TABLE OF CONTENTS In this appendix we give, without proof, three theorems (on fixed points) which have been used in Chapters 3 and 4 of the text. Proofs of these theorems are rather lengthy and can be found in the references cited. Some general references for these as well as other results on fixed points are (Istr?tescu, 1981), (Joshi and Bose, 1985), and (Smart, 1974).
Fixed point theorems
If S is a compact and convex subset of R n and f is a continuous function mapping S into itself, then there exists at least one x ? S such that f(x)=x.
Several proofs exist for this fixed point theorem, one of the most elementary ones being given by Kuga (1974). Its original version has appeared in 1910 (Brouwer, 1910). A generalization of this theorem is the Kakutani fixed point theorem (Kakutani, 1941) given below.
Let f be a function defined on a normed linear space X, and associating with each x ? X a subset f(x) of some (other) normed linear space Y. Then, f is said to be upper semicontinuous (usc) at a point x 0 ? X if, for any sequence {x i } converging to x 0 and any sequence {y i ? f(x i )} converging to y 0 , we have y 0 ? f( x 0). The function f is upper semicontinuous if it is usc at each point of X.
