called an ideal if and
only if
(i) 0 ∈ 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 nontrivial if N ∉ 𝐼 and a nontrivial ideal is
called admissible if {𝑛} ∈ 𝐼 for each 𝑛 ∈ N (see [2]).
N
Definition 2. A family of sets 𝐹 ⊂ 2 is a filter in N if and only
if
(i) 0 ∉ 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 (see
[2]).
𝐹 = 𝐹 (𝐼) = {𝑀 = N \ 𝐴 : 𝐴 ∈ 𝐼}
(1)
is a filter in N (see [2]).
Definition 4. A real sequence 𝑥 = (𝑥𝑘 ) is said to be 𝐼convergent to 𝐿 ∈ R if and only if for each 𝜀 > 0 the set
󵄨
󵄨
𝐴 𝜀 = {𝑘 ∈ N : 󵄨󵄨󵄨𝑥𝑘 − 𝐿󵄨󵄨󵄨 ≥ 𝜀}
(2)
belongs to 𝐼. The number 𝐿 is called the 𝐼-limit of the
sequence 𝑥 (see [2]).
Let 𝑥 = (𝑥𝑘 ) and 𝑦 = (𝑦𝑘 ) be real sequences. Pobyvanets
introduced asymptotic equivalence for 𝑥 and 𝑦 as follows: if
lim
𝑥𝑘
𝑘 → ∞ 𝑦𝑘
= 1,
(3)
then 𝑥 and 𝑦 are called asymptotic equivalent; this is denoted
by 𝑥 ∼ 𝑦 (see [4]). Pobyvanets also introduced asymptotic
regular matrices which preserve the asymptotic equivalence
of two nonnegative number sequences; that is, for the
nonnegative matrix 𝐴 = (𝑎𝑗𝑘 ) if 𝑥 ∼ 𝑦 then 𝐴𝑥 ∼ 𝐴𝑦
(see [5]).
2
Chinese Journal of Mathematics
Theorem 5. Let 𝐴 = (𝑎𝑗𝑘 ) be a nonnegative matrix. 𝐴 is
asymptotic regular if and only if, for each 𝑚,
𝑎𝑗𝑚
∞
𝑗→∞ ∑
𝑘=1 𝑎𝑗𝑘
lim
=1
(4)
Definition 8. Two nonnegative sequences 𝑥 and 𝑦 are said
to be asymptotically statistically equivalent provided that, for
every 𝜀 > 0,
󵄨󵄨
󵄨󵄨 𝑥
1
󵄨󵄨
󵄨󵄨 𝑘
−
1
{the
number
of
𝑘
≤
𝑛
:
lim
󵄨󵄨 ≥ 𝜀} = 0.
󵄨
󵄨
𝑛 𝑛
󵄨󵄨
󵄨󵄨 𝑦𝑘
(see [5]).
𝑆
The frequency of terms having zero values makes a termby-term ratio 𝑥𝑘 /𝑦𝑘 inapplicable in many cases, which motivated Fridy to introduce some related notions. In particular,
he analyzed the asymptotic rates of convergence of the tails
and partial sums of series as well as the supremum of the tails
of bounded sequences (see [6]).
Define 𝑙1 , 𝑃0 , 𝑃, and 𝑃𝛿 spaces as follows:
∞
󵄨 󵄨
𝑙1 = {𝑥 = (𝑥𝑘 ) : ∑ 󵄨󵄨󵄨𝑥𝑘 󵄨󵄨󵄨 < ∞} .
𝑥𝑘 ≥ 𝛿 > 0 ∀𝑘} ,
𝑃0 = {the set of all nonnegative sequences which have
𝑆
𝑆
tistically regular provided that 𝐴𝑥 ∼ 𝐴𝑦 whenever 𝑥 ∼ 𝑦,
𝑥 ∈ 𝑃0 , and 𝑦 ∈ 𝑃𝛿 for some 𝛿 > 0 (see [4]).
Having introduced these ideas, Patterson then offered
characterizations of (i) in Theorem 6 when a nonnegative
summability matrix 𝐴 maps bounded sequences into the
absolutely convergent sequences and has the property that
𝑆
Theorem 10. In order for a summability matrix 𝐴 to be asymptotically statistical regular it is necessary and sufficient that
𝑘
0
𝑎𝑛𝑘 is bounded for each 𝑛;
(i) ∑𝑘=1
at most a finite number of zero terms} ,
(ii) for any fixed 𝑘0 and 𝜀 > 0,
𝑃 = {the set of all sequences 𝑥 such that 𝑥𝑘 > 0 ∀𝑘} .
(5)
Following Fridy, given a sequence 𝑥 = (𝑥𝑘 ), the sequence 𝑅𝑥
is defined to be (𝑅𝑚 𝑥) where, for each 𝑚 ∈ N, 𝑅𝑚 𝑥 = ∑∞
𝑘=𝑚 𝑥𝑘 .
Theorem 6. If 𝐴 is a nonnegative (𝑙∞ , 𝑙1 ) summability matrix,
then the following statements are equivalent:
(i) if 𝑥 and 𝑦 are bounded sequences such that 𝑥 ∼ 𝑦 and
𝑦 ∈ 𝑃𝛿 , for some 𝛿 > 0, then 𝑅𝐴𝑥 ∼ 𝑅𝐴𝑦;
(ii) for each 𝑚,
𝑛→∞
Definition 9. A summability matrix 𝐴 is asymptotically sta-
𝑆
𝑃𝛿 = {The set of all real number sequences such that
lim (
In this case we write 𝑥 ∼ 𝑦 (see [4]).
if 𝑥 ∈ 𝑃0 , 𝑦 ∈ 𝑃𝛿 , and 𝑥 ∼ 𝑦, then 𝑅𝐴𝑥 ∼ 𝑅𝐴𝑦 and
(ii) when a summability matrix is asymptotically statistical
regular summability matrices.
𝑘=1
󵄨
󵄨󵄨 𝑘0
󵄨󵄨 ∑𝑘=1 𝑎𝑛𝑘 󵄨󵄨󵄨
1
󵄨󵄨 ≥ 𝜀} = 0 (9)
󵄨
lim {𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑘 ≤ 𝑛 : 󵄨󵄨󵄨 ∞
󵄨
𝑛→∞𝑛
󵄨󵄨 ∑𝑘=1 𝑎𝑛𝑘 󵄨󵄨󵄨
󵄨
󵄨
(see [4]).
The main results of this paper have a similar focus,
where statistical convergence is replaced by convergence with
respect to an admissible ideal of subsets of N.
2. Main Results
∑∞
𝑘=𝑛 𝑎𝑘𝑚
)=0
∞
∑𝑘=𝑛 ∑∞
𝑗=0 𝑎𝑘𝑗
(6)
In 2003, Patterson extended these concepts by introducing asymptotically statistical equivalent sequences, an analog
of the above definitions, and investigated natural regularity
conditions for nonnegative summability matrices (see [4]).
Definition 7. The sequence 𝑥 has statistical limit 𝐿, denoted
by st-lim 𝑥 = 𝐿 provided that for every 𝜀 > 0,
1
󵄨
󵄨
{the number of 𝑘 ≤ 𝑛 : 󵄨󵄨󵄨𝑥𝑘 − 𝐿󵄨󵄨󵄨 ≥ 𝜀} = 0
lim
𝑛 𝑛
Definition 11. Let 𝑥 and 𝑦 be nonnegative real sequences and
let 𝐼 be an admissible ideal in N. If
󵄨󵄨
󵄨󵄨 𝑥
󵄨
󵄨
{𝑘 ∈ N : 󵄨󵄨󵄨 𝑘 − 1󵄨󵄨󵄨 ≥ 𝜀}
󵄨󵄨
󵄨󵄨 𝑦𝑘
(see [7]).
(see [4]).
(8)
(10)
belongs to 𝐼, for every 𝜀 > 0, then 𝑥 and 𝑦 are called asymp𝐼
totically 𝐼-equivalent sequences; this is denoted by 𝑥 ∼ 𝑦.
Theorem 12. Let 𝐴 = (𝑎𝑗𝑘 ) be a nonnegative (𝑙∞ , 𝑙1 ) summability matrix and let 𝐼 and 𝐽 be admissible ideals in N. Then the
following statements are equivalent:
𝐼
(i) if 𝑥 and 𝑦 are bounded sequences such that 𝑥 ∼ 𝑦, 𝑥 ∈
(7)
𝐽
𝑃0 , and 𝑦 ∈ 𝑃𝛿 , for some 𝛿 > 0, then 𝑅𝐴𝑥 ∼ 𝑅𝐴𝑦;
(ii) 𝐽 − lim𝑛 ((∑𝑗≥𝑛 ∑𝑘∈𝑆 𝑎𝑗𝑘 )/(∑𝑗≥𝑛 ∑∞
𝑘=1 𝑎𝑗𝑘 )) = 0 for each
𝑆 ∈ 𝐼.
Chinese Journal of Mathematics
3
∞
Proof. The proof that statement (ii) implies statement (i) is
+
𝐼
given first. Assuming that 𝑥 ∼ 𝑦, let 𝜀 > 0 be given and set
󵄨󵄨 𝑥
󵄨󵄨󵄨 𝜀
󵄨
𝑆 = {𝑘 : 󵄨󵄨󵄨 𝑘 − 1󵄨󵄨󵄨 ≥ } .
󵄨󵄨 𝑦𝑘
󵄨󵄨 2
−
≥
(12)
∑𝑗≥𝑛 ∑∞
𝑘=1 𝑎𝑗𝑘 𝑦𝑘
∑𝑗≥𝑛 ∑𝑘∈𝑆 𝑎𝑗𝑘
𝛿
𝜀
+ (1 − )
∞
2
sup𝑘 (𝑦𝑘 ) ∑𝑗≥𝑛 ∑𝑘=1 𝑎𝑗𝑘
−
󵄨󵄨 ∑ ∑ 𝑎 󵄨󵄨
󵄨󵄨 𝑗≥𝑛 𝑘∈𝑆 𝑗𝑘 󵄨󵄨 𝜀
𝛿
󵄨󵄨 <
𝑇 = {𝑛 ∈ N : 󵄨󵄨󵄨
󵄨󵄨 2 sup (𝑥 ) } ∈ 𝐹 (𝐽) .
󵄨󵄨 ∑𝑗≥𝑛 ∑∞
𝑎
𝑘
𝑘
𝑘=1 𝑗𝑘 󵄨󵄨
󵄨
(13)
Observe that, for 𝑛 ∈ 𝑇,
(1 − (𝜀/2)) (sup𝑘 (𝑦𝑘 )) ∑𝑗≥𝑛 ∑𝑘∈𝑆 𝑎𝑗𝑘
(16)
By condition (ii), there is a set 𝑈 ∈ 𝐹(𝐽) such that, for each
𝑛 ∈ 𝑈, the first and third terms of the above expression can
be made small in relation to 1 − (𝜀/2) and, in particular,
∑𝑗≥𝑛 (∑∞
𝑘=1 𝑎𝑗𝑘 𝑦𝑘 )
𝐽 − lim
sup𝑘 (𝑥𝑘 ) ∑𝑗≥𝑛 ∑𝑘∈𝑆 𝑎𝑗𝑘
∑𝑗≥𝑛 (∑∞
𝑘=1 𝑎𝑗𝑘 𝑦𝑘 )
+
(14)
∑𝑗≥𝑛 (∑∞
𝑘=1 𝑎𝑗𝑘 𝑦𝑘 )
≤
sup𝑘 (𝑥𝑘 ) ∑𝑗≥𝑛 ∑𝑘∈𝑆 𝑎𝑗𝑘
𝜀
+ (1 + )
𝛿
2
𝑎
∑𝑗≥𝑛 ∑∞
𝑘=1 𝑗𝑘
<
sup𝑘 (𝑥𝑘 ) 𝜀
𝛿
𝜀
) + (1 + )
(
𝛿
2 sup𝑘 (𝑥𝑘 )
2
+
≥
1, if 𝑘 ∉ 𝑆,
0, if 𝑘 ∈ 𝑆,
(1 − (𝜀/2)) ∑𝑗≥𝑛 ∑𝑘∉𝑆 𝑎𝑗𝑘 𝑥𝑘
∑𝑗≥𝑛 ∑∞
𝑘=1 𝑎𝑗𝑘 𝑦𝑘
∑𝑗≥𝑛 ∑𝑘∈𝑆 𝑎𝑗𝑘
𝛿
sup𝑘 (𝑦𝑘 ) ∑𝑗≥𝑛 ∑∞
𝑘=1 𝑎𝑗𝑘
𝐽
𝑦𝑘 = 1.
(19)
Observe that
𝑅𝑛 (𝐴𝑥) ∑𝑗≥𝑛 (∑𝑘∈𝑆 𝑎𝑗𝑘 𝑥𝑘 + ∑𝑘∉𝑆 𝑎𝑗𝑘 𝑥𝑘 )
=
𝑅𝑛 (𝐴𝑦)
∑𝑗≥𝑛 (∑∞
𝑘=1 𝑎𝑗𝑘 𝑦𝑘 )
=
∑𝑗≥𝑛 ∑∞
𝑘=1 𝑎𝑗𝑘 − ∑𝑗≥𝑛 ∑𝑘∈𝑆 𝑎𝑗𝑘
∑𝑗≥𝑛 ∑∞
𝑘=1 𝑎𝑗𝑘
(15)
=1−
∑𝑗≥𝑛 ∑𝑘∈𝑆 𝑎𝑗𝑘
𝛿
sup𝑘 (𝑦𝑘 ) ∑𝑗≥𝑛 ∑∞
𝑘=1 𝑎𝑗𝑘
(18)
𝐼
On the other hand,
≥
𝑅𝑛 (𝐴𝑥)
=1
𝑅𝑛 (𝐴𝑦)
sequences, 𝑥 ∈ 𝑃0 , 𝑦 ∈ 𝑃𝛿 , and 𝑥 ∼ 𝑦, then 𝑅𝐴𝑥 ∼ 𝑅𝐴𝑦. Let 𝑆
be a member of 𝐼 and define the sequences 𝑥 and 𝑦 as follows:
That is,
𝑅𝑛 (𝐴𝑥) ∑𝑗≥𝑛 (∑𝑘∈𝑆 𝑎𝑗𝑘 𝑥𝑘 + ∑𝑘∉𝑆 𝑎𝑗𝑘 𝑥𝑘 )
=
𝑅𝑛 (𝐴𝑦)
∑𝑗≥𝑛 (∑∞
𝑘=1 𝑎𝑗𝑘 𝑦𝑘 )
(17)
and hence (ii) implies (i).
The proof is completed by showing that statement (i)
implies statement (ii). Assume that if 𝑥 and 𝑦 are bounded
𝑥𝑘 = {
= 1 + 𝜀.
𝑅𝑛 (𝐴𝑥)
≤ 1 + 𝜀.
𝑅𝑛 (𝐴𝑦)
.
for each 𝑛 ∈ 𝑈. By (15) and (17), we have
∑𝑗≥𝑛 (∑𝑘∈𝑆 𝑎𝑗𝑘 𝑥𝑘 + ∑𝑘∉𝑆 𝑎𝑗𝑘 𝑥𝑘 )
(1 + (𝜀/2)) ∑𝑗≥𝑛 ∑𝑘∉𝑆 𝑎𝑗𝑘 𝑦𝑘
𝛿 ∑𝑗≥𝑛 ∑∞
𝑘=1 𝑎𝑗𝑘
𝑅𝑛 (𝐴𝑥)
≥1−𝜀
𝑅𝑛 (𝐴𝑦)
∞
𝑅𝑛 (𝐴𝑥) ∑𝑗≥𝑛 (∑𝑘=1 𝑎𝑗𝑘 𝑥𝑘 )
=
𝑅𝑛 (𝐴𝑦) ∑𝑗≥𝑛 (∑∞
𝑘=1 𝑎𝑗𝑘 𝑦𝑘 )
≤
(1 − (𝜀/2)) ∑𝑗≥𝑛 ∑𝑘∈𝑆 𝑎𝑗𝑘 𝑦𝑘
∞
for each 𝑘 ∉ 𝑆. By (ii),
=
∑𝑗≥𝑛 ∑∞
𝑘=1 𝑎𝑗𝑘 𝑦𝑘
∞
(11)
Observe that 𝑆 ∈ 𝐼 and that
𝜀
𝜀
(1 − ) 𝑦𝑘 < 𝑥𝑘 < (1 + ) 𝑦𝑘
2
2
(1 − (𝜀/2)) ∑𝑗≥𝑛 ∑𝑘=1 𝑎𝑗𝑘 𝑦𝑘
(20)
∑𝑗≥𝑛 ∑𝑘∈𝑆 𝑎𝑗𝑘
∑𝑗≥𝑛 ∑∞
𝑘=1 𝑎𝑗𝑘
𝐽
and, as 𝑅A𝑥 ∼ 𝑅𝐴𝑦, it follows that
𝐽 − lim
∑𝑗≥𝑛 ∑𝑘∈𝑆 𝑎𝑗𝑘
∑𝑗≥𝑛 ∑∞
𝑘=1 𝑎𝑗𝑘
= 0.
(21)
Definition 13. Let 𝐴 = (𝑎𝑗𝑘 ) be a nonnegative summability
matrix and let 𝐼 be an admissible ideal in N. Further assume
that 𝑥 and 𝑦 are nonnegative real sequences with 𝑥 ∈ 𝑃 and
𝑦 ∈ 𝑃𝛿 for some 𝛿 > 0. The summability matrix 𝐴 is said to
𝐼
𝐼
be asymptotic 𝐼-regular if 𝑥 ∼ 𝑦 implies 𝐴𝑥 ∼ 𝐴𝑦.
4
Chinese Journal of Mathematics
Lemma 14. Let 𝑥 = (𝑥𝑘 ) and 𝑦 = (𝑦𝑘 ) belong to 𝑃𝛿 ∩ 𝑙∞ and
let 𝐼 be an admissible ideal in N. Then necessary and sufficient
𝐼
condition for 𝑥 ∼ 𝑦 is 𝐼 − lim(𝑥 − 𝑦) = 0.
Theorem 17. Let 𝐼 ⊆ 2N be an admissible ideal in N. Then
a nonnegative summability matrix 𝐴 = (𝑎𝑗𝑘 ) is asymptotic 𝐼regular if and only if
𝐼 − lim
𝐼
Proof. Assuming that 𝑥 ∼ 𝑦, then
󵄨󵄨 𝑥
󵄨󵄨
󵄨
󵄨
𝑇 = {𝑘 : 󵄨󵄨󵄨 𝑘 − 1󵄨󵄨󵄨 < 𝜀} ∈ 𝐹 (𝐼)
󵄨󵄨 𝑦𝑘
󵄨󵄨
(22)
(23)
(1 − 𝜀) 𝑦𝑘 ≤ 𝑥𝑘 ≤ (1 + 𝜀) 𝑦𝑘 󳨐⇒ −𝜀𝑦𝑘 < 𝑥𝑘 − 𝑦𝑘 < 𝜀𝑦𝑘
󵄨 󵄨
󳨐⇒ −𝜀 (sup 󵄨󵄨󵄨𝑦𝑘 󵄨󵄨󵄨) < 𝑥𝑘 − 𝑦𝑘
𝑘
𝑥𝑘 = {
𝑘
(𝐴𝑥)𝑗
=
< 𝜀 for each 𝑘 ∈ 𝑇.
(24)
∑∞
𝑘=1 𝑎𝑗𝑘 𝑦𝑘
∑𝑘∈𝑆 𝑎𝑗𝑘
∑∞
𝑘=1 𝑎𝑗𝑘
(25)
−𝜀 𝑥𝑘
𝜀
<
−1<
𝑦𝑘 𝑦𝑘
𝑦𝑘
∑∞
𝑘=1 𝑎𝑗𝑘
∑𝑘∈𝑆 𝑎𝑗𝑘
∑∞
𝑘=1 𝑎𝑗𝑘
= 0.
(30)
Next we show that condition (27) is sufficient for 𝐴 to be
asymptotic 𝐼-regular. Define the matrix 𝐵 = (𝑏𝑗𝑘 ) by 𝑏𝑗𝑘 =
𝑎𝑗𝑘 / ∑∞
𝑘=1 𝑎𝑗𝑘 . Note that, since the row sums equal 1 and
condition (27) yields criteria (i) of Theorem 16, 𝐵 is an 𝐼regular matrix. Also observe that, since the row sums equal
1, the matrix 𝐵 maps members of 𝑃𝛿 to 𝑃𝛿 . Observe that
𝐼
𝑥 ∼ 𝑦 󳨐⇒ 𝐼 − lim (𝑥 − 𝑦) = 0
(26)
𝐼
and hence 𝑥 ∼ 𝑦.
Definition 15. Let 𝐼 be an admissible ideal in N, 𝑥 a real
sequence, and 𝐴 = (𝑎𝑗𝑘 ) a nonnegative summability matrix.
If 𝐼 − lim 𝑥 = 𝐿 implies 𝐼 − lim(𝐴𝑥) = 𝐿, then 𝐴 is called an
𝐼-regular matrix.
Theorem 16. Let 𝐴 = (𝑎𝑗𝑘 ) be a nonnegative summability
matrix, 𝐴 ∈ (𝑙∞ , 𝑙∞ ), and 𝐼 an admissible ideal. The matrix
𝐴 is 𝐼-regular if and only if
The proof of Theorem 16 is similar to [3, Theorem 2.1].
∑∞
𝑘=1 𝑎𝑗𝑘 − ∑𝑘∈𝑆 𝑎𝑗𝑘
𝐼
󳨐⇒ 𝐼 − lim𝐵 (𝑥 − 𝑦) = 0
󳨐⇒ 1 −
(ii) 𝐼 − lim ∑∞
𝑘=1 𝑎𝑗𝑘 = 1.
=
.
𝐼 − lim
For each 𝑘 ∈ 𝐵𝜀 , we have
(i) {𝑗 : | ∑𝑘∈𝑆 𝑎𝑗𝑘 | ≥ 𝜀} ∈ 𝐼 for all 𝑆 ∈ 𝐼;
(28)
Since 𝐴𝑥 ∼ 𝐴𝑦, it follows that
Hence 𝐼 − lim𝑘 (𝑥𝑘 − 𝑦𝑘 ) = 0.
𝜀
𝜀 𝑥𝑘
<1+
<
𝛿 𝑦𝑘
𝛿
󵄨󵄨 𝑥
󵄨󵄨 𝜀
󵄨
󵄨
󳨐⇒ {𝑘 : 󵄨󵄨󵄨 𝑘 − 1󵄨󵄨󵄨 < } ∈ 𝐹 (𝐼)
󵄨󵄨 𝑦𝑘
󵄨󵄨 𝛿
𝑦𝑘 = 1 ∀𝑘 ∈ N.
(29)
󸀠
−𝜀 𝑥𝑘
𝜀
−1<
<
𝛿
𝑦𝑘
𝛿
1, if 𝑘 ∉ 𝑆,
0, if 𝑘 ∈ 𝑆,
∑𝑘∈𝑆 𝑎𝑗𝑘 𝑥𝑘 + ∑𝑘∉𝑆 𝑎𝑗𝑘 𝑥𝑘
=1−
󳨐⇒ −𝜀󸀠 < 𝑥𝑘 − 𝑦𝑘
󳨐⇒
(27)
Observe that 𝑥 and 𝑦 are bounded sequences, 𝑥 ∈ 𝑃0 , and
𝑦 ∈ 𝑃𝛿 . Hence.
(𝐴𝑦)𝑗
󵄨 󵄨
< 𝜀 (sup 󵄨󵄨󵄨𝑦𝑘 󵄨󵄨󵄨)
−𝜀 < 𝑥𝑘 − 𝑦𝑘 < 𝜀 󳨐⇒
=0
for every 𝑆 ∈ 𝐼.
and hence
Now suppose that 𝐼 − lim𝑘 (𝑥𝑘 − 𝑦𝑘 ) = 0. Then
󵄨
󵄨
𝐵𝜀 = {𝑘 : 󵄨󵄨󵄨𝑥𝑘 − 𝑦𝑘 󵄨󵄨󵄨 < 𝜀} ∈ 𝐹 (𝐼) .
∑∞
𝑘=1 𝑎𝑗𝑘
Proof. Suppose that 𝐴 is asymptotic 𝐼-regular. Let 𝑆 ∈ 𝐼 and
define the sequences 𝑥 and 𝑦 as follows:
and, for each 𝑘 ∈ 𝑇, we have
(1 − 𝜀) 𝑦𝑘 < 𝑥𝑘 < (1 + 𝜀) 𝑦𝑘
∑𝑘∈𝑆 𝑎𝑗𝑘
󳨐⇒ 𝐼 − lim (𝐵𝑥 − 𝐵𝑦) = 0
(31)
𝐼
󳨐⇒ 𝐵𝑥 ∼ 𝐵𝑦
and hence
(𝐵𝑥)𝑗
(𝐵𝑦)𝑗
=
=
∑∞
𝑘=1 𝑏𝑗𝑘 𝑥𝑘
=
∑∞
𝑘=1 𝑏𝑗𝑘 𝑦𝑘
∑∞
𝑘=1 𝑎𝑗𝑘 𝑥𝑘
∑∞
𝑘=1 𝑎𝑗𝑘 𝑦𝑘
∞
∑∞
𝑘=1 (𝑎𝑗𝑘 / ∑𝑘=1 𝑎𝑗𝑘 ) 𝑥𝑘
∞
∑∞
𝑘=1 (𝑎𝑗𝑘 / ∑𝑘=1 𝑎𝑗𝑘 ) 𝑦𝑘
(32)
.
𝐼
𝐼
Since 𝐵𝑥 ∼ 𝐵𝑦, it follows that 𝐴𝑥 ∼ 𝐴𝑦.
Conflict of Interests
The authors declare that there is no conflict of interests
regarding the publication of this paper.
Chinese Journal of Mathematics
References
[1] P. Kostyrko, M. Mácaj, and T. Šalát, “Statistical convergence and
I-convergence,” http://thales.doa.fmph.uniba.sk/macaj/ICON
.pdf.
[2] P. Kostyrko, T. Šalát, and W. Wilezyński, “I-convergence,” Real
Analysis Exchange, vol. 26, no. 2, pp. 669–686, 2000/2001.
[3] J. Connor, J. A. Fridy, and C. Orhan, “Core equality results for
sequences,” Journal of Mathematical Analysis and Applications,
vol. 321, no. 2, pp. 515–523, 2006.
[4] R. F. Patterson, “On asymptotically statistical equivalent
sequences,” Demonstratio Mathematica, vol. 36, no. 1, pp. 149–
153, 2003.
[5] I. P. Pobyvanets, “Asymptotic equivalence of some linear transformation defined by a nonnegative matrix and reduced to
generalized equivalence in the sense of Cesàro and Abel,”
Matematicheskaya Fizika, vol. 28, pp. 83–87, 1980.
[6] J. A. Fridy, “Minimal rates of summability,” Canadian Journal of
Mathematics, vol. 30, no. 4, pp. 808–816, 1978.
[7] M. S. Marouf, “Asymptotic equivalence and summability,” International Journal of Mathematics and Mathematical Sciences, vol.
16, no. 4, pp. 755–762, 1993.
5
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Applied Mathematics
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Mathematical Problems
in Engineering
Journal of
Mathematics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Discrete Dynamics in
Nature and Society
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Abstract and
Applied Analysis
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Journal of
Stochastic Analysis
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014