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

  • 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.