Nonlinear differential equations on bounded and unbounded domains

Resumo

The main purpose of this proyect is the study of the existence, as well as the uniqueness or multiplicity, of solutions of functional equations. We will analize the qualitative properties of the solutions of non linear equations, focusing on the study of constant sign solutions. First, we will work with the Hill's equation. This equation with non constant coefficients, with some important applications in physics, includes any second order differential equation. A second type of considered problems are those in which we work with the average of the studied solution. The considered problem consists of calculating the minimum average that the solution of the equation must have in order to assure its constant sign. On the other hand, we will consider n-th order problems in which the non linear side depends on the function and its derivatives up to n-1 order. The boundary conditions include a wide set in which we can assure the constant sign of both the solution and some of its derivatives. Finally we will work with problems in non bounded intervals. We will study the existence of both heteroclinic and homoclinic solutions of several n-th order problems. The main purpose of this proyect is the study of the existence, as well as the uniqueness or multiplicity, of solutions of functional equations. We will analize the qualitative properties of the solutions of non linear equations, focusing on the study of constant sign solutions. First, we will work with the Hill's equation. This equation with non constant coefficients, with some important applications in physics, includes any second order differential equation. On the other hand, we will consider n-th order problems in which the non linear side depends on the function and its derivatives up to n-1 order. The boundary conditions include a wide set in which we can assure the constant sign of both the solution and some of its derivatives. Finally we will work with problems in non bounded intervals. We will study the existence of both heteroclinic and homoclinic solutions of several n-th order problems.