We consider perturbations of the problem (*) - x'' + bx = lambda ax, x(0) - x(1) = 0 = x'(0) - x'(1) both by changes of the boundary conditions and by addition of nonlinear terms. We assume that at lambda = lambda 0 there are two linearly independent solutions of the unperturbed problem (*) and that a(dot) is bounded away from zero. When only the boundary conditions are perturbed either the Hill’s discriminant or the method of Lyapunov–Schmidt reduces the problem to 0 = det ((lambda - lambda 0)A - epsilon H) + higher order terms, where A and H are real 2 times 2 constant matrices. ...
The method of Lyapunov-Schmidt is used to analyse the full nonlinear problem. In a sequel to this paper we will analyse the bifurcation problem from a "generic" point of view and we will present some numeric examples.
(1984). Perturbation of Periodic Boundary-Conditions. SIAM Journal on Mathematical Analysis, 15 (4), 648-655.