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,
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