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
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