2
PATRICK FITZPATRICK AND JACOBO PEJSACHOWICZ
defined only up to sign. While this is only a minor inconvenience in the
treatment of existence problems by means of the method of a-priori bounds,
such as the Riemann-Hilbert problem considered in that paper, the lack of
additiviy of such a degree makes it inadequate for the study of multipli-
city and bifurcation problems.
The connectedness of the set of all linear isomorphisms of X onto Y
presents an obstruction to the existence, for the whole class of quasilinear
Fredholm mappings, of an additive, integer-valued degree which also has the
property of homotopy invariance. Any such degree must accommodate changes
in sign in the degree along admissible homotopies. In order to be useful in
the analysis of bifurcation and multiplicity problems, these changes cannot
be left indeterminate.
Here, we shall construct an additive, integer-valued degree theory for
quasilinear Fredholm mappings based upon a modification of the well-known
device of Leray and Schauder for formulating the solutions of a quasilinear
second order elliptic boundary value problem as the zeroes of a compact
perturbation of the identity, i.e., of a compact vector field [Le-Sc]. By
the introduction of a homotopy invariant for paths of linear Fredholm
operators with invertible end-points, which we call the parity, we are able
to classify changes in sign of the degree along admissible homotopies, and
so produce a degree useful in the study of multiplicity and bifurcation
problems. Following an idea of Babin [Ba], we show that general elliptic
boundary value problems, which are suitably smooth, induce quasilinear
mappings, both in the Sobolev and the Holder spaces.
Before discussing the construction of the degree, we observe that even
with respect to the question of existence, the formulation of the solutions
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