A note on continuation theory

Recently M. Martelli [6] and M. Furi and M. P. Pera [1] proved some interesting results about the existence and the global topological structure of connected sets of solutions to problems of the form: Lx = N(λ, x) with L:E → F a bounded linear Fredholm operator of index zero (where E, F are real Banach spaces), and N:ℝ × E → F a nonlinear map satisfying suitable conditions. While the existence of solution sets for this kind of problem follows from the Leray–Schauder continuation principle, it is our aim to show in this note that their global topological structure can be obtained as a consequence of the theory developed by J. Ize, I. Massabò, J. Pejsachowicz and A. Vignoli in [3, 4] about parameter dependent compact vector fields in Banach spaces.