ORIENTATION AND DEGREE FOR NONLINEAR BOUNDARY VALUE PROBLEMS 15

Here, ind(f,x0) = deg(f,U,x0), where U is a neighborhood of xQ with

f^CO) n U = {xQ}.

These properties of the degree are sufficient to permit us, in Section

5, to prove, for quasilinear Fredholm mappings, an extension of the classic

Brouwer Invariance of Domain Theorem, to prove a nonlinear version, for odd

mappings, of the Fredholm Alternative, and to extend a theorem of

Caccioppoli [Ca] on the existence of zeroes for a mapping which is homotopic

to a map which has 0 as a regular value and an odd number of zeroes.

As they stand, formulas (1.9) and (1.10) are dependent on choices of

representation which are too particular to be useful in the study of

bifurcation and multiplicity problems.

In order to be useful in the study of bifurcation and multiplicity,

(1.9) and (1.10) need to be reformulated to reflect the topological data

which are encoded in their right-hand sides. To do so, in Section 6 we

study the concept of parity for an admissible path of linear Fredholm

operators. We introduced the parity in the preliminary announcement of the

present paper [Fi-Pe,l]. The parity has since been observed to play an

important role in the study of diverse nonlinear problems ([Fi-Pe: 2,3,4,5],

[F-P-R, 1 and 2]).

The parity is defined by (1.3) if a: [a,b] - $Q(X,Y) is admissible.

Theorem 6.6 is an assertion of several useful properties of the parity,

included among which is its homotopy invariance along homotopies in $n(X,Y)

of admissible paths. We also prove a useful Reduction Lemma, Lemma 6.26,

which allows one to compute the parity of a path in terms of the change in

sign of the determinant of the restriction of the path to a generalized