The fundamental theorem of calculus for Lebesgue–Stieltjes integrals involving non-monotonic derivators
Year: 2026
Type: article
Abstract: In this paper, we extend the concept of the Stieltjes derivative to encompass left-continuous derivators with bounded variation, thereby relaxing the monotonicity constraint. This generalization requires 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. We prove this condition to be optimal by means of 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.
Authors: Lamiae Maia, F. Adrián F. Tojo
Institutions: Mohammed V University, Universidade de Santiago de Compostela
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1050619841, G1175115578, G1187051335, G1193235492, G1300792995, G1496055848, G1622782223, G2046442635, G210041235, G2233490251, G2243422090, G2262748287, G2601397997, G2861531256, G300979063, G3216726442, G34346518, G3480869486, G3681454997, G3694284866, G4042783231, G4092263876, G410571670, G425668148, G451917667, G4526231891, G5080475149, G5195188938, G5340645127, G5539308778, G5967599077, G6062177375, G67688110, G7084143925, G7626925242, G929308097
OpenAlex ID: W4414763057