The evolution of techniques in fixed point theory. Part 1: Brouwer, Banach, and Browder.

Abstract: In 1910, Brouwer proved that any continuous function mapping a compact, convex set 

in Euclidean space back into itself must have a fixed point. In 1922, Banach stated his famous 

Contraction Mapping Theorem, a result distinctly different than Brouwer in the complete metric space setting. 

The next major advance in this vein came in 1965 when Browder proved that any nonexpansive function on 

a Hilbert space which maps a closed, bounded, and convex set back into itself must have a fixed point. 

In the first part of this talk, we will consider the evolution of the questions being asked in fixed point theory 

between 1910 and 1965 and the techniques developed to answer them, giving specific attention to Browder's theorem.