Transversality in Nonlinear Eigenvalue Problems

Abstract

This chapter describes the transversality in nonlinear eigenvalue problems. The use of transversality or general position arguments together with traditional techniques for studying nonlinear eigenvalue problems enables one to obtain new results on the geometric structure and multiplicity of solutions of nonlinear eigenvalue problems. The chapter explains how transversality can be used to obtain a continuum of solutions for an eigenvalue problem involving a cone preserving map. For that, odd multiplicity is not needed, but the cone preservation is used to obtain an unbounded continuum of solutions of an equation u = λ F (u), where F is a compact map that preserves the cone of non-negative functions in an L2 space and has a derivative at zero with a single eigenvalue λ0 corresponding to eigenvectors in the cone. In this case, the continuum emanates from the point (λ0, 0). The result is applied to a simple example involving a pair of linked differential equations.