Topological methods for the search of solutions of nonlinear equations. From planar systems to ordinary and partial differential equations

The thesis focuses on the search of solutions of nonlinear equations. It is divided in five chapters and two appendices. The first chapter deals with an extension of the Poincaré-Bohl Theorem. The second and the third chapter deal respectively with planar systems of differential equations and second order differential equations in Hilbert spaces. Existence results are proved by means of the method of lower and upper solutions. These results are extended to systems of PDEs in the next chapter. The fifth chapter focuses on periodic solutions of nearly integrable infinite-dimensional Hamiltonian systems. In the first appendix we characterize the property of having equals $p$-norms while in the second we investigate the properties of Dini derivatives of real-valued functions.