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.