This book is an introduction to the problem of the existence of solutions to some type of semilinear boundary value problems. The aim of the book is to give the possibility to any good student to reach a research level in this field, starting from the basic knowledge of mathematical analysis which is usually acquired before graduation. To this aim, I will develop some tools which could be used to attack many different boundary value problems, arising from ordinary or partial differential equations. However, I have chosen to deal mainly with the periodic problem for a second-order scalar ordinary differential equation. One reason for this choice is that this apparently simple model already shows so many different aspects, and can be approached by such different techniques, that it seems the ideal starting point to the further understanding of more technical boundary value problems. Another reason comes, of course, from its intrinsic importance in the applications.
