Fixed-point theorems for compact convex sets
Mahlon Marsh Day · Illinois Journal of Mathematics · 1961
proved the well-known Brouwer fixed-point theorem.Let K be an n-cell, that is, a homeomorphic image of an n-dimensional cube.Let f be continuous function from K into itself.Then there is a point P of K such that f(P) P. Schauder [9] extended the domain of validity of this theorem by demon- strating the Schauder fixed-point theorem.If K is a compact convex subset of 8 normed linear space X, and if f is a continuous transformation which carries K into itself, then there is at least one point P of K left fixed by f; that is, f(P) P.This was generalized next by Tyhonov [10] when he showedthat Schauder's proof could be adapted to prove the existence of a fixed point even if X is a