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 ${\displaystyle K}$ is a nonempty convex closed bounded set in uniformly convex Banach space and ${\displaystyle f}$ is a mapping of ${\displaystyle K}$ into itself such that ${\displaystyle \|f(x)-f(y)\|\leq \|x-y\|}$ (i.e. ${\displaystyle f}$ is non-expansive), then ${\displaystyle f}$ has a fixed point.