Browder fixed-point theorem

The Browder fixed-point theorem is a refinement of the Banach fixed-point theorem for uniformly convex Banach spaces. It asserts that if $$K$$ is a nonempty convex closed bounded set in uniformly convex Banach space and $$f$$ is a mapping of $$K$$ into itself such that $$\|f(x)-f(y)\|\leq\|x-y\|$$ (i.e. $$f$$ is non-expansive), then $$f$$ 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 $$f^nx_0$$ of a  non-expansive map $$f$$  has a unique asymptotic center, which is a fixed point of $$f$$. (An asymptotic center of a sequence $$(x_k)_{k\in\mathbb N}$$, if it exists, is a limit of the Chebyshev centers $$c_n$$ for truncated sequences $$(x_k)_{k\ge n}$$.) 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.