ORIENTATION AND DEGREE FOR NONLINEAR BOUNDARY VALUE PROBLEMS 21

is odd. The second assumption is that if un is a nontrivial solution of

2k+2 K

(1.12), then there does not exist a solution in C (Q) of the problem

2(A0)u(x) = TuQ(x), x € fl

B(xQ)u(x) = suQ(x), x e an.

For an interval I = [a,b], there is global bifurcation of nontrivial

solutions of (BVP) from [a,b]x{0 provided that the parity of the path

IK

of linearizations, A i—D F(X,u)| „, X € [a,b], is -1.

x 'u=0

When the boundary data in (BVP) correspond to lower order perturba-

tions of Dirichlet data, we can use our results from Section 9 to determine

the behavior of the solutions of (BVP) directly from data related to the

eigenvalues of (LBVP) . Consider

(DBVP)

2k

f(x,u(x),...,D u(x)) =0, x € Q

u(x) +p. (x, . . . ,D u(x)) =0, x € 3Q, 1 i k.

dT)1 X

Let (£(u),B(u)) denote the linearization of (DBVP) at u, and consider

the eigenvalue problem

£(u)v = nv

(DEVP)

B(u)v =0, u € X.

When (1.8) is induced by (DBVP), we use the convexity and spectral

properties of the uniformly elliptic differential operators with genera-

lized Dirichlet boundary conditions to determine an orientation with

respect to which the degree is homotopy invariant and for which the index

2k+2 r

formula is as follows: If u € C (fl) is a solution of (DBVP) and,