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 1995 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^nx_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\ge n}.) A stronger property than asymptotic center is Delta-limit of T.C. Lim, which in the uniformly convex space coincides with the weak limit if the space has the Opial property.

See also

References


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