f(x) = M(x)(x-C(x)) (1.1)
where M is the restriction to X of a continuous family of isomorphisms
in GL(X,Y) parametrized by X, and C is a compact, possibly nonlinear,
map. From (1.1) it is clear that the zeroes of f coincide with the zeroes
of the compact vector field Id-C. Of course, such a correspondence, by
itself, is insufficient for the purpose of developing a degree theory.
Moreover, in general, the representation (1.1) is not given explicitly.
In particular, it is not given explicitly for the quasilinear Fredholm
mappings induced by fully nonlinear boundary value problems. However,
(1.1) is an assertion of the contact equivalence of f with Id-C. On
this basis, we will define the degree.
If 0 is an open, bounded subset of X on whose boundary f does
not vanish, then Id-C is also nonvanishing on the boundary of 0 and
hence its Leray-Schauder degree, deg (Id-C,0,0), is defined. One cannot
define deg(f,0,0) to be deg (Id-C,0,0), since this is not independent
of the choice of representation (1.1). More precisely, it is independent
only up to sign.
This sign dependency occurs for fundamental reasons. First of all, in
the case of finite dimensional spaces, the degree of f on 0 depends on
the choice of orientations for X and Y, while the degree of Id-C on 0,
which is the fixed-point index of C in 0, is independent of any choice
of orientation. Additionally, in view of Kuiper's Theorem [Ku] on the
contractibility of the general linear group of a Hilbert space, the usual
notion of orientation for finite dimensional vector spaces does not
generalize to infinite dimensional spaces.
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