A generalized-degree homotopy yielding global bifurcation results

We study the nonlinear eigenvalue problem F(x,λ)=Ax-∑j=1kλjBjx-R(x,λ)=0 where F:X×R→Y with X and Y Hilbert spaces such that X⊆Y; i.e., X is imbedded in Y. It is shown that λ0=0 is a global bifurcation point of the eigenvalue problem provided: a standard transversality condition is satisfied, the dimension of the null space of A is an odd number and each Bj,j=1,2,…,k, is a positive operator on the finite-dimensional null space of A. We apply the theory to prove that λ=0 is a global bifurcation point of the periodic boundary-value problem -x″(t)+λx(t)+λ2x′(t)+f(t,x(t),x′(t),x″(t)); x(0)=x(1),x′(0)=x′(1).