ORIENTATION AND DEGREE FOR NONLINEAR BOUNDARY VALUE PROBLEMS 27
where L(z)€ L(X,Y) is defined by
L(z)x = f df(tz)(x)dt for z X and x X.
J0
Thus the algebraic representation (2.3)is not very restrictive. The
crucial point is that each L(x) is in $ (X,Y) and that the family L(x)
is defined and depends continuously on x, for x belonging to a space in
which X is compactly embedded - the latter property, which implies that
x i— L(x) is a compact mapping from X to $n(X,Y), is the reason forthe
adjective "quasi1inear".
We devote the remainder of this section to establishing some general
properties of quasilinear Fredholm mappings. First, we record a useful
observation.
Lemma 2.4 Let L : X L(X,Y) be continuous and x X. Define
h(x) = L(x)(x-x ) for x e X.
Then h : X Y is Frechet differentiable at x and h'(x ) = L(x ).
Proposition 2.5 Two principal parts of a quasi linear Fredholm mapping
f : X Y differ by a family of compact operators.
Proof: Let f : X Y be represented by f(x)= L (x)x + C (x),for
i = 1,2. Fix x X, and set
h.(x) = L1(x)(x-xQ) for x X.
1 2
Lemma 2.4 implies that h'(xn) = L (xn and h'(xQ) = L (xn). On the
other hand, if
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