Existence and Multiplicity of Solutions of Functional Differential Equations

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 -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 -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

Solutions and Green’s function of the first order linear equation with reflection and initial conditions

This work is devoted to the study of the existence and sign of Green’s functions for first order linear problems with constant coefficients and initial (one point) conditions. We first prove a result on the existence of solutions of nth order linear equations with involutions via some auxiliary functions to later prove a uniqueness result in the first order case. We study then different situations for which a Green’s function can be obtained explicitly and derive several results in order to obtain information as regards the sign of the Green’s function. Once the sign is known, optimal maximum and anti-maximum principles followFEDER and Ministerio de Educación y Ciencia, Spain, project MTM2010-15314. FPU scholarship, Ministerio de Educación, Cultura y Deporte, SpainS

Numerical Solution of Stieltjes Differential Equations

This work is devoted to the obtaining of a new numerical scheme based on quadrature formulae for the Lebesgue–Stieltjes integral for the approximation of Stieltjes ordinary differential equations. This novel method allows us to numerically approximate models based on Stieltjes ordinary differential equations for which no explicit solution is known. We prove several theoretical results related to the consistency, convergence, and stability of the numerical method. We also obtain the explicit solution of the Stieltjes linear ordinary differential equation and use it to validate the numerical method. Finally, we present some numerical results that we have obtained for a realistic population model based on a Stieltjes differential equation and a system of Stieltjes differential equations with several derivatorsConsellería de Cultura, Educación e Ordenación Universitaria, Xunta de Galicia: ED431C 2019/2, Ministerio de Economía, Industria y Competitividad, Gobierno de España: MTM2016-75140-PS

A Liouville’s Formula for Systems with Reflection

In this work, we derived an Abel–Jacobi–Liouville identity for the case of two-dimensional linear systems of ODEs (ordinary differential equations) with reflection. We also present a conjecture for the general case and an application to coupled harmonic oscillatorsF. Adrián F. Tojo was partially supported by Xunta de Galicia, project ED431C 2019/02, and by the Agencia Estatal de Investigación (AEI) of Spain under grant MTM2016-75140-P, co-financed by the European Community fund FEDERS

Stieltjes differential systems with nonmonotonic derivators

In this work we study Stieltjes differential systems of which the derivators are allowed to change sign. This leads to the definition of the notion of a function of controlled variation, a characterization of precompact sets of g-continuous functions, and an explicit expression of g-exponential maps. Finally, we prove a Peano-type existence result and apply it to a model of fluid stratification on buoyant miscible jets and plumes.Marlène Frigon was partially supported by NSERC Canada. F. Adrián F. Tojo was partially supported by Ministerio de Economía y Competitividad, Spain, and FEDER, project MTM2013-43014-P, and by the Agencia Estatal de Investigación (AEI) of Spain under grant MTM2016-75140-P, co-financed by the European Community fund FEDERS

Existence and uniqueness of solution for Stieltjes differential equations with several derivators

In this paper, we study some existence and uniqueness results for systems of differential equations in which each of equations of the system involves a different Stieltjes derivative. Specifically, we show that this problems can only have one solution under the Osgood condition, or even, the Montel–Tonelli condition. We also explore some results guaranteeing the existence of solution under these conditions. Along the way, we obtain some interesting properties for the Lebesgue– Stieltjes integral associated to a finite sum of nondecreasing and left–continuous maps, as well as a characterization of the pseudometric topologies defined by this type of mapsIgnacio Márquez Albés was partially supported by Xunta de Galicia under grant ED481A-2017/095 and project ED431C 2019/02. F. Adrián F. Tojo was partially supported by Xunta de Galicia, project ED431C 2019/02, and by the Agencia Estatal de Investigación (AEI) of Spain under Grant MTM2016-75140-P, co-financed by the European Community fund FEDERS

Displacement Calculus

In this work, we establish a theory of Calculus based on the new concept of displacement. We develop all the concepts and results necessary to go from the definition to differential equations, starting with topology and measure and moving on to differentiation and integration. We find interesting notions on the way, such as the integral with respect to a path of measures or the displacement derivative. We relate both of these two concepts by a Fundamental Theorem of Calculus. Finally, we develop the necessary framework in order to study displacement equations by relating them to Stieltjes differential equationsThis research was partially funded by Ministerio de Economía y Competitividad, Spain, and FEDER, project MTM2013-43014-P, and by the Agencia Estatal de Investigación (AEI) of Spain under grant MTM2016-75140-P, co-financed by the European Community fund FEDER. Ignacio Márquez Albés was partially supported by Xunta de Galicia, grant ED481A-2017/095S

A Lipschitz condition along a transversal foliation implies local uniqueness for ODEs

We prove the following result: if a continuous vector field F is Lipschitz when restricted to the hypersurfaces determined by a suitable foliation and a transversal condition is satisfied at the initial condition, then F determines a locally unique integral curve. We also present some illustrative examples and sufficient conditions in order to apply our main resultPartially supported by Xunta de Galicia (Spain), project EM2014/032 (both authors) and MINECO, project MTM2017-85054-C2-1-P (first author)S

On first and second order linear Stieltjes differential equations

This work deals with the obtaining of solutions of first and second order Stieltjes differential equations. We define the notion of Stieltjes derivative on the whole domain of the functions involved, provide a notion of n-times continuously Stieltjes-differentiable functions and prove existence and uniqueness results of Stieltjes differential equations in the space of such functions. We also present the Green's functions associated to the different problems and an application to the Stieltjes harmonic oscillatorThe authors were partially supported by Xunta de Galicia, project ED431C 2019/02, and by the Agencia Estatal de Investigación (AEI) of Spain under grant MTM2016-75140-P, co-financed by the European Community fund FEDER. Ignacio Márquez Albés was partially supported by Xunta de Galicia under grant ED481B-2021-074S

First order differential systems with a nonlinear boundary condition via the method of solution-regions

In this article we extend the known theory of solution regions to encompass nonlinear boundary conditions. We both provide results for new boundary conditions and recover some known results for the linear caseThis work was partially supported by NSERC Canada. Marcos Tella and F. Adrián F. Tojo were partially supported by Ministerio de Economía y Competitividad, Spain, and FEDER, project MTM2013-43014-P, and by the Agencia Estatal de Investigación (AEI) of Spain under grant MTM2016-75140-P, co-financed by the European Community fund FEDERS