The index at infinity for some vector fields with oscillating nonlinearities

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January 2000, 6(1): 165-174. doi: 10.3934/dcds.2000.6.165

The index at infinity for some vector fields with oscillating nonlinearities

Alexander Krasnosel'skii ^1, and Jean Mawhin ^2,

1.	Institute for Information Transmission Problems, Russian Academy of Sciences, 19 Bolshoi Karetny Lane, Moscow 101447
2.	Institut Mathématique Pure et Appliquée, Université Catholique de Louvain, B-1348 Louvain-la-Neuve

Received October 1999 Published December 1999

Abstract
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This paper is devoted to the computation of the index at infinity for some asymptotically linear completely continuous vector fields $x-T(x)$, when the principal linear part $x-Ax$ is degenerate ($1$ is an eigenvalue of $A$), and the sublinear part is not asymptotically homogeneous (in particular do not satisfy Landesman-Lazer conditions). In this work we consider only the case of a one-dimensional degeneration of the linear part, i.e.s $1$ is a simple eigenvalue of $A$. For this case we formulate an abstract theorem and give some general examples for vector fields of Hammerstein type and for a two point boundary value problem.

Keywords: oscillating nonlinearities., Vector fields.

Mathematics Subject Classification: 32S7.

Citation: Alexander Krasnosel'skii, Jean Mawhin. The index at infinity for some vector fields with oscillating nonlinearities. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 165-174. doi: 10.3934/dcds.2000.6.165

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