Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 903736, 13 pages
doi:10.1155/2011/903736
Research Article
Some Geometric Properties of Lacunary Sequence
Spaces Related to Fixed Point Property
Chirasak Mongkolkeha and Poom Kumam
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi
(KMUTT), Bangmod, Thrungkru, Bangkok 10140, Thailand
Correspondence should be addressed to Poom Kumam, poom.kum@kmutt.ac.th
Received 18 June 2011; Accepted 2 November 2011
Academic Editor: Ondřej Došlý
Copyright q 2011 C. Mongkolkeha and P. Kumam. 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.
The main purpose of this paper is considering the lacunary sequence spaces defined by Karakaya
2007, by proving the property β and Uniform Opial property.
1. Introduction
Let X, · be a real Banach space and let BX resp., SX be a closed unit ball resp.,
the unit sphere of X. For any subset A of X, we denote by convA the convex hull of A.
The Banach space X is uniformly convex UC, if for each ε > 0 there exists δ > 0 such
that for x, y ∈ SX the inequality x − y > ε implies x y/2 < 1 − δ see 1. A
Banach space X has the property β if for each ε > 0 there exists δ > 0 such that 1 < x <
1 δ implies α convBX ∪ {x} \ BX < ε, where αA denotes the Kuratowski measure
noncompactness of a subset A of X defined as the infimum of all ε > 0 such that A can be
covered by a finite union of sets of diameter less than ε. The following characterization of
the property β is very useful see 2: A Banach space X has the property β if and only
if for each ε > 0 there exists δ > 0 such that for each element x ∈ BX and each sequence
xn in BX with sepxn ≥ ε there is an index k for which x xk /2 < 1 − δ where
sepxn inf{xn − xm : n / m} > ε. A Banach space X is nearly uniformly convex NUC if
for each ε > 0 and every sequence xn in BX with sepxn ≥ ε, there exists δ ∈ 0, 1 such
/ BX the drop Dx, BX determined
that convxn ∩ 1 − δBX / ∅. Define for any x ∈
by x by Dx, BX convBX ∪ {x}. A Banach space X has the drop property write
D if for every closed set C disjoint with BX there exists an element x ∈ C such that
Dx, BX ∩ C {x}. A point x ∈ SX is an H-point of BX if for any sequence xn in X
such that xn → 1 as n → ∞, the week convergence of xn to x implies that xn −x → 0 as
2
Abstract and Applied Analysis
n → ∞. If every point in SX is an H-point of BX, then X is said to have the property H.
A Banach space is said to have the uniform Kadec-Klee property abbreviated as UKK if
for every ε > 0 there exists δ > 0 such that for every sequence xn in SX with sepxn ≥ ε
w
and xn −→ x as n → ∞, we have x < 1 − δ. Every UKK Banach space has H-property see
3. The following implications are true in any Banach spaces,
(D) =⇒ (Rf x)
⇑
(UC) =⇒ property(β) =⇒ (NUC) =⇒ (UKK) =⇒ property (H),
1.1
where Rfx denotes the property of reflexivity see 3–6. A Banach space X is said to have
the Opial property see 7 if every sequence xn weakly convergent to x0 satisfies
lim inf xn − x0 ≤ lim inf xn − x,
n→∞
n→∞
1.2
for every x ∈ X. Opial proved in 7 that the sequence space lp 1 < p < ∞ have this property
2, 1 < p < ∞ do not have it. A Banach space X is said to have the uniform
but Lp 0, π p /
Opial property see 8, if for each ε > 0 there exists τ > 0 such that for any weakly null
sequence xn in SX and x ∈ X with x ≥ ε there holds
1 τ ≤ lim inf xn x.
n→∞
1.3
For example, the space in 9, 10 has the uniform Opial property. The Opial property is important because Banach spaces with this property have the weak fixed point property see
11 and the geometric property involving fixed point theory can be found, for example, in
9, 12–14.
For a bounded subset A ⊂ X, the set measure of noncompactness was defined in 15
by
αA inf ε > 0 : A can be covered by finitely many sets of diameter ≤ ε .
1.4
The ball measure of noncompactness was defined in 16, 17 by
βA inf ε > 0 : A can be covered by finitely many balls of diameter ≤ ε .
1.5
The functions α and β are called the Kuratowski measure of noncompactness and the Hausdorff measure of noncompactness in X, respectively. We can associate these functions with the
notions of the set-contraction and ball contraction see 18. These notions are very useful
tools to study nonlinear operator propblems see 8, 18. For each ε > 0 define that Δε inf{1 − inf x : x ∈ A: A is closed convex subset of BX with βA ≥ ε}. The function Δ
is called the modulus of noncompact convexity see 16. A Banach space X is said to have
property L if lim ε → 1− Δε 1. It has been proved in 8 that property L is a useful tool in
Abstract and Applied Analysis
3
the fixed point theory and that a Banach space X has property L if and only if it is reflexive
and has the uniform Opial property.
For a real vector space X, a function ρ : X → 0, ∞ is called a modular if it satisfies the
following conditions:
i ρx 0 if and only if x 0;
ii ραx ρx for all scalar α with |α| 1;
iii ραx βy ≤ ρx ρy, for all x, y ∈ X and all α, β ≥ 0 with α β 1;
the modular ρ is called convex if
iv ραx βy ≤ αρx βρy, for all x, y ∈ X and all α, β ≥ 0 with α β 1.
For modular ρ on X, the space
Xρ x ∈ X : ρλx −→ 0 as λ −→ 0
1.6
is called the modular space.
A sequence xn in Xρ is called modular convergent to x ∈ Xρ if there exists a λ > 0 such
that ρλxn − x → 0 as n → ∞.
A modular ρ is said to satisfy the Δ2 -condition ρ ∈ Δ2 if for any ε > 0 there exist
constants K ≥ 2 and a > 0 such that
ρ2u ≤ Kρu ε
1.7
for all u ∈ Xρ with ρu ≤ a.
If ρ satisfies the Δ2 -condition for any a > 0 with K ≥ 2 dependent on a, we say that ρ
satisfies the strong Δ2 -condition ρ ∈ Δs2 .
By a lacunary sequence θ kr , where k0 0, we will mean an increasing sequence
of nonnegative integers with kr − kr−1 → ∞ as r → ∞. The intervals determined by θ will be
denote by Ir kr−1 , kr . We write hr kr − kr−1 and the ratio kr /kr−1 , will be denoted by qr .
The space of lacunary strongly convergent sequence Nθ was defined by Freedman et al. 19
as
Nθ 1 x xk : lim
|xk − l| 0, for some l .
r → ∞ hr
k∈I
1.8
r
It is well known that there is very closed connection between the space of lacunary strongly convergent sequence and the space of strongly Cesaro summability sequences. This connection can be found in 18–23, because a lot of these connection, a lot of geometric property
of Cesaro sequence spaces can generalize the lacunary sequence spaces.
Let w be the space of all real sequences. Let p pr be a bounded sequence of the
positive real numbers. In 2007, Karakaya 24 introduced the new sequence spaces lp, θ
involving lacunary sequence as follows:
l p, θ x xi :
∞
r1
1
|xi|
hr i∈Ir
pr
<∞
1.9
4
Abstract and Applied Analysis
and paranorm on lp, θ is given by
xlp,θ
pr 1/M
∞
1
,
|xi|
hr i∈Ir
r1
1.10
where M supr pr . If pr p for all r ∈ N, we will use the notation lp θ in place of lp, θ. The
norm on lp θ is given by
xlp θ
p 1/p
∞
1
.
|xi|
hr i∈Ir
r1
1.11
By using the properties of lacunary sequence in the space lp, θ, we get the following sequences. If θ 2r , then lp, θ cesp. If θ 2r and pr p for all r ∈ N, then lp, θ cesp .
For x ∈ lp, θ defined the modular on lp, θ by
pr
∞
1
.
x |xi|
hr i∈Ir
r1
1.12
It is easy to see that if limr → ∞ sup pr < ∞ then ∈ Δs2 . The Luxembourg norm on lp, θ is
defined by
x
≤1 .
x inf ε > 0 : ε
1.13
The Luxembourg norm on lp θ can be reduced to a usual norm on lp θ 24, that is,
x xlp θ .
1.14
Throughout this paper, we assume that limr → ∞ inf pr > 1 and limr → ∞ sup pr < ∞ and
for x ∈ w, i ∈ N, we denote
⎛
⎞
i−1 times
⎜
⎟
ei ⎝0, 0, . . . , 0, 1, 0, 0, 0, . . .⎠,
1.15
x|i x1, x2, x3, . . . , xi, 0, 0, 0, . . .,
x|N−i 0, 0, 0, . . . , xi 1, xi 2, . . ..
The following results are very important for our consideration.
Abstract and Applied Analysis
5
Lemma 1.1 see 25, Lemma 2.1. If ρ ∈ Δs2 , then for any L > 0 and ε > 0, there exists δ δL, ε > 0 such that
ρu v − ρu < ε,
1.16
whenever u, v ∈ Xρ with ρu ≤ L, and ρv ≤ δ.
Lemma 1.2 see 25, Lemma 2.3. Convergence in norm and in modular are equivalent in Xρ if
ρ ∈ Δ2 .
Lemma 1.3 see 25, Corollary 2.2, Lemma 2.3. If ρ ∈ Δ2 , then for any sequence xn in Xρ ,
xn → 0 if and only if ρxn → 0 as n → ∞.
Lemma 1.4 see 25, Lemma 2.4. If ρ ∈ Δs2 , then for any ε > 0 there exists δ δε > 0 such
that x ≥ 1 δ whenever ρx ≥ 1 ε.
Lemma 1.5 see 24, Lemma 2.3. The functional is a convex modular on lp, θ.
Lemma 1.6 see 24, Lemma 2.5. i For any x ∈ lp, θ, if x < 1, then x ≤ x.
ii For any x ∈ lp, θ, x 1 if and only if x 1.
2. The Main Results
In this section, we prove the property β and uniform Opial property in lacunary sequence
and connect to the fixed point property. First we shall give some results which are very
important for our consideration.
Lemma 2.1. For any x ∈ lp, θ, there exists k0 ∈ N and λ ∈ 0, 1 such that xk /2 ≤ 1 −
λ/2xk for all k ∈ N with k ≥ k0 , where
⎛
⎞
k−1
⎜
⎟
xk ⎝0, 0, . . . , 0, xk, xk 1, xk 2, . . .⎠.
2.1
Proof. Let k ∈ N be fixed. So there exist rk ∈ N such that k ∈ Irk . Let α be a real number such
that 1 < α ≤ limr → ∞ inf pr , then there exists k0 ∈ N such that α < prk for all k ≥ k0 . Choose
λ ∈ 0, 1 as a real such that 1/2α ≤ 1 − λ/2. Then for each x ∈ lp, θ and k ≥ k0 , we have
xk
2
∞
rrk
pr
1 xi hr i∈Ir 2 ∞ pr
1
rrk
2
1
|xi|
hr i∈Ir
pr
6
α ∞
1
1
≤
|xi|
2 rrk hr i∈Ir
≤
Abstract and Applied Analysis
pr
1 − λ k
x .
2
2.2
Lemma 2.2. For any x ∈ lp, θ and ε ∈ 0, 1 there exists δ ∈ 0, 1 such that x ≤ 1 − ε implies
x ≤ 1 − δ.
Proof. Suppose that the lemma does not hold, then there exist ε > 0 and xn ∈ lp, θ such that
xn ≤ 1 − ε and 1/2 ≤ xn 1. Let an 1/xn − 1. Then an → 0 as n → ∞. Let
L sup{2xn : n ∈ N}. Since ∈ Δs2 there exists K ≥ 2 such that
2u ≤ Ku 1,
2.3
for every u ∈ lp, θ with u < 1. By 2.3, we have 2xn ≤ Kxn 1 ≤ K 1 for all n ∈ N.
Hence 0 < L < ∞. By Lemmas 1.5 and 1.6ii, we have
xn
1
2an xn 1 − an xn xn ≤ an 2xn 1 − an xn 2.4
≤ an L 1 − ε −→ 1 − ε,
which is a contradiction.
Theorem 2.3. The space lp, θ is Banach spaces with respect to the Luxemburg norm.
Proof. Let xn xn i be a Cauchy sequence in lp, θ and ε ∈ 0, 1. Thus there exists
N ∈ N such that xn − xm < εM for all n, m ≥ N. By Lemma 1.6 i, we have
xn − xm ≤ xn − xm < εM
∀n, m ≥ N.
2.5
That is,
∞
r1
1
|xn i − xm i|
hr i∈Ir
pr
< εM
∀n, m ≥ N.
2.6
For fixed r, we get that
|xn i − xm i| < ε
∀n, m ≥ N.
2.7
Abstract and Applied Analysis
7
Thus let xn i be a Cauchy sequence in R for all i ∈ N. Since R is complete, then there exists
xi ∈ R such that xm i → xi as m → ∞ for all i ∈ N. Thus for fixed r, we have
|xn i − xi| < ε
as m −→ ∞, ∀n ≥ N.
2.8
This implies that, for all n ≥ N,
xn − xm −→ xn − x
as m −→ ∞.
2.9
This means that, for all n ≥ N,
∞
r1
pr
pr
∞
1
1
−→
|xn i − xm i|
|xn i − xi|
hr i∈Ir
hr i∈Ir
r1
2.10
as m → ∞. By 2.6, we have
xn − x ≤ xn − xm < εM ≤ ε
∀n ≥ N.
2.11
This implies that xn → x as n → ∞. So we have xN − x ∈ lp, θ. By the linearity of the
sequence space lp, θ, we have x xN − x xN ∈ lp, θ. Therefore the sequence space
lp, θ is Banach space, with respect to the Luxemburg norm and the proof is complete.
Theorem 2.4. The space lp, θ has property β.
Proof. Let ε > 0 and xn ⊂ Blp, θ with sepxn ≥ ε. For each k ∈ N, there exist rk ∈ N such
that k is a minimal element in Irk . Let
⎛
⎞
k−1
⎜
⎟
xnk ⎝0, 0, . . . , 0, xn k, xn k 1, xn k 2, . . .⎠.
2.12
Since for each i ∈ N, xn i∞
n1 is bounded. By using the diagonal method, we have that for
each k ∈ N we can find subsequence xnj of xn such that xnj i converges for each i ∈ N.
Therefore, for any k ∈ N there exists an increasing sequence tk such that sepxnkj j>tk ≥ ε.
Hence for each k ∈ N there exists sequence of positive integers sk ∞
k1 with s1 < s2 < s3 < · · ·
such that xskk ≥ ε/2, and since ∈ Δs2 , by Lemma 1.3 we may assume that there exists η > 0
such that xskk ≥ η for all k ∈ N, that is,
∞
rrk
pr
1 k ≥η
x i
hr i∈Ir sk
2.13
8
Abstract and Applied Analysis
for all k ∈ N. On the other hand by Lemma 2.1, there exist k0 ∈ N and λ ∈ 0, 1 such that
uk
2
≤
1 − λ k
u
2
2.14
for all u ∈ lp, θ and k ≥ k0 . From Lemma 2.2, there exist δ ∈ 0, 1 such that for any y ∈
lp, θ,
λη
⇒ y ≤ 1 − δ.
y ≤1−
4
2.15
Since again ∈ Δs2 , by Lemma 1.1, there exists δ0 such that
u v − u < λη ,
4
2.16
whenever u ≤ 1 and v ≤ δ0 . Since x ∈ Blp, θ, we have that x ≤ 1. Then there exits
k ≥ k0 such that xk ≤ δ0 . We put u xskk and v xk ,
pr
1 xi < δ0 .
hr i∈Ir 2 2.17
pr
u λη 1 − λ u v
λη
1 xi xsk i ≤
≤
.
u hr i∈Ir
2
2
2
4
2
4
2.18
∞
u 2
rrk
pr
1 xsk i < 1,
hr i∈Ir 2 ∞
v 2
rrk
From 2.14 and 2.16, we have
∞
rrk
By 2.13, 2.16, 2.18, and convexity of function ft |t|pr , for all r ∈ N, we have
pr
∞
x xsk
1 xi xsk i 2
hr i∈Ir 2
r1
r
k −1
r1
1
≤
2
pr pr
∞
1 xi xsk i 1 xi xsk i hr i∈Ir 2
hr i∈Ir 2
rrk
pr pr r −1
r −1
k
k
1
1
|xi|
|xs i|
hr i∈Ir
hr i∈Ir k
r1
r1
Abstract and Applied Analysis
pr
∞
λη
1 xsk i hr i∈Ir
2
4
rrk
1
≤
2
9
pr pr r −1
r −1
k
k
1
1
|xi|
|xs i|
hr i∈Ir
hr i∈Ir k
r1
r1
∞
1−λ
2 rrk
1
|xs i|
hr i∈Ir k
pr
λη
4
pr
pr
k −1
k −1
1
1
1 r
1 r
|xi|
|xs i|
2 r1 hr i∈Ir
2 r1 hr i∈Ir k
∞
1−λ
2 rrk
1
|xs i|
hr i∈Ir k
pr
λη
4
pr
pr
∞
k −1
1
1
1 r
1
|xi|
|xs i|
2 r1 hr i∈Ir
2 r1 hr i∈Ir k
∞
λ
−
2 rrk
≤
1
|xs i|
hr i∈Ir k
pr
λη
4
1 1 λη λη
−
2 2
2
4
1−
λη
.
4
2.19
So it follows from 2.15 that
x xsk 2 ≤ 1 − δ.
2.20
Therefore, the space lp, θ has property β.
By the facts presented in the introduction, following results are obtained directly from
Theorem 2.4.
Corollary 2.5. The space lp θ has property β.
Corollary 2.6. The space lp, θ is nearly uniform convexity, has drop property, and is reflexive.
Corollary 2.7. The space lp, θ has property (UKK).
Corollary 2.8 see 24, Theorem 2.9. The space lp, θ has property H.
Corollary 2.9. The space lp θ is nearly uniform convexity, has drop property, and is reflexive.
10
Abstract and Applied Analysis
Corollary 2.10. The space lp θ has property (UKK) and H.
Theorem 2.11. The space lp, θ has uniform Opial property.
Proof. Take any ε > 0 and x ∈ lp, θ with x ≥ ε. Let xn be weakly null sequence in
Slp, θ. By limr → ∞ sup pr < ∞, that is, ∈ Δs2 , hence by Lemma 1.2 there exists δ ∈ 0, 1
independent of x such that x > δ. Also, by ∈ Δs2 , Lemma 1.1 asserts that there exists
δ1 ∈ 0, δ such that
y z − y < δ ,
4
2.21
whenever y ≤ 1 and z ≤ δ1 . Choose r0 ∈ N such that
∞
rr0 1
1
|xi|
hr i∈Ir
pr
<
δ1
.
4
2.22
So, we have
pr
pr
r0
∞
1
1
δ<
|xi|
|xi|
hr i∈Ir
hr i∈Ir
r1
rr0 1
pr
r0
1
δ1
≤
,
|xi|
hr i∈Ir
4
r1
2.23
which implies that
r0
r1
1
|xi|
hr i∈Ir
pr
>δ−
δ 3δ
δ1
>δ− .
4
4
4
2.24
w
Since xn −→ 0, then there exists n0 ∈ N such that
r0
3δ ≤
4
r1
1
|xn i xi|
hr i∈Ir
pr
2.25
w
for all n > n0 , since weak convergence implies coordinatewise convergence. Again, by xn −→
0, then there exists n1 ∈ N such that
δ 1/M
xn|k0 < 1 − 1 −
4
2.26
Abstract and Applied Analysis
11
for all n > n1 where k0 is a minimal element in Ir0 1 and M ∈ N with pr ≤ M for all r ∈ N.
Hence, by the triangle inequality of the norm, we get
δ 1/M
.
xn|N−k0 > 1 −
4
2.27
It follows by the definition of · that we have
xn|N−k0
1<
1 − δ/41/M
p
∞
1/hr i∈Ir |xn i| r
≤
2.28
1 − δ/41/M
rr0 1
M
1
1 − δ/41/M
∞
rr0 1
1
|xn i|
hr i∈Ir
pr
Which implies that
∞
rr0 1
1
|xn i|
hr i∈Ir
pr
>1−
δ
4
2.29
for all n > n1 . By inequality 2.21, 2.25, and 2.29, it yields for any n > n1 that
xn x r0
r1
≥
1
|xn i xi|
hr i∈Ir
∞
3δ
4 rr0 1
≥
δ
3δ
1−
4
4
≥1
pr
pr
1
|xn i|
hr i∈Ir
−
∞
rr0 1
−
1
|xn i xi|
hr i∈Ir
δ
4
pr
2.30
δ
4
δ
.
4
Since ∈ Δs2 and by Lemma 1.4 there exists τ depending on δ only such that xn x ≥ 1 τ,
which implies that limn → ∞ inf xn x ≥ 1 τ, hence the prove is complete.
By the facts presented in the introduction and the reflexivity of lp, θ, we get the
following results.
Corollary 2.12. The space lp θ has uniform Opial property.
Corollary 2.13. The space lp, θ has property L and the fixed point property.
Corollary 2.14. The space lp θ has property L and the fixed point property.
12
Abstract and Applied Analysis
Acknowledgments
The authors are grateful to the referees for their valuable comments. They also wish to
thank the Higher Education Research Promotion and National Research University Project of
Thailand, Office of the Higher Education Commission for financial support under the Project
no. 54000267. Mr. Chirasak Mongkolkeha was supported by the Thailand Research Fund
through the Royal Golden Jubilee Program under Grant PHD/0029/2553 for Ph.D. program
at KMUTT, Thailand. The second author was supported by the Commission on Higher
Education, the Thailand Research Fund, and the King Mongkuts University of Technology
Thonburi Grant no. MRG5380044.
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