Browder fixed-point theorem: Difference between revisions

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{{Short description|Mathematical theorem}}

The '''Browder fixed -point theorem''' is a refinement of the [[Banach fixed -point theorem]] for [[Uniformly convex space|uniformly convex Banach spaces]]. It asserts that if <math>K</math> is a nonempty [[convex set|convex]] closed bounded set in uniformly convex [[Banach space]] and <math>f</math> is a mapping of <math>K</math> into itself such that <math>\|f(x)-f(y)\|\leq\|x-y\|</math> (i.e. <math>f</math> is ''non-expansive''), then <math>f</math> has a [[fixed point (mathematics)|fixed point]].

==History==

Following the publication in 1965 of two independent versions of the theorem by [[Felix Browder]] and by [[William Arthur Kirk|William Kirk]], a new proof of Michael Edelstein showed that, in a uniformly convex Banach space, every iterative sequence <math>f^nx_0</math> of a non-expansive map <math>f</math> has a unique asymptotic center, which is a fixed point of <math>f</math>. (An ''asymptotic center'' of a sequence <math>(x_k)_{k\in\mathbb N}</math>, if it exists, is a limit of the [[Chebyshev center]]s <math>c_n</math> for truncated sequences <math>(x_k)_{k\ge n}</math>.) A stronger property than asymptotic center is [[Delta-convergence|Delta-limit]] of T.C.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 theorem]]s

* [[Banach fixed point theorem]]

==References==

* [[Felix F.Browder|Felix E. Browder]], Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. U.S.A. '''54''' (1965) 1041–1044

* W.[[William Arthur Kirk|William A. Kirk]], A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly '''72''' (1965) 1004–1006.

* M.Michael Edelstein, The construction of an asymptotic center with a fixed-point property, Bull. Amer. Math. Soc. '''78''' (1972), 206-208.

{{Functional Analysis}}