The first part of the memory goes through those discoveries related to Green’s functions.
In order to do that, first we recall some general results concerning involutions which will help
us understand their remarkable analytic and algebraic properties. Chapter 1 will deal about
this subject while Chapter 2 will give a brief overview on differential equations with involutions
to set the reader in the appropriate research framework.
In Chapter 3 we start working on the theory of Green’s functions for functional differential
equations with involutions in the most simple cases: order one problems with constant coefficients
and reflection. Here we solve the problem with different boundary conditions, studying
the specific characteristics which appear when considering periodic, anti-periodic, initial or arbitrary
linear boundary conditions. We also apply some very well known techniques (lower
and upper solutions method or Krasnosel’skiĭ’s Fixed Point Theorem, for instance) in order to
further derive results.
Computing explicitly the Green’s function for a problem with nonconstant coefficients is
not simple, not even in the case of ordinary differential equations. We face these obstacles in
Chapter 4, where we reduce a new, more general problem containing nonconstant coefficients
and arbitrary differentiable involutions, to the one studied in Chapter 3.
To end this part of the work, we have Chapter 5, in which we deepen in the algebraic nature
of reflections and extrapolate these properties to other algebras. In this way, we do not
only generalize the results of Chapter 3 to the case of 𝑛-th order problems and general twopoint
boundary conditions, but also solve functional differential problems in which the Hilbert
transform or other adequate operators are involved.
The last chapters of this part are about applying the results we have proved so far to some
related problems. First, in Chapter 6, setting again the spotlight on some interesting relation
between an equation with reflection and an equation with a 𝜑-Laplacian, we obtain some results
concerning the periodicity of solutions of that first problem with reflection. Chapter 7
moves to a more practical setting. It is of the greatest interest to have adequate computer
programs in order to derive the Green’s functions obtained in Chapter 5 for, in general, the
computations involved are very convoluted. Being so, we present in this chapter such an algorithm,
implemented in Mathematica. The reader can find in the appendix the exact code of
the program.
In the second part of the Thesis we use the fixed point index to solve four different kinds
of problems increasing in complexity: a problem with reflection, a problem with deviated arguments
(applied to a thermostat model), a problem with nonlinear Neumann boundary conditions
and a problem with functional nonlinearities in both the equation and the boundary
conditions.
As we will see, the particularities of each problem make it impossible to take a common
approach to all of the problems studied. Still, there will be important similarities in the different
cases which will lead to comparable results.