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首页> 外文期刊>Real analysis exchange >A NEW UNIFICATION OF CONTINUOUS, DISCRETE, AND IMPULSIVE CALCULUS THROUGH STIELTJES DERIVATIVES
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A NEW UNIFICATION OF CONTINUOUS, DISCRETE, AND IMPULSIVE CALCULUS THROUGH STIELTJES DERIVATIVES

机译：通过钢化衍生物连续，离散和脉冲计算的新统一

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We study a simple notion of derivative with respect to a function that 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 that 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.
机译：我们针对某个函数研究了一个简单的导数概念，我们假设该函数从左到右都是不变的并且是连续的。 Young于1917年和Daniell于1918年已经考虑了这类导数，并将其与Stieltjes积分的微积分基本定理联系在一起。我们表明，在特定情况下，我们的定义包含时间尺度上的增量导数，从而提供了连续和离散演算的新统一。此外，我们可以从新的角度考虑微分方程，并且我们不仅显示时标上的动态方程，而且还包含固定时间带脉冲的常微分方程。我们几乎在所有地方研究单调函数的微分和微积分的基本定理，这些微积分将我们的新导数与Lebesgue-Stieltjes和Kurzweil-Stieltjes积分联系起来。这些基本定理是将微分方程简化为广义积分方程的新导数的关键，这要归功于Kurzweil，Schwabik及其追随者，已经有了许多理论结果。

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