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Zbl 1177.47065
Sun, Jingxian; Liu, Xiaoying
(Sun, Jing-xian; Liu, Xiao-ying)
Computation of topological degree in ordered Banach spaces with lattice structure and its application to superlinear differential equations.
(English)
[J] J. Math. Anal. Appl. 348, No. 2, 927-937 (2008). ISSN 0022-247X

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. \par 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 $\pm\infty$.
[Martin Väth (Praha)]
MSC 2000:
*47H11 Degree theory
46B40 Ordered normed spaces
47H07 Positive operators on ordered topological linear spaces
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H30 Particular nonlinear operators

Keywords: degree zero; ordered Banach space; quasi-additive map; Hammerstein operator; Sturm-Liouville equation; Krasnoselskij theorem; superlinear Sturm-Liouville problems

Cited in: Zbl 1226.34090

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