The index at infinity for some vector fields with oscillating nonlinearities (Q1576891)

This article deals with the problem of calculating the topological index at infinity of a completely continuous vector field \(I-T\) in a Banach space \(E\). The authors consider the case when \(T'(\infty)\) exists and 1 is a simple eigenvalue \(T'(\infty)\). Under the assumption \[ \lim_{ |\xi|\to\infty}\sup_{\|h\|_{\widetilde E}\leq c}\biggl|p \bigl(T-T'(\infty)\bigr)(\xi e+h)-p\bigl(T-T'(\infty)(\xi e)\bigr) \biggr|= 0 \] \((e\) is a normed eigenvector of \(T'(\infty)\) with eigenvalue \(1\), \(p\) is an eigenfunctional of \(T'(\infty)\) with \(p(e)=1\), \(\widetilde E\) is a Banach space embedded in \(E)\), the authors formulate sufficient conditions for the existence and non-existence of the topological index at infinity of \(T\) in terms of the asymptotical behavior of the scalar function \(\Psi(\xi)=p(T-T'(\infty))(\xi e)\) and, in the first case, calculate this index. As applications, Hammerstein integral equations and a two-point boundary value problem for ordinary differential equations of second order are considered.