The Browder fixed-point theorem is a refinement of the Banach fixed-point theorem for 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 (i.e. is non-expansive), then has a fixed point.

History[edit]

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 . (An asymptotic center of a sequence , if it exists, is a limit of the Chebyshev centers for truncated sequences .) 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.

See also[edit]

• Fixed-point theorems
• Banach fixed-point theorem

References[edit]

• Felix E. Browder, Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. U.S.A. 54 (1965) 1041–1044
• William A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965) 1004–1006.
• Michael Edelstein, The construction of an asymptotic center with a fixed-point property, Bull. Amer. Math. Soc. 78 (1972), 206-208.