Real Analysis Exchange

A New Unification of Continuous, Discrete and Impulsive Calculus through Stieltjes Derivatives

Rodrigo López Pouso and Adrián Rodríguez

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Abstract

We study a simple notion of derivative with respect to a function which we assume to be nondecreasing and continuous from the left everywhere. Derivatives of this type were already considered by Young in 1917 and Daniell in 1918, in connection with the fundamental theorem of calculus for Stieltjes integrals. We show that our definition contains as a particular case the delta derivative in time scales, thus providing a new unification of the continuous and the discrete calculus. Moreover, we can consider differential equations in the new sense, and we show that not only dynamic equations on time scales, but also ordinary differential equations with impulses at fixed times are particular cases. We study almost everywhere differentiation of monotone functions and the fundamental theorems of calculus which connect our new derivative with Lebesgue-Stieltjes and Kurzweil-Stieltjes integrals. These fundamental theorems are the key for reducing differential equations with the new derivative to generalized integral equations, for which many theoretical results are already available thanks to Kurzweil, Schwabik and their followers.

Article information

Source
Real Anal. Exchange, Volume 40, Number 2 (2015), 319-354.

Dates
First available in Project Euclid: 4 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.rae/1491271220

Mathematical Reviews number (MathSciNet)
MR3499768

Zentralblatt MATH identifier
1384.26024

Subjects
Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A36: Antidifferentiation
Secondary: 26A45: Functions of bounded variation, generalizations 34A34: Nonlinear equations and systems, general

Keywords
Differentiation Lebesgue--Stieltjes integration Kurzweil--Stieltjes integration Fundamental theorem of calculus

Citation

Pouso, Rodrigo López; Rodríguez, Adrián. A New Unification of Continuous, Discrete and Impulsive Calculus through Stieltjes Derivatives. Real Anal. Exchange 40 (2015), no. 2, 319--354. https://projecteuclid.org/euclid.rae/1491271220


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