arXiv subjects

F. Adrián F. Tojo

Publications and source records attributed to F. Adrián F. Tojo.

At least 19 recordsLinked to original sources

arXiv2026

The Fundamental Theorem of Calculus for Lebesgue-Stieltjes integrals involving non-monotonic derivators

In this work, we extend the concept of the Stieltjes derivative to encompass left-continuous derivators with bounded variation, thereby relaxing the monotonicity constraint. This generalization necessitates a refined definition of the Stieltjes derivative applicable across the entire domain, accommodating derivators that may change sign. We establish a generalized Fundamental Theorem of Calculus for the Lebesgue-Stieltjes integral in this broader context, presenting both "almost-everywhere" and "everywhere" versions. The latter requires a specific condition relating the derivator to its variation function, which we prove to be optimal through a density theorem. Our framework bridges the gap between Stieltjes differential equations and measure differential equations, offering a tool for modeling complex systems with non-monotonic dynamics.

math.CA↗

arXiv2026

Constructive solutions of the heat equation with Stieltjes derivatives

In this work, we investigate the one-dimensional heat equation within the framework of Stieltjes calculus. We first consider the equation associated with two fixed derivators and develop a constructive approach to establish the existence of solutions. We then study the corresponding initial value problem and incorporate several types of boundary conditions. Finally, we introduce a notion of multivariable derivator, suitable for higher-dimensional settings, and obtain explicit solutions of the heat equation for relevant classes of such derivators.

math.AP↗

arXiv2025

Stieltjes differential equations with bounded-variation derivators and application to thermal stress in solar panels

In this work we extend the theory of Stieltjes systems beyond the monotone case, establishing new chain rules, generalized versions of the Fundamental Theorem of Calculus, compactness tools for Peano-type results, and a $g$-exponential for explicit linear solutions. Continuity notions relative to vector-valued derivators further allow us to study everywhere differentiable solutions. As an application, we model thermal stress effects on solar panels and battery health, highlighting the practical value of non-monotone derivators.

math.CA↗

arXiv2025

Stieltjes analytic functions and higher order linear differential equations

In this work we develop a theory of Stieltjes-analytic functions. We first define the Stieltjes monomials and polynomials and we study them exhaustively. Then, we introduce the Stieltjes analytic functions locally, as an infinite series of these Stieltjes monomials and we study their properties in depth and how they relate to higher order Stieltjes differentiation. We define the exponential series and prove that it solves the first order linear problem. Finally, we apply the theory to solve higher order linear homogeneous Stieltjes differential equations with constant coefficients.

math.CA↗

arXiv2025

On the approximation properties of Stieltjes polynomials

We introduce and study the approximation properties of $g$-polynomials, defined as linear combinations of iterated Stieltjes integrals of a constant function. Focusing on the case where the derivator $g$ has finitely many discontinuities, we prove that the space of $g$-polynomials is dense in the space of uniformly $g$-continuous functions. This result establishes a Weierstrass-type approximation theorem within the framework of Stieltjes calculus. The characterization of the closure of the space of $g$-polynomials in the general case, where the derivator may exhibit more complex behavior, remains an open and challenging problem, which we briefly discuss.

math.CA↗

arXiv2025

On the kernel of the Stieltjes derivative and the space of bounded Stieltjes-differentiable functions

We investigate the existence and uniqueness of solutions to first-order Stieltjes differential problems, focusing on the role of the Stieltjes derivative and its kernel. Unlike the classical case, the kernel of the Stieltjes derivative operator is nontrivial, leading to non-uniqueness issues in Cauchy problems. We characterize this kernel by providing necessary and sufficient conditions for a function to have a zero Stieltjes derivative. To address the implications of this nontrivial kernel, we introduce a function space which serves as a suitable framework for studying Stieltjes differential problems. We explore its topological structure and propose a metric that facilitates the formulation of existence and uniqueness results. Our findings demonstrate that solutions to first-order Stieltjes differential equations are, in general, not unique, underscoring the need for a refined analytical approach to such problems.

math.CA↗

arXiv2025

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 maps.

math.CA↗

arXiv2025

Asymptotic properties of PDEs in compact spaces

In this article we combine the study of solutions of PDEs with the study of asymptotic properties of the solutions via compactification of the domain. We define new spaces of functions on which study the equations, prove a version of Ascoli-Arzelà Theorem, develop the fixed point index results necessary to prove existence and multiplicity of solutions in these spaces and also illustrate the applicability of the theory with an example.

math.AP↗

arXiv2025

Stieltjes differential systems with non monotonic 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 \emph{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.

math.CA↗

arXiv2024

When is the composition of functions measurable?

In this article we explore under which conditions on the interior function the composition of functions is measurable. We also study the sharpness of the result by providing a counterexample for weaker hypotheses.

math.FA↗

arXiv2022

Compactness criteria for Stieltjes function spaces and applications

In this work we study some topological aspects of function spaces arising in Stieltjes differential calculus. Chief among them are compactness results related to the Ascoli-Arzelà and Kolmogorov-Riesz theorems, as well as their applications to Stieltjes-Sobolev spaces and decomposable functions.

math.FA↗

arXiv2022

The Wronskian and the variation of parameters method in the theory of linear Stieltjes differential equations of second order

In this work, we define the notions of Wronskian and simplified Wronskian for Stieltjes derivatives and study some of their properties in a similar manner to the context of time scales or the usual derivative. Later, we use these tools to investigate second order linear differential equations with Stieltjes derivatives to find linearly independent solutions, as well as to derive the variation of parameters method for problems with $g$-continuous coefficients. This theory is later illustrated with some examples such as the study of the one-dimensional linear Helmholtz equation with piecewise-constant coefficients.

math.CA↗

arXiv2022

Consequences of the product rule in Stieltjes differentiability

This work revolves around the study of differentiability in the Stieltjes sense of a product of functions. A formula for the first order derivative has been obtained in the past, which is similar to the usual one with some extra terms in its expression. The aim of this paper is to take this behavior into account to study under which conditions we can guarantee the existence of higher order derivatives, while obtaining some other interesting results for the Stieltjes derivative along the way. We also investigate the regularity of the product of two functions.

math.CA↗

arXiv2021

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 notions 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 those spaces. We also present the Green's functions associated to the different problems and an application to the Stieltjes harmonic oscillator.

math.CA↗

arXiv2021

A Digital Twin of a Compartmental Epidemiological Model based on a Stieltjes Differential Equation

We introduce a digital twin of the classical compartmental SIR (Susceptible, Infected, Recovered) epidemic model and study the interrelation between the digital twin and the system. In doing so, we use Stieltjes derivatives to feed the data from the real system to the virtual model which, in return, improves it in real time. As a byproduct of the model, we present a precise mathematical definition of solution to the problem. We also analyze the existence and uniqueness of solutions, introduce the concept of Main Digital Twin and present some numerical simulations with real data of the COVID-19 epidemic, showing the accuracy of the proposed ideas.

q-bio.OT↗

arXiv2020

Stieltjes Bochner spaces and applications to the study of parabolic equations

This work is devoted to the mathematical analysis of Stieltjes Bochner spaces and their applications to the resolution of a parabolic equation with Stieltjes time derivative. This novel formulation allows us to study parabolic equations that present impulses at certain times or lapses where the system does not evolve at all and presents an elliptic behavior. We prove several theoretical results related to existence of solution, and propose a full algorithm for its computation, illustrated with some realistic numerical examples related to population dynamics.

math.AP↗

arXiv2020

Displacements

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 equations.

math.CA↗

arXiv2020

Approximation of Stieltjes ordinary differential equations

This work is devoted to the obtaining of a new numerical scheme based in quadrature formulas for the Lebesgue-Stieltjes integral for the approximation of Stieltjes ordinary differential equations. This novel method allows us to numerically approximate models based in 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 we use it to validate the numerical method. Finally, we present some numerical results that we have obtained for a realistic population model based in a Stieltjes differential equation.

math.NA↗