NOTE ON KKM MAPS AND APPLICATIONS 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

(1)

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 coincidence 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 foundation for many well-known existence results, such as Ky Fan’s minimax inequality theorem, Ky Fan-Browder’s fixed point theorem, Nash’s equilibrium theorem, 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 property provided the resulted family of sets by this mapping has finite intersection property.

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 minimax theorem and coincidence theorem are obtained.

2. The KKM maps

In the sequel, letXbe a set and 2^X 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→2^Xis called a KKM map if

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 53286, Pages1–9 DOI10.1155/FPTA/2006/53286

(2)

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→2^X is called a map 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 2^Y, that is, maps closed set to closed set, and G:F→2^XwithG(yi)⊆G(xi) fori=1, 2,...,nsuch that the composition mappingLG: F→2^Ydefined byLG(f)= ∪x∈G(f)L(x) forf ∈Fis aKKMmap and∩ⁿ_i₌1LG(yi) = ∅ implies that∩ⁿ_i₌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→2^X 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→2^YbyG(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,∞)→2^RbyG(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→2^X 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.

Theorem 2.6. LetXbe a topological space andEbe a set. Suppose thatG:E→2^Xis a closed valued map with the KKM property. Then{G(x)}x∈Ehas a finite intersection property.

Theorem 2.7 (Ky Fan’s theorem). LetXbe a topological space andEbe a subset ofX. If G:E→2^Xis a closed valued map with the KKM property and there is a setG(x) such that G(x) is compact. Then∩x∈EG(x) = ∅.

(3)

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.

Theorem 3.1 (Fan-Browder’s fixed point theorem). LetEbe a convex subset of a vector spaceXandG:E→2^Ebe a map satisfying the following conditions:

(1) there exists {yi: 1≤i≤n} ⊂E such that co{yi: 1≤i≤n} ⊆ ∪ⁿ_i₌1G⁻¹(yi) and G⁻¹(yi)∩co{yi: 1≤i≤n}is open in co{yi: 1≤i≤n}with co{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→2^co^Fby Kyi

=coF\G⁻¹yi

coF (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(yi). Thusyiis a fixed point ofG, and the conclusion holds.) One can easily see that

n i=1

Kyi

=coF\ n i=1

G⁻¹yi

coF. (3.2)

By assumption (1), we have∩ⁿ_i₌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}

∪^k_j₌1K(yij), that is, there existsy∈co{yi1,yi2,...,yik}such thaty /∈K(yij) forj=1, 2,..., k. Thus we have

y∈G⁻¹yij

, 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→2^Ebe 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.

(3.4)

(4)

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⁻¹y1

T⁻¹y2,

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.

Corollary 3.4. LetCbe a nonempty convex subset of a topological vector spaceEand V be an open convex subset with 0∈V. Suppose that a mapT:C→Eis continuous and T(C)⊂ ∪ⁿ_i₌1{yi+V}, where yi∈Cfor i=1, 2,...,n. Then there existsx0∈C such that Tx0∈x0+V.

Proof. Let a mapG:C→2^Cbe 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(yi) is open. Moreover,C= ∪ⁿ_i₌1G⁻¹(yi) and thus

coy1,y2,...,yn

⊆ n i=1

G⁻¹yi

. (3.7)

Therefore, byTheorem 3.1, we know that there existsx0∈Csuch thatx0∈G(x0). This

implies thatTx0∈x0+V.

Corollary 3.5. LetCbe a nonempty convex subset of a locally convex spaceEandKbe a convex compact subset ofE. Suppose thatT:C→Eis continuous andT(C)⊂ ∪ⁿ_i₌1{yi+K}, whereyi∈Cfori=1, 2,...,n. Then there is anx0∈Csuch thatTx0∈x0+K.

4. Coincidence theorem and minimax theorem

Theorem 4.1 (Ky Fan’s coincidence theorem). LetXandYbe nonempty convex subsets of topological vector spacesEandF, respectively. LetA,B:X→2^Ybe two maps satisfying the following conditions:

(1) there existsxi∈Xsuch thatAxiis open fori=1, 2,...,n,Y= ∪ⁿ_i₌1AxiandA⁻¹yis a convex set for eachy∈Y;

(2) there existsyj∈ysuch thatB⁻¹yjis open forj=1, 2,...,m,X= ∪^m_j₌1B⁻¹yjandBx is a convex set for eachx∈Y.

Then there existsx0∈Xsuch thatAx0∩Bx0 = ∅. Proof. Let a mapK:X×Y→2^X^×^Ybe defined by K(x,y)=X×Y\

B⁻¹y×Ax (4.1)

(5)

for all (x,y)∈X×Y. By the assumptions, we have X×Y=

n i=1

m j=1

B⁻¹yj×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 havey0∈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 be lower semi- continuous from above atx0if, for any net (xt)_t_∈_T withxt→x0, f(xt)≤f(xt) fort≥t implies that f(x0)≤lim_tf(xt). Similarly, f is said to upper semi-continuous from below atx0 if, for any net (xt)_t_∈_T withxt→x0, f(xt)≤ f(xt) fort≤timplies that f(x0)≤ lim_tf(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 function f :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

nlim→∞fxn

=1=f(0) (4.7)

(6)

and so f is lower semi-continuous from above at 0. If we takexn= −1/n, then we have

nlim→∞fxn

=0< f(0) (4.8)

and so f cannot be lower semi-continuous at 0.

Lemma 4.5. 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)=min_x_∈_Xf(x) (resp., f(x0)=max_x_∈_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)≤lim_tf(yt) and sof(y0)=inf_y_∈_Cf(y). The proof of upper semi- continuous from below case is similar and hence we omit the detail. This completes the

proof.

Theorem 4.6 (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= ∪ⁿ_i₌1Ai;

(4) for eachr∈R, there existyj, j=1, 2,...,m, such thatBj= {x:f(x,yj)< r}is open andX= ∪^m_j₌1Bj.

Then max_x_∈_Xmin_y_∈_Yf(x,y)=min_y_∈_Ymax_x_∈_X f(x,y).

Proof. By the assumptions (1), (2) andLemma 4.5, we know that max_x_∈_Xmin_y_∈_Yf(x,y) and min_y_∈_Ymax_x_∈_Xf(x,y) both exist. It is obviously that

maxx∈Xmin

y∈Y f(x,y)≤min

y∈Ymax

x∈X f(x,y). (4.9) Now we show that

maxx∈Xmin

y∈Y f(x,y)=min

y∈Ymax

x∈X f(x,y). (4.10)

If this is not true, then there would be a numberr∈Rsuch that maxx∈Xmin

y∈Y f(x,y)< r <min

y∈Ymax

x∈X f(x,y). (4.11)

Define two mapsA,B:X→2^Y 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⁻¹yj. (4.12)

(7)

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.

Theorem 4.7 (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) sup_x_∈_Cf(x,y) is lower semi-continuous from above onC;

(2){y: f(x,y)≤sup_x_∈_Cf(x,x)}is closed for eachx∈C; (3)x→ f(x,y) is quasi-concave onCfor eachy∈C.

Then miny∈Csup_x_∈_Cf(x,y)≤sup_x_∈_Cf(x,x).

Proof. ByLemma 4.5, we know that sup_x_∈_Cf(x,y) obtains its minimum onC.

Now, we may assume that sup_x_∈_Cf(x,x)=μ <∞. Define a mapG:C→2^Cby 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

miny∈Csup

x∈Cf(x,y)≤sup

x∈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 function f : [0, 1]×[0, 1]→Rbe defined by

f(x,y)=

⎧⎨

⎩x+y ify∈[0, 1),

x+ 2 ify=1. (4.15)

Then we have

x∈sup[0,1]

f(x,y)=

⎧⎨

⎩

1 +y ify∈[0, 1),

3 ify=1. (4.16)

Thus it follows that sup_x_∈_[0,1]f(x,y) is not lower semi-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

=[0, 1] (4.17)

(8)

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).

References

[1] J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, New York, 1984.

[2] C. Bardaro and R. Ceppitelli, Some further generalizations of Knaster-Kuratowski-Mazurkiewicz theorem and minimax inequalities, Journal of Mathematical Analysis and Applications 132 (1988), no. 2, 484–490.

[3] F. E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math- ematische Annalen 177 (1968), 283–301.

[4] S.-S. Chang and Y. Zhang, Generalized KKM theorem and variational inequalities, Journal of Mathematical Analysis and Applications 159 (1991), no. 1, 208–223.

[5] J. Dugundji and A. Granas, Fixed Point Theory. I, Monografie Matematyczne, vol. 61, Pa ´nstwowe Wydawnictwo Naukowe (PWN), Warsaw, 1982.

[6] K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proceedings of the National Academy of Sciences of the United States of America 38 (1952), 121–126.

[7] , Sur un th´eor`eme minimax, Comptes Rendus Academie des Sciences Paris 259 (1964), 3925–3928 (French).

[8] , Applications of a theorem concerning sets with convex sections, Mathematische Annalen 163 (1966), 189–203.

[9] , Extensions of two fixed point theorems of F. E. Browder, Mathematische Zeitschrift 112 (1969), 234–240.

[10] A. Granas, KKM-maps and their applications to nonlinear problems, The Scottish Book (R. D.

Mauldin, ed.), Birkh¨auser, Massachusetts, 1981, pp. 45–61.

[11] A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer, New York, 2003.

[12] C. D. Horvath, Contractibility and generalized convexity, Journal of Mathematical Analysis and Applications 156 (1991), no. 2, 341–357.

[13] B. Knaster, C. Kuratowski, and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes f¨urn-dimensionale simplexe, Fundamenta Mathematicae 14 (1929), 132–137 (German).

[14] S. Park and H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, Journal of Mathematical Analysis and Applications 197 (1996), no. 1, 173–187.

[15] S. Singh, B. Watson, and P. Srivastava, Fixed Point Theory and Best Approximation: The KKM- Map Principle, Mathematics and Its Applications, vol. 424, Kluwer Academic, Dordrecht, 1997.

[16] M. Sion, On general minimax theorems, Pacific Journal of Mathematics 8 (1958), 171–176.

[17] G. X.-Z. Yuan, KKM Theory and Applications in Nonlinear Analysis, Monographs and Textbooks in Pure and Applied Mathematics, vol. 218, Marcel Dekker, New York, 1999.

(9)

[18] S. S. Zhang and G. S. Yang, Some further generalizations of Ky Fan’s minimax inequality and its applications to variational inequalities, Applied Mathematics and Mechanics. Yingyong Shuxue he Lixue 11 (1990), no. 11, 961–968 (Chinese), Applied Mathematics and Mechanics (English ed.) 11 (1990), no. 11, 1027–1034.

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]

B. S. Lee: Department of Mathematics, Kyungsung University, Pusan 608-735, Korea E-mail address:[email protected]

(10)

Special Issue on

Time-Dependent Billiards

Call for Papers

This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.

This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under diﬀerent approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many diﬀerent fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).

We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers dis- cussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.

To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.

Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System athttp://

mts.hindawi.com/according to the following timetable:

Manuscript Due March 1, 2009 First Round of Reviews June 1, 2009 Publication Date September 1, 2009

Guest Editors

Edson Denis Leonel,Department of Statistics, Applied Mathematics and Computing, Institute of Geosciences and Exact Sciences, State University of São Paulo at Rio Claro, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil; [email protected]

Alexander Loskutov,Physics Faculty, Moscow State University, Vorob’evy Gory, Moscow 119992, Russia;

[email protected]

Hindawi Publishing Corporation http://www.hindawi.com