Some Sufficient Conditions for Fixed Points of Multivalued Nonexpansive Mappings

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Volume 2009, Article ID 319804,12pages doi:10.1155/2009/319804

Research Article

Some Sufficient Conditions for Fixed Points of Multivalued Nonexpansive Mappings

Zhanfei Zuo

^{1, 2}

and Yunan Cui

^{1, 2}

1Department of Mathematics and Computer Science, Chongqing Three Gorges University, Wanzhou 404000, China

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

Correspondence should be addressed to Zhanfei Zuo,[email protected] Received 2 July 2009; Accepted 3 December 2009

Recommended by Hichem Ben-El-Mechaiekh

We show some sufficient conditions on a Banach space X concerning the generalized James constant, the generalized Jordan-von Neumann constant, the generalized Zbag˘anu constant, the coefficientε0X, the weakly convergent sequence coefficient WCSX, and the coefficient of weak orthogonality, which imply the existence of fixed points for multivalued nonexpansive mappings.

These fixed point theorems improve some previous results in the recent papers.

Copyrightq2009 Z. Zuo 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.

1. Introduction

In 1969, Nadler1established the multivalued version of Banach contraction principle. Since then the metric fixed point theory of multivalued mappings has been rapidly developed.

Some classical fixed point theorems for singlevalued nonexpansive mappings have been extended to multivalued nonexpansive mappings. However, many questions remain open, for instance, the possibility of extending the well-known Kirk’s theorem 2, that is,

“Do Banach spaces with weak normal structure have the fixed point property FPP for multivalued nonexpansive mappings?”

Since weak normal structure is implied by diﬀerent geometric properties of Banach spaces, it is natural to study whether those properties imply the FPP for multivalued mappings. Dhompongsa et al. 3, 4 introduced the DL condition 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 the DL condition or property D. In this setting the following results have been obtained.

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iKaewkhao5proved that a Banach spaceXwith

JX<1 1

μX 1.1

satisfies the DL condition. He also showed that the condition CNJX<1 1

μX² 1.2

implies the DL condition6.

iiSaejung 7 showed that a Banach space X has property D wheneverε0X <

WCSX.

In this paper, we show some sufficient conditions on a Banach space X concerning the generalized James constant, the generalized Jordan-von Neumann constant, the generalized Zb˘aganu constant, the coefficientε0X, the weakly convergent sequence coefficient WCSX, and the coefficient of weak orthogonality, which imply the existence of fixed points for multivalued nonexpansive mappings. These theorems improve the above results.

2. Preliminaries

Before going to the result, let us recall some concepts and results which will be used in the following sections. LetXbe a Banach space with the unit ballBX {x∈X:x ≤1}and the unit sphereSX {x∈X :x 1}. The two constants of a Banach space

CNJX sup

⎧⎨

⎩

xy²x−y² 2

x²y² :x, y∈X andxy>0

⎫⎬

⎭, JX sup

minxy,x−y:x, y∈SX

2.1

are called the von Neumann-Jordan8and James constants9, respectively, and are widely studied by many authors10–20. Recently, both constants are generalized in the following ways foro≤a≤2see12,13:

CNJa, X sup xy²x−z²

2x²y²z² :x, y, z∈X, xyz>0, andy−z≤ax

, Ja, X sup

minxy,x−z

:x, y, z∈SX, andy−z≤ax .

2.2 It is clear thatCNJ0, X CNJXandJ0, X JX.

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Recently, Gao and Saejung in6define a new constant fora≥0:

CZa, X sup

2xyx−z

2x²y²z² :x, y, z∈X, xyz>0, andy−z≤ax

, 2.3

which is inspired by Zb˘aganu paper21. It is clear that

CZ0, X CZX supxyx−z

x²y² :x, y∈X, xy>0

. 2.4

The modulus of convexity ofXsee22is a functionδX:0,2 → 0,1defined by

δX inf

1−xy

2 :x, y∈SX, x−y≥

. 2.5

The functionδXis strictly increasing on0X,2. Here0X sup{:δX 0}is the characteristic of convexity ofX, and the space is called uniformly nonsquare if0X<2.

In23the author introduces a modulus that scales the 3-dimensional convexity of the unit ball: he considers the number

δX sup

ε∈0,2:∃x, y, z∈BX such that minx−y,x−z,y−z≥ε 2.6

and defines the functionδX:0,δX → 0,1by δXε inf

1−

xyz 3

:x, y, z∈BX, minx−y,x−z,y−z≥ε

. 2.7

He also considers the coeﬃcient corresponding to this modulus:

ε0X: sup ε∈

0,δX

:δX 0

. 2.8

It is evident that δXε ≥ δXε for all ε ∈ 0,δX and in consequence ε0X ≤ ε0X.

Moreover this last inequality can be strict, since it was shown in23the existence of Banach spaces withε0X<2 which are not uniformly nonsquare.

The weakly convergent sequence coeﬃcient WCSX of X is defined as follows:

WCSX inf{lim_{n /}mxn−xm}where the infimum is taken over all weakly null sequences {xn}inXsuch that lim_n_{→ ∞}xn 1 and lim_{n,m→ ∞,n /}_mxn−xmexist.

The WORTH property was introduced by Sims in24as follows. A Banach spaceX has the WORTH property if

nlim→ ∞|xnx − xn−x| 0, 2.9

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for all x ∈ X and all weakly null sequences xn. In 25, Jim´enez-Melado and Llorens- Fuster defined the coeﬃcient of weak orthogonality, which measures the degree of WORTH wholeness , by

μX inf

λ: lim sup

n→ ∞ xnx ≤λlim sup

n→ ∞ xn−x

, 2.10

where the infimum is taken over allx∈Xand all weakly null sequencexn. It is known that Xhas the WORTH property if and only ifμX 1.

LetCbe a nonempty subset of a Banach spaceX. We shall denote byCBXthe family of all nonempty closed bounded subsets of X and byKCX the family of all nonempty compact convex subsets of X. A multivalued mapping T : C → CBX is said to be nonexpansive if

H

Tx, Ty

≤x−y, x, y∈C, 2.11 whereH·,·denotes the Hausdorﬀmetric onCBXdefined by

HA, B: max

sup

x∈Ainf

y∈Bx−y, sup

y∈B inf

x∈Ax−y

, A, B∈CBX. 2.12

Let{xn}be a bounded sequence inX. The asymptotic radiusrC,{xn}and the asymptotic centerAC,{xn}of{xn}inCare defined by

rC,{xn} inf

lim sup

n

xn−x:x∈C

, 2.13

AC,{xn}

x∈C: lim sup

n

xn−x rC,{xn}

, 2.14

respectively. It is known thatAC,{xn}is a nonempty weakly compact convex set whenever Cis.

The sequence{xn} is called regular with respect toCif rC,{xn} rC,{xni} for all subsequencesxni of{xn}, and{xn}is called asymptotically uniform with respect toCif AC,{xn} AC,{xni}for all subsequences{xni}of{xn}.

Lemma 2.1. i(See Goebel [26] and Lim [27]) There always exists a subsequence of{xn}which is regular with respect toC.

ii(See Kirk [28]) IfCis separable, then{xn}contains a subsequence which is asymptotically uniform with respect toC.

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

rCD inf

x∈Csup

y∈D

x−y. 2.15

Dhompongsa et al.4introduced the propertyDif there existsλ∈0,1such that for any nonempty weakly compact convex subsetCofX, any sequence{xn} ⊂ Cwhich is regular asymptotically uniform relative toC, and any sequence{yn} ⊂AC,{xn}which is regular asymptotically uniform relative toXwe have

r C,

yn

≤λrC,{xn}. 2.16

The Dom´ınguez-Lorenzo condition, DL condition in short form, introduced in3is defined as follows: if there existsλ∈0,1such that for every weakly compact convex subsetCofX and for every bounded sequence{xn}inCwhich is regular with respect toCwe have,

rCAC,{xn}≤λrC,{xn}. 2.17

It is clear from the definition that propertyDis weaker than the DL condition. The next results show that propertyDis stronger than weak normal structure and also implies the existence of fixed points for multivalued nonexpansive mappings4.

Theorem 2.2. LetXbe a Banach space satisfying property (D). ThenXhas weak normal structure.

Theorem 2.3. LetCbe a nonempty weakly compact convex subset of a Banach spaceXwhich satisfies the property (D). LetT :C → KCCbe a nonexpansive mapping, thenThas a fixed point.

3. Main Results

Theorem 3.1. LetCbe a weakly compact convex subset of a Banach spaceXand let{xn}be a bounded sequence inCregular with respect toC. Then for everya∈0,2,

rCAC,{xn}≤ Ja, X

1|1−a|/μXrC,{xn}. 3.1 Proof. Denoter rC,{xn}andA AC,{xn}. We can assume thatr > 0. By passing to a subsequence if necessary, we can also assume that{xn}is weakly convergent to a pointx∈C.

Since{xn}is regular with respect toC, then passing through a subsequence does not have any eﬀect to the asymptotic radius of the whole sequence{xn}. Letz∈A, then we have that

lim sup

n xn−z r. 3.2

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Denoteμ μX. By the definition ofμ,we have that lim sup

n

xn−2xz lim sup

n

xn−x z−x

≤μlim sup

n xn−x−z−x μr.

3.3

Convexity ofCimplies that2/1μx μ−1/1μz∈C,and thus, we obtain lim sup

n

xn− 2

1μxμ−1 1μz

≥r. 3.4

On the other hand, by the weak lower semicontinuity of the norm, we have that lim inf

n

1−1−a μ

xn−x−

11−a μ

z−x ≥

1|1−a|

μ

z−x. 3.5

For everyε >0,there existsN∈Nsuch that 1xN−z ≤rε,

2xN−2xz ≤μrε,

3xN−2/1μx μ−1/1μz ≥r−ε,

41−1−a/μxN−x−1 1−a/μz−x ≥1|1−a|/μz−xr−ε/r.

Now, putu 1/rεxN−z,v 1/μrεxN−2xz,andω 1−a/μrεxN− 2xz. Using the above estimates, we obtainu, v, ω∈BX,v−ω ≤au,and

uv xN−x

rε − z−x

rε xN−x

μrε z−x μrε

1

rε 1 μrε

xN−x− 1

rε − 1 μrε

z−x 1

rε

1 1 μ

xN−x−1−1/μ 11/μ

z−x 1

rε

1 1 μ

xN− 2

1μxμ−1 1μz

≥

1 1 μ

r−ε rε

, u−ω

xN−x

rε − z−x

rε −1−axN−x

μrε −1−az−x μrε

1

rε

1−1−a μ

xN−x−

11−a μ

z−x

≥

1|1−a|

μ

z−x r

r−ε rε

.

3.6

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Thus,

Ja, X≥ uv ∧ u−ω

≥ 11

μ

r−ε rε

∧

1|1−a|

μ

z−x r

r−ε rε

. 3.7

By the weak lower semicontinuity of the norm again, we conclude thatz−x ≤r,and hence,

1 1 μ

r−ε rε

∧

1|1−a|

μ

z−x r

r−ε rε

1 |1−a|

μ

z−x r

r−ε rε

. 3.8

ThereforeJa, X≥1|1−a|/μz−x/rr−ε/rε.Since the above inequality is true for everyε >0 and everyz∈A, we obtain

sup

z∈Ax−z ≤

Ja, X 1|1−a|/μ

r, 3.9

and therefore,

rCA≤

Ja, X 1|1−a|/μ

r. 3.10

Corollary 3.2. LetCbe a nonempty bounded closed convex subset of a Banach space X such that Ja, X<1|1−a|/μXand letT :C → KCCbe a nonexpansive mapping. ThenThas a fixed point.

Proof. WhenJa, X<1|1−a|/μX, thenXsatisfies the DL condition byTheorem 3.1. So Thas a fixed point byTheorem 2.3.

Remark 3.3. In particular, whena 0, we get the result of Kaewkhao; a Banach spaceXwith

JX<1 1

μX 3.11

satisfies the DL condition.

Theorem 3.4. LetCbe a weakly compact convex subset of a Banach spaceXand let{xn}be a bounded sequence inCregular with respect toC. Then for everya∈0,2,

rCAC,{xn}≤

⎛

⎜⎜

⎝

CNJa, X

2μ⁴μ² 1−a²μ²

−

μ²12

μ²1−a₂

⎞

⎟⎟

⎠rC,{xn}. 3.12

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Proof. Letr, A,{xn}, x, z, andμbe as in the proof of the previous theorem. Thus, lim sup

n

xn−z r, lim sup

n xn−2xz ≤μr.

3.13

Since2/μ²1x μ²−1/μ²1z∈Cand by the definition ofr, we obtain

lim sup

n

xn−

# 2

μ²1xμ²−1 μ²1z$

≥r. 3.14

On the other hand, by the weak lower semicontinuity of the norm, we have that lim inf

n

μ²−1a

xn−x−

μ²1−a

z−x≥

μ²1−a

z−x. 3.15

For everyε >0, there existsN∈Nsuch that 1xN−z ≤rε,

2xN−2xz ≤μrε,

3xN−2/μ²1x μ²−1/μ²1z ≥r−ε,

4μ²−1axN−x−μ²1−az−x ≥μ²1−az−xr−ε/r.

Now, putu μ²xN−z,v xN−2xz,andω 1−axN−2xzand use the above estimates to obtainu ≤μ²rε,v ≤μrε,ω ≤|1−a|μrε, andv−ω ≤au, so that

uv μ²xN−x−z−x xN−x z−x

μ²1

xN−x−μ²−1

μ²1z−x

μ²1 xN−

# 2

μ²1x μ²−1 μ²1z$

≥ μ²1

r−ε,

u−ω μ²xN−x−z−x−1−axN−x z−x

μ²−1a

xN−x−

μ²1−a

z−x

≥

μ²1−a z−x

r−ε r

.

3.16

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By the definition ofCNJa, X, we get

CNJa, X≥ xy²x−z² 2x²y²z²

≥ r−ε

rε 2

μ²12

μ²1−a2z−x/r² 2μ⁴μ² 1−a²μ² .

3.17

Letε → 0; we obtain thatCNJa, X≥μ²1²μ²1−a²z−x/r²/2μ⁴μ²1−a²μ². Then we have

z−x ≤

⎛

⎜⎜

⎝

CNJa, X

2μ⁴μ² 1−a²μ²

−

μ²12

μ²1−a₂

⎞

⎟⎟

⎠r. 3.18

This holds for arbitraryz∈A; hence, we have that

rCA≤

⎛

⎜⎜

⎝

CNJa, X

2μ⁴μ² 1−a²μ²

−

μ²1₂ μ²1−a2

⎞

⎟⎟

⎠r. 3.19

Corollary 3.5. LetCbe a nonempty bounded closed convex subset of a Banach space X such that CNJa, X < μ²1−a² μ²1²/2μ⁴μ² 1−a²μ²and letT : C → KCCbe a nonexpansive mapping. ThenThas a fixed point.

Proof. WhenCNJa, X<μ²1−a² μ²1²/2μ⁴μ² 1−a²μ², thenXsatisfies the DL condition byTheorem 3.4. SoT has a fixed point byTheorem 2.3.

Remark 3.6. In particular, whena 0, we get the result of Kaewkhao; a Banach spaceXwith

CNJX<1 1

μX² 3.20

Repeating the arguments in the proof ofTheorem 3.4, we can easily get the following conclusion.

Theorem 3.7. LetC be a nonempty bounded closed convex subset of a Banach spaceX such that CZa, X < 2μ² 1μ² 1−a/2μ⁴ μ² 1−a²μ² and let T : C → KCCbe a nonexpansive mapping. ThenThas a fixed point.

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Remark 3.8. In particular, whena 0, we get that

CZX<1 1

μX² 3.21

satisfies the DL condition which improves the result of Kaewkhao; a Banach spaceXwith

CNJX<1 1

μX² 3.22

Theorem 3.9. A Banach spaceXhas property (D) wheneverε0X<WCSX.

Proof. Let Cbe a nonempty weakly compact convex subset of X. Suppose that {xn} ⊂ C and {yn} ⊂ AC,{xn} are regular asymptotically uniforms relative to C. Passing to a subsequence, we may assume that {yn} is weakly convergent to a point y0 ∈ C and d lim_{n /}myn −ym exists. Letr rC,{xn}. Again, passing to a subsequence of{xn}, still denoted by{xn}, we assume in addition that

xn−yn≤r 1

n, xn−y_n1≤r 1

n, xn−y_n2≤r1 n, xn− 1

3

ynyn1yn2≥r− 1 n,

3.23

for alln∈N. Now, put

un

1 r1/n

xn−yn

, vn

1 r1/n

xn−y_n1 , ωn

1 r 1

n

xn−y_n2 .

3.24

It is easy to see that lim_nun−vn d/r, lim_nun−ωn d/r, limnωn−vn d/r, and lim_nunvnωn 3. This implies thatδXd/r 0 orε0X≥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

.

3.25

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Hence,

r C,

yn

≤ ε0X

WCSXrC,{xn}. 3.26

Remark 3.10. 1Theorem 3.9strengthens the result of Saejung7and X has propertyD wheneverε0X<WCSX.

2Theorem 3.9also improves the resultε0X<1 implying that the Banach spaceX has normal structure fromTheorem 2.2.

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