Y. Q. CHEN, Y. J. CHO, J. K. KIM, AND B. S. LEE
Received 6 March 2005; Revised 20 July 2005; Accepted 11 August 2005
We apply the KKM technique to study fixed point theory, minimax inequality and coin-cidence theorem. Some new results on Fan-Browder fixed point theorem, Fan’s minimax theorem and coincidence theorem are obtained.
Copyright © 2006 Y. Q. Chen et al. 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 1929, the KKM map was introduced by Knaster et al. [13] and it provides the founda-tion for many well-known existence results, such as Ky Fan’s minimax inequality the-orem, Ky Fan-Browder’s fixed point thethe-orem, Nash’s equilibrium thethe-orem, Hartman-Stampacchia’s variational inequality theorem and many others (see [1,2,5–12,14–17]). The central idea of applying KKM theory to prove that a family of sets has nonempty intersection is to find a suitable space and a mapping defined on that space such that this mapping is a KKM mapping and the original family of sets has finite intersection prop-erty provided the resulted family of sets by this mapping has finite intersection propprop-erty. Based this idea, we first introduce a large class of mappings that can be interpreted as KKM mappings, then we apply the KKM technique to study fixed point theory, minimax inequality and coincidence theorem. A new concept on lower (upper) semi-continuous function is given and some new results on Fan-Browder’s fixed point theorem, Fan’s min-imax theorem and coincidence theorem are obtained.
2. The KKM maps
In the sequel, letXbe a set and 2X be the collection of nonempty subsets ofX. To begin our results, let us first recall the following definition.
Definition 2.1. LetEbe a subset of topological vector spaceX. A mapG:E→2Xis called aKKM mapif
Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 53286, Pages1–9
cox1,x2,...,xn
⊆
n
i=1
Gxi (2.1)
forxi∈E,i=1, 2,...,n.
Definition 2.2. LetEbe a set andXbe a topological space. A mapG:E→2X is called amap with the KKM property if there exists a topological vector spaceY such that, for any{xi: 1≤i≤n} ⊆E, there existF= {yi: 1≤i≤n} ⊆Y, a closed (or closed under
appropriate topology) mappingL:X→Yor 2Y, that is, maps closed set to closed set, and G:F→2XwithG(y
i)⊆G(xi) fori=1, 2,...,nsuch that the composition mappingLG:
F→2Ydefined byLG(f)= ∪x∈G(f)L(x) forf ∈Fis aKKMmap and∩in=1LG(yi) = ∅ implies that∩ni=1G(xi) = ∅.
Remark 2.3. Definition 2.2simply says that the mapGhas the KKM property ifGor the part ofGcan be mapped onto another space such that the composite map is a KKM map. One can easily check that the generalized KKM map in [4,18] is a map with the KKM property.
In the following, we give some examples of maps with the KKM property.
Example 2.4. LetE=[0, 1] be the closed interval ofR,X=R, and letG:E→2X be a map withG(x)=(1, 2 +x) forx∈E. For any (xi)⊂[0, 1],i=1, 2,...,n, putyi=3/2 +xi,
F= {y1,y2,...,yn},Y=R, and defineG:F→2YbyG(yi)=[3/2, 7/4 +xi]. TakeLas the
identity mapping onR. Then the mapLG=Gis aKKMmap and soGis a map with the KKM property.
Example 2.5. Letφ: [0,∞)→Rbe the convex function defined by
φ(x)=
⎧ ⎨ ⎩
1 ifx=0,
(x−1)2−1 ifx >0. (2.2)
DefineG: [0,∞)→2RbyG(x)= {y:φ(y)≤φ(x)}. It is easy to see thatφis not lower semi-continuous at 0 and soG(2)= {y:φ(y)≤φ(2)} is not closed. For {xi: 1≤i≤
n} ⊂[0,∞), ifφ(xi)<0 or φ(xi)≥1, we set yi=xi, otherwise, set yi=xi/2. PutF=
{y1,y2,...,yn},X=Y=R, and defineG:F→2X byG(yi)= {y:φ(y)≤φ(yi)}. TakeL
as the identity mapping onR. ThenLG=Gis a KKM map onF= {yi: 1≤i≤n}, thus Gis a map with the KKM property.
The following results are direct consequences of the KKM theorem.
Theorem2.6. LetXbe a topological space andEbe a set. Suppose thatG:E→2Xis a closed valued map with the KKM property. Then{G(x)}x∈Ehas a finite intersection property.
3. Fan-Browder’s fixed point theorem without compactness condition
The following result is a generalization of Fan-Browder’s fixed point theorem without compactness condition.
Theorem3.1 (Fan-Browder’s fixed point theorem). LetEbe a convex subset of a vector spaceXandG:E→2Ebe a map satisfying the following conditions:
(1)there exists {yi: 1≤i≤n} ⊂E such that co{yi: 1≤i≤n} ⊆ ∪ni=1G−1(yi) and
G−1(y
i)∩co{yi: 1≤i≤n}is open inco{yi: 1≤i≤n}withco{yi: 1≤i≤n}
in-herited with the Euclidean topology, whereG−1(y)= {x∈E:y∈G(x)};
(2)G(y)is convex for ally∈E. ThenGhas a fixed point.
Proof. LetF= {yi: 1≤i≤n}. Define a mapK:F→2coFby
Kyi=coF\G−1yicoF (3.1)
fori=1, 2,...,n. We may assume thatK(yi) = ∅fori=1, 2,....(Otherwise,K(yi)= ∅
for someiand so we have coF⊂G−1(y
i). Thusyiis a fixed point ofG, and the conclusion holds.) One can easily see that
n
i=1
Kyi=coF\ n
i=1
G−1y
icoF. (3.2)
By assumption (1), we have∩ni=1K(yi)= ∅. In view ofTheorem 2.6,Kcannot be a KKM
map on{yi: 1≤i≤n}. Hence there exist yi1,yi2,...,yik such that co{yi1,yi2,...,yik} ∪kj=1K(yij), that is, there existsy∈co{yi1,yi2,...,yik}such thaty /∈K(yij) forj=1, 2,..., k. Thus we have
y∈G−1y ij
, j=1, 2,...,k, (3.3)
that is, yij ∈G(y) for j=1, 2,...,kand the convexity ofG(y) immediately implies that
y∈G(y). This completes the proof.
Remark 3.2. Theorem 3.1only requires the intersectionG−1(y)∩coF fory∈Fis
rela-tively open in the convex hull of some finite subsetFofEand alsoEis not compact, which is different to the result in [3]. See also Theorem 1.2 on page 143 of Granas-Dugundji’s book [11].
Example 3.3. LetE=(0, 1) and a mapT:E→2Ebe defined by
Tx= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x,x+1 2
ifx∈
0,1 2 , 1 3,x+
1 4
ifx∈
1 2, 3 4 , x−1
2,x
otherwise.
It is obvious thatE is not compact and Txis convex for all x∈E. Put y1=1/2 and
y2=3/4. Then it follows that
coy1,y2 = 1 2, 3 4
⊂T−1y 1
T−1y
2,
T−11
2 1 2, 3 4 = 1 2, 3 4 ,
T−13
4 1 2, 3 4 = 1 2, 3 4 (3.5)
are open in [1/2, 3/4]. Therefore, the mapTsatisfies the conditions ofTheorem 2.6. Corollary3.4. Let Cbe a nonempty convex subset of a topological vector spaceEand V be an open convex subset with0∈V. Suppose that a mapT:C→Eis continuous and T(C)⊂ ∪ni=1{yi+V}, where yi∈Cfor i=1, 2,...,n. Then there existsx0∈C such that
Tx0∈x0+V.
Proof. Let a mapG:C→2Cbe defined by
G(x)= {y∈C:Tx−y∈V}. (3.6)
ThenG(x) is convex for allx∈CsinceV is convex. The continuity ofT implies that G−1(y
i) is open. Moreover,C= ∪ni=1G−1(yi) and thus
coy1,y2,...,yn
⊆
n
i=1
G−1y
i. (3.7)
Therefore, byTheorem 3.1, we know that there existsx0∈Csuch thatx0∈G(x0). This
implies thatTx0∈x0+V.
Corollary3.5. LetCbe a nonempty convex subset of a locally convex spaceEandKbe a convex compact subset ofE. Suppose thatT:C→Eis continuous andT(C)⊂ ∪ni=1{yi+K},
whereyi∈Cfori=1, 2,...,n. Then there is anx0∈Csuch thatTx0∈x0+K.
4. Coincidence theorem and minimax theorem
Theorem4.1 (Ky Fan’s coincidence theorem). LetXandYbe nonempty convex subsets of topological vector spacesEandF, respectively. LetA,B:X→2Ybe two maps satisfying the following conditions:
(1)there existsxi∈Xsuch thatAxiis open fori=1, 2,...,n,Y= ∪ni=1AxiandA−1yis a
convex set for eachy∈Y; (2)there existsyj∈ysuch thatB−1y
jis open forj=1, 2,...,m,X= ∪mj=1B−1yjandBx
is a convex set for eachx∈Y.
Then there existsx0∈Xsuch thatAx0∩Bx0 = ∅.
Proof. Let a mapK:X×Y→2X×Ybe defined by
for all (x,y)∈X×Y. By the assumptions, we have
X×Y=
n
i=1 m
j=1
B−1y
j×Axi. (4.2)
Therefore, we have
n
i=1 m
j=1
Kxi,yj= ∅. (4.3)
In view ofTheorem 2.6, we know thatKcannot be a KKM map on{xi: 1≤i≤n} × {yj:
1≤j≤m}. So there existx0,xi1,xi2,...,xilandy0,yj1,yj2,...,yjksuch thatx0∈co{xi1,xi2,
...,xil},y0∈co{yj1,yj2,...,yjk}and
x0,y0
/ ∈
l
s=1 k
t=1
Kxis,yjt
, (4.4)
which implies that
x0,y0∈B−1yjt×Axis
(4.5)
fors=1,...,landt=1, 2,...,k. By the convexities ofA−1xandBy, we havey
0∈Ax0and
y0∈Bx0. This completes the proof.
Remark 4.2. The classical Ky Fan’s coincidence theorem assume that bothXandY are compact. See Theorem 3.12 in Singh-Watson-Srivastava’s book [15]. We do not require this condition inTheorem 4.1.
Definition 4.3. LetXbe a topological space. A functionf :X→Ris said to belower semi -continuous from aboveatx0if, for any net (xt)t∈T withxt→x0, f(xt)≤f(xt) fort≥t implies that f(x0)≤limtf(xt). Similarly, f is said toupper semi-continuous from below
atx0 if, for any net (xt)t∈T withxt→x0, f(xt)≤ f(xt) fort≤timplies that f(x0)≤ limtf(xt).
One can easily see that a lower (resp., upper) semi-continuous function is also a lower (resp., upper) semi-continuous from above (resp., below) function.
The following example shows that the converse is not true.
Example 4.4. Let a functionf :R→Rbe defined by
f(x)=
⎧ ⎨ ⎩x
+ 1 ifx≥0,
x ifx <0. (4.6)
SinceR is a metric space, we consider a sequence{xn} such thatxn→0 with f(x1)≥
f(x2)≥ ··· ≥ f(xn)≥ ···.Then, by the definition of f(x), we know thatxn≥0 for all
n≥1. Therefore, it follows that
lim
n→∞f
and so f is lower semi-continuous from above at 0. If we takexn= −1/n, then we have
lim
n→∞f
xn=0< f(0) (4.8)
and so f cannot be lower semi-continuous at 0.
Lemma4.5. LetXbe a compact topological space and f :X→Rbe a real valued function. If f is lower semi-continuous from above (resp., upper semi-continuous from below), then there existsx0∈Xsuch that f(x0)=minx∈Xf(x)(resp., f(x0)=maxx∈X f(x)).
Proof. Assume thatf is lower semi-continuous from above onX. There exists a net (yt)⊂
Csuch that f(yt)≤ f(yt) ift≥tand f(yt)→infy∈Cf(y). SinceCis compact, without loss of generality, we may assume thatyt→y0. By the lower semi-continuity from above
of f(y), we havef(y0)≤limtf(yt) and sof(y0)=infy∈Cf(y). The proof of upper
semi-continuous from below case is similar and hence we omit the detail. This completes the
proof.
Theorem4.6 (von Neuman’s minimax principle). LetXandYbe two nonempty compact convex subsets of topological vector spacesEandF, respectively. Suppose that f :X×Y→R is a real valued function satisfying the following conditions:
(1)y→ f(x,y) is lower semi-continuous from above and quasi convex for each fixed x∈X, that is,{y:f(x,y)< r}is convex for eachx∈X;
(2)x→ f(x,y)is upper semi-continuous from below and quasi concave for each fixed y∈Y, that is,{x:f(x,y)> r}is convex for eachy∈Y;
(3)for eachr∈R, there existxi,i=1, 2,...,n, such thatAi= {y:f(xi,y)> r}is open
andY= ∪ni=1Ai;
(4)for eachr∈R, there existyj, j=1, 2,...,m, such thatBj= {x:f(x,yj)< r}is open
andX= ∪mj=1Bj.
Thenmaxx∈Xminy∈Yf(x,y)=miny∈Ymaxx∈X f(x,y).
Proof. By the assumptions (1), (2) andLemma 4.5, we know that maxx∈Xminy∈Yf(x,y)
and miny∈Ymaxx∈Xf(x,y) both exist. It is obviously that
max
x∈Xminy∈Y f(x,y)≤miny∈Ymaxx∈X f(x,y). (4.9)
Now we show that
max
x∈Xminy∈Y f(x,y)=miny∈Ymaxx∈X f(x,y). (4.10)
If this is not true, then there would be a numberr∈Rsuch that
max
x∈Xminy∈Y f(x,y)< r <miny∈Ymaxx∈X f(x,y). (4.11)
Define two mapsA,B:X→2Y byAx= {y: f(x,y)> r}andBx= {y: f(x,y)< r}for x∈X. It is obvious that
Y=
n
i=1
Axi, X= m
j=1
B−1y
It is direct to check thatA−1yis convex for y∈Y andBxis convex for eachx∈Xand,
byTheorem 4.1, there existsx0∈Xandy0∈Y such thaty0∈Ax0∩Bx0 = ∅. Hence we
have f(x0,y0)< r < f(x0,y0), which is a contradiction. This completes the proof.
Theorem4.7 (Ky Fan’s minimax inequality). LetCbe a compact convex subset of a topo-logical vector spaceX. Let f :C×C→Rbe a real valued function satisfying the following conditions:
(1) supx∈Cf(x,y)is lower semi-continuous from above onC; (2){y: f(x,y)≤supx∈Cf(x,x)}is closed for eachx∈C; (3)x→ f(x,y)is quasi-concave onCfor eachy∈C. Thenminy∈Csupx∈Cf(x,y)≤supx∈Cf(x,x).
Proof. ByLemma 4.5, we know that supx∈Cf(x,y) obtains its minimum onC. Now, we may assume that supx∈Cf(x,x)=μ <∞. Define a mapG:C→2Cby
G(x)=y∈C:f(x,y)≤μ (4.13)
for allx∈C. The quasi-concavity ofx→ f(x,y) onCfor eachy∈Cimplies thatGis a KKM map. By the assumption (2), we know thatG(x) is compact. Therefore, it follows fromTheorem 2.7that∩x∈CG(x) = ∅, thus there existsy0∈Csuch thaty0∈G(x) for
allx∈C, that is, f(x,y0)≤μfor allx∈C. This immediately implies that
min
y∈Csupx∈Cf(x,y)≤supx∈Cf(x,x). (4.14)
To end this paper, we give a function f which satisfies all the conditions ofTheorem 4.6.
Example 4.8. Let a functionf : [0, 1]×[0, 1]→Rbe defined by
f(x,y)=
⎧ ⎨ ⎩x
+y ify∈[0, 1),
x+ 2 ify=1. (4.15)
Then we have
sup
x∈[0,1]
f(x,y)=
⎧ ⎨ ⎩
1 +y ify∈[0, 1),
3 ify=1. (4.16)
Thus it follows that supx∈[0,1]f(x,y) is not lower continuous, but lower
semi-continuous from above. It is obvious that the set
y:f(x,y)≤ sup
x∈[0,1]
f(x,x)=3
is closed and
x:f(x, 1)> r=x:x > r−2, x:f(x,y)> r= {x:x > r−y} (4.18)
for ally∈[0, 1) are convex sets, that is,x→f(x,y) is quasi-concave onCfor eachy∈C. Therefore, the function f satisfies all the conditions ofTheorem 4.6.
Acknowledgments
The authors are grateful to the referees for their valuable suggestions which help the revi-sion of this paper. The second and fourth authors were supported by the Korea Research Foundation Grant (KRF-2000-DP0013).
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Y. Q. Chen: Department of Mathematics, Foshan University, Foshan, Guangdong 528000, China E-mail address:[email protected]
Y. J. Cho: Department of Mathematics Education and the RINS, College of Education, Gyeongsang National University, Chinju 660-701, Korea
E-mail address:[email protected]
J. K. Kim: Department of Mathematics Education, College of Education, Kyungnam University, Masan 631-701, Korea
E-mail address:[email protected]
