Suppose that $f: \mathbb{D} \subseteq \mathbb{R}^n \to \mathbb{R}^n$, where $\mathbb{D}$ is a half-space or a convex cone. We wish to show the existence of a fixed point $x = f(x)$.
Suppose we know that $f$ is continuous, single-valued and always maps into this half space (or convex cone), given any point or subset of it. Then the only problem is the lack of compactness for Brouwer's fixed point theorem.
Is there any way to deal with these cases?
(Motivation: rather large classes of functions satisfy this property, such as $\exp(-x): [0, \infty) \to \mathbb{R}$).
Suppose we're in $\mathbb R^2$ and $\mathbb D$ is the right half-plane. Then the map $(x,y)\to (x+1,y)$ takes $\mathbb D$ t0 $\mathbb D$ and has no fixed points.
In any convex cone in $\mathbb R^2$ with vertex at the origin, we can do the analogous thing: Define the map $z\to z+v,$ where $v$ points in same direction as the angle bisector of the two angles defining the cone.
Examples in higher dimensions are also not hard to come by.