A degree theory for a class of perturbed Fredholm maps
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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. 1. Introduction In this paper, we define a concept of degree for a special class of perturbations of (nonlinear) Fredholm maps of index zero between (infinite-dimensional real) Banach spaces, called α-Fredholm maps. The definition is based on the following two numbers (see, e.g., [10]) associated with a map f : Ω → F from an open subset of a Banach space E into a Banach space F:
α f (A) α( f ) = sup : A ⊆ Ω bounded, α(A) > 0 , α(A) α f (A) ω( f ) = inf : A ⊆ Ω bounded, α(A) > 0 , α(A)
(1.1)
where α is the Kuratowski measure of noncompactness (in [10] ω( f ) is denoted by β( f ), however, since ω is the last letter of the Greek alphabet, we prefer the notation ω( f ) as in [8]). Roughly speaking, the α-Fredholm maps are of the type f = g − k, where g is Fredholm of index zero and k satisfies, locally, the inequality α(k) < ω(g).
(1.2)
These maps include locally compact perturbations of Fredholm maps (called quasiFredholm maps, for short) since, when g is Fredholm and k is locally compact, one has Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:2 (2005) 185–206 DOI: 10.1155/FPTA.2005.185
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A degree theory for a class of perturbed Fredholm maps
α(k) = 0 and ω(g) > 0, locally. Moreover, they also contain the α-contractive perturbations of the identity (called α-contractive vector fields), where, following Darbo [5], a map k is α-contractive if α(k) < 1. The degree obtained in this paper is a generalization of the degree for quasi-Fredholm maps defined for the first time in [14] by means of the Elworthy-Tromba theory. The latter degree has been recently redefined in [3] avoiding the use of the Elworthy-Tromba construction and using as a main tool a natural concept of orientation for nonlinear Fredholm maps introduced in [1, 2]. Our construction is based on this new definition. The paper ends by showing that for α-contractive vector fields, our degree coincides with the degree defined by Nussbaum in [12, 13]. 2. Orientability for Fredholm maps In this section, we give a summary of the notion of orientability for nonlinear Fredholm maps of index zero between Banach spaces introduced in [1, 2]. The starting point is a preliminary definition of a concept of orientation for linear Fredholm operators of index zero between real vector spaces (at this level no topological structure is needed). Recall that, given two real vector spaces E and F, a linear operator L : E → F is said to be (algebraic) Fredholm if the spaces Ker L and coKer L = F/ Im L are finite-dimensional. The index of L is the integer indL = dimKer L − dimcoKerL.
(2.1)
Given a Fredholm operator of index z
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