Volume 2008, Article ID 162037,10pages doi:10.1155/2008/162037
Research Article
The Locally Uniform Nonsquare in Generalized Ces `aro Sequence Spaces
Narin Petrot
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
Correspondence should be addressed to Narin Petrot,[email protected] Received 20 August 2008; Accepted 10 November 2008
Recommended by Martin J. Bohner
We show that the generalized Ces`aro sequence spaces possess the locally uniform nonsquare and have the fixed point property, but they are not uniformly nonsquare. This result is related to the result of the paper by J. Falset et al.2006by giving the examples and the motivation to find the geometric properties that are weaker than uniformly nonsquare but still possess the fixed point property in any Banach spaces.
Copyrightq2008 Narin Petrot. 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
In the whole paper, N and R stand for the sets of natural numbers and of real numbers, respectively. The space of all real sequencex xi∞i1is denoted by0.For a real normed spaceX,·,we denote bySXthe unit sphere ofX.We now give some definitions and basic concepts which will be used in this paper.
A Banach spaceX,·which is a subspace of0is said to be a K¨othe sequence space, if ifor anyx ∈0andy∈X such that|xi| ≤ |yi|for alli∈N,we havex∈Xand
x ≤ y;
iithere isx∈Xwithxi/0 for alli∈N.
An element x from a K ¨othe sequence space X is called order continuous if for any sequencexnin X the positive cone of X such that xn ≤ |x| for alln ∈ N and xn → 0coordinatewise , we havexn → 0. It is easy to see thatxis order continuous if and only if 0,0, . . . ,0, xn1, xn2, . . . → 0 asn → ∞.
1.1. Modular spaces
For a real vector space X,a function : X → 0,∞is called a modular if it satisfies the following conditions:
ix 0 if and only ifx0;
iiαx xfor all scalarαwith|α|1;
iiiαxβy≤x yfor allx, y∈Xand allα, β≥0 withαβ1.
The modularis called convex if
ivαxβy≤αx βy, for allx, y∈Xand allα, β≥0 withαβ1.
For any modularonX, the space
X{x∈X:λx−→0 asλ−→0} 1.1
is called the modular space.
Ifis a convex modular, the function
xinf
λ >0 : x
λ
≤1
1.2
is a norm onX, which is called the Luxemburg normsee1.
A modular is said to satisfy theΔ2-condition ∈ Δ2 if for anyε > 0 there exist constantsK≥2 anda >0 such that
2x≤Kx ε 1.3
for allx∈Xwithx≤a.
Ifsatisfies theΔ2-condition for alla > 0 withK ≥2 dependent ona,we say that satisfies the strongΔ2-condition∈Δs2.
Lemma 1.1. If∈Δs2,then for anyL >0 andε >0,there existsδ >0 such that
|uv−u|< ε, 1.4
wheneveru, v∈Xwithu≤Landv≤δ.
Proof. See2, Lemma 2.1.
Lemma 1.2. If∈Δs2,then for anyx∈X, x1 if and only ifx 1.
Proof. See2, Corollary 2.2.
1.2. Generalized Ces `aro sequence spaces
For 1≤p <∞, the Ces`aro sequence spacewrite cesp, for shortis defined by
cesp
x∈0: ∞ j1
1 j
j i1
|xi|
p
<∞
, 1.5
equipped with the norm
x ∞
j1
1 j
j i1
|xi|
p 1/p
. 1.6
Ces`aro sequence spaces cesp appeared in 1968 as the problem in the Dutch Mathematical Society to find their dualssee3,4. Regular investigation of these spaces was done by Shiue5in 1970, while Leibowitz6and Jagers7proved that ces1{0}, cespare separable reflexive Banach spaces for 1< p <∞andpspaces are in cespfor 1< p≤ ∞.
Letp pj be a sequences of positive real numbers withpj ≥ 1 for allj ∈ N, the generalized Ces`aro sequence space, cesp, is defined by
cesp{x∈l0:ρλx<∞, for someλ >0}, 1.7 where
ρx ∞
j1
1 j
j i1
|xi|
pj
1.8
is a convex modular on cesp see8. Observe that if the space cespis nontrivial, then it belongs to the class of K ¨othe sequence spaces. We also have some observations on cesp as follows.
Remark 1.3. i In the case whenpj p, 1 ≤ p < ∞ for allj ∈ N,the generalized Ces`aro sequence space cespis nothing but the Ces`aro sequence space cespand the Luxemburg norm is expressed by the formula1.6.
iiCondition limj→ ∞infpj > 1 is obviously sufficient for cesp/{0} but it is not a necessity condition, for example, whenpj12ln lnj/lnj, j ≥2. However, if cesp/{0}, we have∞
j11/jpj <∞.
iiiIt is easy to see that if limj→ ∞suppj < ∞thenρ ∈Δs2,andAp cesp, where Ap {x∈l0 :ρλx<∞for allλ >0}, but unfortunately we do not know whether it is a necessity condition forApcesp.
ivFor eachx∈Ap, whereApis defined as inii, we havexis an order continuous element. Indeed, for given ε > 0 byx ∈ Ap, we can find a natural number i0 such that ρx−xi/ε<1−εfor alli > i0. This implies thatx−xip< εfor alli > i0.
Other investigations to generalized Ces`aro spaces can be found in7,9–14.
1.3. Nonsquareness
Now, we give the basic definitions related to the nonsquareness in Banach space.
Definition 1.4. A Banach spaceX,·is said to be
iuniformly nonsquare in the sense of James or uniformly non-l1nwrite UN-l1n,n∈N, n≥ 2, if there isδ >0 such that for anyx1, x2, . . . , xn∈SX,
min{x1ε2x2· · ·εnxn:εi±1, i2, . . . , n} ≤n−δ; 1.9
iilocally uniform nonsquare in the sense of James or locally uniform non-l1nwrite LUN-l1n, n∈N, n≥2, if for everyx∈SXthere existsδ >0 such that for anyx1, x2, . . . , xn∈ SX,
min{xε2x2· · ·εnxn:εi±1, i2, . . . , n} ≤n−δ; 1.10
iiinonsquare in the sense of James or non-l1n write N-l1n, n ∈ N, n ≥ 2, if for every x∈SXthere existsδ >0 such that for anyx1, x2, . . . , xn∈SX,
min{x1ε2x2· · ·εnxn:εi±1, i2, . . . , n}< n. 1.11
The spaces UN-ln1, LUN-l1n, and N-ln1were considered by many authorssee15–18.
On the other hand, in 1976, Sch¨affer see 19 introduced the other definitions of various kind nonsquareness.
Definition 1.5. A Banach spaceXis said to be
inonsquarewrite N-Sif for anyx, y∈SX,
max{xy,x−y}>1; 1.12
iilocally uniform nonsquarewrite LUN-Sor if for anyx∈SXthere existsδ >0 such that for ally∈SX,
max{xy,x−y}>1δ 1.13
for anyx, y∈SX;
iiiuniformly nonsquarewrite UN-Sif there existsδ >0 such that
max{xy,x−y}>1δ 1.14
for anyx, y∈SX.
Remark 1.6. It is well known that N-S⇔N-l12and UN-S⇔UN-l21but LUN-S is not equivalent to LUN-l12see15or18.
Recall that the Banach spaceXis said to be strictly convex if for anyx, y ∈SXwith x /ywe must havexy/2<1. The next results can be found in20, but for the sake of completeness we present here a proof.
Theorem 1.7. IfXis a strictly convex Banach space thenXis nonsquare in the sense of Sch¨affer.
Proof. Suppose thatXis not nonsquare in the sense of Sch¨affer. Then there existx, y ∈SX such thatx±y 1. Putz xy x−y/2 thenz ∈ SXbut it is not an extreme point. Hence,Xis not strictly convex.
Also, let us recall that the Banach spaceXis said to be locally uniform convex if{xn}and {yn}are any sequences inSXsuch that limn→ ∞xnyn 2 we must have limn→ ∞xn− yn 0. The next theorem shows the relation between locally uniform convex and locally uniform nonsquare in any Banach spaces.
Theorem 1.8. IfXis a locally uniform convex Banach space thenXis locally uniform nonsquare in the sense of James.
Proof. Suppose thatXis not locally uniform nonsquare in the sense of James. Then there exists x∈SXand{xn} ⊂SXsuch thatxn±x → 2. On the other hand, byXis locally uniform convex, we must have
nlim→ ∞xn−x0, 1.15
which is a contradiction.
In21, it was showed that every uniformly nonsquare Banach space must have the fixed point propertythat is for any nonempty closed, convex, and bounded subsetAofX and any nonexpansive mappingP fromAinto itself has a fixed pointzinA. On the other hand, under some suitable conditions, it is well known that the generalized Ces`aro sequence spaces have the fixed point property, however, they are not uniform nonsquaresee14,22.
The main purpose of this this paper is to find the conditions for locally uniform nonsquare in the sense of James and Sch¨affer in these spaces. As consequently, we have the examples of Banach spaces that agree with a weaker geometric property than uniformly nonsquare but still possess the fixed point property.
In the sequel, we will assume that limj→ ∞suppj < ∞, saypj ≤ β for allj ∈ N. The following result is quite useful for our purpose.
Theorem 1.9. cespis locally uniform convex.
Proof. See8.
2. Main results
We begin by obtaining the first main result.
Theorem 2.1. cespis locally uniform nonsquare in the sense of James.
Proof. The result is an immediate consequence of Theorems1.8and1.9.
Next, we will show that the space cesp is locally uniform nonsquare in the sense of Sch¨affer. To do this, we need the following lemma.
Lemma 2.2. A closed bounded setK⊂cespis compact if for anyε >0 there existsN∈Nsuch that ∞
kN
1 k
k i1
|yi|
pk
< ε 2.1
for anyy∈K.
Proof. Let{xn}be a sequence inK. Define
πk: cesp−→R byπkx xk, 2.2
we haveπk is a continuous function for eachk ∈N.So,πk{xn}is a bounded subset ofR.
Then, by using the orthogonal method, a subsequence{xnj} ⊂ {xn}andx∈0can be found such thatxnjk → xkasj → ∞for allk∈N. We claim thatx∈Kandxnj → xasj → ∞.
Indeed, by our hypothesis, for eachε∈0,1there existsN Nε∈Nsuch that for allj ∈N, we have
M kN
1 k
k i1
|xnji|
pk
< εβ, 2.3
whenM > N. Lettingj → ∞andM → ∞,we get ∞
kN
1 k
k i1
|xi|
pk
< εβ, 2.4
which implies thatx ∈ cesp. Moreover, since there existsβ ∈ Rwhich 1 ≤ pk ≤ βfor all k∈N,2.3gives
ρ
xnj−xNnj ε
∞
kN1
1/kk
i1|xnji|
ε
pk
< 1 εβ
∞ kN1
1 k
k i1
|xnji|
pk
<1, 2.5
that is, for eachj ∈N
xnj−xNnj
p< ε, 2.6
and similarly,2.4gives
x−xN
p< ε. 2.7
Next, since for eachk∈Nwe havexnjk → xkasj → ∞, there existsjo∈Nsuch that
|xnjk−xk|< εβ1
δN3 2.8
for allj > joandk∈ {1,2,3, . . . , N}, whereδmax{1,∞
jN11/jpj}. Therefore,
ρ
xNnj−xN ε
N
k1
1/kk
i1|xnji−xi|
ε
pk
∞
kN1
1/kN
i1|xnji−xi|
ε
pk
< 1 εβ
εβ1 δN3·N3
2 εβ1 δN3·N·δ
< ε 2 ε
2 ε
2.9
for allj > jo, that is,
xNnj −xN
p< ε 2.10
for allj > jo.Hence, by2.6,2.7, and 2.10we can conclude thatxnj → xasj → ∞.
Finally, sinceKis closed, we must havex∈K.
Now, we are in position to prove the other main result.
Theorem 2.3. cespis locally uniform nonsquare in the sense of Sch¨affer.
Proof. Assume on the contrary that cesp is not locally uniform nonsquare in the sense of Sch¨affer. Then, there arex∈Scespand{yn} ⊂Scespsuch that
x±yn −→1 as n−→ ∞. 2.11
First, we claim that there isεo>0 such that for anyj∈Nthere existsnj> jfor which
∞ kj
1 k
k i1
|ynji|
pk
≥εo. 2.12
Otherwise, byLemma 2.2the set{yn}is compact. So there is a subsequence{ynk} ⊂ {yn}andy∈Scespsuch that
klim→ ∞ynk−y0. 2.13
Therefore,x±y 1, that is, cespis not nonsquare. This contradicts to Theorems1.9and 1.7, respectively. Hence, the claim holds true.
Note that byLemma 1.1, a real numberδ∈0, εo/4can be found such that
|ρuv−ρu|< εo
4, 2.14
wheneveru, v∈cespwithρu≤1 andρv≤δ.Using this positive real numberδ, sincex is an order continuous elementseeRemark 1.3iiiandiv, there existsio∈Nsuch that
∞ kjo1
1 k
k i1
|xi|
pk
≤δ. 2.15
Hence, byLemma 1.2, we get
1−εo
4 < ρx− ∞
kjo1
1 k
k i1
|xi|
pk
jo
k1
1 k
k i1
|xi|
pk
. 2.16
Now, by2.11, for a real numberε1∈0, εowhich satisfies the inequality1ε1β<1εo/2, a numberno∈Ncan be found such that
max{x±yn}<1ε1 2.17
for alln > no. Takingj1 max{jo, no}, so there existsnj1 > j11 satisfying2.12. Thus, by 2.14and2.16, we would have
max
ρ
x±ynj1
1ε1
max
∞
k1
1 k
k i1
xi±ynj1i 1ε1
pk
≥ 1
1ε1βmax j
1
k1
1 k
k i1
xi±ynj1i pk
∞
kj11
1 k
k i1
xi±ynj1i pk
≥ 1
1ε1βmax j
1
k1
1 k
k i1
xi±ynj1i pk
∞
kj11
1 k
k i1
ynj1i pk−εo
4
≥ 1
1ε1βmax j
1
k1
1 k
k i1
xi±ynj1i pk3εo 4
≥ 1 1ε1β
j 1
k1
1 2
1 k
k i1
xi ynj1i pk 1
2 1
k k i1
xi−ynj1i pk 3εo
4
≥ 1 1ε1β
j 1
k1
1 2k
k i1
xi ynj1ixi−ynj1i pk3εo
4
≥ 1 1ε1β
j 1
k1
1 k
k i1
xi pk3εo
4
> 1ε0/2 1ε1β >1.
2.18 This implies that max{x±ynj1}>1ε1, which is a contradiction to2.17. Hence, cespis locally uniform nonsquare in the sense of Sch¨affer.
Remark 2.4. In Theorems2.1and2.3, we have shown that, under some suitable conditions, the generalized Ces`aro sequence spaces are locally uniform nonsquare in the sense of James and Sch¨affer. This result gives the motivation to consider the geometric properties that are weaker than uniform nonsquare but still possess the fixed point property in any Banach spaces.
Acknowledgments
The author would like to thank Professor Suthep Suantai for making good suggestions and comments while preparing the manuscript. This work was supported by The Thailand Research FundProject no. MRG5180178.
References
1 J. Musielak, Orlicz Spaces and Modular Spaces, vol. 1034 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1983.
2 Y. A. Cui and H. Hudzik, “On the uniform Opial property in some modular sequence spaces,”
Functiones et Approximatio Commentarii Mathematici, vol. 26, pp. 93–102, 1998.
3 “Programma van Jaarlijkse PrijsvragenAnnual Problem Section,” Niew Archief voor Wiskunde, vol.
16, pp. 47–51, 1968, problem 2 on page 49.
4 “Programma van Jaarlijkse PrijsvragenAnnual Problem Section,” Niew Archief voor Wiskunde, vol.
21, pp. 70–76, 1971, problem 1968.2 on page 70.
5 J.-S. Shiue, “On the Ces`aro sequence spaces,” Tamkang Journal of Mathematics, vol. 1, no. 1, pp. 19–25, 1970.
6 G. M. Leibowitz, “A note on the Ces`aro sequence spaces,” Tamkang Journal of Mathematics, vol. 2, no.
2, pp. 151–157, 1971.
7 A. A. Jagers, “A note on Ces`aro sequence spaces,” Nieuw Archief voor Wiskunde, vol. 22, pp. 113–124, 1974.
8 S. Suantai, “On some convexity properties of generalized Ces`aro sequence spaces,” Georgian Mathematical Journal, vol. 10, no. 1, pp. 193–200, 2003.
9 V. Karakaya, “Some geometric properties of sequence spaces involving lacunary sequence,” Journal of Inequalities and Applications, vol. 2007, Article ID 81028, 8 pages, 2007.
10 L. Leindler, “Factorization of inequalities,” Journal of Inequalities in Pure and Applied Mathematics, vol.
5, no. 1, article 12, pp. 1–13, 2004.
11 L. Maligranda, N. Petrot, and S. Suantai, “On the James constant andB-convexity of Ces`aro and Ces`aro-Orlicz sequences spaces,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp.
312–331, 2007.
12 N. Petrot and S. Suantai, “On uniform Kadec-Klee properties and rotundity in generalized Ces`aro sequence spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 2, pp.
91–97, 2004.
13 N. Petrot and S. Suantai, “Uniform Opial properties in generalized Ces`aro sequence spaces,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 8, pp. 1116–1125, 2005.
14 W. Sanhan and S. Suantai, “Onk-nearly uniform convex property in generalized Ces`aro sequence spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 57, pp. 3599–3607, 2003.
15 S. Chen, “Geometry of Orlicz spaces,” Dissertationes Mathematicae, vol. 356, p. 204, 1996.
16 J. Diestel, Geometry of Banach Spaces: Selected Topics, Springer, Berlin, Germany, 1984.
17 R. Grza¸´slewicz, H. Hudzik, and W. Orlicz, “Uniform non-ln1 property in some normed spaces,”
Bulletin of the Polish Academy of Sciences. Mathematics, vol. 34, no. 3-4, pp. 161–171, 1986.
18 M. Kato, L. Maligranda, and Y. Takahashi, “On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces,” Studia Mathematica, vol. 144, no. 3, pp. 275–295, 2001.
19 J. J. Sch¨affer, Geometry of Spheres in Normed Spaces, vol. 20 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1976.
20 Y. A. Cui, L. Jie, and R. Pluciennik, “Local uniform nonsquareness in Ces`aro sequence spaces,”
Commentationes Mathematicae, vol. 27, pp. 47–58, 1997.
21 J. Garc´ıa-Falset, E. Llorens-Fuster, and E. M. Mazcu ˜nan-Navarro, “Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings,” Journal of Functional Analysis, vol.
233, no. 2, pp. 494–514, 2006.
22 Y. Q. Lui, B. E. Wu, and P. Y. Lee, Method of Sequence Spaces, Guangdong of Science and Technology Press, Guangzhou, China, 1996.
