ORIENTATION AND DEGREE FOR NONLINEAR BOUNDARY VALUE PROBLEMS 13
invariance of such a degree has motivated several constructions of a homo-
topy invariant oriented degree for particular classes of smooth nonlinear
Fredholm mappings (cf.[Fe],[Is],[Ki] and [Tr].) A detailed discussion of
the behavior of the oriented degree for smooth nonlinear Fredholm mappings,
based on the parity, will be found in [F-P-R,l and 2], Of course, quasi-
linear Fredholm mappings need not be smooth.
Particular classes of nonlinear boundary value problems may be formu-
lated as critical points of nonlinear functionals. In the early sixties,
certain Galerkin approximation techniques which were useful in such a
variational context were extended to the study of the broader class of
quasilinear boundary value problems in divergence form (cf. [Vi], [Le-Li],
[Br,l]). A functional analytic framework was developed on the basis of
generalized Galerkin approximation schemes, and existence theorems were
developed for classes of mappings which included monotone, pseudomonotone,
(S+), A-proper and others (cf. [Br,4], [Pet], [Sk]). Generalized degrees
were introduced for these classes in [Br-Pe], [Br,4], [Fi,l] and [Sk]. In
[Sk], Skrypnik deduced various existence results for fully nonlinear ellip-
tic problems, based upon the formulation of such problems as the zeroes of
(S+) mappings.
We now outline the contents of the sections. In the second section,
we consider various properties of the representation of a quasilinear
Fredholm mapping. If a quasilinear Fredholm mapping f: X Y is repre-
sented by f(x) = L(x)x + C(x), we call the family L: X $Q(X,Y) a
principal part of f. We show that the principal part is unique, modulo
families of compact maps, and that if f: X Y is Frechet differentiable
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