Differential problems with Stieltjes derivatives and applications

This Thesis is a collection of the research work developed by the author during his predoctoral stage. As the title suggests, this thesis revolves around the concept of Stieltjes derivative and the differential problems associated with it. Roughly speaking, this derivative is a modification of the usual derivative through a nondecreasing and left--continuous map, called derivator. After exploring this concept, we look for conditions ensuring the existence and uniqueness of solution of differential problems with this type of derivative. In particular, we consider differential equations with initial value conditions, differential equations with functional arguments subject to more general boundary conditions, and differential inclusions. For these problems, we look at some classical results for the corresponding problems with the usual derivative, and adapt them to this new setting, accounting for the differences that arise naturally from the Stieltjes derivative. In this case, and since we are basing our results on the classical setting, we only consider one derivator. However, we later explore similar problems in a framework that makes sense in the context of differential problems with Stieltjes derivatives, namely, differential problems with several different derivators. In other words, we consider systems of equations in which each of the equations is differentiated with respect to a different derivator. This, of course, offers a more general setting than the previous case, not only from a theoretical point of view, but also from the applications point of view, as we show with many examples

The logistic equation in the context of Stieltjes differential and integral equations

In this paper, we introduce logistic equations with Stieltjes derivatives and provide explicit solution formulas. As an application, we present a population model which involves intraspecific competition, periods of hibernation, as well as seasonal reproductive cycles. We also deal with various forms of Stieltjes integral equations, and find the corresponding logistic equations. We show that our work extends earlier results for dynamic equations on time scales, which served as an inspiration for this paper

Existence of extremal solutions for discontinuous Stieltjes differential equations

Stieltjes differential equations, which contain equations with impulses and equations on time scales as particular cases, simply consist in replacing usual derivatives by derivatives with respect to a nondecreasing function. In this paper we prove new results for the existence of extremal solutions for discontinuous Stieltjes differential equations. In particular, we prove that the pointwise infimum of upper solutions of a Stieltjes differential equation is the minimal solution under certain hypotheses. These results can be adapted to the context of both difference equations and impulsive differential equationsRodrigo López Pouso was partially supported by Ministerio de Economía y Competitividad, Spain, and FEDER, Project MTM2016-75140-P and Xunta de Galicia under grant ED431C 2019/02. Ignacio Márquez Albés was supported by Ministerio de Economía y Competitividad, Spain, and FEDER, Project MTM2016-75140-P and Xunta de Galicia under grants ED481A-2017/095 and ED431C 2019/02S

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

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

Resolution methods for mathematical models based on differential equations with Stieltjes derivatives

Stieltjes differential equations, i.e. differential equations with usual derivatives replaced by derivatives with respect to given functions (derivators), are useful to model processes which exhibit dead times and/or sudden changes. These advantages of Stieltjes equations are exploited in this paper in the analysis of two real life models: first, the frictionless motion of a vehicle equipped with an electric engine and, second, the evolution of populations of cyanobacteria Spirullina plantensis in semicontinuous cultivation processes. Furthermore, this is not only a paper on applications of known results. For the adequate analysis of our mathematical models we first deduce the solution formula for Stieltjes equations with separate variables. Finally, we show that differential equations with Stieltjes derivatives reduce to ODEs when the derivator is continuous, thus obtaining another resolution method for more general cases

Resolution methods for mathematical models based on differential equations with Stieltjes derivatives

Stieltjes differential equations, i.e. differential equations with usual derivatives replaced by derivatives with respect to given functions (derivators), are useful to model processes which exhibit dead times and/or sudden changes. These advantages of Stieltjes equations are exploited in this paper in the analysis of two real life models: first, the frictionless motion of a vehicle equipped with an electric engine and, second, the evolution of populations of cyanobacteria Spirullina plantensis in semicontinuous cultivation processes. Furthermore, this is not only a paper on applications of known results. For the adequate analysis of our mathematical models we first deduce the solution formula for Stieltjes equations with separate variables. Finally, we show that differential equations with Stieltjes derivatives reduce to ODEs when the derivator is continuous, thus obtaining another resolution method for more general cases

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

Extremal solutions of systems of measure differential equations and applications in the study of Stieltjes differential problems

We use lower and upper solutions to investigate the existence of the greatest and the least solutions for quasimonotone systems of measure differential equations. The established results are then used to study the solvability of Stieltjes differential equations; a recent unification of discrete, continuous and impulsive systems. The applicability of our results is illustrated in a simple model for bacteria population