The Browder fixed-point theorem is a refinement of the

Banach fixed-point theorem In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certai ...

for uniformly convex Banach spaces. It asserts that if

K

is a nonempty convex closed bounded set in uniformly convex

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...

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 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^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)_

, if it exists, is a limit of the

Chebyshev center In geometry, the Chebyshev center of a bounded set Q having non-empty Interior (topology), interior is the center of the minimal-radius ball enclosing the entire set Q, or alternatively (and non-equivalently) the center of largest inscribed ball ...

c_n

for truncated sequences

(x_k)_

.) 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.

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. {{Functional Analysis Fixed-point theorems

History

See also

References