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The index at infinity for some vector fields with oscillating nonlinearities

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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.
    Mathematics Subject Classification: 32S70.

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