# Banach-Saks Type and Gurariǐ Modulus of Convexity of Some Banach Sequence Spaces

Authors: Henryk Hudzik; Vatan Karakaya; Mohammad Mursaleen; Necip Simsek
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Banach-Saks type is calculated for two types of Banach sequence spaces and Gurariǐ modulus of convexity is estimated from
above for the spaces of one type among them.


---

## Body

## 1. Introduction

Recently, there has been a lot of interest in investigating geometric properties of sequence spaces besides topological and some other usual properties. In the literature, there are many papers concerning geometric properties of various Banach sequence spaces. For example, geometry of Orlicz spaces and of Musielak-Orlicz spaces has been studied in [1–25]. Several authors including Cui and Hudzik [26–29], Cui and Meng [30], Suantai [31], and Lee [32] investigated the geometric properties of Cesàro sequence space ces
(
p
). Also Cesàro-Orlicz sequence spaces equipped with Luxemburg norm have been studied in [5, 33–36]. Additionally, geometry of Orlicz-Lorentz sequence spaces and of generalized Orlicz-Lorentz sequence space were studied in [37, 38]. Furthermore, Mursaleen et al. [39] studied some geometric properties of normed Euler sequence space. Additionally, Hudzik and Narloch [40] have studied relationships between monotonicity and complex rotundity properties with some consequences. Besides, some geometrical properties of Calderon-Lozanovskii sequence spaces have been investigated in [41–43].Quite recently, Karakaya [20] defined a new sequence space involving lacunary sequence space equipped with the Luxemburg norm and studied Kadec-Klee (
H
) and rotundity (
R
) properties of these spaces. Further information on topological and geometric properties of sequence spaces can be found in [39, 44–62].LetX be a real Banach space and S
(
X
) and B
(
X
) be the unit sphere and the unit ball of X, respectively. Let ℓ
0, c
0, c, ℓ
∞, and ℓ
1 be the spaces of all real sequences, null, convergent, and bounded sequences and absolutely convergent series, respectively, and let c
00 be the space of those real sequences which have only a finite number of nonzero coordinates and ℓ
p
=
{
x
=
x
(
i
)
:
∑
i
=
1
∞
|
x
(
i
)
|
p
<
∞
}.Note thatc
0, c, and ℓ
∞ are Banach spaces with the sup-norm ∥
x
∥
∞
=
sup
⁡
i
|
x
(
i
)
| and ℓ
p
(
1
≤
p
<
∞
) are Banach spaces with the norm ∥
x
∥
p
=
(
∑
‍
|
x
(
i
)
|
p
)
1
/
p, while c
00 is not a Banach space with respect to any norm.Let us recall that a sequence{
v
(
i
)
}
i
=
1
∞ in a Banach space X is called S
c
h
a
u
d
e
r
b
a
s
i
s of X (or b
a
s
i
s for short) if for each x
∈
X there exists a unique sequence {
λ
(
i
)
}
i
=
1
∞ of scalars such that x
=
∑
i
=
1
∞
‍
λ
(
i
)
v
(
i
), that is, lim
⁡
n
→
∞
∑
i
=
1
n
‍
λ
(
i
)
v
(
i
)
=
x.A sequence spaceX with a linear topology is called a K-s
p
a
c
e if each of the projection maps P
j
:
X
→
ℂ defined by P
j
(
x
)
=
x
(
i
) for x
=
(
x
(
i
)
)
i
=
1
∞
∈
X is continuous for each natural j. A F
r
e
´
c
h
e
t
s
p
a
c
e is a complete metric linear space and the metric is generated by an F-norm and a Fréchet space which is a K-space is called an F
K-s
p
a
c
e; that is, a K-space X is called an F
K-space if X is a complete linear metric space. In other words, X is an F
K-space if X is a Fréchet space with continuous coordinate projections. All the sequence spaces mentioned above are F
K spaces except the space c
00.AnF
K-space X which contains the space c
00 is said to have the p
r
o
p
e
r
t
y
A
K if, for every sequence {
x
(
i
)
}
∈
X, x
=
∑
i
=
1
∞
‍
x
(
i
)
e
(
i
), where e
(
i
)
=
(
0,0
,
…
,
1
(
i
th-place
)
,
0,0
,
…
). The spaces c
0 equipped with the sup-norm,
ℓ
p
(
1
≤
p
<
∞
) equipped with the norm ∥
x
∥
=
(
∑
i
=
1
∞
‍
|
x
(
i
)
|
p
)
1
/
p and ℓ
0 endowed with the metric d
(
x
,
y
)
=
∑
i
=
1
∞
‍
2
-
i
(
|
x
(
i
)
-
y
(
i
)
|
/
(
1
+
|
x
(
i
)
-
y
(
i
)
|
)
) have property A
K, while c and ℓ
∞ do not have property A
K. Banach spaces with continuous coordinate projections are called B
K-spaces.A Banach spaceX is said to be a K
o
¨
t
h
e
s
e
q
u
e
n
c
e
s
p
a
c
e (see [28, 63]) if X is a subspace of ℓ
0 such that(i)
ifx
∈
ℓ
0
,
y
∈
X, and |
x
(
i
)
|
≤
|
y
(
i
)
|, for all i
∈
ℕ, then x
∈
X and ∥
x
∥
≤
∥
y
∥;
(ii)
there exists an elementx
∈
X such that x
(
i
)
>
0 for all i
∈
ℕ.We say thatx
∈
X is order continuous if, for any sequence (
x
n
) in X
+ (the positive cone in X) such that x
n
(
i
)
≤
|
x
(
i
)
| and x
n
(
i
)
→
0
(
n
→
∞
) for each i
∈
ℕ
(
n
→
∞
), we have ∥
x
n
∥
→
0
(
n
→
∞
) holds.A Köthe sequence spaceX is said to be order continuous that if all sequences in X are order continuous. It is easy to see that x
∈
X is order continuous if and only if ∥
(
0,0
,
…
,
0
,
x
(
n
+
1
)
,
x
(
n
+
2
)
,
…
)
∥
→
0 as n
→
∞.A Köthe sequence spaceX is said to have the F
a
t
o
u
p
r
o
p
e
r
t
y if, for any real sequence x
∈
ℓ
0 and any {
x
n
} in X such that x
n
↑
xcoordinatewisely and sup
⁡
n
∥
x
n
∥
<
∞, we have the fact that x
∈
X and ∥
x
n
∥
→
∥
x
∥.A Banach spaceX is said to have the B
a
n
a
c
h-S
a
k
s
p
r
o
p
e
r
t
y if every bounded sequence {
x
n
} in X admits a subsequence {
z
n
} such that the sequence {
t
k
(
z
)
} is convergent in X with respect to the norm, where
(1)
t
k
(
z
)
=
1
k
(
z
1
+
z
2
+
⋯
+
z
k
)
,
∀
k
∈
ℕ
.
A Banach space X is said to have the w
e
a
k
B
a
n
a
c
h
-
S
a
k
s
p
r
o
p
e
r
t
y whenever given any weakly null sequence {
x
n
} in X there exists its subsequence {
z
n
} such that the sequence {
t
k
(
z
)
} converges to zero strongly.Given anyp
∈
(
1
,
∞
), we say that a Banach space (
X
,
∥
·
∥
) has the B
a
n
a
c
h
-
S
a
k
s
p
r
o
p
e
r
t
y
o
f
t
y
p
e
p if there exists a constant c
>
0 such that every weakly null sequence {
x
k
} has a subsequence {
x
k
ℓ
} such that (see [22])
(2)
∥
∑
ℓ
=
1
n
‍
x
k
ℓ
∥
≤
c
n
1
/
p
(
∀
n
∈
ℕ
)
.
The Banach-Saks property of type p
∈
(
1
,
∞
) and the weak Banach-Saks property for Cesa
`ro sequence spaces have been considered in [28]. These properties and stronger property (
S
p
) for Musielak-Orlicz and Nakano sequence spaces have been studied in [17].We say that a Banach spaceX has the w
e
a
k
f
i
x
e
d
p
o
i
n
t
p
r
o
p
e
r
t
y if every nonexpansive self-mapping defined on a nonempty weakly compact convex subset A of X has a fixed point in A.In [64], Garcia-Falset introduced the following coefficient for a Banach space (
X
,
∥
·
∥
):
(3)
R
(
X
)
=
sup
⁡
{
liminf
⁡
n
→
∞
∥
x
n
-
x
∥
:
x
∈
B
(
X
)
,
{
x
n
}
⊂
B
(
X
)
,
x
n
⟶
0
weakly
liminf
⁡
n
→
∞
}
and he proved (see [64, 65]) that a Banach space X with R
(
X
)
<
2 has the weak fixed point property.Clarksonm
o
d
u
l
u
s
o
f
c
o
n
v
e
x
i
t
y of a normed space (
X
,
∥
·
∥
) is defined (see Clarkson [66] and Day [67]) by the formula
(4)
δ
X
(
ε
)
=
inf
⁡
{
1
-
∥
x
+
y
∥
2
;
x
,
y
∈
S
(
X
)
,
∥
x
-
y
∥
=
ε
}
for any ε
∈
[
0,2
]. The inequality δ
X
(
ε
)
>
0 for all ε
∈
(
0,2
] characterizes the uniform convexity of X and the equality δ
X
(
2
)
=
1 characterizes strict convexity (=rotundity) of X.The Gurariǐ modulus of convexity of a normed spaceX is defined (see [68, 69]) by
(5)
β
X
(
ε
)
=
inf
⁡
{
1
-
inf
⁡
α
∈
[
0,1
]
∥
α
x
+
(
1
-
α
)
y
∥
;
x
,
y
∈
S
(
X
)
,
∥
x
-
y
∥
=
ε
1
-
inf
⁡
α
∈
0,1
]
∥
α
x
+
(
1
-
α
)
y
∥
;
}
for any ε
∈
[
0,2
]. It is obvious that δ
X
(
ε
)
≤
β
X
(
ε
) for any Banach space X and any ε
∈
[
0,2
]. It is also known that β
X
(
ε
)
≤
2
δ
X
(
ε
) for any ε
∈
[
0,2
] and that X is rotund if and only if β
X
(
ε
)
=
2 and as well as that X is uniformly convex if and only if β
X
(
ε
)
>
0 for any ε
∈
[
0,2
]. Gurariǐ [68] proved that if X
=
c
0 is renormed by the norm
(6)
∥
x
∥
=
∥
x
∥
∞
+
(
∑
n
=
0
‍
(
P
x
n
2
n
)
2
)
1
/
2
,
∀
{
x
n
}
∈
c
0
,
then β
X
(
ε
)
=
0 for any ε
∈
[
0,2
) and β
X
(
2
)
=
1.Gurariǐ and Sozonov [70] proved that a normed linear space (
X
,
∥
·
∥
) is an inner product space if and only if, for every x
,
y
∈
S
(
X
)
(7)
inf
⁡
α
∈
[
0,1
]
∥
α
x
+
(
1
-
α
)
y
∥
=
∥
x
+
y
∥
2
.
Zanco and Zucchi [71] showed an example of a normed space X with δ
X
(
2
)
≠
β
X
(
2
).Now, we will define Köthe sequence spacesm
(
ϕ
,
p
) and ℓ
p
(
u
,
v
) that will be considered in this paper.Let𝒞 denote the set whose elements are finite sets of distinct positive integers. Given any element σ of 𝒞, we denote by c
(
σ
) the sequence {
c
n
(
σ
)
} such that c
n
(
σ
)
=
1 for n
∈
σ, and c
n
(
σ
)
=
0 otherwise. Further, we define
(8)
𝒞
τ
=
{
σ
∈
𝒞
:
∑
n
=
1
∞
‍
c
n
(
σ
)
≤
τ
}
;
that is, 𝒞
τ is the set of those σ whose support has cardinality at most τ. Let us define
(9)
Φ
=
{
Δ
ϕ
k
≥
0
and
Δ
(
ϕ
k
k
)
≤
0
,
∀
k
∈
ℕ
ϕ
=
{
ϕ
n
}
n
=
1
∞
∈
ℓ
0
:
ϕ
1
>
0
,
Δ
ϕ
k
≥
0
,
Δ
(
ϕ
k
k
)
≤
0
,
∀
k
∈
ℕ
}
,
where Δ
ϕ
n
=
ϕ
n
-
ϕ
n
-
1.Given anyϕ
∈
Φ, we define the following sequence space, introduced in [55]:
(10)
m
(
ϕ
)
=
{
x
=
{
x
n
}
n
=
1
∞
∈
ℓ
0
:
sup
⁡
τ
≥
1
(
sup
⁡
σ
∈
𝒞
τ
(
1
ϕ
τ
∑
n
∈
σ
‍
|
x
n
|
)
)
<
∞
}
.
Sargent [55] established the relationship of this space to the space ℓ
p
(
1
≤
p
≤
∞
) and characterized some matrix transformations. In [49], matrix classes (
X
,
m
(
ϕ
)
) have been characterized, where X is assumed to be any F
K-space.Recently in [52], some of the geometric properties of m
(
ϕ
) have been investigated. In [61], Tripathy and Sen extended the space m
(
ϕ
) to m
(
ϕ
,
p
) as follows:
(11)
m
(
ϕ
,
p
)
:
=
{
x
=
{
x
n
}
n
=
1
∞
∈
ℓ
0
:
sup
⁡
τ
≥
1
(
sup
⁡
σ
∈
𝒞
τ
(
1
ϕ
τ
∑
n
∈
σ
‍
|
x
n
|
p
)
)
<
∞
}
,
for ϕ
∈
Φ and p
>
0.It has been proved in [61] that, for 1
≤
p
<
∞, m
(
ϕ
,
p
) is a Banach space if it is endowed with the norm
(12)
∥
x
∥
m
(
ϕ
,
p
)
=
sup
⁡
τ
≥
1
(
sup
⁡
σ
∈
𝒞
τ
(
1
ϕ
τ
∑
n
∈
σ
‍
|
x
n
|
p
)
1
/
p
)
,
and that one has the following.(i)
Ifϕ
n
=
1, for all n
∈
ℕ, then m
(
ϕ
,
p
)
=
ℓ
p. Moreover, ℓ
p
⊆
m
(
ϕ
,
p
)
⊆
ℓ
∞.
(ii)
Ifp
=
1, then m
(
ϕ
,
p
)
=
m
(
ϕ
). Also, for any p
≥
1
,
m
(
ϕ
)
⊆
m
(
ϕ
,
p
).
(iii)
m
(
ϕ
,
p
)
⊆
m
(
ψ
,
p
) if and only if sup
⁡
τ
≥
1
(
ϕ
τ
/
ψ
τ
)
<
∞.It is easy to see thatm
(
ϕ
,
p
) is a Köthe sequence space, indeed a B
K-space with respect to its natural norm (see [55]). Note that throughout the present paper we will study the space m
(
ϕ
,
p
) except the case ϕ
n
=
n, for which it is reduced to the space ℓ
∞.Now we will introduce the spaceℓ
p
(
u
,
v
). Let u
=
{
u
n
}
n
=
0
∞ and let v
=
{
v
n
}
n
=
0
∞ be arbitrary real sequences with all coordinates u
k and v
k different from zero and let, for any p
∈
[
1
,
∞
),
(13)
ℓ
p
(
u
,
v
)
=
{
x
=
{
x
k
}
k
=
1
∞
:
∑
n
=
1
∞
‍
|
∑
k
=
1
n
‍
u
n
v
k
x
(
k
)
|
p
<
∞
}
.
It is obvious that this is a linear space. It is known (see [48]) that the functional
(14)
∥
x
∥
ℓ
p
(
u
,
v
)
=
(
∑
n
‍
|
∑
k
=
1
n
‍
u
n
v
k
x
(
k
)
|
p
)
1
/
p
is a norm in ℓ
p
(
u
,
v
) and that the couple (
ℓ
p
(
u
,
v
)
,
∥
·
∥
ℓ
p
(
u
,
v
)
) is a Banach space. The space ℓ
p
(
u
,
v
) is a generalization of three spaces. Namely, one has the following.(i)
If(
v
k
)
=
(
1,1
,
1
,
…
) and (
u
n
)
=
(
1
/
n
), then ℓ
p
(
u
,
v
) is the Cesáro sequence space X
p of nonabsolute type (see [32]) and ∥
x
∥
ℓ
p
(
u
,
v
)
=
∥
x
∥
X
p.
(ii)
Let{
p
n
} be a real sequence with p
1
>
0 and p
n
≠
0 for all n
∈
ℕ. If (
v
k
)
=
{
p
k
} and (
u
n
)
=
(
1
/
Q
n
), where Q
n
=
p
1
+
p
2
+
⋯
+
p
n for any n
∈
ℕ. Then we obtain that ℓ
p
(
u
,
v
) is the Riesz sequence space of nonabsolute type denoted by γ
p
(
v
) and that ∥
x
∥
ℓ
p
(
u
,
v
)
=
∥
x
∥
γ
p
(
v
) (see [44]).
(iii)
Let{
p
n
} be a real sequence with p
1
>
0 and p
n
≠
0, for all n
∈
ℕ, n
≥
2. If (
v
k
)
=
(
p
n
-
k
+
1
) and (
u
n
)
=
(
1
/
Q
n
), where Q
n
=
p
n
+
p
n
-
1
+
⋯
+
p
1 for any n
∈
ℕ. Then ℓ
p
(
u
,
v
) reduces to the Nörlund sequence space of nonabsolute type denoted by N
p and ∥
x
∥
ℓ
p
(
u
,
v
)
=
∥
x
∥
N
p(see [60]).


## 2. Banach-Saks Type of Sequence Spacem
(
ϕ
,
p
)

In this section, we investigate some properties of the spacem
(
ϕ
,
p
) such as the Fatou property, the Banach-Saks property of type p, and the weak fixed point property. Let us start with the following lemma.Lemma 1.
If aB
K-space X containing c
00 has the p
r
o
p
e
r
t
y
A
K, then it is order continuous, that is, ∥
(
0,0
,
…
,
x
(
n
)
,
x
(
n
+
1
)
,
…
)
∥
→
0 as n
→
∞ for any x
∈
X.Proof.
From the definition of propertyA
K, we have that every x
=
{
x
(
i
)
}
∈
X has the unique representation x
=
∑
i
=
1
∞
‍
x
(
i
)
e
(
i
)
,
that
is
,
x
[
n
]
=
∑
i
=
1
n
‍
x
(
i
)
e
(
i
)
→
x as n
→
∞. Hence ∥
x
-
x
[
n
]
∥
X
→
0 as n
→
∞, that is, ∥
(
0,0
,
…
,
x
(
n
)
,
x
(
n
+
1
)
,
…
)
∥
→
0 as n
→
∞, which means that X is order continuous.Corollary 2.
The spacem
(
ϕ
,
p
) is order continuous.Proof.
It is easy to see thatm
(
ϕ
,
p
) contains c
00 and that every x
=
{
x
(
i
)
}
∈
m
(
ϕ
,
p
) has the unique representation x
=
∑
i
=
1
∞
‍
x
(
i
)
e
(
i
)
,
that
is
,
x
[
n
]
=
∑
i
=
1
n
‍
x
(
i
)
e
(
i
)
→
x as n
→
∞, which means that m
(
ϕ
,
p
) has the property A
K. Hence, from the above lemma, m
(
ϕ
,
p
) is order continuous.Theorem 3.
The spacem
(
ϕ
,
p
) has the Fatou property.Proof.
Let x be any real sequence from(
l
0
)
+ and {
x
n
} be any non-decreasing sequence of non-negative elements from m
(
Φ
,
p
) such that x
n
(
i
)
→
x
(
i
) as n
→
∞ coordinatewisely and su
p
n
∥
x
n
∥
m
(
Φ
,
p
)
<
∞.
Let us denotes
=
su
p
n
∥
x
n
∥
m
(
Φ
,
p
). Then, since the supremum is homogeneous, we have
(15)
1
s
sup
⁡
τ
≥
1
(
sup
⁡
σ
∈
𝒞
τ
(
1
Φ
τ
∑
i
∈
σ
‍
|
x
n
(
i
)
|
p
)
1
/
p
)
=
sup
⁡
τ
≥
1
(
sup
⁡
σ
∈
𝒞
τ
(
1
Φ
τ
∑
i
∈
σ
‍
|
x
n
(
i
)
s
|
p
)
1
/
p
)
≤
sup
⁡
τ
≥
1
(
sup
⁡
σ
∈
𝒞
τ
(
1
Φ
τ
∑
i
∈
σ
‍
|
x
n
(
i
)
∥
x
n
∥
m
(
Φ
,
p
)
|
p
)
1
/
p
)
=
1
∥
x
n
∥
m
(
Φ
,
p
)
∥
x
n
∥
m
(
Φ
,
p
)
=
1
.
Moreover, by the assumptions that {
x
n
} is non-decreasing and convergent to x coordinatewisely and by the Beppo-Levi theorem, we have
(16)
1
s
lim
⁡
n
→
∞
[
sup
⁡
τ
≥
1
(
sup
⁡
σ
∈
𝒞
τ
(
1
Φ
τ
∑
i
∈
σ
‍
|
x
n
(
i
)
|
p
)
1
/
p
)
]
=
sup
⁡
τ
≥
1
(
sup
⁡
σ
∈
𝒞
τ
(
1
Φ
τ
∑
i
∈
σ
‍
|
x
(
i
)
s
|
p
)
1
/
p
)
=
∥
x
s
∥
m
(
Φ
,
p
)
≤
1
,
whence
(17)
∥
x
∥
m
(
Φ
,
p
)
≤
s
=
sup
⁡
n
∥
x
n
∥
m
(
Φ
,
p
)
=
lim
⁡
n
→
∞
∥
x
n
∥
m
(
Φ
,
p
)
<
∞
.
Therefore, x
∈
m
(
Φ
,
p
). On the other hand, since 0
≤
x
n
≤
x for any natural number n and the sequence {
x
n
} is non-decreasing, we obtain that the sequence {
∥
x
n
∥
m
(
Φ
,
p
)
} is bounded from above by ∥
x
∥
m
(
Φ
,
p
). In consequence lim
⁡
n
→
∞
∥
x
n
∥
m
(
Φ
,
p
)
≤
∥
x
∥
m
(
Φ
,
p
), which together with the opposite inequality proved already, yields that ∥
x
∥
m
(
Φ
,
p
)
=
lim
⁡
n
∥
x
n
∥
m
(
Φ
,
p
).Theorem 4.
The spacem
(
ϕ
,
p
) has the Banach-Saks property of the type p.Proof.
Let{
ϵ
n
} be a sequence of positive numbers with ∑
n
=
1
∞
‍
ϵ
n
≤
1
/
2. Let {
x
n
} be a weakly null sequence in B
(
m
(
ϕ
,
p
)
). Let us set x
0
=
0 and z
1
=
x
n
1
=
x
1. Then, there exists τ
1
∈
ℕ such that
(18)
∥
∑
i
∈
s
1
‍
z
1
(
i
)
e
(
i
)
∥
m
(
ϕ
,
p
)
<
ϵ
1
,
where s
1 consist of the elements of σ which exceed τ
1. Since x
n
→
ω
0 implies that x
n
→
0 coordinatewise, there is n
2
∈
ℕ such that
(19)
∥
∑
i
=
1
τ
1
‍
x
n
(
i
)
e
(
i
)
∥
m
(
ϕ
,
p
)
<
ϵ
1
,
for all n
≥
n
2. Set z
2
=
x
n
2. Then there exists τ
2
>
τ
1 such that
(20)
∥
∑
i
∈
s
2
‍
z
2
(
i
)
e
(
i
)
∥
m
(
ϕ
,
p
)
<
ϵ
2
,
where s
2 consist of all elements of σ which exceed τ
2. Using again the fact that x
n
→
0 coordinatewise, there exists n
3
>
n
2 such that
(21)
∥
∑
i
=
1
τ
2
‍
x
n
(
i
)
e
(
i
)
∥
m
(
ϕ
,
p
)
<
ϵ
2
,
for all n
≥
n
3.
Continuing this process, we can find two increasing sequences{
τ
j
} and {
n
j
} such that
(22)
∥
∑
i
=
1
τ
j
‍
x
n
(
i
)
e
(
i
)
∥
m
(
ϕ
,
p
)
<
ϵ
j
,
for
each
n
≥
n
j
+
1
,
∥
∑
i
∈
s
j
‍
z
j
(
i
)
e
(
i
)
∥
m
(
ϕ
,
p
)
<
ϵ
j
,
where z
j
=
x
n
j and s
j consist of the elements of σ which exceed τ
j. Since ϵ
j
-
1
+
ϵ
j
<
1, we have
(23)
(
1
ϕ
τ
∑
n
∈
σ
‍
|
z
j
(
n
)
|
)
≤
ϵ
j
-
1
+
ϵ
j
<
1
,
for all j
∈
ℕ and τ
≥
1. Hence
(24)
∥
∑
j
=
1
n
‍
z
j
∥
m
(
ϕ
,
p
)
=
∥
∑
j
=
1
n
‍
(
∑
i
=
1
τ
j
-
1
‍
z
j
(
i
)
e
(
i
)
+
∑
i
=
τ
j
-
1
+
1
τ
j
‍
z
j
(
i
)
e
(
i
)
+
∑
i
∈
s
j
‍
z
j
(
i
)
e
(
i
)
)
∥
m
(
ϕ
,
p
)
≤
∥
∑
j
=
1
n
‍
(
∑
i
=
1
τ
j
-
1
‍
z
j
(
i
)
e
(
i
)
)
∥
m
(
ϕ
,
p
)
+
∥
∑
j
=
1
n
‍
(
∑
i
=
τ
j
-
1
+
1
τ
j
‍
z
j
(
i
)
e
(
i
)
)
∥
m
(
ϕ
,
p
)
≤
∑
j
=
1
n
‍
∥
(
∑
i
=
τ
j
-
1
+
1
τ
j
‍
z
j
(
i
)
e
(
i
)
)
∥
m
(
ϕ
,
p
)
+
2
∑
j
=
1
n
‍
ϵ
j
.
By using the norm of the space m
(
ϕ
,
p
), we have
(25)
∑
j
=
1
n
‍
∥
(
∑
i
=
τ
j
-
1
+
1
τ
j
‍
z
j
(
i
)
e
(
i
)
)
∥
m
(
ϕ
,
p
)
=
∑
j
=
1
n
‍
sup
⁡
τ
≥
1
sup
⁡
s
j
-
1
∈
𝒞
τ
(
1
ϕ
τ
∑
n
∈
s
j
-
1
‍
|
z
j
(
n
)
|
p
)
≤
∑
j
=
1
n
‍
sup
⁡
τ
≥
1
sup
⁡
σ
∈
𝒞
τ
(
1
ϕ
τ
∑
n
∈
σ
‍
|
z
j
(
n
)
|
p
)
≤
n
.
Therefore,
(26)
∥
∑
j
=
1
n
‍
z
j
∥
m
(
ϕ
,
p
)
≤
n
1
/
p
+
1
≤
2
n
1
/
p
.
This completes the proof of the theorem.Theorem 5.
For1
<
p
<
∞, the space m
(
ϕ
,
p
) has the weak fixed point property, if K
>
2
1
-
p, where K
=
sup
⁡
τ
≥
1
ϕ
τ
<
∞.Proof.
Ifψ
τ
=
1, for all τ
∈
ℕ, it follows that
(27)
m
(
ϕ
,
p
)
⊆
ℓ
p
iff
sup
⁡
τ
≥
1
(
ϕ
τ
)
1
/
p
<
∞
,
∥
x
∥
m
(
ϕ
,
p
)
=
sup
⁡
τ
≥
1
(
1
ϕ
τ
)
1
/
p
∥
x
∥
ℓ
p
.
Hence
(28)
R
(
m
(
ϕ
,
p
)
)
=
sup
⁡
τ
≥
1
(
1
ϕ
τ
)
1
/
p
R
(
ℓ
p
)
=
(
2
K
)
1
/
p
<
2
,
since
R
(
ℓ
p
)
=
2
1
/
p
,
where R
(
X
) stands for the Garcia-Falset coefficient of X. Therefore, m
(
ϕ
,
p
) has in this case the weak fixed point property.


## 3. Banach-Saks Type and Gurariǐ Modulus of Sequence Spacesℓ
p
(
u
,
v
)

Theorem 6.
The spaceℓ
p
(
u
,
v
) has the Banach-Saks property of the type p.Proof.
Let(
ε
n
) be a sequence of positive numbers for which ∑
n
=
1
∞
ε
n
≤
(
1
/
2
). Let {
x
n
} be a weakly null sequence in B
(
ℓ
p
(
u
,
v
)
). Set t
0
=
x
0
=
0 and t
1
=
x
n
1
=
x
1. Then there exists r
1
∈
ℕ such that
(29)
∥
∑
i
=
r
1
+
1
∞
‍
t
1
(
i
)
e
(
i
)
∥
ℓ
p
(
u
,
v
)
<
ε
1
.
Since the fact that {
x
n
} is a weakly null sequence implies that x
n
→
0, coordinatewise, there is an n
2
∈
ℕ such that
(30)
∥
∑
i
=
1
r
1
‍
x
n
(
i
)
e
(
i
)
∥
ℓ
p
(
u
,
v
)
<
ε
1
,
for all n
≥
n
2. Set t
2
=
x
n
2. Then there exists an r
2
>
r
1 such that
(31)
∥
∑
i
=
r
2
+
1
∞
‍
t
2
(
i
)
e
(
i
)
∥
ℓ
p
(
u
,
v
)
<
ε
2
.
By using the fact that x
n
→
0 coordinatewise, there exists an n
3
>
n
2 such that
(32)
∥
∑
i
=
1
r
2
‍
x
n
(
i
)
e
(
i
)
∥
ℓ
p
(
u
,
v
)
<
ε
2
,
for all n
≥
n
3. Continuing this process, we can find by induction two increasing subsequences (
r
i
) and (
n
i
) of natural numbers such that
(33)
∥
∑
i
=
1
r
j
‍
x
n
(
i
)
e
(
i
)
∥
ℓ
p
(
u
,
v
)
<
ε
j
,
for all n
≥
n
j
+
1 and
(34)
∥
∑
i
=
r
j
+
1
∞
‍
t
j
(
i
)
e
(
i
)
∥
ℓ
p
(
u
,
v
)
<
ε
j
,
where t
j
=
x
n
j. Hence,
(35)
∥
∑
j
=
0
n
‍
t
j
∥
ℓ
p
(
u
,
v
)
=
∥
∑
j
=
0
n
‍
(
+
∑
i
=
r
j
+
1
∞
‍
t
j
(
i
)
e
(
i
)
)
∑
i
=
0
r
j
-
1
‍
t
j
(
i
)
e
(
i
)
+
∑
i
=
r
j
-
1
+
1
r
j
‍
t
j
(
i
)
e
(
i
)
+
∑
i
=
r
j
+
1
∞
‍
t
j
(
i
)
e
(
i
)
)
∥
ℓ
p
(
u
,
v
)
≤
∥
∑
j
=
0
n
‍
(
∑
i
=
r
j
-
1
+
1
r
j
‍
t
j
(
i
)
e
(
i
)
)
∥
ℓ
p
(
u
,
v
)
+
∥
∑
j
=
0
n
‍
(
∑
i
=
0
r
j
-
1
‍
t
j
(
i
)
e
(
i
)
)
∥
ℓ
p
(
u
,
v
)
+
∥
∑
j
=
0
n
‍
(
∑
i
=
r
j
+
1
∞
‍
t
j
(
i
)
e
(
i
)
)
∥
ℓ
p
(
u
,
v
)
≤
∥
∑
j
=
0
n
‍
(
∑
i
=
r
j
-
1
+
1
r
j
‍
t
j
(
i
)
e
(
i
)
)
∥
ℓ
p
(
u
,
v
)
+
2
∑
j
=
0
n
‍
ε
j
.
On the other hand, since ∥
x
n
∥
=
(
∑
i
=
0
∞
|
∑
k
=
0
i
u
i
v
k
x
n
j
(
k
)
|
p
)
1
/
p, it can be easily seen that ∥
x
n
∥
<
1. Therefore, ∥
x
n
∥
p
<
1 and
(36)
∥
∑
j
=
0
n
‍
(
∑
i
=
r
j
-
1
+
1
r
j
‍
t
j
(
i
)
e
(
i
)
)
∥
ℓ
p
(
u
,
v
)
p
=
∑
j
=
0
n
‍
∑
i
=
r
j
-
1
+
1
r
j
‍
|
∑
k
=
0
i
‍
u
i
v
k
t
j
(
k
)
|
p
≤
∑
j
=
0
n
‍
∑
i
=
0
∞
‍
|
∑
k
=
0
i
‍
u
i
v
k
t
j
(
k
)
|
p
≤
(
n
+
1
)
.
Hence we obtain
(37)
∥
∑
j
=
0
n
‍
(
∑
i
=
r
j
-
1
+
1
r
j
‍
t
j
(
i
)
e
(
i
)
)
∥
≤
(
∑
j
=
0
n
‍
1
)
1
/
p
=
(
n
+
1
)
1
/
p
.
By using the inequality 1
≤
(
n
+
1
)
1
/
p for all n
∈
ℕ and 1
≤
p
<
∞, we have
(38)
∥
∑
j
=
0
n
‍
t
j
∥
l
p
(
u
,
v
)
≤
(
n
+
1
)
1
/
p
+
1
≤
2
(
n
+
1
)
1
/
p
.
Therefore, the space ℓ
p
(
u
,
v
) has the Banach-Saks type p, which completes the proof of the theorem.Let us define the matrixG
=
G
(
u
,
v
)
=
{
g
n
k
} by
(39)
g
n
k
=
{
u
n
v
k
;
0
≤
k
≤
n
,
0
;
k
>
n
for all k
,
n
∈
ℕ, where u
n depends only on n and v
k depends only on k. The matrix G is called generalized weighted mean or factorable matrix. By H
=
H
(
v
,
u
)
=
(
h
n
k
), we denote the inverse of the matrix G
(
u
,
v
) as follows:
(40)
h
n
k
=
{
(
-
1
)
n
-
k
v
n
u
k
;
n
-
1
≤
k
≤
n
,
0
;
0
≤
n
<
k
or
n
>
k
+
1
.Theorem 7.
Forx
∈
ℓ
p
(
u
,
v
), by (39), one has the fact that the Gurariǐ modulus of convexity for the normed space ℓ
p
(
u
,
v
) satisfies the inequality
(41)
β
ℓ
p
(
u
,
v
)
(
ε
)
≤
1
-
(
1
-
(
ε
2
)
p
)
1
/
p
for any 0
≤
ε
≤
2.Proof.
Letx
∈
ℓ
p
(
u
,
v
). By using (39), we have
(42)
∥
x
∥
ℓ
p
(
u
,
v
)
=
∥
G
(
u
,
v
)
x
∥
ℓ
p
=
(
∑
n
‍
|
(
G
(
u
,
v
)
x
)
n
|
p
)
1
/
p
.
Let 0
≤
ε
≤
2. Then using (40), let us consider the following sequences:
(43)
x
=
(
x
n
)
=
(
H
(
(
1
-
(
ε
2
)
p
)
1
/
p
)
,
H
(
ε
2
)
,
0,0
,
…
)
,
t
=
(
t
n
)
=
(
H
(
(
1
-
(
ε
2
)
p
)
1
/
p
)
,
H
(
-
ε
2
)
,
0,0
,
…
)
.
Since y
n
=
(
G
x
)
n and z
n
=
(
G
t
)
n, we have
(44)
y
=
(
y
n
)
=
(
(
1
-
(
ε
2
)
p
)
1
/
p
,
(
ε
2
)
,
0,0
,
…
)
,
z
=
(
z
n
)
=
(
(
1
-
(
ε
2
)
p
)
1
/
p
,
(
-
ε
2
)
,
0,0
,
…
)
.
By using the sequences given above, we obtain the following equalities:
(45)
∥
x
∥
ℓ
p
(
u
,
v
)
p
=
∥
G
(
u
,
v
)
x
∥
ℓ
p
p
=
|
(
1
-
(
ε
2
)
p
)
1
/
p
|
p
+
|
ε
2
|
p
=
1
-
(
ε
2
)
p
+
(
ε
2
)
p
=
1
;
∥
t
∥
ℓ
p
(
u
,
v
)
p
=
∥
G
(
u
,
v
)
t
∥
ℓ
p
p
=
|
(
1
-
(
ε
2
)
p
)
1
/
p
|
p
+
|
-
ε
2
|
p
=
1
-
(
ε
2
)
p
+
(
ε
2
)
p
=
1
;
∥
x
-
t
∥
ℓ
p
(
u
,
v
)
=
∥
G
(
u
,
v
)
x
-
G
(
u
,
v
)
t
∥
ℓ
p
=
(
|
(
1
-
(
ε
2
)
p
)
1
/
p
-
(
1
-
(
ε
2
)
p
)
1
/
p
|
p
+
|
ε
2
-
(
-
ε
2
)
|
p
|
(
1
-
(
ε
2
)
p
)
1
/
p
-
(
1
-
(
ε
2
)
p
)
1
/
p
|
p
)
1
/
p
=
ε
.
To complete the upper estimate of the Gurariǐ modulus of convexity, it remains to calculate the infimum of ∥
α
x
+
(
1
-
α
)
t
∥
ℓ
p
(
u
,
v
) for 0
≤
α
≤
1. We have
(46)
inf
⁡
0
≤
α
≤
1
∥
α
x
+
(
1
-
α
)
t
∥
ℓ
p
(
u
,
v
)
=
inf
⁡
0
≤
α
≤
1
∥
α
G
(
u
,
v
)
x
+
(
1
-
α
)
G
(
u
,
v
)
t
∥
ℓ
p
=
inf
⁡
0
≤
α
≤
1
[
|
α
(
ε
2
)
+
(
1
-
α
)
(
-
ε
2
)
|
p
[
|
α
(
1
-
(
ε
2
)
p
)
1
p
]
|
α
(
1
-
(
ε
2
)
p
)
1
/
p
+
(
1
-
α
)
(
1
-
(
ε
2
)
p
)
1
/
p
|
p
+
|
α
(
ε
2
)
+
(
1
-
α
)
(
-
ε
2
)
|
p
[
|
α
(
1
-
(
ε
2
)
p
)
1
p
]
1
/
p
=
inf
⁡
0
≤
α
≤
1
[
1
-
(
ε
2
)
p
+
|
2
α
-
1
|
p
(
ε
2
)
p
]
1
/
p
=
(
1
-
(
ε
2
)
p
)
1
/
p
.
Consequently, we get for p
≥
1 the inequality
(47)
β
l
p
(
u
,
v
)
(
ε
)
≤
1
-
(
1
-
(
ε
2
)
p
)
1
/
p
,
which is the desired result.



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*Source: 427382-2014-03-26.xml*