On Some Geometric Constants and the Fixed Point Property for Multivalued Nonexpansive Mappings

Volume 2010, Article ID 596952,12pages doi:10.1155/2010/596952

Research Article

On Some Geometric Constants and the Fixed Point Property for Multivalued Nonexpansive Mappings

Jingxin Zhang

and Yunan Cui

1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

2Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, China

Correspondence should be addressed to Jingxin Zhang,zhjx [email protected] Received 30 July 2010; Accepted 5 October 2010

Academic Editor: L. G ´orniewicz

Copyrightq2010 J. Zhang and Y. Cui. 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.

We show some geometric conditions on a Banach space X concerning the Jordan-von Neumann constant, Zb˘aganu constant, characteristic of separation noncompact convexity, and the coefficient R1, X, the weakly convergent sequence coefficient, which imply the existence of fixed points for multivalued nonexpansive mappings.

1. Introduction

Fixed point theory for multivalued mappings has many useful applications in Applied Sciences, in particular, in game theory and mathematical economics. Thus it is natural to try of extending the known fixed point results for single-valued mappings to the setting of multivalued mappings.

In 1969, Nadler 1 established the multivalued version of Banach’s contraction principle. One of the most celebrated results about multivalued mappings was given by Lim2in 1974. Using Edelstein’s method of asymptotic centers, he proved the existence of a fixed point for a multivalued nonexpansive self-mappingT :C → KCwhereCis a nonempty bounded closed convex subset of a uniformly convex Banach space. Since then the metric fixed point theory of multivalued mappings has been rapidly developed. Some other classical fixed point theorems for single-valued mappings have been extended to multivalued mappings. However, many questions remain open, for instance, the possibility of extending the well-known Kirk’s theorem, that is, do Banach spaces with weak normal structure have the fixed point propertyFPP, in shortfor multivalued nonexpansive mappings?

Since weak normal structure is implied by different geometrical properties of Banach spaces, it is natural to study if those properties imply the FPP for multivalued mappings.

Dhompongsa et al.3,4introduced the Domnguez-Lorenzo conditionDLcondition, in short and property D which imply the FPP for multivalued nonexpansive mappings.

A possible approach to the above problem is to look for geometric conditions in a Banach space X which imply either theDLcondition or propertyD. In this setting the following results have been obtained.

1Dhompongsa et al. 3 proved that uniformly nonsquare Banach spaces with property WORTH satisfy theDLcondition.

2Dhompongsa et al.4showed that the condition

CNJX<1WCSX²

4 1.1

implies propertyD.

3Satit Saejung5proved that the conditionε0X<WCSXimplies propertyD.

4Gavira6showed that the condition

JX<1 1

R1, X 1.2

impliesDLcondition.

In 2007, Dom´ınguez Benavides and Gavira7have established FFP for multivalued nonexpansive mappings in terms of the modulus of squareness, universal infinite-dimension- al modulus, and Opia modulus. Attapol Kaewkhao8has established FFP for multivalued nonexpansive mappings in terms of the James constant, the Jordan-von Neumann Constants, weak orthogonality.

Besides, In 2010, Dom´ınguez Benavides and Gavira9have given a survey of this subject and presented the main known results and current research directions.

In this paper, in terms of the Jordan-von Neumann constant, Zb˘aganu constant,εβX and the coefficient R1, X, the weakly convergent sequence coefficient, we show some geometrical properties which imply the propertyDorDLcondition and so the FPP for multivalued nonexpansive mappings.

2. Preliminaries

LetXbe a Banach space andCbe a nonempty subset ofX; we denote all nonempty bounded closed subsets ofXbyCBXand all nonempty compact convex subsets ofXbyKCX.

A multivalued mappingT :C → CBXis said to be nonexpansive if the inequality H

Tx, Ty

≤x−y 2.1

holds for everyx, y∈C, whereH·,·is the Hausdorffdistance onCBX, that is, HA, B: max

sup

x∈A inf

y∈Bx−y,sup

y∈B inf

x∈Ax−y

, A, B∈CBX. 2.2

LetC ⊂ X be a nonempty bounded closed convex subset and{xn} ∈ X a bounded sequence; we use rC,{xn} and AC,{xn} to denote the asymptotic radius and the asymptotic center of{xn}inC, respectively, that is,

rC,{xn} inf

lim sup

n

xn−x:x∈C

, AC,{xn}

x∈C: lim sup

n

xn−x rC,{xn}

.

2.3

It is known thatAC,{xn}is a nonempty weakly compact convex asCis.

Let{xn}and Cbe as above; then{xn}is called regular relative toC ifrC,{xn} rC,{xni}for all subsequence{xni}of {xn}; further,{xn} is called asymptotically uniform relative toCifAC,{xn} AC,{xni}for all subsequence{xni}of{xn}. In Banach spaces, we have the following results:

1 Goebel 10 and Lim 2 there always exists a subsequence of {xn} which is regular relative toC;

2 Kirk11ifCis separable, then{xn}contains a subsequence which is asymptoti- cally uniform relative toC.

IfDis a bounded subset ofX, the Chebyshev radius ofDrelative toCis defined by rCD inf

x∈Csup

y∈D

x−y. 2.4

In 2006, Dhompongsa et al.3introduced the Domnguez-Lorenzo condition DL condition, in shortin the following way.

Definition 2.1 see3. We say that a Banach spaceX satisfies the DL condition if there existsλ ∈ 0,1 such that for every weakly compact convex subsetC ofX and for every bounded sequence{xn}inCwhich is regular with respect toC,

rCAC,{xn}≤λrC,{xn}. 2.5

TheDLcondition implies weak normal structure3. We recll that a Banach spaceX is said to have a weak normal structurew-NSif for every weakly compact convex subsetC ofXwith diamC: sup{x−y:x, y∈C}>0 there existx∈Csuch that sup{x−y:y∈ C}<diamC.

The DL condition also implies the existence of fixed points for multivalued nonexpansive mappings.

Theorem 2.2see3. LetCbe a weakly compact convex subset of Banach spaceX; ifCsatisfies (DL) condition, then multivalued nonexpansive mappingT :C → KCChas a fixed point.

Definition 2.3see4. A Banach spaceXis said to have propertyDif there existsλ∈0,1 such that for every weakly compact convex subsetCofX and for every sequence{xn} ⊂C and for every{yn} ⊂AC,{xn}which are regular asymptotically uniform relative toC,

r C, yn

≤λrC,{xn}. 2.6

It was observed that propertyD is weaker than theDL condition and stronger than weak normal structure, and Dhompongsa et al.4proved that propertyDimplies the w-MFPP.

Theorem 2.4see4. LetCbe a weakly compact convex subset of Banach spaceX; ifCsatisfies property (D), then multivalued nonexpansive mappingT :C → KCChas a fixed point.

Before going to the results, let us recall some more definitions. LetXbe a Banach space.

The Benavides coefficientR1, Xis defined by Dom´ınguez Benavides [12] as

R1, X sup

lim inf

n→ ∞ {xnx}

, 2.7

where the supremum is taken over allx∈XwithX ≤1 and all weakly null sequence{xn}inBX

such that

Dxn: lim sup

n→ ∞ lim sup

m→ ∞ xn−xm ≤1. 2.8

Obviously, 1≤R1, X≤2.

The weakly convergent sequence coefficient WCSXis equivalently defined bysee 13

WCSX inf

lim_{n /m}xn−xm lim sup_nxn

, 2.9

where the infimum is taken over all weakly not strongly null sequences {xn} with lim_{n /m}xn−xmexisting.

The ultrapower of a Banach space has proved to be useful in many branches of mathematics. Many results can be seen more easily when treated in this setting.

First we recall some basic facts about ultrapowers. LetFbe a filter on an index setN and letXbe a Banach space. A sequencexninXconvergers toxwith respect toF, denoted by lim_Fx_n x, if for each neighborhoodUofx,{i∈I :xi∈U} ∈ F. A filterUonNis called an ultrafilter if it is maximal with respect to the set inclusion. An ultrafilter is called trivial if it is of the form{A⊂N, i0∈A}for some fixedi0∈N; otherwise, it is called nontrivial. Letl∞X denote the subspace of the product spaceΠ_i∈NXiequipped with the norm

xn: sup

n∈Nxn<∞. 2.10

LetUbe an ultrafilter onNand let NU

xn∈l∞X: lim

U xn 0

. 2.11

The ultrapower ofX, denoted byX, is the quotient space l∞X/NU equipped with the quotient norm. Writexn_U to denote the elements of ultrapower. It follows from the definition of the quotient norm that

xn_U lim

U xn. 2.12

Note that ifUis nontrivial, thenX can be embedded intoX isometrically. For more details see14.

3. Main Results

We first give some sufficient conditions which imply DL condition. The Jordan-von Neumann constantCNJXwas defined in 1937 by Clarkson15as

CNJX sup

⎧⎨

⎩

xy²x−y² 2

x²y² : x, y∈X, xy/0

⎫⎬

⎭. 3.1

Theorem 3.1. LetX be a Banach space and Ca weakly compact convex subset ofX. Assume that {xn}is a bounded sequence inCwhich is regulary relative toC. Then

rCAC,{xn}≤ R1, X

2CNJX

R1, X 1 rC,{xn}. 3.2

Proof. Denoter rC,{xn}andA AC,{xn}. We can assume thatr >0. Since{xn} ⊂Cis bounded andCis a weakly compact set, by passing through a subsequence if necessary, we can also assume thatxnconverges weakly to some element inx∈Candd lim_{n /m}xn−xm exists. We note that since{xn}is regular,rC,{xn} rC,{yn}for any subsequence{yn}of {xn}. Observe that, since the norm is weak lower semicontinuity, we have

lim inf

n xn−x ≤lim inf

n lim inf

m xn−xm lim inf

n /m xn−xm d. 3.3 Letη >0; taking a subsequence if necessary, we can assume thatxn−x<dηfor alln.

Letz∈A. Then we have lim sup_nxn−z randx−z ≤lim infnxn−z ≤r. Denote R R1, X; by definition, we have

R≥lim inf

n

xn−x

dη z−x r

lim inf

n

xn−x

dη −x−z r

. 3.4

On the other hand, observe that the convexity ofCimpliesR−1/R1x 2/R 1z∈C; since the norm is weak lower semicontinuity, we have

lim inf

n

1

rxn−z 1 R

xn−x

dη −x−z r

lim inf

n

1

r 1

R dη

xn−

1 R

dη 1 Rr

x−

1 r − 1

Rr

z

≥

1 r − 1

Rr

x 2 Rrz−

1 r 1

Rr

z

1 r 1

Rr

R−1 R1x 2

R1z−z

≥ 1

r 1 Rr

rCA, lim inf

n

1

rxn−z− 1 R

xn−x

dη −x−z r

lim inf

n

1

r − 1 R

dη

xn−x− 1

r 1 Rr

z−x

≥ 1

r 1 Rr

z−x ≥ 1

r 1 Rr

rCA.

3.5

In the ultrapowerX ofX, we consider

u 1

r{xn−z}_U ∈S_X, v 1 R

xn−x

dη −x−z r

U∈B_X. 3.6

Using the above estimates, we obtain uv lim

U

1

rxn−z 1 R

xn−x

dη −x−z r

≥ 1

r 1 Rr

rCA, u−v lim

U

1

rxn−z− 1 R

xn−x

dη −x−z r

≥ 1

r 1 Rr

rCA.

3.7

Therefore, we have

CNJ

X

≥ uv ²u−v ² 2

u²v ²

≥ 21/r1/Rr²rCA² 211

1 2

1 r 1

Rr ₂

rCA².

3.8

Since Jordan-von Neumann constantCNJX ofXequals toCNJXofX, we obtain

CNJX≥ 1 2

1 r 1

Rr ₂

rCA². 3.9

Hence we deduce the desired inequality.

By Theorems2.2and3.1, we have the following result.

Corollary 3.2. LetCbe a nonempty bounded closed convex subset of a Banach space X such that CNJX< 1/R1, X 1²/2 andT :C → KCCa nonexpansive mapping. ThenT has a fixed point.

Proof. sinceR1, X ≥ 1, ifCNJX < 1/R1, X 1²/2, then we haveCNJX < 2 which implies thatXis uniformly nonsquare; henceX is reflexive. Thus by Theorems2.2and3.1, the result follows.

Remark 3.3. Note that JX²/2 ≤ CNJX; it is easy to see that Theorem 3.1 includes 6, Theorem 3andCorollary 3.2includes6, Corollary 2.

To characterize Hilbert space, Zb˘aganu defined the following Zb˘aganu constant:see 16

CZX supxyx−y

x²y² : x, y∈X, xy>0

. 3.10

We first give the following tool.

Proposition 3.4. CZX CZX.

Proof. Clearly,CZX≤CZX. To show CZX ≤CZX, supposex, y∈X are not all zero.

Without loss of generality, we assumex a >0.

Let us chooseη∈0, a. Sincex lim_Uxn aand

c: xy xy x²y ² lim

U

xnynxn−yn xn²yn² : lim

U cn, 3.11

the setA: {n∈N: |cn−c|< η and|xn −a|< η}belongs toU. In particular, noticing that xn/0 forn∈A, there existsnsuch that

xy xy

x ²y² < xnynxn−yn xn²yn² η

≤CZX η.

3.12

Hence, the inequalityCZX ≤CZXfollows from the arbitrariness ofη.

Theorem 3.5. LetX be a Banach space and Ca weakly compact convex subset ofX. Assume that {xn}is a bounded sequence inCwhich is regulary relative toC. Then

rCAC,{xn}≤ R1, X

2CZX

R1, X 1 rC,{xn}. 3.13

Proof. Letu, vbe as inTheorem 3.1. Then

uv ≥ 1

r 1 Rr

rCA, u−v ≥ 1

r 1 Rr

rCA. 3.14

Therefore, by the definition of Zb˘aganu constant, we have

CZ

X

≥ utvu−tv u²v ²

≥ 1 2

1 r 1

Rr ₂

rCA².

3.15

Since Zb˘aganu constantCZX ofXequals toCZXofX, we obtain

CZX≥ 1 2

1 r 1

Rr 2

rCA². 3.16

Hence we deduce the desired inequality.

UsingTheorem 2.2, we obtain the following corollary.

Corollary 3.6. LetCbe a nonempty weakly compact convex subset of a Banach spaceX such that CZX <11/R1, X²/2 and letT : C → KCCbe a nonexpansive mapping. ThenT has a fixed point.

In the following, we present some properties concerning geometrical constants of Banach spaces which also imply the propertyD.

Theorem 3.7. LetXbe a Banach space. IfCZX<WCSX; thenXhas property (D).

Proof. LetC be a weakly compact convex subset ofX; suppose that{xn} ⊂ Cand{yn} ⊂ AC,{xn}are regular and asymptotically uniform relative toC. Passing to a subsequence of {yn}, still denoted by{yn}, we may assume thatyn−→w y0∈Candd lim_{n /m}yn−ymexists.

Letr rC,{xn}. Again passing to a subsequence of{xn}, still denoted by{xn}, we assume in addition that

nlim→ ∞xn−y2n lim

n→ ∞xn−y_2n1 lim

n→ ∞

xn− 1

2

y2ny_2n1

r. 3.17

Let us consider an ultrapowerX ofX. Put

u 1

r xn−y2n

U, v 1

r xn−y2n1

U; 3.18

then we know thatu∈S_X,v∈S_X. We see that uv lim

U

xn−y2n

r xn−y2n1

r

2, 3.19 u−v lim

U

xn−y2n

r − xn−y2n1

r

lim

U

y2n−y2n1

r

d

r. 3.20

Thus, By the definition of Zb˘aganu constant, we have CZ

X

≥ uv u−v u ²v² ≥ d

r. 3.21

Since the Zb˘aganu constants ofXand ofXare the same, we obtainCZX≥d/r. Now we estimatedas follows:

d lim

n /myn−ym lim

n /myn−y0

−

ym−y0

≥WCSXlim sup

n

yn−y0

≥WCSXr C, yn

.

3.22

HencerC,{yn} ≤ CZX/WCSXrC,{xn}and the assertion follows by the definition of propertyD.

Using Theorems2.4and3.7, we obtain the follwing corollary.

Corollary 3.8. LetCbe a nonempty bounded closed convex subset of a reflexive Banach spaceXsuch thatCZX < WCSXand letT : C → KCCbe a nonexpansive mapping. ThenT has a fixed point.

The separation measure of noncompactness is defined by

βB sup ε: there exists a sequence{xn}inBsuch that sep{xn}≥ε

3.23 for any bounded subsetBof a Banach spaceX, where

sep{xn} inf{xn−xm: n /m}. 3.24 The modulus of noncompact convexity associated toβis defined in the following way:

ΔX,βε inf 1−d0, A: A⊂BX is convex, βA≥ε

. 3.25

The characteristic of noncompact convexity of X associated with the measure of noncompactnessβis defined by

εβX sup ε≥0 : ΔX,βε 0

. 3.26

WhenX is a reflexive Banach space, we have the following alternative expression for the modulus of noncompact convexity associated withβ,

εβX inf

1− x: {xn} ⊂BX, x w−lim

n xn, sep{xn}≥ε

. 3.27

It is known that X is NUC if and only ifεβX 0. The above-mentioned definitions and properties can be found in17.

Theorem 3.9. LetXbe a reflexive Banach space. IfεβX<WCSX, thenXhas property (D).

Proof. LetC be a weakly compact convex subset of X; suppose that{xn} ⊂ Cand {yj} ⊂ AC,{xn}are regular and asymptotically uniform relative toC. Passing to a subsequence of {yj}, still denoted by{yj}, we may assume thatyj −→w y0∈Candd lim_{k /}lyk−ylexists.

Letr rC,{xn}.

Since{y0, yj} ⊂AC,{xn}, we have lim sup

n

xn−y0 r, lim sup

n

xn−yj r, ∀j ∈N. 3.28 So for anyη≥0, there existsN∈Nsuch thatxN−y0 ≥r−ηandxN−yi ≤rη, for all j∈N.

Without loss of generality, we suppose thatyk−yl ≥ d−η for allk /l. Now we consider sequence{xN−yj/rη} ⊂BX; notice that

β

xN−yj

rη

≥ d−η

rη, xN−yj

rη

−→w xN−y0

rη . 3.29

By the definition ofΔX,β·, we have ΔX,β

d−η rη

≤1−

xN−y0

rη

≤1−r−η

rη. 3.30

Since the last inequality is true for anyη > 0, we obtainΔX,βd/r 0; thusεβX ≥ d/r.

Now we estimatedas follows:

d lim

k /lyk−yl lim

k /lyk−y0

−

yl−y0

≥WCSXlim sup

n

yn−y0

≥WCSXr C, yn

.

3.31

Hence,

r C, yn

≤ εβX

WCSXrC,{xn}. 3.32

Remark 3.10. Since εβX ≤ ε0X,Theorem 3.9 implies the 5, Theorem 3. Furthermore, it is easy to seeCNJX ≥ 1 ε0X²/4 ≥ 1 εβX²/4; thenTheorem 3.9also includes 4, Theorem 3.7.

ByTheorem 3.9, we obtain the following Corollary.

Corollary 3.11. LetCbe a nonempty bounded closed convex subset of a reflexive Banach spaceX such thatεβX<WCSXand letT:C → KCCbe a nonexpansive mapping. ThenT has a fixed point.

Noticing WCSX ≥ 1, obviously,Corollary 3.11 extends the following well-known result.

Theorem 3.12see18, Theorem 3.5. LetCbe a nonempty bounded closed convex subset of a reflexive Banach spaceXsuch thatεβX<1 and letT :C → KCCbe a nonexpansive mapping.

ThenT has a fixed point.

Acknowledgments

The authors would like to thank the anonymous referee for providing some suggestions to improve the manuscript. This work was supported by China Natural Science Fund under grant 10571037.

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