Boundary value problems associated with nonlinear differential systems. Existence and multiplicity results

This thesis consists of eight chapters, divided into three main parts. The first part is on the application of lower and upper solutions, the second examines periodic problems, and the final part addresses antiperiodic problems. The first chapter provides the existence of a solution to some two-dimensional nonlinear boundary value problems, in the presence of lower and upper solutions. More precisely, we consider the following nonlinear system with mixed boundary conditions: \begin{equation*} \begin{cases} \dot{u}=f(t,v)\,,\quad \frac{d}{dt}(a(t)v)=g(t,u)\,,\quad t\in (0,1)\,,\\ v(0)=0=u(1)\sin\theta + v(1)\cos\theta \,, \end{cases} \end{equation*} where $\theta \in \,]-\frac{\pi}{2}, \frac{\pi}{2}] $, and the function $a:[0,1]\to R$ satisfies some specific conditions detailed within the chapter. In Chapters 2 and 3, we establish multiplicity results for coupled Hamiltonian systems under Neumann-type boundary conditions. The coupling occurs between a Hamiltonian system with periodic properties and another that admits a well-ordered pair of strict lower and upper solutions. We used the Szulkin critical point theory to obtain the multiplicity results. In Chapter 4, we extend the Poincar\'e--Birkhoff theorem to systems involving Landesman-Lazer conditions. To be more precise, we will assume a Poincar\'e--Birkhoff setting for the planar system \begin{equation*} \dot{q}=\partial_p \mathcal{H}(t,q,p)\,, \qquad \dot{p}=-\partial_q \mathcal{H}(t,q,p)\,, \end{equation*} while for the scalar equation \begin{equation*} \ddot{u}+g(t,u)=0\,, \end{equation*} the nonlinearity $g$ will have an asymmetric behaviour combined with some Landesman--Lazer conditions. We investigate nonresonance, simple resonance and double resonance situations, by implementing some kind of Landesman--Lazer conditions. Chapter 5 is concerned with the existence of multiple periodic solutions for the periodic problem associated with a system of the form \begin{equation*} \begin{cases} \dot{x}= \nabla_y H(t,x,y)+\nabla_y P(t,x,y,w)\,,\\ \dot{y}= -\nabla_x H(t,x,y)-\nabla_xP(t,x,y,w)\,, \\ J\dot{w} = F(t,w)+ \,\nabla_{w} P(t,x,y,w)\,, \end{cases} \end{equation*} where $F(t,w)$ is the gradient of a Hamiltonian function $K: \R \times \R^{2} \to \R$, i.e., $$ F(t,w) = \nabla_{w} K(t,w)\,, $$ and $H: \R \times \R^{2M} \to \R$. Moreover, all the involved functions are continuous, $T$-periodic in the variable $t$ and continuously differentiable with respect to $(x,y,w)$. To be more precise, we couple a system having twist condition (a core condition in the Poincar\'e--Birkhoff theorem) with another one whose nonlinearity lies between the gradients of two positive and positively $2$-homogeneous Hamiltonain functions. In addition to the periodic problem, we also discuss the Neumann-type problem. In Chapter 6, we extend the results obtained in Chapter 4 under the assumption that $g$ exhibits a singular behavior near the origin. In this setting, we successfully retained the multiplicity results obtained in that chapter. Chapter 7 is focused on perturbed Hamiltonian systems, demonstrating the existence of both periodic and bounded solutions. We imposed the Frederickson--Lazer-type nonresonance condition for the nonlinearity to be far away from resonance. The last chapter addresses a $T$-antiperiodic problem associated with a scalar second order equation whose nonlinearity has an asymptotic linear growth structure. We successfully demonstrate the existence of a solution using a topological degree argument. The discussion thoroughly covers both the resonant (with Landesman--Lazer conditions) and nonresonant scenarios.