Consider the following (less general than possible) statement of Schauder's fixed point theorem:
Suppose that $X$ is a Banach space, that $B_1$ is the unit ball of $X$ and that $f: X \to X$ is a continuous function. If $f(B_1)$ is a compact subset of $B_1$, then $f$ has a fixed point in $B_1$.
Now let $B_{1+\epsilon}$ denote the ball around $0$ of radius $1+\epsilon$ and suppose that $f(B_1)$ is a compact subset of $B_{1+\epsilon}$ for some $\epsilon > 0$. Is there anything that can be said about possible fixed points of $f$? For example, is it possible to prove the existence of a point $x$ such that $\|f(x) - x\| < \epsilon$? Can anything at all be said about this scenario? Does anyone know of any works in which such functions have been examined in some detail?
Thanks in advance.