Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 485730, 17 pages
doi:10.1155/2011/485730
Research Article
On Some Generalized B m-Difference Riesz
Sequence Spaces and Uniform Opial Property
Metin Başarır and Mahpeyker Öztürk
Department of Mathematics, Sakarya University, 54187 Sakarya, Turkey
Correspondence should be addressed to Metin Başarır, basarir@sakarya.edu.tr
Received 29 November 2010; Accepted 18 January 2011
Academic Editor: Radu Precup
Copyright q 2011 M. Başarır and M. Öztürk. 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.
q
q
q
We define the new generalized difference Riesz sequence spaces r∞ p, B m , rc p, B m , and r0 p, B m q
which consist of all the sequences whose B m -transforms are in the Riesz sequence spaces r∞ p,
q
q
rc p, and r0 p, respectively, introduced by Altay and Başar 2006. We examine some topological
q
q
q
properties and compute the α-, β-, and γ-duals of the spaces r∞ p, B m , rc p, B m , and r0 p, B m .
Finally, we determine the necessary and sufficient conditions on the matrix transformation
q
q
q
from the spaces r∞ p, B m , rc p, B m , and r0 p, B m to the spaces l∞ and c and prove that
q
q
m
m
sequence spaces r0 p, B and rc p, B have the uniform Opial property for pk ≥ 1 for all
k ∈ .
1. Introduction
Let w be the space of real sequences. We write l∞ , c, c0 for the sequence spaces of
all bounded, convergent and null sequences, respectively. Also, by bs, cs, and l1 , we
denote the sequence spaces of all bounded, convergent and absolutely convergent series,
respectively.
A linear topological space X over the real field R is said to be a paranormed space
if there is a subadditive function g : X → R such that gθ 0, gx g−x and scalar
multiplication is continuous; that is, |αn − α| → 0 and gxn − x → 0 imply gαn xn − αx → 0
for all α’s in R and all x’s in X, where θ is the zero vector in the linear space X. Assume
here and after that p pk is a bounded sequence of strictly positive real numbers with
sup pk H and M max{1, H}. Then, the linear space l∞ p, cp, c0 p, and lp were
defined by Maddox 1, 2, Nakano 3 and Simons 4 as follows:
2
l∞ p Journal of Inequalities and Applications
x xk ∈ w : sup|xk |pk < ∞ ,
k∈
x xk ∈ w : lim |xk − l|pk 0 for some l ∈ Ê ,
k→∞
pk
c0 p x xk ∈ w : lim |xk | 0 ,
c p 1.1
k→∞
which are the complete spaces paranormed by
g1 x sup|xk |pk /M ,
k∈
pk
l p x xk ∈ w :
|xk | < ∞ , with 0 < pk ≤ H < ∞ ,
1.2
k
which is the complete space paranormed by
g2 x 1/M
|xk |pk
.
1.3
k
For simplicity notation, here and in what follows, the summation without limits runs from 0
to ∞. We assume throughout that pk −1 pk −1 1 provided 1 < inf pk ≤ H < ∞ and denote
the collection of all finite subsets of Æ by F, where Æ {0, 1, 2, . . .}.
Let λ and μ be two sequence spaces and A ank be an infinite matrix of real numbers
ank , where n, k ∈ Æ . Then, we say that A defines a matrix mapping from λ into μ and we
denote it by writing A : λ → μ if for every sequence x xk ∈ λ the sequence Ax {Axn },
the A-transform of x, is in μ, where
Axn ank xk
n ∈ Æ .
k
1.4
By λ : μ, we denote the class of all matrices A such that A : λ → μ. Thus, A ∈ λ : μ
if and only if the series on the right side of 1.4 converges for each n ∈ λ. A sequence x is
said to be A-summable to α if Ax converges to α which is called as the A-limit of x.
Let qk be a sequence of positive numbers and
Qn n
qk ,
n ∈ Æ .
1.5
k0
q
Then, the matrix Rq rnk of the Riesz mean is given by
q
rnk
⎧
q
⎪
⎨ k,
Qn
⎪
⎩0,
0 ≤ k ≤ n,
k > n.
1.6
Journal of Inequalities and Applications
3
The Riesz sequence spaces introduced by Altay and Başar in 5, 6 are
⎧
⎫
pk
k
⎨
⎬
1 r q p x xk ∈ w :
qj xj < ∞ ,
⎩
⎭
Q
k k j0
with 0 < pk ≤ H < ∞ ,
⎫
⎧
pk
k
⎬
⎨
1
q
rc p x xk ∈ w : lim qj xj − l 0, for some l ∈ R ,
⎭
⎩
k → ∞ Qk
j0
⎫
⎧
pk
k
⎬
⎨
1
q
r0 p x xk ∈ w : lim qj xj 0 ,
⎭
⎩
k → ∞ Qk
j0
1.7
⎫
⎧
pk
k
⎬
⎨
1
q
r∞ p x xk ∈ w : sup
qj xj < ∞ ,
⎭
⎩
k∈N Qk j0
which are the sequence spaces of the sequences x whose Rq -transforms are in lp, cp, c0 p,
and l∞ p, respectively.
Also, Altay and Başar 7 introduced the generalized difference matrix B bnk by
bnk
⎧
⎪
r,
⎪
⎪
⎨
s,
⎪
⎪
⎪
⎩
0,
k n,
k n − 1,
1.8
0 ≤ k < n − 1 or k > n
for all k, n ∈ Æ , r, s ∈ Ê − {0}. The matrix B can be reduced the difference matrix Δ in case
r 1, s −1 by, where Δ denotes the matrix Δ Δnk defined by
Δnk ⎧
⎨−1n−k ,
n − 1 ≤ k ≤ n,
⎩0,
k < n − 1 or k > n.
1.9
The results related to the matrix domain of the matrix B are more general and more
comprehensive than the corresponding consequences of matrix domain of Δ and include
them 6, 8–13.
m
which reduced the difference
Başarır and Kayikçi 14 defined the matrix Bm bnk
m
m−1
matrix Δ ΔΔ in case r 1, s −1 by
m
bnk
⎞
⎧⎛
m
⎪
⎪
⎨⎝
⎠r m−n k sn−k ;
n
−
k
⎪
⎪
⎩
0;
max{0, n − m} ≤ k ≤ n,
0 ≤ k < max{0, n − m} or k > n,
1.10
4
Journal of Inequalities and Applications
and introduced the generalized Bm -difference Riesz sequence space which is the sequence
space of the sequences x whose Rq Bm -transforms are in lp.
The main purpose of this paper is to introduce the Bm -difference Riesz sequence spaces
q
q
q
r∞ p, Bm , rc p, Bm , and r0 p, Bm of the sequences whose Rq Bm -transform are in l∞ p, cp,
and c0 p, respectively, and to investigate some topological and geometric properties of them.
For simplicity, we take the matrix Rq Bm T.
2. Bm -Difference Riesz Sequence Spaces
Let us define the sequence y {yn q}, which is used, as the Rq Bm T-transform of a
sequence x xk , that is,
n
n−1 m
1 m−i k i−k
yn q Txn r
s qi xk
Qn k0 ik i − k
q
rm
qn xn ,
Qn
q
n ∈ Æ .
2.1
q
We define the Bm -difference Riesz sequence spaces r∞ p, Bm , rc p, Bm , and r0 p, Bm by
q r∞ p, Bm x xj ∈ w : Txn ∈ l∞ p ,
q
rc p, Bm x xj ∈ w : Txn ∈ c p ,
2.2
q
r0 p, Bm x xj ∈ w : Txn ∈ c0 p .
q
q
q
If m 1 then they are reduced the spaces r∞ p, B, rc p, B, and r0 p, B defined by Başarır in
q
q
q
15. If we take B Δ then we have r∞ p, Δm , rc p, Δm , and r0 p, Δm . If we take B Δ and
q
q
q
m 1 then we have r∞ p, Δ, rc p, Δ, and r0 p, Δ. If we take pk p for all k then we have
q
q
q
r∞ Bm , rc Bm , and r0 Bm .
We have the following.
q
Theorem 2.1. r0 p, Bm is a complete linear metric space paranormed by gB , defined by
gB x sup|Txk |pk /M ,
k∈
q
q
2.3
gB is a paranorm for the spaces r∞ p, Bm and rc p, Bm only in the trivial case with inf pk > 0 when
q
q
q
q
r∞ p, Bm r∞ Bm and rc p, Bm rc Bm .
Journal of Inequalities and Applications
5
q
q
Proof. We prove the theorem for the space r0 p, Bm . The linearity of r0 p, Bm with respect to
the coordinatewise addition and scalar multiplication that follow from the inequalities which
q
are satisfied for u, v ∈ r0 p, Bm 16
⎡ ⎡
k−1 k
m
1 r m−i j si−j qi uj
sup ⎣ ⎣
Q
i−j
k∈ k j0 ij
⎤
r m qk
uk
Qk
vj ⎦
⎤pk /M
⎦
vk ⎡ ⎡
⎤
k−1 k
m
1 ⎣ ⎣
r m−i j si−j qi uj ⎦
≤ sup
Q
k
i
−
j
k∈
j0 ij
⎤pk /M
r m qk
uk ⎦
Qk
⎡ ⎡
⎤
k−1 k
m
1 r m−i j si−j qi vj ⎦
sup ⎣ ⎣
i−j
k∈ Qk j0 ij
⎤pk /M
r m qk
⎦
vk ,
Qk
2.4
and for any α ∈ Ê 1,
|α|pk ≤ max 1, |α|M .
2.5
q
It is clear that gB θ 0 and gB −x gB x for all x ∈ r0 p, Bm . Again, the inequalities
2.4 and 2.5 yield the subadditivity of gB and
gB αu ≤ max{1, |α|}gB u.
2.6
q
Let {xn } be any sequence of the elements of the space r0 p, Bm such that
gB xn − x −→ 0,
2.7
and λn also be any sequence of scalars such that λn → λ, as n → ∞. Then, since the
inequality
gB xn ≤ gB x
gB xn − x
2.8
6
Journal of Inequalities and Applications
holds by subadditivity of gB , {gB xn } is bounded, and thus we have
gB λn xn − λx
⎡ ⎡
⎤
k−1 k
!
"
m
1 r m−i j si−j qi λn xjn − λxj ⎦
sup ⎣ ⎣
Q
k
i
−
j
k∈
j0 ij
|λn − λ|
1/M
⎡ ⎡
⎤
k−1 k
m
1 sup ⎣ ⎣
r m−i j si−j qi xjn ⎦
i−j
k∈ Qk j0 ij
⎤pk /M
r m qk λn xkn − λxk ⎦
Qk
⎤pk /M
r m qk n ⎦
x
Qk k ⎡ ⎡
⎤
k−1 k
!
"
m
1
⎣ ⎣
r m−i j si−j qi xjn − xj ⎦
|λ|1/M sup
i−j
k∈ Qk j0 ij
≤ |λn − λ|1/M gB xn 2.9
⎤pk /M
r m qk n
xk − xk ⎦
Qk
|λ|1/M gB xn − x,
which tends to zero as n → ∞. Hence, the scalar multiplication is continuous. Finally, it
q
is clear to say that gB is a paranorm on the space r0 p, Bm . Moreover, we will prove the
q
q
completeness of the space r0 p, Bm . Let xi be a Cauchy sequence in the space r0 p, Bm , where
q
i
xi {xk } {x0i , x1i , x2i , . . .} ∈ r0 p, Bm . Then, for a given ε > 0, there exists a positive integer
n0 ε such that
!
"
gB xi − xj < ε,
2.10
for all i, j ≥ n0 ε. If we use the definition of gB , we obtain for each fixed k ∈ Æ that
!
!
!
!
"
" "
" pk /M
< ε,
Txi − Txj ≤ sup Txi − Txj k
k
k
k
k∈
2.11
for i, j ≥ n0 ε which leads us to the fact that
!
Tx0
" !
" !
"
, Tx1 , Tx2 , . . .
k
k
k
2.12
is a Cauchy sequence of real numbers for every fixed k ∈ Æ . Since Ê is complete, it converges,
so we write Txi k → Txk as i → ∞. Hence, by using these infinitely many limits
Tx0 , Tx1 , Tx2 , . . ., we define the sequence {Tx0 , Tx1 , Tx2 , . . .}. From 2.11 with
j → ∞, we have
!
"
Txi − Txk < ε,
k
2.13
i ≥ n0 ε for every fixed k ∈ Æ . Since xi {xk } ∈ r0 p, Bm ,
i
q
!
" pk /M
< ε,
Txi k
2.14
Journal of Inequalities and Applications
7
for all k ∈ Æ . Therefore, by 2.13, we obtain that
!
" pk /M
|Txk |pk /M ≤ Txk − Txi !
" pk /M
< ε,
Txi k
2.15
k
for all i ≥ n0 ε. This shows that the sequence Tx belongs to the space c0 p. Since {xi } was
q
an arbitrary Cauchy sequence, the space r0 p, Bm is complete.
# m
Theorem 2.2. Let kij i−j r m−i j si−j qi / 0 for all k, m and 0 ≤ j ≤ k − 1. Then the Bm -difference
q
q
q
m
m
Riesz sequence spaces r∞ p, B , rc p, B , and r0 p, Bm are linearly isomorphic to the spaces l∞ p,
cp, and c0 p, respectively, where 0 < pk ≤ H < ∞.
q
Proof. We establish this for the space r∞ p, Bm . For the proof of the theorem, we should show
q
the existence of a linear bijection between the space r∞ p, Bm and l∞ p for 0 < pk ≤ H < ∞.
q
With the notation of 2.1, define the transformation S from r∞ p, Bm to l∞ p by x → y Sx.S is a linear transformation, morever; it is obviuos that x θ whenever Sx θ and hence
S is injective.
Let y yk ∈ l∞ p and define the sequence x xk by
k−1 n 1
sk−i
xk −1k−n m k−i
r
n0 in
k−i−1
m
k−i
1
Qn yn
qi
Qk
yk ,
r m qk
∀k ∈ Æ .
2.16
Then,
⎡
⎤
k
m
1 k−1 ⎣
gB x sup
r m−i j si−j qi xj ⎦
i−j
k∈ Qk j0 ij
pk /M
r m qk xk Qk
2.17
pk /M
p /M
k
sup δkj yj supyk k
g1 y < ∞,
k∈ j0
k∈
where
δkj ⎧
⎨1,
k j,
⎩0,
k/
j.
2.18
q
Thus, we have that x ∈ r∞ p, Bm . Consequently, S is surjective and is paranorm preserving.
q
Hence, S is linear bijection, and this explains that the spaces r∞ p, Bm and l∞ p are linearly
isomorphic.
q
q
3. The Basis for the Spaces rc p, Bm and r0 p, Bm q
q
In this section, we give two sequences of the points of the spaces r0 p, Bm and rc p, Bm which form the basis for those spaces.
8
Journal of Inequalities and Applications
If a sequence space λ paranormed by h contains a sequence bn with the property that
for every x ∈ λ, there is a unique sequence of scalars αn such that
lim h x −
n→∞
n
αk βk
0,
3.1
k0
#
then bn is called a Schauder basis or briefly basis for λ. The series αk βk which has the
#
sum x is then called the expansion of x with respect to bn and written as x αk βk .
Because of the isomorphism S is onto, defined in the proof of Theorem 2.2, the inverse
q
image of the basis of the spaces, c0 p and cp are the basis of the new spaces r0 p, Bm and
q
m
rc p, B , respectively.
We have the following.
Theorem 3.1. Let μk q Txk for all k ∈ Æ and 0 < pk ≤ H < ∞. Define the sequence bk q q
k
{bn q}n∈ of the elements of the space r0 p, Bm for every fixed k ∈ Æ by
⎛
⎞
⎧
k 1
n−i
m
n
−
i
−
1
⎪
s
⎪
⎪
⎠ 1 Qk ,
⎪
−1n−k m n−i ⎝
⎪
⎪
qi
r
⎨
n−i
ik
k bn q Qn
⎪
⎪
,
⎪
mq
⎪
r
⎪
n
⎪
⎩
0,
n > k,
3.2
k n,
k > n.
Then, one has the following.
q
q
a The sequence {bk q}k∈ is a basis for the space r0 p, Bm , and any x ∈ r0 p, Bm has a
unique representation of the form
x
μk q bk q .
3.3
k
b The set {z T−1 e, bk q} is a basis for the space rc p, Bm and any x ∈ rc p, Bm has
a unique representation of the form
q
x le
k μk q − lb
q ,
k
q
3.4
where
l lim Txk .
3.5
k→∞
q
Proof. It is clear that {bk q} ⊂ r0 p, Bm , since
Tbk q ek ∈ c0 p ,
for k ∈ Æ ,
3.6
Journal of Inequalities and Applications
9
for 0 < pk ≤ H < ∞, where ek is the sequence whose only non-zero term is 1 in kth place for
q
each k ∈ Æ . Let x ∈ r0 p, Bm be given. For every nonnegative integer m, we put
xm m
μk q bk q .
3.7
k0
Then, we obtain by applying T to 3.7 with 3.6 that
Txm m
m
μk q Tbk q Tk ek ,
k0
!
q
R
!
x−x
M
""
i
k0
⎧
⎨0,
3.8
0 ≤ i ≤ m,
⎩Tx , i > m.
i
Given ε > 0, then there exists an integer m0 such that
sup|Txi |pk /M <
i≥m
ε
,
2
3.9
for all m ≥ m0 . Hence,
!
"
ε
gB x − xm sup|Txi |pk /M ≤ sup|Txi |pk /M < < ε,
2
i≥m
i≥m0
3.10
q
for all m ≥ m0 , which proves that x ∈ r0 p, Bm is represented as in 3.3.
To show the uniqueness of this representation, we suppose that there exists a
representation
x
λk q bk q .
3.11
k
q
Since the linear transformation S from r0 p, Bm to c0 p, used in Theorem 2.2, is continuous
we have
Txn λk q Tbk q
k
n
k
k
λ k q en λ n q ;
n ∈ Æ,
3.12
which contradicts the fact that Txn μk q for all n ∈ Æ . Hence, the representation 3.3 of
q
x ∈ r0 p, Bm is unique. Thus, the proof of the part a of theorem is completed.
q
q
b Since {bk q} ⊂ r0 p, Bm and e ∈ cp, the inclusion {e, bk q} ⊂ rc p, Bm q
trivially holds. Let us take x ∈ rc p, Bm . Then, there uniquely exists an l satisfying 3.5. We
q
thus have the fact that u ∈ r0 p, Bm whenever we set u x − le. Therefore, we deduce by part
a of the present theorem that the representation of x given by 3.4 is unique and this step
concludes the proof of the part b of theorem.
10
Journal of Inequalities and Applications
q
q
q
4. The α-, β-, and γ -Duals of the Spaces rc p, Bm , r0 p, Bm , and r∞ p, Bm In this section, we prove the theorems determining the α-, β-, and γ -duals of the sequence
q
q
spaces rc p, Bm and r0 p, Bm .
For the sequence spaces λ and μ, define the set Sλ, μ by
S λ, μ z zk ∈ w : xz xk zk ∈ μ ∀x ∈ λ .
4.1
With the notation 4.1, the α-, β-, γ -duals of a sequence space λ, which are, respectively,
denoted by λα , λβ , and λγ are defined by
λα Sλ, l1 ,
λβ Sλ, cs,
λγ Sλ, bs.
4.2
Now, we give some lemmas which we need to prove our theorems
Lemma 4.1 see 17. A ∈ l∞ p : l1 if and only if
1/pk sup ank K
< ∞,
K∈F n k∈K
∀ integers K > 1.
4.3
Lemma 4.2 see 18. Let pk > 0 for every k ∈ Æ . Then, A ∈ l∞ p : l∞ if and only if
sup
n∈
|ank |K 1/pk < ∞,
∀ integers K > 1.
k
4.4
Lemma 4.3 see 18. Let pk > 0 for every k ∈ Æ . Then, A ∈ l∞ p : c if and only if
|ank |K 1/pk converges uniformly in n,
k
lim αnk ,
n→∞
∀k ∈ Æ .
∀ integers K > 1,
4.5
4.6
Journal of Inequalities and Applications
Theorem 4.4. For each m ∈
follows:
Æ,
11
define the sets R1 p, R2 p, R3 p, R4 p, R5 p, and R6 p as
k 1
$
a ak ∈ w : sup ∇i, n, kQk an
R1 p N∈F n k∈N ik
K>1
Qn an
1/pk K
< ∞, ∀K > 1 ,
r m qn
⎧
⎛
⎞
n
1/p
$⎨
a
k
k
⎝
⎠
∇i,
n,
k
a
converges
a ak ∈ w :
Q
R2 p j
k K
m
⎩
r qk
K>1
k jk 1
⎫
%
&
⎬
ak Qk 1/pk
uniformly in n and m K
∈ c0 , ∀K > 1 ,
⎭
r qk
⎧
⎛
$⎨
⎝ ak
R3 p a ak ∈ w :
r mq
⎩
k
K>1
k ∇i, n, k
aj ⎠Qk K 1/pk < ∞, ∀K > 1,
jk 1
⎫
n
⎬
∇i, n, k
ai Qk ∈ l∞ ,
⎭
ik 1
ak
m
r qk
n
⎞
k 1
'
R4 p ∇i, n, kQk ai
a ak ∈ w : sup N∈F n k∈N ik
K>1
Qn an
−1/pk K
< ∞, ∀K > 1 ,
r m qn
k 1
Qn an ∇i, n, kQk ai
R5 p a ak ∈ w :
<∞ ,
k ik
r m qn n
⎧
⎫
⎛
⎞ n
⎬
−1/p
'⎨
a
k
k
⎠Qk K
⎝
a ak ∈ w :
∇i,
n,
k
a
<
∞,
∀K
>
1
,
R6 p j
r mq
⎩
⎭
k
K>1
k jk 1
4.7
where
∇i, n, k −1n−k
n−i−1
m
sn−i
r m n−i
n−i
1
.
qi
4.8
Then,
q r∞ p, Bm
α
R1 p ,
q
rc p, Bm
α
q r∞ p, Bm
R4 p ∩ R5 p ,
q
rc p, Bm
q
r0 p, Bm
α
R4 p ,
γ
β
R2 p ,
q
rc p, Bm
q
r∞ p, Bm
β
γ
R3 p ,
R6 p ∩ cs,
R6 p ∩ bs,
q
r0 p, Bm
β
q
r0 p, Bm
γ
R6 p .
4.9
12
Journal of Inequalities and Applications
q
Proof. We give the proof for the space r∞ p, Bm . Let us take any a an ∈ w. We easily
derive with the notation
⎡
⎤
k−1 k
m
1 ⎣
yk r m−i j si−j qi xj ⎦
Qk j0 ij i − j
rm
qk xk ,
Qk
4.10
that
an xn n−1 k 1
k0
n
an Qn yn
r m qn
∇i, n, kan Qk yk
ik
4.11
unk yk Uy n ,
k0
n ∈ Æ , where U unk is defined by
unk
⎧k 1
⎪
⎪
⎪
∇i, n, kan Qi , 0 ≤ k ≤ n − 1,
⎪
⎪
⎨ ik
an Qn
⎪
,
k n,
⎪
⎪
r m qn
⎪
⎪
⎩
0,
k > n,
4.12
for all k, n ∈ Æ . Thus we deduce from 4.6 that ax an xn ∈ l1 whenever x xk ∈
q
r∞ p, Bm if and only if Uy ∈ l1 whenever y yk ∈ l∞ p. From Lemma 4.1, we obtain the
desired result that
)α
( q
R1 p .
r p, Bm
4.13
Consider the equation
⎛
n−1
⎝ ak
ak xk r m qk
k0
k0
n
V y n,
k 1
∇i, n, k
ik
n
⎞
aj ⎠Qk yk
jk 1
ak Qk yk
r m qk
4.14
n ∈ Æ ,
where V vnk defined by
vnk
⎧⎛
⎪
⎪
⎪
⎝ ak
⎪
⎪
⎪
⎨ r m qk
ak Qk
⎪
⎪
,
⎪
⎪
r m qk
⎪
⎪
⎩
0,
⎞
k 1
n
∇i, n, k
aj ⎠Qk ,
ik
0 ≤ k ≤ n − 1,
jk 1
k n,
k > n,
4.15
Journal of Inequalities and Applications
13
for all k, n ∈ Æ . Thus, we deduce by with 4.11 that ax ak xk ∈ cs whenever x q
xk ∈ r∞ p, Bm if and only if V y ∈ c whenever y yk ∈ l∞ p. Therefor, we derive
from Lemma 4.3 that
⎛
⎝ ak
r mq
k
k ∇i, n, k
aj ⎠Qk K 1/pk converges uniformly in n ∀K > 1,
1
⎞
n
jk
lim
ak Qk
k → ∞ r m qk
4.16
K 1/pk 0,
which shows that r q p, Bm β R2 p.
q
As this, we deduce by 4.11 that ax ak xk ∈ bs whenever x xk ∈ r∞ p, Bm if and only if V y ∈ l∞ whenever y yk ∈ l∞ p. Therefore, we obtain by Lemma 4.2 that
r q p, Bm γ R3 p and this completes proof.
q
q
Now we characterize the matrix mappings from the spaces r∞ p, Bm , rc p, Bm , and
and c. Since the following theorems can be proved by using standart
methods, we omit the detail.
q
r0 p, Bm to the spaces l∞
q
Theorem 4.5. (i) A ∈ r∞ p, Bm : l∞ if and only if
lim
ank
Qk M1/pk 0, ∀n, M ∈ Æ ,
qk
n
∇i, n, k
anj Qk M1/pk < ∞, ∀M ∈ Æ jk 1
k→∞
ank
sup m
n∈ k r qk
hold.
4.17
4.18
q
(ii) A ∈ rc p, Bm : l∞ if and only if 4.14,
⎛
ank
sup ⎝ m
r
qk
n∈ k ∇i, n, k
anj ⎠Qk M1/pk 0, ∃M ∈ Æ ,
jk 1
⎛
⎞ n
a
nk
∇i, n, k
anj ⎠Qk < ∞
sup ⎝ m
r qk
n∈ k jk 1
hold.
q
n
⎞
(iii) A ∈ r0 p, Bm : l∞ if and only if 4.14 and 4.18 hold.
4.19
14
Journal of Inequalities and Applications
q
Theorem 4.6. (i) A ∈ r∞ p, Bm : c if and only if 4.14,
⎛
ank
sup ⎝ m
r qk
n∈ k ∇i, n, k
anj ⎠Qk M1/pk < ∞,
1
n
jk
⎛
ank
such that lim ⎣ ⎝ m
n→∞
r qk
k ⎞
⎡
∃αk ⊂ Ê
∇i, n, k
4.20
⎤
1/p
anj ⎠Qk − αk M k ⎦ 0,
1
⎞
n
jk
∀M ∈ Æ ,
∀M ∈ Æ hold.
q
(ii) A ∈ rc p, Bm : c if and only if 4.14, 4.18,
∃αk ⊂ Ê
⎛
⎞
n
a
nk
⎠
⎝
∇i, n, k
anj Qk − α 0,
∃α ⊂ Ê such that lim m
n → ∞
r qk
jk 1
⎛
⎞
n
a
nk
⎠
⎝
such that lim ∇i, n, k
anj Qk − αk 0, ∀k ∈ Æ ,
n → ∞
r m qk
jk 1
∃αk ⊂ Ê
⎛
ank
such that sup⎝ m
r
qk
n∈ ∇i, n, k
4.22
⎠
anj Qk − αk M−1/pk < ∞,
1
n
jk
4.21
⎞
4.23
∃M ∈ Æ hold.
q
(iii) A ∈ r0 p, Bm : c if and only if 4.14, 4.18, 4.21, and 4.22 hold.
5. Uniform Opial Property of Bm -Difference Riesz Sequence Spaces
q
In this section, we investigate the uniform Opial property of the sequence spaces r0 p, Bm q
and rc p, Bm .
The Opial property plays an important role in the study of weak convergence of
iterates of mapping of Banach spaces and of the asymptotic behavior of nonlinear semigroup.
The Opial property is important because Banach spaces with this property have the weak
fixed point property 19 see 20, 21.
We give the definition of uniform Opial property in a linear metric space and use the
q
q
method in 22, and obtain that r0 p, Bm and rc p, Bm have uniform Opial property for
pk ≥ 1.
q
q
For a sequence x xn ∈ r0 p, Bm or x xn ∈ rc p, Bm and for i ∈ Æ , we use the
notation x|i x1, x2, . . . , xi, 0, 0, . . . and x|−i 0, 0, . . . , 0, xi 1, xi 2, . . ..
We know that every total paranormed space becomes a linear metric space with the
q
q
q
metric given by dx, y gx − y. It is clear that r∞ p, Bm , r0 p, Bm , and rc p, Bm are total
paranormed spaces with dx, y gB x − y.
Now, we can give the definition of uniform Opial property in a linear metric space.
Journal of Inequalities and Applications
15
A linear metric space X, d has the uniform Opial property if for each ε > 0 there
exists τ > 0 such that for any weakly null sequence {xn } in S0, r and x ∈ X with dx, 0 ≥ ε
the following inequality holds:
τ ≤ lim inf dxn
r
n→∞
5.1
x, 0.
Now, we give following lemma which we need it to prove our main theorem. It can be
proved by using same method in 14 we omit the detail.
Lemma 5.1. If lim infk → ∞ pk > 0 then for any L > 0 and ε > 0, there exists δ δε, L > 0 for
u, v ∈ X such that
dM u
v, 0 < dM u, 0
5.2
ε
q
q
whenever dM u, 0 ≤ L and dM v, 0 ≤ δ, where X r0 p, Bm or rc p, Bm .
q
q
Theorem 5.2. If pk ≥ 1, then r0 p, Bm and rc p, Bm have uniform Opial property.
q
q
Proof. We prove the theorem for r0 p, Bm . rc p, Bm can be proved by similiar way. For any
ε > 0, we can find a positive number ε0 ∈ 0, ε such that
rM
εM
> r
4
5.3
ε0 M .
q
Take any x ∈ r0 p, Bm with dM x, 0 ≥ εM and xn to be weakly null sequence in S0, r. By
this, we write
dM xn , 0 r M .
5.4
∞
! ε "M ε M
0
.
<
dM x|−q0 , 0 |Txk|pk <
4
4
kq 1
5.5
There exists q0 ∈ Æ such that
0
Furthermore, we have
εM ≤ dM x, 0 q0
|Txk|pk
k0
εM ≤
q0
|Txk|pk
k0
0
3εM ≤
|Txk|pk .
4
k0
q
∞
|Txk|pk ,
kq0 1
εM
,
4
5.6
16
Journal of Inequalities and Applications
By xn → 0, weakly, this implies that xn → 0, coordinatewise, hence there exists n0 ∈
that with 5.6
0
3εM ≤
|Txn k
4
k0
Æ such
q
xk|pk ,
5.7
for all n ≥ n0 . Lemma 5.1 asserts that
dM y
z, 0 ≤ dM y, 0
εM
,
4
5.8
whenever dM y, 0 ≤ r M and dM z, 0 ≤ ε0 . Again by xn → 0, weakly, there exists n1 > n0
such that dM xn|q0 , 0 < ε0 for all n > n1 , so by 5.8, we obtain that
xn|q0 , 0 < dM xn|−q0 , 0
dM xn|−q0
εM
,
4
5.9
hence,
dM xn , 0 −
∞
εM
< dM xn|−q0 , 0 |Txn k|pk ,
4
kq 1
0
r
M
∞
εM
<
−
|Txn k|pk ,
4
kq 1
5.10
0
for all n > n1 . This, together with 5.5, 5.6, implies that for any n > n1 ,
dM xn
x, 0 q0
|Txn k
xk|pk
k0
∞
|Txn k
xk|pk
kq0 1
≥
q0
|Txn k
xk|pk
k0
∞
pk |Txn k| kq0 1
∞
p
− |Txk| k kq0 1
>
3εM
4
> r
rM
ε0 M .
εM
4
−
εM
rM
4
εM
4
5.11
Journal of Inequalities and Applications
This means that dM xn x, 0 > r
uniform Opial property for pk ≥ 1.
17
q
ε0 , so we get that the sequence space r0 p, Bm has
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