Two-Dimensional Problems

Abstract

As the computation of the Brouwer degree is especially developed for two-dimensional mappings, the chapter collects a number of results in this direction. It starts with the method of lower and upper solutions for periodic solutions of second order differential equations, developed from the method of Stampacchia. It is followed by the results of Ortega about the use of the Brouwer index in the study of the stability of periodic solutions of second order equations of Duffing type, with convex or with periodic nonlinearities. Positive solutions of some perturbations of positively homogeneous Hamiltonian systems are then considered, in the line of Fabry and Fonda, and have for special case the perturbed asymmetric piecewise-linear second order equations considered by Fučik and Dancer. Special techniques for the computation of the Brouwer degree (or of the Kronecker index) are then developed for planar multilinear mappings, including the use of Sturm sequences when the mappings are not factorized. The case of mappings defined by holomorphic functions is also considered, and applied to questions of stability and control for linear ordinary differential and difference equations, associated to the Routh–Hurwitz and Schur–Cohn stability criteria for linear differential and difference systems.