0471069 - MÚ 2018 RIV HU eng J - Journal Article
Monteiro, Giselle Antunes - Satco, B. R.
Distributional, differential and integral problems: Equivalence and existence results.
Electronic Journal of Qualitative Theory of Differential Equations. Roč. 2017, č. 7 (2017), s. 1-26. ISSN 1417-3875. E-ISSN 1417-3875
Institutional support: RVO:67985840
Keywords : derivative with respect to functions * distribution * Kurzweil-Stieltjes integral
OECD category: Pure mathematics
Impact factor: 0.881, year: 2017 ; AIS: 0.34, rok: 2017
Result website:
http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=4753DOI: https://doi.org/10.14232/ejqtde.2017.1.7

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).
Permanent Link: http://hdl.handle.net/11104/0268530