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