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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 ±.

47H11Degree theory (nonlinear operators)
46B40Ordered normed spaces
47H07Monotone and positive operators on ordered topological linear spaces
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H30Particular nonlinear operators