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Let X and Y be two nonempty compact convex subsets of two linear topological spaces Ex and E y . c. c. for each x E X. 41 §3. Results Based on Convexity Then: (A) there is a point (xo, YO) such that f(x, Yo) (B) maxminf(x,y) xEX yEY E X x Y (called a saddle point for J) :s f(xo, y) for all (x, y) E X x Y, = f(xo,yo) = minmaxf(x,y). yEY xEX PROOF.

PROOF. We can assume z = O. Let Ilxll = (J 2: R and let X = An V(x, K); clearly, F maps X into itself; we shall estimate Ily - F(y) lion X. ) ::; t((J + r) (J+r ::; R - r (Ilyll - IIF(y) II)· Therefore, applying the theorem of Caristi with (J+r 'P(x) = R _ r Ilxll o yields the result. 2) THEOREM (Supporting drops theorem). Let A be a closed set in a Banach space E, and z E E - A a point with d(z, A) = R > O. Then for any r < R < (J there is an Xo E 8A with Ilz-xoll::; (J and AnV(xo,K(z,r)) = {xo}.

THEOREM. 5) ; for if d(Fx , Fy) :S ad( x, y) with constant a < 1, then the condition (**) is satisfied with 'P (x) = d(x, Fx) because (1 - a)d(x, y) :S d(x, y) - d(Fx , Fy) :S d(x, Fx) + d(y, Fy) and infxEx d(x , Fx) = 0, since we have seen that d(Fnx, Fn+lx) ---+ 0 for each x E X. 6. Miscellaneous Results and Examples A. Fixed point theorems in complete metric spaces (A. I) Let (X, d) be complete and F : X ~ X a map such that FN : X ~ X is contractive for some N (easy examples show that F itself need not be continuous).