FACTA UNIVERSITATIS (NI ˇ S) Ser. Math. Inform. Vol. 22, No. 2 (2007), pp. 123–153 HOMEOMORPHISMS AND FINITE SOLVABILITY OF THEIR PERTURBATIONS FOR FREDHOLM MAPS OF INDEX ZERO WITH APPLICATIONS * Petronije S. Milojevi´c Abstract. We prove a number of homeomorphism results for nonlinear Fredholm maps of index zero and their perturbations. Moreover, we show that k-ball and k-set pertur- bations of these homeomorphisms are again homeomorphisms or that the corresponding equations are finitely solvable. Various generalized first Fredholm theorems are given and finite solvability of general (odd) Fredholm maps is also studied. We apply these results to finite solvability of quasilinear elliptic equations on R N as well as on bounded domains. The basic tool used are the recent degree theories for nonlinear C 1 -Fredholm maps of index zero and their perturbations. 1. Introduction In the early nineties, a rather complete degree theory for nonlinear C 2 -Fredholm maps of index zero has been developed by Fitzpatrick-Pejsachowisz-Rabier [6]. Sub- sequently, their degree has been extended to C 1 Fredholm maps of index zero and to their (non) compact perturbations by Pejsachowicz, Rabier, Salter, Benevieri, Calamai and Furi ([12], [15], [2] and [1]). They have also given applications to global bifurcation problems for equations envolving these maps and to quasilinear elliptic equations. Part of this paper is devoted to proving various homeomorphism and finite solvability results for these maps using these degrees and applying them to quasilinear elliptic equations. New types of generalized first Fredholm theorems are proven. We also study the stability of these homeomorphisms under k-ball and k-set perturbations. Let us describe our main results in more detail. Throughout the paper, we assume that X and Y are infinite dimensional Banach spaces. In Section 2, we establish a number of homeomorphism results for nonlinear Fredholm maps of index zero T : X → Y and their perturbations. Using the recent open mapping theorem for such maps of Calamai [4] and Rabier-Salter [15], we establish first a number Received September 8, 2007. 2000 Mathematics Subject Classification. Primary 47H15, 35L70, 35L75; Secondary 35J40. * This research has been done while the author was on the sabbatical leave at Rutgers University, New Brunswick, NJ. 123