Perturbation of Periodic Boundary-Conditions

Abstract

We consider perturbations of the problem (*) - x\u27\u27 + bx = lambda ax, x(0) - x(1) = 0 = x\u27(0) - x\u27(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