M. FURI (Firenze) -M. )/~ARTt~LLI (Firenze) -A. VIG~OLI (L'Aquila)Sunto. -Si introduce una de]inizione di spettro ~(]) per applieazioni continue de]inite in uno spazio di Banaeh. Tale definizione coincide con ~uetla etassica net caso in eui ] sia lineare e continua. Alcuni de~ ~'isultati pi~ noti della teo~'ia spettrale lineare vengono estesi al easo q~o~ lineare. I~ particolare si dimostra ehe a(]) ~ eIduso e ehela sua #ontiera ~ eontenuta q~ello spettro puntuale approssimato %(]).
In [1] we introduced a concept of orientation and topological degree for nonlinear Fredholm maps between real Banach manifolds. In this paper we study properties of this notion of orientation and we compare it with related results due to Elworthy-Tromba and Fitzpatrick-Pejsachowicz-Rabier.
We apply topological methods to the study of the set of harmonic solutions of periodically perturbed autonomous ordinary differential equations on differentiable manifolds, allowing the perturbing term to contain a fixed delay. In the crucial step, in order to cope with the delay, we define a suitable (infinite dimensional) notion of Poincaré T -translation operator and prove a formula that, in the unperturbed case, allows the computation of its fixed point index.
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