A degree theory for a class of perturbed Fredholm maps

Abstract

We define a notion of degree for a class of perturbations of nonlinear Fredholm maps of index zero between infinite-dimensional real Banach spaces. Our notion extends the degree introduced by Nussbaum for locally α-contractive perturbations of the identity, as well as the recent degree for locally compact perturbations of Fredholm maps of index zero defined by the first and third authors

A degree theory for a class of perturbed Fredholm maps II

“…To emphasize the fact that the set C ∞ is uniquely determined by the covering ᐂ, 12 A degree theory for a class of perturbed Fredholm maps II sometimes it will be denoted by C ᐂ ∞ . In addition C ∞ verifies the following two properties (see [1] for the proof):…”

“…As proved in [1], the above definition is well posed since the right-hand side of formula (5.8) is independent of the choice of the α-pair (ᐂ,C), of the retraction r and of the open set W.…”

“…In this section we sketch the construction of the degree for α-Fredholm maps introduced in [1]. These maps are special noncompact perturbations of Fredholm maps, defined in terms of the numbers α p and ω p .…”

“…Actually, in [1] only the fundamental properties (i.e., normalization, additivity and homotopy invariance) were stated and proved. The excision and existence properties are easy consequences of the additivity.…”

“…In a recent paper [1] we defined a concept of degree for a special class of noncompact perturbations of nonlinear Fredholm maps of index zero between (infinite dimensional real) Banach spaces, called α-Fredholm maps. The definition of these maps is based on the following two numbers (see, e.g., [12]) associated with a map f : Ω → F from an open subset of a Banach space E to a Banach space F:…”