Journal of Applied Mathematics and Stochastic Analysis, 13:3 (2000), 299-302. G-H-KKM SELECTIONS WITH APPLICATIONS TO MINIMAX THEOREMS

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


j-1
This generalizes the notion of a KKM selection in a pseudoconvex space by Joo' and Kassay [2].

G-H-KKM Theorems and Applications
In this section we first recall and obtain some auxiliary results and then establish some minimax theorems.
Proposition 2.1: Let X be a nonempty set and (Y,H,{p)) a G-H-space.Let f: X x Y---.R, e:YR and h:XR be functions.Then the following statements are equivalent: (a) A mapping T:X-P(Y) defined by T(x) {y e Y: f(x, y) + e(y) h(x) <_ O} (rasp.T(x) {y E Y: f(x,y) + e(y)-h(x) >_ 0}), is a G-H-KKM mapping.
(iv) There exists an element x o 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.Since f is 0-generalized G-H-quasiconcave, it implies that T(x) is nonempty.It follows from Proposition 2.1 that T is a G-H-KKM mapping.By (i) and (ii), each T(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 that xEX 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.