arXiv subjects

Ignacio Márquez Albés

Publications and source records attributed to Ignacio Márquez Albés.

5 recordsLinked to original sources

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.

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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.

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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.

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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.

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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.

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