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Abstract and Applied Analysis
Volume 2011, Article ID 950364, 9 pages
doi:10.1155/2011/950364
Research Article
The Cesáro Core of Double Sequences
Kuddusi Kayaduman1 and Celal Çakan2
1
2
Department of Mathematics, Faculty of Science and Arts, Gaziantep University, 27310 Gaziantep, Turkey
Faculty of Education, Inönü University, 44280 Malatya, Turkey
Correspondence should be addressed to Celal Çakan, ccakan@inonu.edu.tr
Received 24 March 2011; Accepted 24 May 2011
Academic Editor: Malisa R. Zizovic
Copyright q 2011 K. Kayaduman and C. Çakan. 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 have characterized a new type of core for double sequences, PC -core, and determined the
necessary and sufficient conditions on a four-dimensional matrix A to yield PC -core{Ax} ⊆ αP core{x} for all 2∞ .
1. Introduction
A double sequence x xjk ∞
j,k0 is said to be convergent in the Pringsheim sense or P convergent if for every > 0 there exists an N ∈ N such that |xjk − | < ε whenever
j, k > N, 1. In this case, we write P − lim x . By c2 , we mean the space of all P -convergent
sequences.
A double sequence x is bounded if
x supxjk < ∞.
j,k≥0
1.1
2
By ∞
, we denote the space of all bounded double sequences.
Note that, in contrast to the case for single sequences, a convergent double sequence
need not be bounded. So, we denote by c2∞ the space of double sequences which are bounded
and convergent.
A double sequence x xjk is said to converge regularly if it converges in
Pringsheim’s sense and, in addition, the following finite limits exist:
lim xjk j ,
k→∞
lim xjk tj ,
j →∞
j 1, 2, 3, . . . ,
k 1, 2, 3, . . ..
1.2
2
Abstract and Applied Analysis
∞ be a four-dimensional infinite matrix of real numbers for all m, n 0, 1, . . ..
Let A amn
jk j,k0
The sums
∞ ∞
ymn j0 k0
amn
jk xjk
1.3
are called the A-transforms of the double sequence x xjk . We say that a sequence x xjk is A-summable to the limit if the A-transform of x xjk exists for all m, n 0, 1, . . . and is
convergent to in the Pringsheim sense, that is,
lim
q
p p,q → ∞
j0 k0
amn
jk xjk ymn ,
1.4
lim ymn .
m,n → ∞
We say that a matrix A is bounded-regular if every bounded-convergent sequence x
is A-summable to the same limit and the A-transforms are also bounded. The necessary and
sufficient conditions for A to be bounded-regular or RH-regular cf., Robison 2 are
lim amn
m,n → ∞ jk
lim
m,n → ∞
j0 k0
∞ mn ajk 0,
m,n → ∞
lim
j, k 0, 1, . . . ,
∞ ∞
amn
jk 1,
m,n → ∞
lim
0,
k 0, 1, . . .,
j0
∞ mn ajk 0,
1.5
j 0, 1, . . . ,
k0
∞ ∞
mn ajk ≤ C < ∞ m, n 0, 1, . . ..
j0 k0
A double sequence x xjk is said to be almost convergent see 3 to a number L if
p
lim sup
p,q → ∞ s,t≥0
q
1 xsj,tk L.
pq j0 k0
1.6
Let σ be a one-to-one mapping from N into itself. The almost convergence of double
sequences has been generalized to the σ-convergence in 4 as follows:
p
q
1 lim sup
xσ j s,σ k t ,
p,q → ∞ s,t≥0 pq
j0 k0
1.7
Abstract and Applied Analysis
3
where σ j s σσ j−1 s. In this case, we write σ − lim x . By Vσ2 , we denote the set of
all σ-convergent and bounded double sequences. One can see that in contrast to the case
for single sequences, a convergent double sequence need not be σ-convergent. But every
bounded convergent double sequence is σ-convergent. So, c2∞ ⊂ Vσ2 ⊂ 2∞ . In the case σi i 1, σ-convergence of double sequences reduces to the almost convergence. A matrix A ∞ is said to be σ-regular if Ax ∈ V2σ for x xjk ∈ c2∞ with σ − lim Ax lim x,
amn
jk j,k0
and we denote this by A ∈ c2∞ , V2σ reg , see 5, 6. Mursaleen and Mohiuddine defined and
characterized σ-conservative and σ-coercive matrices for double sequences 6.
A double sequence x xjk of real numbers is said to be Cesáro convergent or C1 convergent to a number L if and only if x ∈ C1 , where
C1 x ∈ 2∞ : lim Tpq x L; L C1 − lim x ,
p,q → ∞
p q
1
Tpq x xmn .
p 1 q 1 j1 k1 jk
1.8
We shall denote by C1 the space of Cesáro convergent C1 -convergent double sequences.
is said to be C1 -multiplicative if Ax ∈ C1 for x xjk ∈ c2∞
A matrix A amn
jk
with C1 − lim Ax α lim x, and in this case we write A ∈ c2∞ , C1 α . Note that if α 1, then
C1 -multiplicative matrices are said to be C1 -regular matrices.
Recall that the Knopp core or K-core of a real number single sequence x xk is
defined by the closed interval x, Lx, where x lim inf x and Lx lim sup x. The
well-known Knopp core theorem states cf., Maddox 7 and Knopp 8 that in order that
LAx ≤ Lx for every bounded real sequence x, it is necessary and sufficient that A ank should be regular and limn → ∞ ∞
k0 |ank | 1. Patterson 9 extended this idea for double
sequences by defining the Pringsheim core or P-core of a real bounded double sequence
x xjk as the closed interval P − lim inf x, P − lim sup x. Some inequalities related to the
these concepts have been studied in 5, 9, 10. Let
L∗ x lim sup sup
p,q → ∞
s,t
p
q
1 xjs,kt ,
pq j0 k0
p
q
1 Cσ x lim sup sup
xσ j s,σ k t .
p,q → ∞ s,t pq j0 k0
1.9
Then, MR- Moricz-Rhoades and σ-core of a double sequence have been introduced by the
closed intervals −L∗ −x, L∗ −x and −Cσ −x, Cσ x, and also the inequalities
LAx ≤ L∗ x, L∗ Ax ≤ Lx, L∗ Ax ≤ L∗ x, LAx ≤ Cσ x, Cσ Ax ≤ Lx
have been studies in 3–5, 11.
1.10
4
Abstract and Applied Analysis
In this paper, we introduce the concept of C1 -multiplicative matrices and determine
to belong to the class c2∞ , C1 α .
the necessary and sufficient conditions for a matrix A amn
jk
Further we investigate the necessary and sufficient conditions for the inequality
C1∗ Ax ≤ αLx
1.11
2
.
for all x ∈ ∞
2. Main Results
Let us write
p q
1
xjk .
p 1 q 1 j0 k0
C1∗ x lim sup p,q → ∞
2.1
Then, we will define the PC -core of a realvalued bounded double sequence x xjk by the
closed interval −C1∗ −x, C1∗ x. Since every bounded convergent double sequence is Cesáro
convergent, we have C1∗ x ≤ P − lim sup x, and hence it follows that PC -corex ⊆ P -corex
for a bounded double sequence x xjk .
Lemma 2.1. A matrix A amn
is C1 -multiplicative if and only if
jk
lim β j, k, p, q 0 j, k 0, 1, . . . ,
p,q → ∞
lim
∞ ∞
β j, k, p, q α,
p,q → ∞
lim
p,q → ∞
lim
p,q → ∞
2.2
2.3
j0 k0
∞ β j, k, p, q 0 k 0, 1, . . .,
2.4
j0
∞ β j, k, p, q 0 j 0, 1, . . . ,
2.5
k0
∞ ∞ mn ajk ≤ C < ∞,
m, n 0, 1, . . .,
2.6
j0 k0
where the lim means P − lim and
p q
1
amn .
p 1 q 1 j0 k0 jk
β j, k, p, q 2.7
Proof. Sufficiency. Suppose that the conditions 2.2-2.6 hold and x xjk ∈ c2∞ with P −
limj,k xjk L, say. So that for every > 0 there exists N > 0 such that |xjk | < || whenever
j, k > N.
Abstract and Applied Analysis
5
Then, we can write
∞ N ∞ N−1
∞
N
β j, k, p, q xjk β j, k, p, q xjk β j, k, p, q xjk
j0 k0
j0 k0
jN k0
∞
N−1
∞
β j, k, p, q xjk j0 kN
∞
β j, k, p, q xjk .
2.8
jN1 kN1
Therefore,
∞
N N ∞ N−1
∞
β j, k, p, q xjk ≤
β
j,
k,
p,
q
β
j,
k,
p,
q
x
x
x
jk j0 k0
j0 k0
jN k0
∞ N−1
β j, k, p, q x
j0 kN
2.9
∞
∞ |L| β j, k, p, q .
jN1 kN1
Letting p, q → ∞ and using the conditions 2.2–2.6, we get
∞
∞ lim
β
j,
k,
p,
q
x
jk ≤ |L| α.
p,q → ∞
j0 k0
2.10
Since is arbitrary, C1 − lim Ax αL. Hence A ∈ c2∞ , C1 α , that is, A is C1 -multiplicative.
Necessity 1. Suppose that A is C1 -multiplicative. Then, by the definition, the A-transform of
x exists and Ax ∈ C1 for each x ∈ c2∞ . Therefore, Ax is also bounded. Then, we can write
sup
∞ ∞ mn ajk xjk < M < ∞,
m,n j0 k0
2.11
for each x ∈ c2∞ . Now, let us define a sequence y yjk by
yjk ⎧
⎨sgn amn , 0 ≤ j ≤ r, 0 ≤ k ≤ r,
jk
⎩0,
otherwise,
2.12
m, n 0, 1, 2, . . .. Then, the necessity of 10 follows by considering the sequence y yjk in
2.11.
6
Abstract and Applied Analysis
Also, by the assumption, we have
lim
p,q → ∞
∞
∞ β j, k, p, q xjk α lim xjk .
j,k → ∞
j0 k0
2.13
Now let us define the sequence eil as follows:
1,
e 0,
il
j, k i, l,
otherwise,
2.14
and write sl i eil l ∈ N, r i l eil i ∈ N. Then, the necessity of 2.2, 2.4, and 2.5
follows from C1 − lim Aeil , C1 − lim Ar j and C1 − lim Ask , respectively.
Note that when α 1, the above theorem gives the characterization of A ∈ c2∞ , C1 reg .
Now, we are ready to construct our main theorem.
Theorem 2.2. For every bounded double sequence x,
C1∗ Ax ≤ αLx,
2.15
or PC − core{Ax} ⊆ αP − core{x} if and only if A is C1 -multiplicative and
lim sup
∞ ∞ β j, k, p, q α.
p,q → ∞ j0 k0
2.16
2
2
Proof. Necessity. Let 2.15 hold and for all x ∈ ∞
. So, since c2∞ ⊂ ∞
, then, we get
α−L−x ≤ −C1∗ −Ax ≤ C1∗ Ax ≤ αLx.
2.17
α lim inf x ≤ −C1∗ −Ax ≤ C1∗ Ax ≤ α lim sup x,
2.18
That is,
where
−C1∗ −Ax lim inf
p,q → ∞
∞ ∞
β j, k, p, q xjk .
2.19
j0 k0
2
, we get from 2.17 that
By choosing x xjk ∈ c∞
−C1∗ −Ax C1∗ Ax C1 − lim Ax α lim x.
This means that A is C1 -multiplicative.
2.20
Abstract and Applied Analysis
7
2
with ||y|| ≤ 1 such that
By Lemma 3.1 of Patterson 9, there exists a y ∈ ∞
∞
∞ β j, k, p, q .
C1∗ Ay lim sup
2.21
p,q → ∞ j0 k0
If we choose y v vjk , it follows
vjk 1
if j k,
0,
elsewhere.
2.22
Since vjk ≤ 1, we have from 2.15 that
α C1∗ Av lim sup
∞ ∞ β j, k, p, q ≤ αL vjk ≤ αv ≤ α.
p,q → ∞ j0 k0
2.23
This gives the necessity of 2.16.
Sufficiency 1. Suppose that A is C1 -regular and 2.16 holds. Let x xjk be an arbitrary
bounded sequence. Then, there exist M, N > 0 such that xjk ≤ K for all j, k ≥ 0. Now, we can
write the following inequality:
∞
∞
∞
β j, k, p, q β j, k, p, q
∞
β
j,
k,
p,
q
x
jk 2
j0 k0
j0 k0
−
≤
xjk β j, k, p, q − β j, k, p, q
2
∞ ∞ β j, k, p, q xjk j0 k0
∞ ∞ β j, k, p, q − β j, k, p, q xjk j0 k0
M
N β j, k, p, q ≤ x
j0 k0
x
N ∞ β j, k, p, q jM1 k0
M ∞ β j, k, p, q x
j0 kN1
8
Abstract and Applied Analysis
∞
sup xjk j,k≥M,N
∞ β j, k, p, q jM1 kN1
∞ ∞ β j, k, p, q − β j, k, p, q .
x
j0 k0
2.24
Using the condition of C1 -multiplicative and condition 2.16, we get
C1∗ Ax ≤ αLx.
2.25
This completes the proof of the theorem.
Theorem 2.3. For x, y ∈ 2∞ , if C1 − lim |x − y| 0, then C1 − core{x} C1 − core{y}.
Proof. Since C2 − lim |x − y| 0, we have C1 − limx − y 0 and C1 − lim−x − y 0. Using
definition of C1 − core, we take C1∗ x − y −C1∗ −x − y 0. Since C1∗ is sublinear,
0 −C1∗ − x − y ≤ −C1∗ −x − C1∗ y .
2.26
Therefore, C1∗ y ≤ −C1∗ −x. Since −C1∗ −x ≤ C1∗ x, this implies that C1∗ y ≤ C1∗ x. By an
argument similar as above, we can show that C1∗ x ≤ C1∗ y. This completes the proof.
Acknowledgment
The authors would like to state their deep thanks to the referees for their valuable suggestions
improving the paper.
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Abstract and Applied Analysis
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