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2015 A New Unification of Continuous, Discrete and Impulsive Calculus through Stieltjes Derivatives
Rodrigo López Pouso, Adrián Rodríguez
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Real Anal. Exchange 40(2): 319-354 (2015).


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.


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Rodrigo López Pouso. Adrián Rodríguez. "A New Unification of Continuous, Discrete and Impulsive Calculus through Stieltjes Derivatives." Real Anal. Exchange 40 (2) 319 - 354, 2015.


Published: 2015
First available in Project Euclid: 4 April 2017

zbMATH: 1384.26024
MathSciNet: MR3499768

Primary: 26A24 , 26A36
Secondary: 26A45 , 34A34

Keywords: differentiation , fundamental theorem of calculus , Kurzweil--Stieltjes integration , Lebesgue--Stieltjes integration

Rights: Copyright © 2015 Michigan State University Press

Vol.40 • No. 2 • 2015
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