The <Schauder fixed-point theorem> states that a continuous compact self-map of a nonempty closed bounded <convex set> in a <Banach space> has a <fixed point>. Equivalently in this bounded-domain version, it suffices for the image of the map to be relatively compact. The theorem guarantees existence, without requiring a contraction estimate or promising uniqueness.

Solution

Past exam of the fixed point methods for nonlinear partial differential equations course of the University of Oxford / 2008 / 2 / b / Solution

 Solution (source code)