ORIENTATION AND DEGREE FOR NONLINEAR BOUNDARY VALUE PROBLEMS 29

\jj(x) = K(x)x- R(x)C(x) for x € X.

Since X is compactly embedded in X and both R : X — L(X,Y) and

K : X — L(X,Y) are continuous, the compactness of i/i : X — Y follows from

the compactness of C : X — Y and of each K(x) for x € X. •

Proposition 2.8 Let f : X —- » Y be quasi linear Fredholm. Then f may be

represented as

f(x) = M(x)(x - tff(x)) for x € X, (2.9)

where M : X — GL(X,Y) is a family of isomorphisms and \fi : X —- » X is

compact.

Proof: Let f : X —-»Y be represented by (2.3). According to Theorem

2.1, we may select R : X —- GL(Y,X) to be a parametrix for L. If weset

M(x) = [R(x)] for x € X and apply Lemma 2.7, we obtain the representa-

tion (2.9). a

Recall that if T € $ (X,Y), then the restriction of T to closed,

bounded sets is proper. The following is a generalization of this

assertion to nonlinear mappings, which is of independent interest as a

quite general criterion for establishing properness.

Proposition 2.10 Let f: X — Y be quasilinear Fredholm. If D Q X is

closed and bounded, then f:D — Y is proper.

Proof: Let f : X — Y be represented by (2.3). Then the properness of

f : D — Y follows from the compactness of the embedding of X in X, the

compactness of C : X —- Y and the continuity of L : X — L(X,Y), together

with the properness of L(x) : D — Y for each x € X. •