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 [math]\displaystyle{ K }[/math] is a nonempty convex closed bounded set in uniformly convex Banach space and [math]\displaystyle{ f }[/math] is a mapping of [math]\displaystyle{ K }[/math] into itself such that [math]\displaystyle{ \|f(x)-f(y)\|\leq\|x-y\| }[/math] (i.e. [math]\displaystyle{ f }[/math] is non-expansive), then [math]\displaystyle{ f }[/math] 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 [math]\displaystyle{ f^nx_0 }[/math] of a non-expansive map [math]\displaystyle{ f }[/math] has a unique asymptotic center, which is a fixed point of [math]\displaystyle{ f }[/math]. (An asymptotic center of a sequence [math]\displaystyle{ (x_k)_{k\in\mathbb N} }[/math], if it exists, is a limit of the Chebyshev centers [math]\displaystyle{ c_n }[/math] for truncated sequences [math]\displaystyle{ (x_k)_{k\ge n} }[/math].) 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
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.
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