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

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

Fixed-point theorems

References

Felix E. Browder, Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci.