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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

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