ORIENTATION AND DEGREE FOR NONLINEAR BOUNDARY VALUE PROBLEMS 17
X h— DxF(A,0), and r|D F(A,0),A 1 s lim r|D F(A,0), [AQ-T),AQ+T)]1 = -1,
then (An,0) is a bifurcation point of the equation F(A,x) = 0. This
latter condition is a necessary condition in order that (An,0) be a bi-
furcation point. Using the corresponding result of Ize [Iz] in finite
dimensions and a stability property of the parity, we showed in [Fi-Pe,2]
that if a : I R —-» $ (X,Y) has A as an isolated singular point and r(a,A )
= +1, then there is a C mapping F : I R x X Y such that D F(A,0) =
a(A) and F(A,0) = 0 for all A IR, and (*00) is not a bifurcation
point for the nontrivial solutions of F(A,x) = 0.
We close Section 8 by proving a surprising continuation result for a
family of quasilinear Fredholm maps parametrized by S . If F : S xX Y
is represented by F(A,x) = L.(x)x + C(x) and there is some A S with
A U
[F(A ,*)]~1(0) bounded and deg(F(A , •),X,0) * 0, then if r(L (O^S1) = -1,
F (0) is unbounded. Thus the behavior of F(An,«) and of the "top-order
terms" of F(A,0), over A S , precludes the existence of a-priori bounds
for the solutions of the equations F(A,x) = 0, (A,x) S xX.
In Section 9, we study particular choices of orientations and the
corresponding degree. Given G £ $ (X,Y), an orientation e of GL(X,Y)
is said to strongly orient G provided that for each admissible path
a : [a,b] G,
r(a, [a,b]) = e(oc(0))e(x(b)).
Observe that if GL(X,Y) is connected, it is not possible to find a strong
orientation for all of $n(X,Y). If e strongly orients G, then the
associated degree is constant along homotopies whose principal part belongs
to G.
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