Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 581728,4pages doi:10.1155/2010/581728
Research Article
Fixed Point Properties Related to Multivalued Mappings
Hidetoshi Komiya
Faculty of Business and Commerce, Keio University, 223-8521 Yokohama, Japan
Correspondence should be addressed to Hidetoshi Komiya,[email protected] Received 12 January 2010; Accepted 2 April 2010
Academic Editor: Tomonari Suzuki
Copyrightq2010 Hidetoshi Komiya. 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 discuss fixed point properties of convex subsets of locally convex linear topological spaces.
We derive equivalence among fixed point properties concerning several types of multivalued mappings.
1. Introduction
We present fundamental definitions related to multivalued mappings in order to fix our terminology. We assume Hausdorffseparation axiom for all of the topological spaces which appear hereafter. LetXandYbe topological spaces. A multivalued mappingF:XY from XtoYis a function which attains a nonempty subset ofYfor each pointxofXand the subset is denoted byFx. For any subsetBofY, the upper inverseFuBand the lower inverseFlB are defined byFuB {x ∈ X : Fx ⊂ B}andFlB {x ∈ X : Fx∩B /∅}, respectively.
A multivalued mappingF :X Y is said to be upper semicontinuouslower semicontinuous, resp.ifFuG FlG, resp.is open inXfor any open subsetGofY. Moreover,Fis said to be upper demicontinuous ifFuHis open inX for any open half-spaceHofY in caseY is a linear topological space.
We are interested in fixed point properties of convex subsets of locally convex linear topological spaces. A topological space is said to have a fixed point property if every continuous functions from the topological space to itself has a fixed point. Following to this terminology, we define several fixed point properties depending on types of multivalued mappings we concern.
We always deal with convex-valued multivalued mappings defined on a convex subset of a locally convex topological linear space in this paper. Such situations appear often in arguments on fixed point theory for multivalued mappings, for example, Kakutani fixed point theorem1, Browder fixed point theorem2, and so forth. LetX be a convex
2 Fixed Point Theory and Applications subset of a locally convex topological linear space and letF : X X be a convex-valued multivalued mapping fromXtoX. We callF Kakutani-type ifF is closed-valued and upper semicontinuous and weak Kakutani-type ifFis closed-valued and demicontinuous. Similarly Fis said to be Browder-type ifFhas open lower sections; that is,F−1y {x∈X :Fxy}is open for ally∈X. We callF open graph-type if it has an open graph.
A convex subset X of a locally convex linear topological space is said to have a Kakutani-type fixed point property if every Kakutani-type multivalued mapping fromX toX has a fixed point. Similarly, we define weak Kakutani-type fixed point property, Browder-type fixed point property, and open graph-type fixed point property.
2. Result
Our main result is the following.
Theorem 2.1. LetXbe a paracompact convex subset of a locally convex linear topological spaceY. Then each of the following statements is mutually equivalent.
1Xhas a fixed point property.
2Xhas a Browder-type fixed point property.
3Xhas an open graph-type fixed point property.
4Xhas a weak Kakutani-type fixed point property.
5Xhas a Kakutani-type fixed point property.
Proof. The proofs of2⇒3and4⇒5⇒1are obvious.
(1)⇒(2). The method of the proof is similar to that of2, Theorem 1. LetF :X X be Browder-type. The family {F−1y}y∈X is an open cover of X because any point xof X belongs to an open set F−1y with y ∈ Fx. Therefore, there is a partition of unity{fα}α∈A subordinated to {F−1y}y∈X. That is, each function fα : X → 0,1 is continuous, the family{{x∈X:fαx>0}}α∈A of open sets is a locally finite refinement of{F−1y}y∈X, and
α∈Afαx 1 for allx∈X. For eachα∈A, takeysuch that{x∈X :fαx>0} ⊂F−1y, and we denote it byyα. Then define a functionf :X → Xby
fx
α∈A
fαxyα. 2.1
Here the summation
α∈Ais well defined because there are only a finite number of indicesα withfαx>0. The functionfis continuous because the family{x∈X :fαx>0}of open sets is locally finite. On the other hand, it follows thatfX⊂Xsince X is convex. Thusfhas a fixed pointx0 ∈Xby the hypothesis. That is, we have
x0
α∈A
fαx0yα. 2.2
It follows thatx0 ∈F−1yαfor eachαwithfαx0>0, and hence we haveyα ∈Fx0. SinceFx0
is convex, we havex0∈Fx0, and it is proved thatx0is a fixed point ofF.
Fixed Point Theory and Applications 3 (3) ⇒ (4). The method of this proof is inspired by the discussions found in 3, 4.
Suppose that F : X X is weak Kakutani-type but it has no fixed point; that is, x /∈Fx for anyx∈X. SinceFxis closed and convex, there is a continuous linear functionalf onY which separatesxandFxstrictly. Thus there is a real numberαsuch that
x∈Ix
y∈Y :f y
< α
, Fx⊂Jx
y∈Y :f y
> α
. 2.3
Put
UxIx∩FuJx. 2.4
ThenUx is a neighborhood ofxinX, and we haveFUx ⊂ Jx. Since{Ux}x∈X is an open cover ofX, there is an open cover{Wα}α∈AofXsuch that{Wα}α∈Ais locally finite and refines {Ux}x∈XbecauseXis paracompact. For eachα∈A, take anxsuch thatWα⊂Uxand denote it byxα. For eachx∈X, defineGxby
Gx
Wαx
Jxα ∩X. 2.5
Sincex ∈ Uxα for any αwith Wα x, we haveFx ⊂ FUxα ⊂ Jxα. Thus we haveFx ⊂
WαxJxα Gx. Therefore, we have Gx /∅ for all x ∈ X, and the definition of Gx above defines a multivalued mapping G : X X. It is easily seen that G is open and convex valued.
Next we show thatGhas an open graph. Take any elementx0, y0of the graph GrG ofGand fix it. Define
Mx0
x0/∈Wα
X\Wα
, 2.6
then Mx0 is a neighborhood of x0 because {Wα}α∈A is locally finite. ThusMx0 ×Gx0 is a neighborhood ofx0, y0. We show thatMx0 ×Gx0 ⊂ GrG. Take anyx, y ∈ Mx0 ×Gx0. Sincex ∈ Mx0, we havex /∈Wα for anyαwithx0/∈Wα. Therefore, we have{α ∈ A : x ∈ Wα} ⊂ {α∈A:x0∈Wα}. From this inclusion, we have
y∈Gx0⊂Gx. 2.7
That is,Mx0×Gx0⊂GrG. Therefore,Ghas an open graph.
On the other hand, take anyx∈X. There isα∈Asuch thatx∈Wα. Sincex∈Uxα, we havex /∈Jxα, and hencex /∈Gx. ThusGhas no fixed point and this contradicts the assumption thatXhas open graph-type fixed point property.
Klee5proved that a convex subset of a locally convex metrizable linear topological space is compact if and only if it has a fixed point property. Since any metrizable topological space is paracompact, we have the following corollary ofTheorem 2.1.
4 Fixed Point Theory and Applications Corollary 2.2. LetXbe a convex subset of a locally convex metrizable linear topological space. Then the following statements are mutually equivalent.
1Xis compact.
2Xhas a fixed point property.
3Xhas a Browder-type fixed point property.
4Xhas an open graph-type fixed point property.
5Xhas a weak Kakutani-type fixed point property.
6Xhas a Kakutani-type fixed point property.
Acknowledgment
This paper is written with support from Research Center of Nonlinear Analysis and Discrete Mathematics, National Sun Yat-Sen University.
References
1 S. Kakutani, “A generalization of Brouwer’s fixed point theorem,” Duke Mathematical Journal, vol. 8, pp. 457–459, 1941.
2 F. E. Browder, “The fixed point theory of multi-valued mappings in topological vector spaces,”
Mathematische Annalen, vol. 177, pp. 283–301, 1968.
3 H. Komiya, “Inverse of the Berge maximum theorem,” Economic Theory, vol. 9, no. 2, pp. 371–375, 1997.
4 T. Yamauchi, “An inverse of the Berge maximum theorem for infinite dimensional spaces,” Journal of Nonlinear and Convex Analysis, vol. 9, no. 2, pp. 161–167, 2008.
5 V. L. Klee Jr., “Some topological properties of convex sets,” Transactions of the American Mathematical Society, vol. 78, pp. 30–45, 1955.
