22

PATRICK FITZPATRICK AND JACOBO PEJSACHOWICZ

when u = u and | i = 0, (DEVP) has only trivial solutions, then

ind(F,uQ) = (-l)m,

where m is the sum, over JI 0, of the multiplicities of (DEVP) at

U = UQ.

We note that in the case when k = 1 and the boundary condition is

u = 0 on 8Q, in [Li], Yan Yan Li modified our construction in [Fi-Pe,l]

to also develop a homotopy invariant degree theory for (DBVP).

For families of (DBVP)'s parametrized by R, if u = 0 is a solution

for all A, there is global bifurcation of nontrivial solutions from

[a,b] x {0} provided that ind(F (0)) * ind(Fw(0)), and the index is

a b

computed as above in terms of the multiplicities of the eigenvalue problem

(DEVP).