12 PATRICK FITZPATRICK AND JACOBO PEJSACHOWICZ

Dirichlet problems. Since [Le-Sc], much progress has been made in formu-

lating existence and degree theoretic results for maps having such

diagonal, or intertwining, representations. In particular, we mention

Leray and Lions [Le-Li], Browder and Nussbaum [Br-Nu], Zabreiko and

Krasnosel'skii [Za-Ka], Browder [Br,4] and Krasnosel*skii and Zabreiko

[Kr-Za]. Quasilinear Fredholm mappings, as (1.1) shows, have intertwining

representations. However, the family of permissible homeomorphisms is

GL(X,Y) and since GL(X,Y) is not convex one cannot base our degree on the

theory developed in [Br-Nu] and [Br,4] for maps having intertwining

representations.

In [Ma], Mawhin introduced a coincidence degree for nonlinear pertur-

bations of closed Fredholm operators (see also [Pe-Vi]). This construction

is also based on an intertwining representation.

A radically different approach to degree theory for nonlinear elliptic

problems was invented by Caccioppoli [Ca], almost contemporaneously with

the Leray-Schauder paper. Caccioppoli introduced the class of proper C -

Fredholm mappings, i.e., C mappings whose derivative at each point is

Fredholm of index 0. For this class, he formulated a global version of the

Lyapunov-Schmidt reduction by considering a finite dimensional subspace V

of the range of f which is transversal to f on a neighborhood of the

zeroes of f. Then, if M is the inverse image under f of V, he

observed that M is a manifold for which the degree of f: M— V, when

reduced mod 2, is independent of the choice of V. This mod 2 degree was

rediscovered by Smale in [Sm] and then improved to an oriented degree by

Elworthy and Tromba [El-Tr]. Along different lines, the lack of homotopy