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Distributional, differential and integral problems: Equivalence and existence results
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SYSNO ASEP 0471069 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title Distributional, differential and integral problems: Equivalence and existence results Author(s) Monteiro, Giselle Antunes (MU-W) RID, SAI, ORCID
Satco, B. R. (RO)Source Title Electronic Journal of Qualitative Theory of Differential Equations. - : University of Szeged - ISSN 1417-3875
Roč. 2017, č. 7 (2017), s. 1-26Number of pages 26 s. Language eng - English Country HU - Hungary Keywords derivative with respect to functions ; distribution ; Kurzweil-Stieltjes integral Subject RIV BA - General Mathematics OECD category Pure mathematics Institutional support MU-W - RVO:67985840 UT WOS 000393037600001 EID SCOPUS 85011019650 DOI https://doi.org/10.14232/ejqtde.2017.1.7 Annotation We are interested in studying the matter of equivalence of the following problems: Dx = f (t, x)Dg x(0) = x0 (1) where Dx and Dg stand for the distributional derivatives of x and g, respectively, x'g(t) = f (t, x(t)), mg-a.e. x(0) = x0 (2) where x'g denotes the g-derivative of x (in a sense to be specified in Section 2) and mg is the variational measure induced by g, and x(t) = x0 + ...t 0 f (s, x(s))dg(s), (3) where the integral is understood in the Kurzweil-Stieltjes sense. We prove that, for regulated functions g, (1) and (3) are equivalent if f satisfies a bounded variation assumption. The relation between problems (2) and (3) is described for very general f, though, more restrictive assumptions over the function g are required. We provide then two existence results for the integral problem (3) and, using the correspondences established with the other problems, we deduce the existence of solutions for (1) and (2). Workplace Mathematical Institute Contact Jarmila Štruncová, struncova@math.cas.cz, library@math.cas.cz, Tel.: 222 090 757 Year of Publishing 2018
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