\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
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.
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