Computation of topological degree in ordered Banach spaces with lattice structure and its application to superlinear differential equations (Q950484)

An approach is presented to prove that a compact map has fixed point index (degree) zero in Banach space with a cone. In contrast to Krasnoselskii's compression/expansion theorem on a cone, no cone invariance is assumed for the map, but only some inequalities. The main hypothesis is that the map is of some abstract Hammerstein type, i.e., the composition of a linear map (satisfying a certain monotonicity condition w.r.t.\ some functional) with a so-called quasi-additive map like a superposition operator. The results are illustrated by proving the existence of nontrivial solutions of a Sturm--Liouville problem under some growth assumptions on the nonlinearity at \(0\) and \(\pm\infty\).