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