Browder fixed-point theorem


The Browder fixed-point theorem is a refinement of the Banach fixed-point theorem for convex space|uniformly convex Banach spaces]. It asserts that if is a nonempty convex closed bounded set in uniformly convex Banach space and is a mapping of into itself such that, then has a fixed point.

History

Following the publication in 1965 of two independent versions of the theorem by Felix Browder and by William Kirk, a new proof of Michael Edelstein showed that, in a uniformly convex Banach space, every iterative sequence of a non-expansive map has a unique asymptotic center, which is a fixed point of. A stronger property than asymptotic center is Delta-limit of Teck-Cheong Lim, which in the uniformly convex space coincides with the weak limit if the space has the Opial property.