TheBrowder fixed-point theorem is a refinement of theBanach fixed-point theorem foruniformly convex Banach spaces. It asserts that if${\displaystyle K}$ is a nonemptyconvex closed bounded set in uniformly convexBanach space and${\displaystyle f}$ is a mapping of${\displaystyle K}$ into itself such that${\displaystyle \Vert f(x)-f(y)\Vert \le \Vert x-y\Vert }$ (i.e.${\displaystyle f}$ isnon-expansive), then${\displaystyle f}$ has afixed point.

Following the publication in 1965 of two independent versions of the theorem byFelix Browder and byWilliam Kirk, a new proof of Michael Edelstein showed that, in a uniformly convex Banach space, every iterative sequence${\displaystyle f^n{x}_0}$ of a non-expansive map${\displaystyle f}$ has a unique asymptotic center, which is a fixed point of${\displaystyle f}$. (Anasymptotic center of a sequence${\displaystyle (x_k{)}_{k\in \mathbb{N}}}$, if it exists, is a limit of theChebyshev centers${\displaystyle c_n}$ for truncated sequences${\displaystyle (x_k{)}_{k\ge n}}$.) A stronger property than asymptotic center isDelta-limit of Teck-Cheong Lim, which in the uniformly convex space coincides with the weak limit if the space has theOpial property.

• Fixed-point theorems

• 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. Monthly72 (1965) 1004–1006.
• Michael Edelstein, The construction of an asymptotic center with a fixed-point property, Bull. Amer. Math. Soc.78 (1972), 206-208.

• Fixed-point theorems

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