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

TL;DR

This work analyzes the kernel of the Stieltjes derivative and constructs the ${\mathcal{BD}}$-spaces to provide a robust framework for first-order Stieltjes differential problems. It develops multiple Mean Value Theorems across $g$-continuity, left-continuity, and general Stieltjes differentiability, yielding precise kernel characterizations and showing nonuniqueness of solutions in general. The authors establish Banach and complete-metric structures for ${\mathcal{BD}}$-spaces under suitable conditions and connect these spaces to Stieltjes-Sobolev spaces, enabling a decomposition of solutions into kernel and absolutely continuous components. The results underscore the necessity of refined analytical tools when dealing with Stieltjes differential equations, and they provide explicit constructions and a unifying framework for existence, uniqueness, and representation of solutions.

Abstract

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.

Figures (5)

• Figure 4.1: The $F_3$ function.
• Figure 4.2: Geometric interpretation of the function $l$ in two dimensions (without the cross product). In this figure we are assuming $y>x>0$. Observe that $\alpha=\tan x$ and $y=\tan \beta$.
• Figure 4.3: Representation of the function $f$.
• Figure 5.1: $v(t)=\exp_g(\beta,t)$.
• Figure 5.2: $\widetilde{v}(t)=h(t)\, v(t)$.

Theorems & Definitions (103)

• Definition 2.1: Fernandez2022
• Remark 2.2
• Proposition 2.3: Fernandez2022
• Proposition 2.4: Fernandez2022
• Remark 2.5
• Definition 2.6: FriLo17
• Definition 2.7
• Definition 2.8: LoRo14
• Theorem 2.9: LoRo14
• Theorem 2.10: Fundamental Theorem of Calculus for the Lebesgue-Stieltjes Integral LoRo14