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

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.