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