ORIENTATION AND DEGREE FOR NONLINEAR BOUNDARY VALUE PROBLEMS 9

For certain classes M of quasilinear Fredholm mappings it is

possible to choose an orientation so that, for homotopies whose sections

are mappings in M, the corresponding degree is homotopy invariant. For

instance, if G is a subset of $n(X,Y) which is closed under compact

perturbations and which has the property that the parity of each admissible

path in G depends only on the end-points, then there is an orientation of

GL(X,Y) which induces a homotopy invariant degree for quasilinear Fredholm

mappings whose principal part lies in G. One example of such a choice,

which is important in application to elliptic boundary value problems, is

the following: Let X be compactly embedded in Y, and G be a convex

subset of $ (X,Y) having the property that for each pair of isomorphisms

T and T in X there exists Xm 0 such that tT + (l-t)T + XI is

an isomorphism if t € [0,1] and A Xm; here, I denotes the inclusion

of X into Y. There is an orientation e of GL(X,Y) having the

property that if T is an isomorphism in G, then e(T) = (-1) , where m

is the sum of the algebraic multiplicities of the negative eigenvalues of

T. Moreover, for a C quasilinear Fredholm mapping whose derivatives at

each point lie in G, (1.5) is sharpened to the classic regular value

formula

deg(f,0,O) = Y (-l)n(x), (1.6)

x€0nf-1(O)

where n(x) is the number, counted with algebraic multiplicity, of negative

eigenvalues of f'(x). For quasilinear Fredholm homotopies having a

representation whose principal part, when restricted to the origin, is an

admissible path in G, the degree is homotopy invariant.