Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 385029, 18 pages
doi:10.1155/2009/385029
Research Article
On the Generalized Bm -Riesz Difference Sequence
Space and β-Property
Metin Başarir1 and Mustafa Kayikçi2
1
2
Department of Mathematics, Sakarya University, 54187 Sakarya, Turkey
Duzce MYO, Duzce University, 81010 Duzce, Turkey
Correspondence should be addressed to Metin Başarir, basarir@sakarya.edu.tr
Received 1 May 2009; Accepted 17 July 2009
Recommended by Ramm Mohapatra
We introduce the generalized Riesz difference sequence space r q p, B m which is defined by
r q p, B m {x xk ∈ w : B m x ∈ r q p} where r q p is the Riesz sequence space defined by
Altay and Başar. We give some topological properties, compute the α , β duals, and determine the
Schauder basis of this space. Finally; we study the characterization of some matrix mappings on
this sequence space. At the end of the paper, we investigate some geometric properties of r q p, B m and we have proved that this sequence space has property β for pk ≥ 1.
Copyright q 2009 M. Başarir and M. Kayikçi. 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.
1. Introduction
Let w be the space of all real valued sequences. We write l∞ , c, c0 for the sequence spaces
of all bounded, convergent, and null sequences, respectively. Also by cs, l1 , and lp , we
denote the sequence spaces of all convergent, absolutely and p-absolutely, convergent series,
respectively; where 1 < p < ∞.
Let qk be a sequence of positive numbers and
Qn n
qk ,
n ∈ N.
1.1
k0
q
Then the matrix Rq rnk of the Riesz mean R, qn , which is triangle limitation matrix, is
given by
⎧
q
⎪
⎨ k , 0 ≤ k ≤ n,
q
1.2
rnk Qn
⎪
⎩0,
k > n.
q
It is well known that the matrix Rq rnk is regular if and only if Qn → ∞ as n → ∞.
2
Journal of Inequalities and Applications
q
q
Altay and Başar 1, 2 introduced the Riesz sequence space r q p, r∞ p, rc p, and
of nonabsolute type which is the set of all sequences whose Rq -transforms are in the
space lp, l∞ p, cp, and c0 p; respectively. Here and afterwards, p pk will be used as a
bounded sequence of strictly positive real numbers with sup pk H and M max{1, H} and
F denotes the collection of all finite subsets of N, where N {0, 1, 2, . . .}. The Riesz sequence
space introduced in 1 by Altay and Başar is
q
rc0 p
⎧
∞
k
⎨
1 qj xj
r q p x xk ∈ w :
⎩
Qk j0
k0
pk
⎫
⎬
<∞ ;
⎭
with 0 < pk ≤ H < ∞ .
1.3
The difference sequence spaces l∞ Δ, cΔ, and c0 Δ were first defined and studied
by Kızmaz in 3 and studied by several authors, 4–9. Başar and Altay 10 have studied the
sequence space bvp as the set of all sequences such that their Δ-transforms are in the space lp ;
that is,
bvp x xk ∈ w :
∞
p
|xk − xk−1 | < ∞ ,
1 ≤ p < ∞,
1.4
k0
where Δ denotes the matrix Δ Δnk defined by
Δnk ⎧
⎨−1n−k ;
n − 1 ≤ k ≤ n,
⎩0;
k < n − 1 or k > n.
1.5
The idea of difference sequences is generalized by Çolak and Et 11. They defined the
sequence spaces:
Δm λ {x xk ∈ w : Δm x ∈ λ},
1.6
where m ∈ N, Δ1 x xk −xk 1 , and Δm x ΔΔm−1 x, where Δm denotes the matrix Δm Δm
nk
defined by
Δm
nk
⎛
⎞
⎧
m
⎪
⎪
n−k
⎨−1 ⎝
⎠;
n
−
k
⎪
⎪
⎩
0;
max{0, n − m} ≤ k ≤ n,
1.7
0 ≤ k < max{0, n − m} or k > n,
for all k, n ∈ N and for any fixed m ∈ N.
Recently, Başarir and Öztürk 12 introduced the Riesz difference sequence space as
follows:
r q p, Δ x xk ∈ w : Δx xk − xk−1 ∈ r q p ;
with 0 < pk ≤ H < ∞ .
1.8
Journal of Inequalities and Applications
3
Başar and Altay defined the matrix B bnk which generalizes the matrix Δ Δnk . Now
m
and if we take r 1, s −1, then it corresponds to the matrix
we define the matrix Bm bnk
m
m
Δ Δnk . We define
m
bnk
⎧
⎪
m
⎪
⎪
⎨
r m−n k sn−k ;
n
−
k
⎪
⎪
⎪
⎩
0;
max{0, n − m} ≤ k ≤ n,
1.9
0 ≤ k < max{0, n − m} or k > n.
The results related to the matrix domain of the matrix Bm are more general and more
comprehensive than the corresponding consequences of matrix domain of Δm .
Our main subject in the present paper is to introduce the generalized Riesz difference
sequence space r q p, Bm which consists of all the sequences such that their Bm -transforms are
in the space r q p and to investigate some topological and geometric properties with respect
to paranorm on this space.
2. Basic Facts and Definitions
In this section we give some definitions and lemmas which will be frequently used.
Definition 2.1. Let λ and μ be two sequence spaces and let A ank be an infinite matrix of
real numbers ank , where n, k ∈ N. 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 ∈ N.
2.1
k0
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 2.1 converges for each n ∈ N and every x ∈ λ, and we
have Ax {Axn }n∈N ∈ μ for all x ∈ 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.
Definition 2.2. For any sequence space λ, the matrix domain λA of an infinite matrix A is
defined by
λA {x xk ∈ w : Ax ∈ λ}.
2.2
Definition 2.3. 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 bk
0,
2.3
k0
then bn is called a Schauder basis or briefly basis for λ. The series ∞
has the
k0 αk bk which
sum x is then called the expansion of x with respect to bn and is written as x ∞
k0 αk bk .
4
Journal of Inequalities and Applications
Definition 2.4. 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 .
2.4
With the notation of 2.2, the α-, β-, γ-duals of a sequence space λ, which are, respectively,
denoted by λα , λβ , λγ , are defined by
λα Sλ, l1 ,
λβ Sλ, cs,
λγ Sλ, bs.
2.5
Now we give some lemmas which we need to prove our theorems.
Lemma 2.5 see 13. i Let 1 < pk ≤ H < ∞ for every k ∈ N. Then A ∈ lp : l1 if and only if
there exists an integer K > 1 such that
sup
∞ ank K
−1
pk
< ∞.
2.6
K∈F k0 n∈K
ii Let 0 < pk ≤ 1 for every k ∈ N. Then A ∈ lp : l1 if and only if
sup sup
pk
< ∞.
ank
2.7
K∈F k∈N n∈K
Lemma 2.6 see 14. i Let 1 < pk ≤ H < ∞ for every k ∈ N. Then A ∈ lp : l∞ if and only if
there exists an integer K > 1 such that
sup
∞
n∈N k0
−1
a−1
nk K
pk
< ∞.
2.8
ii Let 0 < pk ≤ 1 for every k ∈ N. Then A ∈ lp : l∞ if and only if
sup |ank |pk < ∞.
2.9
n,k∈N
Lemma 2.7 see 14. Let 0 < pk ≤ H < ∞ for every k ∈ N. Then A ∈ lp : c if and only if
2.8, 2.9 hold, and
lim ank βk
n→∞
also holds.
for k ∈ N
2.10
Journal of Inequalities and Applications
5
3. Some Topological Properties of Generalized
Bm -Riesz Difference Sequence Space
Let us define the sequence y {yn q}, which will be used for the Rq Bm -transform of a
sequence x xk , that is,
n
n−1 1 m
q m
m−i k i−k
yn q R B xn s qi xk
r
Qn k0 ik i − k
rm
qn xn .
Qn
3.1
After this, by Rq Bm , we denote the matrix Rq Bm rnk m, q, r, s defined by
⎧
n
n−1 ⎪
m
1 ⎪
m−i k i−k
⎪
⎪
r
s qi , k < n,
⎪
⎪
Qn k0 ik i − k
⎪
⎪
⎪
⎨
rnk m, q, r, s r m
qn ,
k n,
⎪
⎪
⎪
Qn
⎪
⎪
⎪
⎪
⎪
⎪
⎩0,
k > n,
3.2
for all n, k, m ∈ N. Then we define
r q p, Bm x xk ∈ w : yn q ∈ l p
∞
k
k−1 1 m
m−i n i−n
s qi xn
x xk ∈ w :
r
Qk n0 in i − n
k0
pk
rm
qk xk
Qk
<∞ .
3.3
If we take m 1, then we have
⎧
⎨
⎡
∞
k−1
1 ⎣
qj r
r p, B x xk ∈ w :
⎩
Qk j0
k0
q
qj 1 s xj
⎤
qk rxk ⎦
⎫
⎬
pk
<∞ .
⎭
3.4
Here are some topological properties of the generalized Riesz difference sequence
space.
Theorem 3.1. The sequence space r q p, Bm is a complete linear metric space paranormed by
⎛
⎡ ⎤
k−1 ∞
k
m
1
⎣
r m−i j si−j qi xj ⎦
gx ⎝
Q
k
i
−
j
j0 ij
k0
where H supk pk and M max1, H.
r m qk
xk
Qk
pk ⎞1/M
⎠
,
3.5
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Journal of Inequalities and Applications
Proof. The linearity of r q p, Bm with respect to the co-ordinatewise addition and scalar
multiplication follows from the inequalites which are satisfied for u, v ∈ r q p, Bm 15:
∞
1/M
|R B uk
q
m
R B vk |
q
m
pk
k0
1/M
∞
pk
q m
≤
|R B uk |
1/M
∞
pk
q m
,
|R B vk |
k0
k0
3.6
and for any α ∈ R 16, we have
!
|α|pk ≤ max 1, |α|M .
3.7
It is obvious that gθ 0 and g−u gu for all u ∈ r q p, Bm . Let uk , vk ∈ r q p, Bm :
gu
⎡
⎛
∞
k k−1 1
m
⎣
r m−i j si−j qi uj
v ⎝
Qk j0 ij i − j
k0
⎤
vj ⎦
⎛
⎤
⎡
∞
k k−1 1
m
⎣
≤⎝
r m−i j si−j qi uj ⎦
i
−
j
Q
k
j0 ij
k0
r m qk
uk
Qk
⎛
⎤
⎡
∞
k k−1 1
m
⎝
⎣
r m−i j si−j qi vj ⎦
i
−
j
Q
k
j0
ij
k0
gu
v ≤ gu
r m qk
uk
Qk
pk ⎞1/M
vk ⎠
pk ⎞1/M
r m qk
vk
Qk
⎠
3.8
pk ⎞1/M
⎠
,
3.9
gv.
Again the inequalities 3.7 and 3.9 yield the subadditivity of g and
gαu ≤ max{1, |α|}gu.
3.10
Let {xn } be any sequence of the elements of the space r q p, Bm such that
gxn − x −→ 0,
3.11
and λn also be any sequence of scalars such that λn → λ. Then, since the inequality
gxn ≤ gx
gxn − x
3.12
Journal of Inequalities and Applications
7
holds by subadditivity of g, {gxn } is bounded, and thus we have
gλn xn − λx
⎛
⎡ ⎤
∞
k
k−1 "
#
m
1
⎣
⎝
r m−i j si−j qi λn xjn − λxj ⎦
Q
k
i
−
j
j0
ij
k0
r m qk λn xkn − λxk
Qk
⎛
⎡ ⎤
∞
k
k−1 " #
m
1
⎣
r m−i j si−j qi xjn ⎦
≤ |λn − λ|1/M ⎝
Qk j0 ij i − j
k0
⎛
1/M ⎝
|λ|
r m qk n
x
Qk k
⎡ ⎤
∞
k
k−1 "
#
m
1 m−i
j
i−j
n
⎣
r
s qi xj − xj ⎦
Qk j0 ij i − j
k0
≤ |λn − λ|1/M gxn pk ⎞1/M
⎠
,
pk ⎞1/M
⎠
r m qk n
xk − xk
Qk
pk ⎞1/M
⎠
,
|λ|1/M gxn − x,
3.13
which tends to zero as n → ∞. Hence the continuity of the scalar multiplication has shown.
Finally; it is clear to say that g is a paranorm on the space r q p, Bm .
Moreover; we will prove the completeness of the space r q p, Bm . Let xi be any Cauchy
sequence in the space r q p, Bm where xi {xki } {x0i , x1i , . . .} ∈ r q p, Bm . Then, for a given
ε > 0, there exists a positive integer n0 ε such that
"
#
g xi − xj < ε,
3.14
for all i, j ≥ n0 ε. If we use the definition of g, we obtain for each fixed k ∈ N that
"
q
m i
R B x
#
"
k
− R B x
q
m j
#
≤
k
∞ "
q
m i
R B x
k0
#
"
k
− R B x
q
m j
#
pk
1/M
k
< ε,
3.15
for i, j ≥ n0 ε which leads us to the fact that
"
Rq Bm x0
# "
#
!
, Rq Bm x1 , . . . ,
k
k
3.16
is a Cauchy sequence of real numbers for every fixed k ∈ N. Since R is complete, it converges,
so we write Rq Bm xi k → Rq Bm xk as i → ∞. Hence by using these infinitely many limits
Rq Bm x0 , Rq Bm x1 , . . ., we define the sequence {Rq Bm x0 , Rq Bm x1 , . . .}. Since 3.14 holds
for each p ∈ N and i, j ≥ n0 ε,
p "
Rq Bm xi
k0
#
k
#
"
− Rq Bm xj
pk
k
#%M
$ "
≤ g xi − xj
< εM .
3.17
8
Journal of Inequalities and Applications
Take any i ≥ n0 ε, first let j → ∞ in 3.17 and then p → ∞, to obtain gxi − x ≤ ε. Finally,
taking ε 1 in 3.17 and letting i ≥ n0 1, we have Minkowski’s inequality for each p ∈ N,
that is,
p
"
q
m i
R B x
#
pk
1/M
k
k0
"
#
≤ g xi − x
" #
g xi ≤ 1
" #
g xi ,
3.18
which implies that x ∈ r q p, Bm . Since gxi − x ≤ ε for all i ≥ n0 ε it follows that xi → x as
i → ∞, so r q p, Bm is complete.
"m#
Theorem 3.2. Let kij i−j r m−i j si−j qi /
0 for each k ∈ N. Then the difference sequence space
r q p, Bm is linearly isomorphic to the space lp where 0 < pk ≤ H < ∞.
Proof. For the proof of the theorem, we should show the existence of a linear bijection between
the spaces r q p, Bm and lp for 0 < pk ≤ H < ∞. With the notation of
⎡
⎤
k k−1 1 m
⎣
yk r m−i j si−j qi xj ⎦
Qk j0 ij i − j
rm
qk xk ,
Qk
3.19
define the transformation T from r q p, Bm to lp by x → y T x. However, T is a linear
transformation, moreover; it is obviuos that x θ whenever T x θ and hence T is injective.
Let y ∈ lp and define the sequence x xk by
xk k−1 n 1
n0
−1
k−n
k−i−1 1
Qn yn
k−i
qi
m
sk−i
r m k−i
in
Qk
yk ,
r m qk
for k ∈ N.
3.20
Then,
⎛
⎡
⎤
∞
k k−1 1
m
m−i
j
i−j
⎣
s qi xj ⎦
gx ⎝
r
Qk j0 ij i − j
k0
⎛
∞ k
⎝
δkj yj
rm
qk xk
Qk
pk ⎞1/M
⎠
pk ⎞1/M
3.21
⎠
k0 j0
∞
1/M
yk
pk
g1 y < ∞,
k0
where
δkj ⎧
⎨1,
k j,
⎩0,
k/
j,
3.22
Journal of Inequalities and Applications
9
and g1 y is a paranorm on lp. Thus, we have that x ∈ r q p, Bm . Consequently; T is
surjective and is paranorm preserving. Hence, T is a linear bijection and this explains that
the spaces r q p, Bm and lp are linearly isomorphic.
Now, the Schauder basis for the space r q p, Bm will be given in the following theorem.
k
Theorem 3.3. Define the sequence bk q {bn q}n∈N of the elements of the space r q p, Bm for
every fixed k ∈ N by
⎧
k 1
n−i
⎪
m n−i−1 1
⎪
⎪ −1n−k s
⎪
Qk ,
⎪
⎪
qi
r m n−i
⎪
n−i
⎪
ik
⎪
⎨
k Qk
bn q ,
⎪
⎪
⎪
r m qk
⎪
⎪
⎪
⎪
⎪
⎪
⎩0,
n > k,
n k,
3.23
k > n.
Then; the sequence {bk q}k∈N is a basis for the space r q p, Bm and any x ∈ r q p, Bm has
a unique representation of the form
x
∞
μk q b k q ,
3.24
k0
where μk q Rq Bm xk for all k ∈ N and 0 < pk ≤ H < ∞.
Proof. This can be easily obtained by 12, Theorem 5 so we omit the proof.
Theorem 3.4. i Let 1 < pk ≤ H < ∞ for every k ∈ N. Define the set Q1 p as follows:
Q1 p
⎧
k 1
∞ n−i &⎨
an
m n−i−1 1
n−k s
a ak ∈ w:sup
an Qk m Qn K −1
−1
⎩
n−i
qi
r qn
r m n−i
N∈F k0 n∈N ik
K>1
pk
<∞ .
3.25
Then; r q p, Bm α Q1 p.
10
Journal of Inequalities and Applications
ii Let 0 < pk ≤ 1 for every k ∈ N. Define the set Q2 p by
Q2 p
&
K>1
k 1
n−i an
m n−i−1 1
n−k s
a ak ∈ w : sup sup
an Qk m Qn K −1
−1
n−i
qi
r qn
r m n−i
N∈F k∈N n∈N ik
pk
<∞ .
3.26
Then; r q p, Bm α Q2 p.
Proof. i Let a ak ∈ w. We easily derive with the notation
⎡
⎤
k k−1 1 m
⎣
yk r m−i j si−j qi xj ⎦
Qk j0 ij i − j
1
qk xk ,
Qk
3.27
and the matrix U unk which is defined by
unk
⎧
k 1
n−i
⎪
m n−i−1 1
⎪
n−k s
⎪
⎪
an Qk , 0 ≤ k ≤ n − 1,
−1
⎪
⎪
qi
r m n−i
⎪
n−i
⎪
ik
⎪
⎨
an Qn
,
k n,
⎪
⎪
⎪
r m qn
⎪
⎪
⎪
⎪
⎪
⎪
⎩0,
k > n,
3.28
for all k, n ∈ N, thus, by using the method in 1,12 we deduce that ax an xn ∈ l1
whenever x xk ∈ r q p, Bm if and only if Uy ∈ l1 whenever y yk ∈ lp. From
Lemma 2.5i, we obtain the desired result that
(α
' q
Q1 p .
r p, Bm
3.29
ii This is easily obtained by proceeding as in the proof of i, above by using the
second part of Lemma 2.5. So we omit the detail.
Journal of Inequalities and Applications
11
Theorem 3.5. i Let 1 < pk ≤ H < ∞ for every k ∈ N. Define the set Q3 p as follow:
Q3 p
⎧
⎡⎛
∞
⎨
&⎪
⎣⎝ ak
a ak ∈ w :
⎪
r m qk
K>1⎩
k0
⎞ ⎤
n
k 1
n−i m n−i−1 1 n−k s
aj ⎠Qk ⎦K −1
−1
n−i
qi jk 1
r m n−i
ik
pk
<∞ .
3.30
Then; r q p, Bm β Q3 p ∩ cs.
ii Let 0 < pk ≤ 1 for every k ∈ N. Define the set Q4 p by
Q4 p
⎧
⎡⎛
⎨
ak
a ak ∈ w : sup ⎣⎝ m
⎩
r qk
k∈N
⎞ ⎤
k 1
n
n−i s
1
m n−i−1
aj ⎠Qk ⎦
−1n−k m n−i
n−i
qi jk 1
r
ik
pk
⎫
⎬
<∞ .
⎭
3.31
Then; r q p, Bm β Q4 p ∩ cs.
Proof. i If we take the matrix T tnk by
tnk
⎧⎛
⎪
⎪
a
⎪
⎨⎝ k
m
r qk
⎪
⎪
⎪
⎩
0,
k 1
−1n−k
ik
sn−i
rm
m
n−i
⎞
n
n−i−1 1 aj ⎠Qk ,
qi jk 1
n−i
0 ≤ k ≤ n,
3.32
k > n,
for k, n ∈ N and if we carry out the method which is used in 1, 12, we get that ax an xn ∈
cs whenever x xk ∈ r q p, Bm if and only if T y ∈ c whenever y yk ∈ lp. Hence we
deduce from Lemma 2.7 that
∞
k0
⎡⎛
⎣⎝ ak
r m qk
k 1
ik
−1n−k
sn−i
rm
m
n−i
⎞ ⎤
n
n−i−1 1 aj ⎠Qk ⎦K −1
n−i
qi jk 1
pk
< ∞,
3.33
and limn tnk exists which is shown that
(β
' q
Q3 p ∩ cs.
r p, Bm
3.34
12
Journal of Inequalities and Applications
ii This may be obtained in the similar way as in the proof of i above by using the
second part of Lemmas 2.6 and 2.7. So we omit the detail.
Now we will characterize the matrix mappings from the space r q p, Bm to the space
l∞ . It can be proved by applying the method in 1, 12. So we omit the proof.
Theorem 3.6. i Let 1 < pk ≤ H < ∞ for every k ∈ N. Then A ∈ r q p, Bm ; l∞ if and only if there
exists an integer K > 1 such that
QK sup
∞
n∈N k0
⎡⎛
⎣⎝ ank
r m qk
⎞ ⎤
n
k 1
n−i m n−i−1 1 n−k s
anj ⎠Qk ⎦K −1
−1
m n−i
n
−
i
q
r
i
ik
jk 1
{ank }k∈N ∈ cs,
pk
< ∞,
3.35
for each n ∈ N.
ii Let 0 < pk ≤ 1 for every k ∈ N. Then A ∈ r q p, Bm ; l∞ if and only if
⎡⎛
ank
sup ⎣⎝ m
r qk
n,k∈N
⎞ ⎤
k 1
n
n−i s
1
m n−i−1
anj ⎠Qk ⎦
−1n−k m n−i
n−i
qi jk 1
r
ik
pk
< ∞,
3.36
{ank }k∈N ∈ cs ,
for each n ∈ N.
4. β-Property of Generalized Riesz Difference Sequence Space
In the previous section; we show that the sequence space r q p, Bm , which is the space of all
pk
q m
< ∞, is a complete paranormed space.
real sequences x xn such that ∞
k0 |R B xk |
∞
1/M
It is paranormed by gx k0 |Rq Bm xk |pk for all x xn ∈ r q p, Bm , where M max{1, H}; H supk pk . We recall that a paranormed space is total if gx 0 implies x 0. Every total paranormed space becomes a linear metric space with the metric given by
dx, y gx − y. It is clear that r q p, Bm is a total paranormed space.
In this section, we investigate some geometric properties of r q p, Bm . First we give the
definition of the property β in a paranormed space and we will use the method in 17 to
prove the property β. Consequently, we obtain that r q p, Bm has property β for pk ≥ 1.
From here, for a sequence x xn ∈ r q p, Bm and for i ∈ N, we use the notation
x|i x1, x2, . . . , xi, 0, 0, . . . and x|N−i 0, 0, . . . , 0, xi 1, xi 2, . . ..
Now we give the definition of the property β in a linear metric space.
Definition 4.1. A linear metric space X, d is said to have the property β if for each ε > 0
and r > 0, there exists δ > 0 such that for each element x ∈ B0, r and each sequence xn in
n, there is an index k for which dx xk /2, 0 ≤ r − δ.
B0, r with dxn , xm ≥ ε for all m /
Journal of Inequalities and Applications
13
Lemma 4.2. If lim infk → ∞ pk > 0, then for any L > 0 and ε > 0, and for any u, v ∈ r q p, Bm , there
exists δ δε, L > 0 such that
dM u
v, 0 < dM u, 0
4.1
ε,
whenever dM u, 0 ≤ L and dM v, 0 ≤ δ.
Proof. Let ε > 0 and L > 0 be given. Let 0 < α0 < lim infk → ∞ pk and α0 < 1, there exists k0 ∈ N
such that 0 < α0 ≤ pk for all k ≥ k0 . Let α min{pk : k 1, 2, . . . , k0 ; α0 }. Thus α ≤ pk for all
k ∈ N. There exists K0 ≥ 2 such that
4.2
dM 2u, 0 ≤ K0 dM u, 0,
for all u ∈ r q p, Bm . Set β 2α ε/2K0 L1/α . There exists K1 ≥ K0 such that
dM
2
u, 0 ≤ K1 dM u, 0,
β
4.3
for all u ∈ r q p, Bm . Set δ 2α ε/2βα K1 . Assume that dM u, 0 ≤ L and dM v, 0 ≤ δ. We
recall that x|i x1, x2, . . . , xi, 0, 0, . . . and x|N−i 0, 0, . . . , 0, xi 1, xi 2, . . .. With
these notations, let A {k ∈ N − i : pk < 1} and C {k ∈ N − i : pk ≥ 1}. By using convexity
of the function ft |t|pk for all pk ≥ 1 and the fact that a bpk ≤ apk bpk for pk < 1 and
0 < βpk < βα where β ∈ 0, 1 and k ∈ N, we have
dM u
$
v, 0 dM 1 − β u
∞
# %
β−1 v , 0
"
β u
$
Rq Bm 1 − β ui
"
β ui
#%
β−1 vi
pi
i0
≤
∞
'
(
Rq Bm 1 − β ui
$ "
Rq Bm β ui
#%
β−1 vi
pi
'
(
Rq Bm 1 − β ui
$ "
Rq Bm β ui
#%
β−1 vi
pi
i0
i∈A
$ "
Rq Bm β ui
'
(
Rq Bm 1 − β ui
i∈C
q m
≤ 1−β
|R B ui|pi
i∈A
i∈A
q m
1−β
|R B ui|pi
i∈C
≤
|Rq Bm ui|pi
i∈A
|Rq Bm ui|pi
i∈C
$
Rq Bm β ui
#%
β−1 vi
%
β−1 vi
$
Rq Bm β ui
i∈C
βα
$
Rq Bm ui
i∈A
βα
i∈C
$
Rq Bm ui
β−1 vi
pi
β−1 vi
%
pi
%
β−1 vi
pi
pi
%
pi
14
Journal of Inequalities and Applications
≤
∞
|Rq Bm ui|pi
βα
i0
∞
$
Rq Bm ui
β−1 vi
%
pi
i0
≤ dM u, 0
βα
∞
i0
≤ dM u, 0
βα
βα
pi
"
$
2−1 2Rq Bm ui
%#
β−1 vi
pi
%#
β−1 vi
$
2−1 2Rq Bm ui
≤ dM u, 0
βα
%#
β−1 vi
i∈A
"
i∈C
"
$
2−1 2Rq Bm ui
βα
$
2−1 2Rq Bm ui
i∈A
$
2−1 2Rq Bm ui
pi
#%
2Rq Bm β−1 vi
"
#%
2Rq Bm β−1 vi
"
pi
pi
i∈C
≤ dM u, 0
βα
βα
%
2−1 2Rq Bm β−1 vi
i∈A
1
β
2
α |2R B ui|
1
β
2
pi
q
m
pi
pi
i∈C
≤ dM u, 0
dM u
2−1 2Rq Bm ui
i∈A
$
α ∞
1
β
2
α 2Rq Bm β−1 vi
pi
i∈C
∞
1 α
β
|2Rq Bm ui|pi
2
i0
2Rq Bm β−1 vi
pi
i0
≤ dM u, 0
1 2α ε M
d 2u, 0
2α 2K0 L
≤ dM u, 0
ε
2
v, 0 ≤ dM u, 0
ε.
#
1 α M " −1
β d 2β v, 0
α
2
1 α
2α ε
β
K
,
1
2α
2βα K1
4.4
Lemma 4.3. If lim infn → ∞ pn > 0, then for any x ∈ r q p, Bm , there exists k0 ∈ N and θ ∈ 0, 1
such that
dM
for all k ∈ N with k ≥ k0 .
x|N−k
1 − θ M ,0 ≤
d x|N−k , 0
2
2
4.5
Journal of Inequalities and Applications
15
Proof. Let α be a real number such that 1 < α < lim infn → ∞ pn . Then there exists k0 ∈ N such
that α ≤ pk for all k ≥ k0 . Let θ ∈ 0, 1 be a real number such that 1/2α < 1 − θ/2. Then for
each x ∈ r q p, Bm and k ≥ k0 , we have
dM
∞
x|N−k
Rq Bm xi
,0 2
2
ik 1
≤
pi
α ∞
1
|Rq Bm xi|pi
2 ik 1
4.6
∞
1 − θ ≤
|Rq Bm xi|pi
2 ik 1
1 − θ M d x|N−k , 0 .
2
Theorem 4.4. If pk ≥ 1, then r q p, Bm has property β.
Proof. Let ε > 0 and xn ⊂ B0, r with dxn , xm ≥ ε for m /
n. Take 0 < ε0 < εM . There exists
M
δ > 0 such that ε − δ ≥ ε0 . Let x ∈ B0, r. Since for each j ∈ N, xn j∞
n1 is bounded, by
using the diagonal method, we have that for each q ∈ N, we can find a subsequence xna of
xn such that xna j converges for all j ∈ N with 1 ≤ j ≤ q. Since xna j is Cauchy sequence
for all 1 ≤ j ≤ q, there exists tq ∈ N such that
q
|Rq Bm xna k − Rq Bm xnb k|pk k0
q
|Rq Bm xna k − xnb k|pk < δ,
4.7
k0
for all na , nb ≥ tq . Then we see that
ε < dxna , xnb ∞
|Rq Bm xna k − xnb k|pk
1/M
,
k0
εM ≤
q
|Rq Bm xna k − xnb k|pk
k0
∞
|Rq Bm xna k − xnb k|pk ,
4.8
kq 1
εM ≤ δ
∞
|Rq Bm xna k − xnb k|pk .
kq 1
Therefore, for each q ∈ N, there exists tq ∈ N such that
"
#
dM xna |N−q , xnb |N−q ≥ εM − δ ≥ ε0 ,
4.9
16
Journal of Inequalities and Applications
with σ1 < σ2 < · · ·
for all na , nb ≥ tq . Hence, there is a sequence of positive integers σq ∞
q1
such that
∞
#
"
"
#
Rq Bm xσq k
dM xσq |N−q , 0 pk
≥
kq 1
ε0
,
2
4.10
for all q ∈ N. By Lemma 4.3, there exists q0 ∈ N and θ ∈ 0, 1 such that
dM
u
#
1 − θ M "
|N−q
,0 ≤
d u|N−q , 0 ,
2
2
4.11
for all u ∈ r q p, Bm and q ≥ q0 . Let δ0 be a real number corresponding to Lemma 4.2 with
ε
θ ε0
· ,
4 2
4.12
and L r M , that is
dM u
v, 0 < dM u, 0
θ ε0
· ,
4 2
4.13
whenever dM u, 0 ≤ r M and dM v, 0 ≤ δ0 . Since x ∈ B0, r, we have that dM x, 0 ≤ r M .
Let q ≥ q0 be such that
"
#
dM x|N−q , 0 ≤ δ0 .
4.14
Put u xσq |N−q and v x|N−q . Then
dM
x
∞
#
"
#
σq |N−q
Rq Bm xσq k
, 0 dM
,0 2
2
kq 1
"u
d
M
"v
2
pk
< rM,
#
"
#
∞
, 0 dM x|N−q , 0 |Rq Bm xk|pk < δ0 .
kq 1
4.15
Journal of Inequalities and Applications
17
Hence;
dM
"u
"
q m
∞
R
xσq k
B
#
v
,0 2
2
kq 1
≤
xk
∞
Rq Bm xσq k
pk
Rq Bm xk
pk
4.16
2
kq 1
"u
#
#
,0
θ ε0
·
2
4 2
θ ε0
1 − θ M
d u, 0
· ,
≤
2
4 2
≤ dM
dM
"u
∞
# 1 − θ ,0 Rq Bm xσq k
2
2 kq 1
v
pk
θ ε0
· .
4 2
4.17
By using 4.17 and convexity of the function ft |t|pk , k ∈ N, we have
dM
x
xσq
2
"
∞ Rq B m xσq k
,0 2
k0
∞ Rq B m x k
σq
q
Rq Bm xσq k
pk
Rq Bm xk
pk
Rq Bm xk
pk
2
k0
∞
Rq Bm xσq k
kq 1
1 q m
|R B xk|pk
2 k0
q
≤
#
2
k0
≤
xk
Rq Bm xk
2
1 q m
R B xσq k
2 k0
q
∞
1 − θ Rq Bm xσq k
2 kq 1
1 q m
|R B xk|pk
2 k0
q
≤
−
rM
2
pk
pk
θ ε0
·
4 2
θ ε0
·
4 2
pk
θ ε0
·
4 2
∞
1
Rq Bm xσq k
2 k0
∞
θ Rq Bm xσq k
2 kq 1
r M θ ε0
− ·
2
2 2
θ ε0
≤ rM − · .
4 2
≤
pk
pk
4.18
18
Hence dM x
Journal of Inequalities and Applications
xσq /2, 0 ≤ r M − θ/4 · ε0 /2
dM
""
x
1/M
. So this implies that
#
#
xσq /2, 0 ≤ r − δ
4.19
for some δ > 0. Finally; we can say that the sequence space r q p, Bm has property β.
Acknowledgment
We would like to express our gratitude to the reviewer for his/her careful reading and
valuable suggestions which improved the presentation of the paper.
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