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 formula_1 is a nonempty convex closed bounded set in uniformly convex Banach space and formula_2 is a mapping of formula_1 into itself such that formula_4 (i.e. formula_2 is "non-expansive"), then formula_2 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 formula_7 of a non-expansive map formula_2 has a unique asymptotic center, which is a fixed point of formula_2. (An "asymptotic center" of a sequence formula_10, if it exists, is a limit of the Chebyshev centers formula_11 for truncated sequences formula_12.) 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.