NONLINEAR FUNCTIONAL ANALYSIS AND PERIODIC SOLUTIONS OF SEMILINEAR WAVE EQUATIONS

Abstract

This chapter focuses on nonlinear functional analysis and periodic solutions of semilinear wave equations. It provides the proof and the application of a Leray–Schauder type continuation theorem for semilinear equations in a Hilbert space H of the form Lu + Nu = 0, where L belongs to a class of linear operators having possibly an infinite-dimensional kernel and N is a (not necessarily small) nonlinear operator. The assumptions made on L and N are motivated by some periodic-Dirichlet problems for semilinear hyperbolic equations. The approach is a combination of Galerkin arguments, Leray–Schauder theory, and monotone-like operators.