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 ) ∥ ≤ ∥ x − y ∥ (i.e. f is non-expansive), then f has a fixed point.
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 ∈ N , if it exists, is a limit of the Chebyshev centers c n for truncated sequences ( x k ) k ≥ 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.
