4 - The basic fixed point theorems for nonexpansive mappings

Summary

We have already seen that some bounded, closed and convex sets K in certain Banach spaces have the property that every nonexpansive self-mapping T: K→K must have a fixed point. When this is the case we say that K has the fixed point property (f.p.p.) for nonexpansive mappings and, if it is clear that only nonexpansive mappings are being considered, we shall simply say that K has the fixed point property or that K has f.p.p. Also, unless otherwise specified, we shall always assume that K is nonempty, bounded, closed and convex.

The problem of determining conditions on K (or on the space X containing K) which always insure that K has the f.p.p. has its origins in four papers which appeared in 1965. In the first of these (Browder, 1965a), F. Browder proved that a bounded, closed, convex set K ⊂ X has f.p.p. if X is a Hilbert space. Almost immediately, both Browder (1965b) and Göhde (1965) proved that the same is true if X belongs to the much wider class of ‘uniformly convex’ spaces (discussed in the next chapter). At the same time Kirk (1965) observed that the presence of a geometric property called ‘normal structure’ guarantees that K ⊂ X has f.p.p. if X is reflexive. The concept of normal structure was introduced in 1948 by Brodskii and Milman (1948) to study fixed points of isometries, and it is a property shared by all uniformly convex spaces.