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Criterions for absolute continuity on time scales. (English) Zbl 1081.39011

This nice paper is devoted to the study of the absolutely continuous functions on an arbitrary bounded time scale \(\mathbb{T} \subset \mathbb{R}\) with the property that \(\min \mathbb{T} = a\) and \(\max \mathbb{T}=b\). The authors introduce the concept of absolutely continuous function on \(\mathbb{T}\) and give characterizations for this concept, extending the classical Banach-Zarecki theorem. They prove the equivalence between absolute continuity on the time scale \(\mathbb{T}\) and absolute continuity on the interval \([a, b]\). The authors establish a version of the Fundamental Theorem of Calculus for this class of functions. As consequences of their techniques, they deduce conditions for the absolute continuity of the inverse function of any strictly monotone and absolutely continuous function on \(\mathbb{T}\).

MSC:

39A12 Discrete version of topics in analysis
39A10 Additive difference equations
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
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References:

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[2] DOI: 10.1007/978-0-8176-8230-9
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