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Characterization of
-Core and Absolute Equivalence of Double Sequences
Journal of Inequalities and Applications volume 2010, Article number: 432059 (2010)
Abstract
The -core of a double sequence has been defined and it is studied by many authors. In this paper, we have determined two permutations
and
on the set of natural numbers for which
for all
.
1. Introduction and Preliminaries
A double sequence is said to be convergent in the Pringsheim sense or P-convergent if for every
there exists
such that
whenever
[1]. In this case, we write
. By
, we mean the space of all P-convergent sequences.
A double sequence is said to be bounded if there exists a positive number
such that
for all
, that is, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F432059/MediaObjects/13660_2009_Article_2148_Equ1_HTML.gif)
By we will denote the set of all bounded double sequences. We note that in contrast to the case for single sequences, a convergent double sequence needs not to be bounded. So, by
we will denote the space of all real bounded and convergent double sequences.
Let be a four-dimensional infinite matrix of real numbers for all
The sums
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F432059/MediaObjects/13660_2009_Article_2148_Equ2_HTML.gif)
are called the -transforms of the double sequence
and we will denote it by
. We say that a sequence
is
-summable to the limit
if the
-transform of
exists for all
and convergent to
in the Pringsheim sense, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F432059/MediaObjects/13660_2009_Article_2148_Equ3_HTML.gif)
Móricz and Rhoades [2] have defined almost convergence of a double sequence as follows.
A double sequence of real numbers is said to be almost convergent to a limit
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F432059/MediaObjects/13660_2009_Article_2148_Equ4_HTML.gif)
uniformly in . By
we denote the set of all almost convergent double sequences.
Recall that Knopp's Core of a single bounded sequence is the closed interval
in [3, page 138]. In the sense of Knopp's Core,
-core of a double sequence was introduced by Patterson as the closed interval
in [4], where the definitions of
(Pringsheim limit inferior) and
(Pringsheim limit superior) are given as follows. Let
and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F432059/MediaObjects/13660_2009_Article_2148_Equ5_HTML.gif)
After this definition, this concept has been studied by many authors. For example see in [5–11] and the others.
Let denote the set of all natural numbers. A bijective function
is said to be a permutation. In this paper we have determined two permutations
and
for which
-core
-core
for all
.
A two-dimensional matrix transformation is said to be regular (see [3, page 64]) if it maps every convergent sequence into a convergent sequence with the same limit. In 1926, Robison presented a four-dimensional analogue of regularity for double sequences in which he added an additional assumption of boundedness. A four-dimensional matrix is said to be
-regular if it maps every bounded P-convergent sequence into a P-convergent sequence with the same P-limit.
The four-dimensional matrix is bounded-regular or
-regular if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F432059/MediaObjects/13660_2009_Article_2148_Equ6_HTML.gif)
Lemma (see [4]).
If is a four-dimensional matrix, then for all real-valued double sequences
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F432059/MediaObjects/13660_2009_Article_2148_Equ7_HTML.gif)
if and only if is RH-regular and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F432059/MediaObjects/13660_2009_Article_2148_Equ8_HTML.gif)
Now let us state the definition given in [14] for absolutely equivalent -regular matrices.
Definition 1.3.
Two
-regular matrices
and
are said to be absolutely equivalent for a given class of sequences
whenever
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F432059/MediaObjects/13660_2009_Article_2148_Equ9_HTML.gif)
This means that and
have the same limit or neither
nor
have a limit but their difference goes to zero.
The following proposition, lemma, and theorem characterizing the relationship between absolutely equivalent matrices and the -core are given in [14].
Proposition 1.4.
A necessary and sufficient condition for the -regular matrices
and
to be absolutely equivalent for all bounded sequences is that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F432059/MediaObjects/13660_2009_Article_2148_Equ10_HTML.gif)
Lemma.
If two double sequences and
are such that
, then
-core
-core
.
Theorem.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F432059/MediaObjects/13660_2009_Article_2148_IEq78_HTML.gif)
-core -core
for all bounded sequences
if and only if
is
-regular and is absolutely equivalent to a nonnegative matrix
for all bounded sequences.
If , then the four-dimensional matrix
is said to be strongly
-regular. In [2] the characterization of strongly
-regular has been given as follows.
Theorem.
A matrix is strongly
-regular if and only if
is
-regular and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F432059/MediaObjects/13660_2009_Article_2148_Equ11_HTML.gif)
2. The Main Results
Theorem 2.1.
If is a strongly
-regular matrix and
are two permutations such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F432059/MediaObjects/13660_2009_Article_2148_Equ12_HTML.gif)
then -core
-core
for all
, where
is a positive integer and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F432059/MediaObjects/13660_2009_Article_2148_Equ13_HTML.gif)
Proof.
In the light of Lemma 1.5, it is enough to show that .
Let and
. Then, it is clear that
for each
. Now, for
, we can write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F432059/MediaObjects/13660_2009_Article_2148_Equ14_HTML.gif)
On the other hand, since is strongly
-regular, by an easy calculation it can be seen that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F432059/MediaObjects/13660_2009_Article_2148_Equ15_HTML.gif)
By the same way, one can also see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F432059/MediaObjects/13660_2009_Article_2148_Equ16_HTML.gif)
Now, conditions (1.11) imply that This completes the proof.
Here, let us specialize the permutations and
. Let
and
be a permutation on
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F432059/MediaObjects/13660_2009_Article_2148_Equ17_HTML.gif)
for (see in [15, 16]). Also, let choose the permutation
such that
for all
. Then, we have the following theorem.
Theorem 2.2.
If is a strongly
-regular nonnegative matrix and
are two permutations as above, then
-core
-core
for all
.
Proof.
By the same technique used in Theorem 2.1, one can see that . So, Lemma 1.5 implies that
-core
-core
. But, since
is an
-regular nonnegative matrix,
-core
-core
for all
. This step completes the proof.
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Furkan, H., Çakan, C. Characterization of -Core and Absolute Equivalence of Double Sequences.
J Inequal Appl 2010, 432059 (2010). https://doi.org/10.1155/2010/432059
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DOI: https://doi.org/10.1155/2010/432059