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 ${\displaystyle K}$ is a nonempty convex closed bounded set in uniformly convex Banach space and ${\displaystyle f}$ is a mapping of ${\displaystyle K}$ into itself such that ${\displaystyle \|f(x)-f(y)\|\leq \|x-y\|}$ (i.e. ${\displaystyle f}$ is non-expansive), then ${\displaystyle f}$ 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 ${\displaystyle f^{n}x_{0}}$ of a non-expansive map ${\displaystyle f}$ has a unique asymptotic center, which is a fixed point of ${\displaystyle f}$. (An asymptotic center of a sequence ${\displaystyle (x_{k})_{k\in \mathbb {N} }}$, if it exists, is a limit of the Chebyshev centers ${\displaystyle c_{n}}$ for truncated sequences ${\displaystyle (x_{k})_{k\geq n}}$.) 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.

See also

• Fixed-point theorems
• Banach fixed-point theorem

References

• 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.