The paper is devoted to general elliptic operators in Hölder spaces in bounded or unbounded domains. We discuss the Fredholm property of linear operators and properness of nonlinear operators. We construct a topological degree for Fredholm and proper operators of index zero.

We prove that every Banach space containing an isomorphic copy of $c_0$ fails to have the fixed-point property for asymptotically nonexpansive mappings with respect to some locally convex topology which is coarser than the weak topology. If the copy of $c_0$ is asymptotically isometric, this result can be improved, because we can prove the failure of the fixed-point property for nonexpansive mappings.