Existence and multiplicity of solutions for boundary value problems associated with Hamiltonian systems

Our initial aim of this study is to improve the results by Fonda and Gidoni (where they studied the Poincar\'e--Birkhoff theorem for Hamiltonian systems, coupling twist with linear Hamiltonian systems), and extend the Poincar\'e--Birkhoff theorem for Hamiltonian systems, coupling twist with generalized lower/upper solutions, or with systems having an isochronous center, or with systems having singularities. The details regarding these works can be found in Chapters 1, 2, and 3. Denoting by $J= \Big( \begin{array}{cc} 0 & -I \\ I & \;0 \end{array} \Big) $ the standard symplectic matrix, our Hamiltonian system \begin{equation} J\dot z=\nabla_z H(t,z)\,, \end{equation} when writing $z=((x,y),(u,v))\in\mathbb{R}^{2M}\times\mathbb{R}^{2L}$, is driven by a Hamiltonian function of the type \begin{equation} H(t,z)=\mathcal H(t,x,y)+\mathscr H(t, u,v)+ \varepsilon P(t,x,y,u,v)\,, \end{equation} where $\varepsilon$ is a small parameter. All the involved functions are assumed to be continuous, and $T$-periodic in their first variable $t$. The next focus of our study in Chapter 4 is to explore coupled Hamiltonian systems, where the first system is periodic in the space variables $u_i$ and the second involves a quadratic term or a positively homogeneous Hamiltonian system. We proved multiplicity results for a Neumann-type problem, analogous to the results by Fonda and Gidoni for the periodic problem. We then extend our study to more general systems and see that results are preserved for both periodic and Neumann-type problems. We next study in Chapter 5, the multiplicity of solutions for a Hamiltonian system coupling two systems associated with mixed boundary conditions, i.e., corresponding to the first system, we impose periodic boundary conditions and assume the usual twist condition, while for the second one, we consider a two-point boundary conditions of Neumann type. The final goal of this work is to study systems which are at resonance, given in Chapters 6 and 7. For such type of systems, we impose some nonresonance conditions of Landesman--Lazer-type and Frederickson--Lazer-type which are crucial for the nonlinearity to be kept sufficiently far from resonance.