ORIENTATION AND DEGREE FOR NONLINEAR BOUNDARY VALUE PROBLEMS 19

LQu = nu

B u = 0, u € X.

In the case when X is compactly and densely embedded in Y and K

is the inclusion, Theorem 9.37 asserts that T € G^ if and only if there is

some A* 0 and c 0 such that

||Tx + Ax|| cA||x|| for all A A*, x € X. (1.9)

In the final section, we turn to (BVP) and, when A is a parameter

space, to parametrized families of such problems:

?k

f(A,x,u(x),...,D u(x)) =0, x € Q

gi(A,x,u(x),...,Dmiu(x)) =0, x e dQ, 1 i k.

(BVP)A

As (BVP) had a formal linearization (LBVP), so (BVP) has a family of

formal linearizations

J? (u)v = V f (x,u(x),...,D2ku(x))Dav(x), x € Q

lafe*a (LBVP

A

B. .(u)v = V g. (A,x,u(x),..

.,Dmiu(x))D%(x),

x € 6fl, 1 i k.

1, A / . « 1, Ot

|a|2k

We prescribe analytical conditions under which (1.7) and (1.8), and

their parametrized correspondents, are quasilinear Fredholm. It should be

stressed that the mapping F does not have a principal part given by

L(u) = (£(u),B1(u),...fBk(u));

any principal part involves integro-differential operators. However, for a