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 $f^{n}x_{0}$ 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 $(x_{k})_{k\geq n}$.) 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

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.