Fixed Points of Non-expansive Maps

Abstract

Let X be a Banach space, $C\subset X$ nonempty, closed, bounded and convex, and let $f:C\rightarrow C$ be a non-expansive map, namely, such that

$$d(f(x),f(y))\le d(x,y),\qquad \forall x, y\in C.$$

The problem is whether f admits a fixed point in C. The answer, in general, is false.