The index at infinity for some vector fields with oscillating nonlinearities

Alexander Krasnosel'skii; Jean Mawhin

doi:10.3934/dcds.2000.6.165

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2000, Volume 6, Issue 1: 165-174. Doi: 10.3934/dcds.2000.6.165

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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: September 30, 1999

Published: December 1999

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Abstract

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:

Vector fields,
oscillating nonlinearities.

Mathematics Subject Classification: 32S70.

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