A degree theory for a class of perturbed Fredholm maps II
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In a recent paper we gave a notion of degree for a class of perturbations of nonlinear Fredholm maps of index zero between real infinite dimensional Banach spaces. Our purpose here is to extend that notion in order to include the degree introduced by Nussbaum for local α-condensing perturbations of the identity, as well as the degree for locally compact perturbations of Fredholm maps of index zero recently defined by the first and third authors. Copyright © 2006 Pierluigi Benevieri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction 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:
α f (A) : A ⊆ Ω bounded, α(A) > 0 , α( f ) = sup α(A) α f (A) : A ⊆ Ω bounded, α(A) > 0 , ω( f ) = inf α(A)
(1.1)
where α is the Kuratowski measure of noncompactness (in [12] ω( f ) is denoted by β( f ), however, we prefer here the more recent notation ω( f ) as in [9]). Roughly speaking, an α-Fredholm map is of the type f = g − k, with the inequality α(k) < ω(g)
(1.2)
satisfied locally. These maps include locally compact perturbations of Fredholm maps (quasi-Fredholm maps for short) since, when g is Fredholm and k is locally compact, Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 27154, Pages 1–20 DOI 10.1155/FPTA/2006/27154
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A degree theory for a class of perturbed Fredholm maps II
one has α(k) = 0 and ω(g) > 0, locally. Moreover, they also contain local α-contractive perturbations of the identity, where, following Darbo [6], a map k is α-contractive if α(k) < 1. The purpose of this paper is to give an extension of the notion of the degree for αFredholm maps to a more general class of noncompact perturbations of Fredholm maps, still defined in terms of the numbers α and ω. This class of maps, that we call weakly α-Fredholm, includes local α-condensing perturbations of the identity, where a map k is α-condensing if α(k(A)) < α(A), for every A such that 0 < α(A) < +∞. We show that, for local α-condensing perturbations of the identity, our degree coincides with the degree defined by Nussbaum in [14, 15]. For an interesting, although partial, extension of the Leray-Schauder degree to a large class of maps (called quasi-ruled Fredholm maps) we mention the work of Efendiev (see [10, 11] and references therein). This class of maps has nonempty intersection with our class of weakly α-Fredholm maps. However, our degree is integer valued and, as said before, extends completely the Nussbaum degree (and, consequently, the Leray-Schauder degree). This is no
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