Volume 2010, Article ID 432059,6pages doi:10.1155/2010/432059
Research Article
Characterization of
P
-Core and Absolute
Equivalence of Double Sequences
Hasan Furkan
1and Celal C
¸ akan
21Faculty of Education, S ¨utc¸ ¨u ˙Imam University, 46100 Kahramanmaras¸, Turkey
2Faculty of Education, ˙In¨on ¨u University, 44280 Malatya, Turkey
Correspondence should be addressed to Celal C¸ akan,[email protected]
Received 13 November 2009; Accepted 31 January 2010
Academic Editor: Ram N. Mohapatra
Copyrightq2010 H. Furkan and C. C¸ akan. 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.
TheP-core of a double sequence has been defined and it is studied by many authors. In this paper, we have determined two permutationsπ1andπ2on the set of natural numbers for which
P-coreAπ1x P-coreAπ2xfor allx∈
2
∞.
1. Introduction and Preliminaries
A double sequence x xjk∞j,k0 is said to be convergent in the Pringsheim sense or P-convergent if for everyε >0 there existsN ∈Nsuch that|xjk−l|< εwheneverj, k > N1. In this case, we writeP−limxl. Byc2, we mean the space of all P-convergent sequences.
A double sequencexis said to be bounded if there exists a positive numberMsuch that|xjk|< Mfor allj, k, that is, if
xsup j,k
xjk<∞. 1.1
By2
∞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, byc∞2 we will denote the space of all real bounded and convergent double sequences.
LetA amnjk ∞
j,k0be a four-dimensional infinite matrix of real numbers for allm, n
0,1, . . . .The sums
ymn
∞
j0
∞
k0
amn
are called theA-transforms of the double sequencexand we will denote it byAx. We say that a sequencexisA-summable to the limitlif theA-transform ofxexists for allm, n 0,1, . . .and convergent tolin the Pringsheim sense, that is,
lim p,q→ ∞
p
j0
q
k0
amn
jk xjkymn,
lim
m,n→ ∞ymnl.
1.3
M ´oricz and Rhoades2have defined almost convergence of a double sequence as follows.
A double sequencex xjk∞j,k0of real numbers is said to be almost convergent to a limitlif
lim p,q→ ∞
pq1
p
j0
q
k0
xjs,kt−l
0 1.4
uniformly ins, t. Byf2we denote the set of all almost convergent double sequences.
Recall that Knopp’s Core of a single bounded sequence x is the closed interval
lim infx,lim supx in 3, page 138. In the sense of Knopp’s Core, P-core of a double sequence was introduced by Patterson as the closed intervalP−lim infx, P−lim supxin4, where the definitions ofP−lim infxPringsheim limit inferiorandP−lim supxPringsheim limit superiorare given as follows. Letαnsupn{xjk:j, k≥n}andβninfn{xjk:j, k≥n}. Then
P−lim supx ⎧ ⎨ ⎩
∞, αn ∞for eachn,
inf
n αn, αn <∞for somen,
P−lim infx ⎧ ⎪ ⎨ ⎪ ⎩
−∞, βn ∞for eachn,
sup
n βn, βn <∞for somen.
1.5
After this definition, this concept has been studied by many authors. For example see in5–11and the others.
LetNdenote the set of all natural numbers. A bijective functionπ :N → Nis said to be a permutation. In this paper we have determined two permutationsπ1 andπ2 for which
P-coreAπ1x P-coreAπ2xfor allx∈2
∞.
Lemma 1.1see12,13. The four-dimensional matrixAis bounded-regular orRH-regular if and only if
P−lim m,na
mn jk 0
j, k0,1, . . . ,
P−lim m,n
j,k amn
jk 1,
P−lim m,n
j amn
jk 0 k0,1, . . .,
P−lim m,n
k amn
jk 0
j0,1, . . . ,
j,k amn
jk ≤C <∞ m, n0,1, . . ..
1.6
Lemma 1.2see4. IfAis a four-dimensional matrix, then for all real-valued double sequencesx,
P−lim supAx≤P−lim supx, 1.7
if and only ifAis RH-regular and
P−lim m,n
j,k amn
jk 1. 1.8
Now let us state the definition given in 14 for absolutely equivalent RH-regular matrices.
Definition 1.3. Two RH-regular matricesAand Bare said to be absolutely equivalent for a given class of sequencesx xjkwhenever
P−limAx−Bx 0. 1.9
This means thatAxandBxhave the same limit or neitherAxnorBxhave a limit but their difference goes to zero.
The following proposition, lemma, and theorem characterizing the relationship between absolutely equivalent matrices and theP-core are given in14.
Proposition 1.4. A necessary and sufficient condition for theRH-regular matricesA amnjk and B bmn
jk to be absolutely equivalent for all bounded sequences is that
P−lim m,n
j,k amn
jk −bmnjk 0. 1.10
Theorem 1.6. P-coreAx⊆P-corexfor all bounded sequencesx xjkif and only ifAis RH-regular and is absolutely equivalent to a nonnegative matrixB bmnjk for all bounded sequences.
If A ∈ f2, c2∞, then the four-dimensional matrixA amnjk is said to be strongly RH-regular. In2the characterization of stronglyRH-regular has been given as follows.
Theorem 1.7. A matrixA amnjk is stronglyRH-regular if and only ifAisRH-regular and
P−lim m,n
j,k amn
jk −amnj1,k0,
P−lim m,n
j,k amn
jk −amnj,k10.
1.11
2. The Main Results
Theorem 2.1. IfA amnjk is a stronglyRx-regular matrix andπ1, π2are two permutations such
that
1≤ |π1k−π2k| ≤M ∀k∈N, 2.1
thenP-coreAπ1x P-coreAπ2xfor allx∈2
∞, whereMis a positive integer and
Aπ1x j,k
amn
π1j,π1kxjk, Aπ2x
j,k amn
π2j,π2kxjk. 2.2
Proof. In the light ofLemma 1.5, it is enough to show thatP−lim|Aπ1x−Aπ2x|0. Letmk min{π1k, π2k}and Mk max{π1k, π2k}. Then, it is clear that |π1k−π2k|Mk−mkfor eachk∈N. Now, forj, k, m, n∈N, we can write
amn
π1j,π1k−a
mn π2j,π2k
amn
Mj,Mk−ammnj,mjk
≤amn
Mj,Mk−amnMj−1,Mk−1
amn
Mj−1,Mk−1−amnMj−2,Mk−2
· · ·amn
mj1,mk1−amnmj,mk
Mj−mj
r1
Mk−mk
s1
amn
mjr,mks−amnmjr−1,mks−1.
On the other hand, sinceAis stronglyRH-regular, by an easy calculation it can be seen that
|Aπ1x−Aπ2x| ≤ x j,k
amn
π1j,π1k−a
mn π2j,π2k
≤ x j,k
Mj−mj
r1
Mk−mk
s1
amn
mjr,mks−amnmjr−1,mks−1
≤ x j,k
M
r1
M
s1
amn
mjr,mks−amnmjr−1,mks−1
xMj−mj r1
Mk−mk
s1
j,k amn
mjr,mks−amnmjr−1,mks−1
≤4x M
r1
M
s1
j,k amn
j,k −amnj1,k.
2.4
By the same way, one can also see that
|Aπ1x−Aπ2x| ≤4x M
r1
M
s1
j,k amn
j,k −amnj,k1. 2.5
Now, conditions1.11imply thatP −lim|Aπ1x−Aπ2x| 0.This completes the proof.
Here, let us specialize the permutationsπ1 andπ2. LetIr 2r−1,2r −1 {k ∈ N : 2r−1≤k≤2r−1}, r∈N,andπ
1be a permutation onI1such thatπ11 1 and
π1
2r−1z
⎧ ⎨ ⎩
2r−1z1 ifzis even,
2r−1z−1 ifzis odd, 2.6
for z 0,1,2, . . . ,2r−1 −1 see in 15, 16. Also, let choose the permutation π
2 such that
π2k kfor allk∈N. Then, we have the following theorem.
Theorem 2.2. If A amnjk is a strongly RH-regular nonnegative matrix and π1, π2 are two
permutations as above, thenP-coreAπ1x P-corexfor allx∈2
∞.
Proof. By the same technique used inTheorem 2.1, one can see thatP−lim|Aπ1x−Aπ2x| 0. So,Lemma 1.5implies thatP-coreAπ1x P-coreAπ2x. But, sinceAis an RH -regular nonnegative matrix,P-coreAx P-corexfor allx ∈2
∞. This step completes
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