H-7 - Fixed Point Theorems

Abstract

A fixed point of a self-map ƒ: X →X of a topological space X is a point x of X such that ƒ(x) is equal to x. The set of all fixed points of ƒ is denoted by Fix(ƒ). A topological space X is said to have the fixed-point property if every continuous self-map of X has a fixed point. This chapter focuses on the various generalizations of the Brouwer Fixed Point Theorem on an elementary level. The Brouwer Fixed Point Theorem is the most basic and fundamental result and serves as a prototype for the theory of fixed points. It states that every n-cell (n ≥ 1) has the fixed-point property. The theorem relies on the fact that there is no retraction of an n-cell onto its boundary, which is not the case for infinite dimensional Hilbert spaces. Some compactness assumption must be imposed to obtain an infinite-dimensional analogue of the Brouwer Fixed Point Theorem. This is where the Schauder–Tychonoff Fixed Point Theorem comes into consideration. The Lefschetz Fixed Point Theorem provides an elegant generalization of the Brouwer Fixed Point Theorem to non-contractible spaces.