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Computation of topological degree in ordered Banach spaces with lattice structure and its application to superlinear differential equations. (English) Zbl 1177.47065

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 $±\infty$.

##### MSC:
 47H11 Degree theory (nonlinear operators) 46B40 Ordered normed spaces 47H07 Monotone and positive operators on ordered topological linear spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H30 Particular nonlinear operators