A generalized-degree homotopy yielding global bifurcation results

Abstract

We study the nonlinear eigenvalue problem F ( x , λ ) = Ax - ∑ j = 1 k λ j B j x - 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 λ 0 = 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 B j , 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 ) + λ x ( t ) + λ 2 x ′ ( t ) + f ( t , x ( t ) , x ′ ( t ) , x ″ ( t ) ) ; x ( 0 ) = x ( 1 ) , x ′ ( 0 ) = x ′ ( 1 ) .