Abstract:
We prove a global bifurcation result for an equation of the type , where is a linear Fredholm operator of index zero between Banach spaces, and, given an open subset of , are and continuous, respectively. Under suitable conditions, we prove the existence of an unbounded connected set of nontrivial solutions of the above equation, that is, solutions with , whose closure contains a trivial solution . The proof is based on a degree theory for a special class of noncompact perturbations of Fredholm maps of index zero, called -Fredholm maps, which has been recently developed by the authors in collaboration with M. Furi.

Abstract:
We prove a global bifurcation result for an equation of the type Lx+ (h(x)+k(x))=0, where L:E ￠ € ‰ ￠ € ‰ ￠ ’ ￠ € ‰ ￠ € ‰F is a linear Fredholm operator of index zero between Banach spaces, and, given an open subset of E, h,k: —[0,+ ￠ ) ￠ € ‰ ￠ € ‰ ￠ ’ ￠ € ‰ ￠ € ‰F are C1 and continuous, respectively. Under suitable conditions, we prove the existence of an unbounded connected set of nontrivial solutions of the above equation, that is, solutions (x, ) with ￠ ‰ 0, whose closure contains a trivial solution (x ˉ,0). The proof is based on a degree theory for a special class of noncompact perturbations of Fredholm maps of index zero, called ±-Fredholm maps, which has been recently developed by the authors in collaboration with M. Furi.

Abstract:
We develop a nonlinear Fredholm alternative theory involving k-ball and k-set perturbations of general homeomorphisms and of homeomorphisms that are nonlinear Fredholm maps of index zero. Various generalized first Fredholm theorems and finite solvability of general (odd) Fredholm maps of index zero are 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 basic tools used are the Nussbaum degree and the degree theories for nonlinear $C^1$-Fredholm maps of index zero and their perturbations.

Abstract:
We obtain an estimate for the covering dimension of the set of bifurcation points for solutions of nonlinear elliptic boundary value problems from the principal symbol of the linearization of the problem along the trivial branch of solutions.

Abstract:
We define a nonoriented coincidence index for a compact, fundamentally restrictible, and condensing multivalued perturbations of a mapwhich is nonlinear Fredholm of nonnegative index on the set of coincidence points. As an application, we consider an optimal controllability problem for a system governed by a second-order integro-differential equation.

Abstract:
We study transversality for Lipschitz-Fredholm maps in the context of bounded Fr\'{e}chet manifolds. We show that the set of all Lipschitz-Fredholm maps of a fixed index between Fr\'{e}chet spaces has the transverse stability property. We give a straightforward extension of the Smale transversality theorem by using the generalized Sard's theorem for this category of manifolds. We also provide an answer to the well known problem concerning the existence of a submanifold structure on the preimage of a transversal submanifold.

Abstract:
We present an integer valued degree theory for locally compactperturbations of Fredholm maps of index zero between (open setsin) Banach spaces (quasi-Fredholm maps, for short). Theconstruction is based on the Brouwer degree theory and on thenotion of orientation for nonlinear Fredholm maps given by theauthors in some previous papers. The theory includes in a naturalway the celebrated Leray-Schauder degree.

Abstract:
We present an integer valued degree theory for locally compact perturbations of Fredholm maps of index zero between (open sets in) Banach spaces quasi-Fredholm maps, for short). The construction is based on the Brouwer degree theory and on the notion of orientation for nonlinear Fredholm maps given by the authors in some previous papers. The theory includes in a natural way the celebrated Leray-Schauder degree.

Abstract:
Let $\Y$ be a smooth connected manifold, $\Sigma\subset\C$ an open set and $(\sigma,y)\to\scrP_y(\sigma)$ a family of unbounded Fredholm operators $D\subset H_1\to H_2$ of index 0 depending smoothly on $(y,\sigma)\in \Y\times \Sigma$ and holomorphically on $\sigma$. We show how to associate to $\scrP$, under mild hypotheses, a smooth vector bundle $\kerb\to\Y$ whose fiber over a given $y\in \Y$ consists of classes, modulo holomorphic elements, of meromorphic elements $\phi$ with $\scrP_y\phi$ holomorphic. As applications we give two examples relevant in the general theory of boundary value problems for elliptic wedge operators.