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The index bundle for Fredholm morphisms  [PDF]
Nils Waterstraat
Mathematics , 2012,
Abstract: We extend the index bundle construction for families of bounded Fredholm operators to morphisms between Banach bundles.
Bifurcation Results for a Class of Perturbed Fredholm Maps  [cached]
Benevieri Pierluigi,Calamai Alessandro
Fixed Point Theory and Applications , 2008,
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.
Bifurcation Results for a Class of Perturbed Fredholm Maps  [cached]
Pierluigi Benevieri,Alessandro Calamai
Fixed Point Theory and Applications , 2008, DOI: 10.1155/2008/752657
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.
Homeomorphisms and Fredholm theory for perturbations of nonlinear Fredholm maps of index zero with applications
Petronije S. Milojevic
Electronic Journal of Differential Equations , 2009,
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.
Bifurcation of Fredholm Maps II; The Dimension of the Set of Bifurcation Points  [PDF]
Jacobo Pejsachowicz
Mathematics , 2010,
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.
On coincidence index for multivalued perturbations of nonlinear Fredholm maps and some applications  [PDF]
Valeri Obukhovskii,Pietro Zecca,Victor Zvyagin
Abstract and Applied Analysis , 2002, DOI: 10.1155/s1085337502002853
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.
Transversality and Lipschitz-Fredholm maps  [PDF]
Kaveh Eftekharinasab
Mathematics , 2015,
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.
A degree theory for locally compact perturbations of Fredholm maps in Banach spaces  [PDF]
Pierluigi Benevieri,Massimo Furi
Abstract and Applied Analysis , 2006, DOI: 10.1155/aaa/2006/64764
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.
A degree theory for locally compact perturbations of Fredholm maps in Banach spaces
Pierluigi Benevieri,Massimo Furi
Abstract and Applied Analysis , 2006,
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.
The kernel bundle of a holomorphic Fredholm family  [PDF]
Thomas Krainer,Gerardo A. Mendoza
Mathematics , 2013,
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.
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