Homeomorphisms and Fredholm Theory for Perturbations of Nonlinear Fredholm Maps of Index Zero and of A-proper Maps with Applications

1. Summary. In Part I, we develop a nonlinear Fredholm alternative theory involving k-ball and k-set perturbations of general homeomorphisms as well as of homeomorphisms that are nonlinear Fredholm maps of index zero. Various generalized first Fredholm theorems are given and finite solvability of general (odd) Fredholm maps of index zero is also studied. We apply these results to the unique and finite solvability of potential and semilinear problems with strongly nonlinear boundary conditions and to quasilinear elliptic equations. The results of Section 2 are based on the Browder and Banach-Mazur homeomorphism theorems. The basic tools used in section 3 are the recent degree theories for nonlinear C-Fredholm maps of index zero and their perturbations as defined by Fitzpatrick, Pejsachowicz-Rabier ([13],[22]), Benevieri-Furi [2,3], Rabier-Salter [24] and Benevieri-Calamai-Furi [4]. The results discussed in this part are based on Milojević [31,32].