Nonlinear periodic perturbations of linear boundary value problems at resonance
by
A. Cañada
University of Granada, Spain
Coauthors: A. Ureña (University of Granada, Spain)

The Fredholm Alternative Theorem is a very useful tool to have a precise description of the range of many linear operators which arise from the theory of boundary value problems in Differential Equations. However, the mathematical models have usually a nonlinear character. For example, this is the case of pendulum-type equations, which may be considered as nonlinear periodic perturbations of linear boundary value problems at resonance. In this talk we present some new results about the solvability of nonlinear periodic perturbations of linear resonance boundary value problems, with linear damping and homogeneous Dirichlet boundary conditions. In particular, we show that if the perturbation is nontrivial, the range of the corresponding nonlinear operator contains, in a strict manner, the range of the corresponding linear one (which, due to the presence of damping, is a nonselfadjoint operator). This result points out another important qualitative difference with respect to the case of periodic boundary conditions. In the proofs, we use the Liapunov-Schmidt reduction (or Alternative Method), the Schauder fixed point theorem and some ideas about connectivity, together with a careful study of the oscillatory properties of the bifurcation equation.

(T)

Date received: March 27, 2000