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.
