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
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