ORIENTATION AND DEGREE FOR NONLINEAR BOUNDARY VALUE PROBLEMS 3
of general nonlinear elliptic boundary value problems as the fixed-points
of a compact mapping has, in itself, been a controversial matter. In fact,
while, in the second part of their paper, Leray and Schauder already
introduced a method of reduction for fully nonlinear second order elliptic
equations with Dirichlet boundary conditions, which under the name of
"intertwined representation" was further developed by Browder and Nussbaum
[Br-Nu], by Krasnosel*skii and Zabreiko [Kr-Za] and especially in the mono-
graph of Browder [Br], the approaches to degree theory based on the Sard-
Smale Theorem or through Galerkin-approximation are frequently motivated in
the literature by the widely shared belief that general boundary value
problems for nonlinear elliptic equations cannot be reduced to compact per-
turbations of the identity. Probably much of this misunderstanding
originates in the example given in the book of Ladyzhenskaja and Ural*tseva
[La-Ur] in which it is shown that a direct application of the Leray-
Schauder technique to the oblique derivative problem produces a map which
is not compact. Also, in the review paper [Ni], Nirenberg raises the
question about the existence of a reduction for the solutions of general
elliptic boundary value problems to the zeros of a compact vector field.
Our approach to topological degree may be described as follows: A
parametrix for a continuous family L : A $n(X,Y) of linear Fredholm
operators of index zero parametrized by a topological space A is a
continuous map R : A GL(Y,X) (the set of all isomorphisms from X to
Y) such that R(X)L(A) = Id + K(X) with each K(X) compact. Families
parametrized by contractible spaces admit parametrices, and from this it
follows that any quasilinear Fredholm map f : X Y can be represented
in the form
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