Bifurcation from Simple Eigenvalue for Functional-Differential Equations

Abstract

A method to determine bifurcation points of complicated boundary value problems for functional-differential equations is developed, which provides sufficient conditions on exis-tence of intervals containing bifurcation points in terms of some simple estimates rather than in terms of spectral properties of the linearized problem. In the case of Frechèt-differentiable non-linearity the method reduces to the study of bifurcation from simple eigenvalues of the lin-earized problem. It still works, if the linearized problem contains homogeneous but nonlinear operators. The heart of the approach consists in the study of branching equations obtained by Lyapunov-Schmidt reduction and a special non-equivalent change of variables. Construction of such equations based on the choice of generalized Green’s operator is discussed. At last, some applications to second order equations with deviating argument are provided.