© Hindawi Publishing Corp.
SOME RESULTS ON MAXIMAL ELEMENTS
ANTONIO CARBONE
(Received 31 December 1998)
Abstract.We prove some results on maximal elements using the KKM-map principle.
Keywords and phrases. Maximal elements, KKM-map principle.
2000 Mathematics Subject Classification. Primary 47H10; Secondary 54H25.
The well-known KKM-map principle has various applications in nonlinear analysis, mathematical economics, applied mathematics, game theory, and minimax theory. The aim of this paper is to prove some results on maximal elements by using the KKM-map principle.
Recall that a binary relationFon a setCis a subset ofC×Cor a mapping ofCinto itself. The notationyFx ory∈Fxmeansy stands in relation F tox. A maximal element ofF is a pointxsuch that no pointysatisfiesy∈Fx, i.e.,Fx= ∅. Thus the set of maximal elements is
{x∈C:Fx= ∅} =
x∈C
C\F−1x, (1)
where
F−1x= {y∈C:x∈Fy}. (2)
The following well-known theorem is due to Sonnenschein [7] proved by using the KKM-principle.
Theorem1. LetCbe a compact convex subset ofRnandF a binary relation onC satisfying
(i) x∉coFxfor allx∈C(costands for convex hull), (ii) ify ∈F−1x, then there exists some x
1∈C (possiblyx1=x) such thaty ∈ intF−1x
1.
ThenF has a maximal element.
In [1], Bergstrom proved the following by using selection theorem and fixed point theorem.
Theorem2. LetCbe a nonempty, compact convex subset ofRn andF:C→2C a preference map satisfying
We note that the results on maximal elements are useful in fixed point theory, vari-ational inequalities and complementarity problems (see [2]).
Definition3. LetCbe a nonempty subset of a topological vector spaceX. A map
F:C→2Xis said to be a KKM-map if co(x1,x2,...,xn)⊆ ∪n
i=1Fxifor each finite subset {x1,x2,...,xn}ofC.
Theorem4[3]. LetC be a subset of a Hausdorff topological vector spaceX and F : C →2X a closed-valued map. If Fx
0 is compact for at least one x0∈ C, then ∩x∈CFx∉∅.
Theorem5[5]. LetC be a subset of a Hausdorff topological vector spaceX and F:C→2Xan open-valued KKM-map. Then∩n
i=1Fxi≠∅.
Ky Fan’s lemma[3]. LetCbe a nonempty compact convex subset ofRnandF:C→
2C a multifunction such that
(i) Fxis convex for eachx∈C, (ii) F has open graph,
(iii) x∉Fxfor eachx∈C. ThenF has a maximal element.
We prove the following theorem for maximal element.
Theorem6. LetC be a nonempty closed convex subset of a Hausdorff topological vector spaceXandF:C→2C satisfy
(i) x∉Fxfor eachx∈C, (ii) Fxis closed for eachx∈C,
(iii) F−1(y)= {x∈C:y∈Fx}is convex for eachy∈C,
(iv) Ccan be covered by some finite number of closed setsFx1,Fx2,...,Fxn. ThenF has a maximal element.
Proof. LetFx≠∅for eachx∈C. DefineG(x)=C\Fx, thenG(x)is open for each
x∈C sinceFxis closed. By (iv)C= ∪n
i=1Fxi, so∩Gxi= ∩(C\Fxi)c=(∪Fxi)c= ∅. ThereforeG is not a KKM-map. Hence, there exists a finite set{x1,x2,...,xk}ofC such that
z=
k
i=1
λixi
k
1
Gxi, (3)
whereλi≥0 andλi=1. Soz∈ ∩n
i=1Fxiandxi∈F−1z,i=1,2,...,k. SinceF−1zis convex so
z=
k
i=1
λixi∈F−1z, (4)
implying thatz∈Fz, this contradicts hypothesis (i). SoFx= ∅.
Proof. LetG:C→2C be defined by
Gx=F−1x= {y∈C:x∈Fy}. (5)
SinceF is upper semicontinuous, eachG(x)is closed. Now,G−1(y)=(F−1y)−1=
Fyis convex. AlsoC can be covered by{G(x):x∈B}, finitely many closed sets. By hypothesisx∉FxForx∈C. So, by Theorem 5, we get thatFhas a maximal element.
In the following, closed-valued KKM-map is applied [6].
Theorem8. LetC be a nonempty convex subset of a topological vector spaceX, andF:C→2Csatisfy
(i) x∉coFxfor eachx∈C,
(ii) ify∈F−1x, then there exists somex
1∈C(possiblyx1=x) such thaty∈intF−1x1, (iii) Chas a nonempty compact convex subsetDsuch that the set
B= {x∈C:y∉Fxfor ally∈D} (6)
is compact.
ThenF has a maximal element.
Proof. DefineG(x)=C\intF−1(x)for eachx∈C. ThenG(x)is closed for each
x∈C. We claim thatGis a KKM-map. Letz∈co(x1,x2,...,xn). Ifz∉∪ni=1Gxi, then
z∉Gxi,i=1,2,...,n, that is,z∈F−1xi,i=1,2,...,n. Thusxi∈Fzandz∈coFz, this contradicts (i). HenceGis a KKM-map.
Now,
B=x∈C:y∉Fxfor ally∈D
=x∈C:x∉F−1yfor ally∈D =x∈C:x∈Gy for ally∈D
=
y∈D
Gy= ∅,
(7)
i.e.,x0∈ y∈DGy. Thusx0∉F−1y for ally∈C, that is,y∉Fx0for ally∈C and
Fx0= ∅.
Corollary9. LetCbe a nonempty convex subset of a topological vector spaceE andF:C→2Csatisfy
(i) x∉Fx,
(ii) Fxis convex for eachx∈C,
(iii) F−1(y)= {x∈C:y∈Fx}is open inCfor eachy∈C, (iv) Chas a nonempty compact convex subsetDsuch that the set
B= {x∈C:y∉Fxfor ally∈D} (8)
is compact.
Corollary10(see [6]). LetCbe a nonempty convex subset of a topological vector spaceXandF:C→2C satisfy
(i) x∉coF(x)for eachx∈C,
(ii) F−1(y)= {x∈C:y∈Fx}is open for eachy∈C,
(iii) Chas a nonempty compact convex subsetDsuch that the set
B= {x∈C:y∉Fxfor ally∈D} (9)
is compact.
ThenF has a maximal element.
In caseCis a nonempty compact convex subset of a topological vector spaceXand
F:C→2C satisfy
(i) x∉coF(x)for eachx∈C,
(ii) F−1(y)= {x∈C:y∈Fx}is open for eachy∈C, thenF has a maximal element.
We conclude by giving a couple of applications of maximal elements.
Theorem11(Ky Fan best approximation theorem [4]). LetCbe a compact, convex subset ofRnandf:C→Rna continuous function. Then there is anx
0∈Csuch that f x0−x0 ≤ f x0−yfor ally∈C. (10)
Proof. DefineF byyFxif and only if
f x−y<f x−xfor allx∈C. (11)
It is easy to see thatFx is convex and x∉ Fx. F has open graph because f is continuous. Therefore by Fan’s lemma,F has a maximal element, that is, there is an
¯
x∈C (12)
such thatFx¯= ∅. This implies thatfx¯−x¯ ≤ fx¯−yfor ally∈C.
Note. In casef:C→C, thenfhas fixed point, that is, there is anx0∈Csuch that
f x0=x0.
The following result deals with the existence of zero for a continuous function.
Theorem12. LetCbe a closed bounded convex subset ofRandg:C→Ra con-tinuous function. If f x=gx+x andf :C →C, then there is an x0∈C such that
gx0=0.
Proof. DefineFonCbyy∈Fxif and only if|f x−y|<|f x−x|for eachx∈C. As in the above theorem we getFx0= ∅. For somex0∈C, that is,|f x0−x0| ≤ |f x0−y| for ally∈C. Thus|f x0−x0| =d(f x0,C). Sincef :C→C sof x0=x0=gx0+x0. Hencegx0=0.
References
[1] T. Bergstrom,The Existence of Maximal Elements and Equilibria in the Absence of Transi-tivity, University of Michigan, 1975, Lecture note.
[2] K. C. Border,Fixed point theorems with applications to economics and game theory, Cam-bridge University Press, CamCam-bridge, 1985. MR 86j:90002. Zbl 558.47038.
[3] K. Fan,A generalization of Tychonoff’s fixed point theorem, Math. Ann.142(1961), 305– 310. MR 24#A1120. Zbl 093.36701.
[4] ,Extensions of two fixed point theorems of F. E. Browder, Math. Z.112(1969), 234– 240. MR 40#4830. Zbl 185.39503.
[5] W. K. Kim,Some applications of the Kakutani fixed point theorem, J. Math. Anal. Appl.121 (1987), no. 1, 119–122. MR 88h:55002. Zbl 612.54055.
[6] T. C. Lin,Convex sets, fixed points, variational and minimax inequalities, Bull. Austral. Math. Soc.34(1986), no. 1, 107–117. MR 87h:47128. Zbl 597.47038.
[7] H. F. Sonnenschein,Demand theory without transitive preferences, with applications to the theory of competitive equilibrium, Preferences, utility and demand (a Minnesota sym-posium) (New York), Harcourt Brace Jovanovich, 1971, pp. 215–223. MR 56#17746. Zbl 277.90012.
Carbone: Dipartimento di Matematica, Università della Calabria,87036 Arcava-cata di Rende (Cosenza), Italy