SELECTIONS WITH TO MINIMAX THEOREMS

Journal of Applied Mathematics and Stochastic Analysis, 13:3 (2000), 299-302.

G-H-KKM SELECTIONS WITH APPLICATIONSTO MINIMAX THEOREMS

RAM U. VERMAInternational Publications

Division of Mathematical Sciences1206 Coed Drive

Orlando, FL 32825 USA

(Received September, 1998; Revised November, 1999)

Based on the G-H-KKM selections, some nonempty intersection theoremsand their applications to minimax inequalities are presented.

Key words: G-H-KKM Selections, Intersection Theorems, MinimaxTheorems.

AMS subject classifications: 49J40.

1. Introduction

Minimax inequalities have numerous applications to variational inequalities, whilevariational inequalities turn out to be a powerful tool to the solvability of problemsin elasticity and plasticity theory, heat conduction, diffusion theory, optimizationtheory, mathematical economics, and others.

Chang and Zhang [1] introduced the notion of the generalized quasiconcavity andobtained some nonempty intersection theorems and their applications to minimaxinequalities in a linear topological space setting. Recently, Tan [4] extended thisnotion to the case of a G-convex space with applications to minimax theorems andsaddle points. Our aim here is to present some G-H-KKM selection theorems andrelated applications to minimax inequalities in a G-H-space setting.

Let X be a topological space, P(X) denote the power set of X, and (X), a familyof all nonempty finite subsets of X. Let An denote a standard (n-1) simplex{el, e2,..., en} of Rn.

Definition 1.1: A triple (X,H,{p}) is called a G-H-space [6] if X is a topologicalspace and H: (X)P(X)\{O} is a mapping such that:

(i)

(ii)

For each F,G E (X), there exists F1 C F such that F1 C G impliesH(F1) CH(G)..For F {xi, x2,..., xn} (X), there is a continuous mapping p: An-H(F)such that for {il,i2,...,ik}C{1,2,...,n}, we have p({eii,ei2,...,eik})CH({Xil,Xi2,...,Xik}) where {Xil,Xi2,...,Xik C F.

Printed in the U.S.A. ()2000 by North Atlantic Science Publishing Company 299

300 R.U. VERMA

A subset D of X is called finitely G-H-closed in X if for each A E (X), there exists

A C A such that D(- )H(A) is closed in H(A ).A subset K of X is said to be compactly closed in X if K(-)L is closed in L for

all compact subsets L of X.Definition 1.2: Let (X,H,{p}) be a G-H-space and T:X-P(X) a multivalued

mapping. T is called a G-H-KKM mapping if for each {Xl, x2,..., Xn} (X), thereexists {Xil Xi2,..., Xik} C {X1, X2,..., Xn} such that

H({Xil,Xi2,...,xik}) C k)T(xij for {il,i2,...,ik} C {1,2,...,n}.

j=l

Definition 1.3: Let (X,H, {p}) be a G-H-space and let M1,M2,...,Mn be subsetsof X. A subset {Xl,X2,...,Xn} (X) is said to be a G-H-KKM selection forM1,M2,...,Mn if for any {Xil, Xi2,...,Xik} C {Xl,X2,...,Xn}, we have

H({Xil, Xi2,...,Xik} Ck

)Mij for {il, i2,...,ik} C {1,2,...,n}.j-1

This generalizes the notion of a KKM selection in a pseudoconvex space by Joo’and Kassay [2].

Definition 1.4: Let X be nonempty set and (Y,H,{p}) a G-H-space. Letf:X Y---,R, e:Y---R and h:XR be functions. The function f is said to be 0-generalized G-H-quasiconcave (rasp. O-generalized G-H-quasiconvex) in its firstvariable x if for each {Xl,X2,...,Xn} (X), there exists {Vl, V2,... Vn} (Y) such thatfor each {vii v12,... vik} C {v1, v2,..., vn} and for any Yo H({Vil, vi2,’", Vik}), wehave

min [f(xij Yo) + e(Yo)- h(xij)] < 0l_j_k

(rasp.1 _<maxj<_ k[f(xij, YO) + e(Yo) h(xij)] >- 0),

where {il, i2,..., ik) C {1, 2,..., n}.This generalizes the notion of a 0-generalized quasiconcavity (0-generalized

quasiconvexity) by Chang and Zhang [1].

2. G-H-KKM Theorems and Applications

In this section we first recall and obtain some auxiliary results and then establishsome minimax theorems.Lemma 2.1: [7] Let (X,H, {p}) be a G-H-space and M1,M2,...,Mn be finitely

G-H-closed subsets of X. Suppose that M1,M2,...,Mn have a G-KKM selection.Then )= 1Mi O.

Proposition 2.1: Let X be a nonempty set and (Y,H,{p)) a G-H-space. Letf: X x Y---.R, e:YR and h:XR be functions. Then the following statements areequivalent:

(a) A mapping T:X-P(Y) defined by

T(x) {y e Y: f(x, y) + e(y) h(x) <_ O}

G-H-KKM Selections With Applications to Minimax Theorems 301

(rasp. T(x) {y E Y: f(x,y) + e(y)- h(x) >_ 0}),

is a G-H-KKM mapping.(b) f is O-generalized G-H-quasiconcave (rasp. O-generalized G-H-quasiconvex)

in its first variable x.Proof: (a)=:v(b) Since T is G-H-KKM, it implies for each {Xl,X2,...,xn}

and corresponding {vl,v2,...,Vn} (YI that there exist {Xil,Xi2,...,xik} C {xl,x2,...,xn}, {vii, vi2,..., vik} C {v1, v2,..., vk} and any Yo H({Vil, vi2,"’, vik}) such that

H({Vil, Vi2,..., vik} Ck

.)T(xij)"j=l

This implies Yo (. k)T(xij), and as a result, there exists some index m (i < m < k)

j=lsuch that Yo T(Xim)" Hence, f(Xim Yo) + e(Yo) h(xim) - 0 (rasp. f(Xim Yo) +e(Yo)- h(xim >_ 0). It follows that

min k[f(xij Yo) + e(Yo) h(xij)] < 0l<_j<_

(resp. max k[f(xij Yo) + e(Yo) h(xij)] > 0).<j<

(b)::v(a) Since f is 0-generalized G-H-quasiconcave (resp. 0-generalized G-H-quasiconvex) in x, it implies for any {Xl, X2,...,z,} (X) and {Vl, V2,...,v,}there exist {vii vi2,... Vik} C {v1, v2,..., v}, and any Yo H({Vil, vi2,"’, Vik}) suchthat

min k[f(xij Yo) + e(Yo) h(xij)] < 0l<_j<_

(resp. max k[f(xij Yo) + e(yo) h(x) > 0).l<j<

It follows that there exists some ind.x m l _< m _< ) such that 0 ta,) C

T(ai). This completes the proof. 1

Theorem 2.1: Let X be a nonempty set and (Y,H,{p}) a G-H-space such thatH(F) is compact for all F (Y). Suppose that I:X x Y---R, e:Y---,R and h:X-Rare functions satisfying the following assumptions:

(i) f is lower semicontinuous in y on compact subsets of Y.(ii) e is lower semicontinuous on compact subsets of Y.(iii) f is O-generalized G-H-quasiconcave in x.

(iv) There exists an element xo X such that the set

{y Y: f(xo, y) + e(y) h(xo) <_ 0}

is a compact subset of Y.Then there exists an element y Y such that

f(x,y )+ e( )-h(x) <_ 0 for all x e X.

Proof." Let us define a mapping T: X---,P(Y) by

T(x) {y Y: f(x, y) + e(y) h(x) <_ 0} for all x X.

302 R.U. VERMA

Since f is 0-generalized G-H-quasiconcave, it implies that T(x) is nonempty. Itfollows from Proposition 2.1 that T is a G-H-KKM mapping. By (i) and (ii), eachT(x) is finitely G-H-closed, that is, for each A C A E (Y), we have

T(x)( )H(A {y H(A )" f(x, y)+ e(y)- h(x) < O}

{y H(A ): f(x, y) + e(y) < h(x)},

is closed in H(A) by the lower semicontinuity of f and e, so the family{T(x)’x X} has the finite intersection property by Lemma 2.1. Now applying (iv),we find that {T(x)(- )T(x0): x X} is a family of compact subsets of Y. Hence,

)T(x) # O. That means, there exists an element y Y such thatxEX

f(x,y )+ e(y )-h(x) 0 for all x E X.

This completes the proof.For Y compact, Theorem 2.1 reduces to:Theorem 2.2: Let X be a nonempty set and (Y,H,{p}) a compact G-H-space with

H(F) compact for all F (Y). Suppose that f:X x YR, e" Y--,R and h" X---,R are

functions such that:(i) f is lower semicontinnous in second variable y.(ii) e is lower semicontinuous.

(iii) f is O-generalized G-H-quasiconcave in first variable x.Then there is an element y Y such that

)+ )-h(.) <_ 0 for art X.

References

[1]

[2]

[3]

[5]

[6]

[7]

[8]

[9]

Chang, S.S. and Zhang, Y., Generalized KKM theorem and variational inequali-ties, J. Math. Anal. Appl. 159 (1991), 208-223.Joo’, I. and Kassay, G., Convexity, minimax theorems and their applications,Annales Univ. Sci. Budapest 38 (1995), 71-93.Lin, L.J. and Chang, T.H., S-KKM theorems, saddle points and minimax in-equalities, Nonlinear Analysis 34 (1998), 73-86.Tan, K.K., G-KKM theorem, minimax inequalities and saddle points, NonlinearAnalysis 32 (1997), 4151-4160.Verma, R.U., Some generalized KKM selection theorems and minimax inequa-lities, Math. Sci. Res. Hot-Line 2:6 (1998), 13-21.Verma, R.U., G-H-KKM selections and generalized minimax theorems, Math.Sci. Res. Hot-line 2:8 (1998), 5-10.Verma, R.U., Role of generalized KKM type selections in a class of minimax in-equalities, Appl. Math. Lett. (to appear).Neumann, J.V. and Morgenstern, O., Theory of Games and Economic Behavior,University Press, Princeton 1944.Zeidler, E., Nonlinear Functional Analysis and its Applications I, Springer-Verlag, New York 1986.

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SELECTIONS WITH TO MINIMAX THEOREMS