[Paper Review] The Riemann-Stieltjes integral on time scales

Dorota Mozyrska, Ewa Pawłuszewicz+1 more|arXiv (Cornell University)|Mar 6, 2009

Nonlinear Differential Equations AnalysisMathematics15 references22 citations

TL;DR

This paper introduces the Riemann-Stieltjes integral on time scales, unifying delta and nabla integration through a common framework using the box symbol (□). It establishes fundamental properties such as existence, linearity, and integration by parts, proving that the Riemann-Stieltjes □-integral of f with respect to g equals the Riemann □-integral of f times g□, enabling generalization of classical inequalities and paving the way for Henstock-Kurzweil-Stieltjes integration on time scales.

ABSTRACT

We study the process of integration on time scales in the sense of Riemann-Stieltjes. Analogues of the classical properties are proved for a generic time scale, and examples are given.

Motivation & Objective

• To formalize the Riemann-Stieltjes integral on time scales, filling a gap in the literature where Stieltjes integration had not been systematically studied.
• To unify delta and nabla integration by using a common notation (□) for both, reducing redundancy in proofs.
• To establish foundational properties such as existence, linearity, and bounds for the Riemann-Stieltjes □-integral on bounded intervals.
• To prove integration by parts formulas for both delta and nabla Riemann-Stieltjes integrals, linking them to standard Riemann integrals via the derivative of the integrator function.
• To lay the groundwork for extending current integration theories on time scales, including potential generalization to Henstock-Kurzweil-Stieltjes integrals.

Proposed method

• The paper defines partitions of a time scale interval I=[a,b]𝕋 and constructs upper and lower Darboux–Stieltjes □-sums using infimum and supremum of f on subintervals.
• It introduces the concept of a strictly increasing function g on I and defines Δgⱼ = g(tⱼ)−g(tⱼ₋₁) to weight the sums.
• A key technical tool is Lemma 2.1, which ensures that for any δ>0, a partition exists such that each subinterval's g-increment is ≤δ or corresponds to a jump (ρ(tⱼ)=tⱼ₋₁).
• The Riemann-Stieltjes □-integral is defined as the common value of upper and lower integrals when they coincide, analogous to the classical Riemann-Stieltjes integral.
• The main result (Theorem 4.3) establishes that ∫_a^b f □g = ∫_a^b f g□ □t, proving equivalence between Riemann-Stieltjes and weighted Riemann □-integrals.
• Integration by parts is derived using the □-version of the mean value theorem and properties of σ and ρ operators, yielding ∫f □g = [fg]_a^b − ∫g^σ □f for delta and ∫f □g = [fg]_a^b − ∫g^ρ □f for nabla.

Experimental results

Research questions

• RQ1 Can the classical Riemann-Stieltjes integral be generalized to arbitrary time scales, preserving key properties like existence and integration by parts?
• RQ2 How can the Riemann-Stieltjes integral be defined consistently for both delta and nabla integration on time scales using a unified notation?
• RQ3 Under what conditions does the Riemann-Stieltjes □-integral of f with respect to g equal the Riemann □-integral of f times g□?
• RQ4 What are the implications of this equivalence for extending inequalities and dynamic equations on time scales?
• RQ5 Can this framework be extended to more general integrals, such as Henstock-Kurzweil-Stieltjes integrals on time scales?

Key findings

• The Riemann-Stieltjes □-integral exists and is well-defined for bounded functions f and strictly increasing functions g on a closed time scale interval I=[a,b]𝕋.
• The paper proves that ∫_a^b f(t) □g(t) = ∫_a^b f(t)g^□(t) □t, showing equivalence between the Riemann-Stieltjes □-integral and a weighted Riemann □-integral.
• The integration by parts formula holds: ∫_a^b f □g = [fg]_a^b − ∫_a^b g^σ □f for the delta case and ∫_a^b f □g = [fg]_a^b − ∫_a^b g^ρ □f for the nabla case.
• The existence of a partition P_δ with subintervals satisfying Δgⱼ ≤ δ or Δgⱼ > δ with ρ(tⱼ)=tⱼ₋₁ ensures uniform approximation of the integral.
• The results generalize classical Riemann-Stieltjes theory to time scales, enabling extension of inequalities from ℝ to arbitrary time scales.
• The framework provides a foundation for future work on Henstock-Kurzweil-Stieltjes integration on time scales.