On the relationships between Stieltjes type integrals of Young, Dushnik and Kurzweil
| Autor: | Umi Mahnuna Hanung, Milan Tvrdý |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
dushnik-stieltjes integral
Pure mathematics Mathematics::Classical Analysis and ODEs young-stieltjes integral Of the form Type (model theory) Lebesgue integration Mathematical proof symbols.namesake Linear differential equation Classical Analysis and ODEs (math.CA) FOS: Mathematics Mathematics Mathematics::Functional Analysis convergence theorem lcsh:Mathematics 26A39 28B05 Riemann–Stieltjes integral Absolute continuity lcsh:QA1-939 Integral equation kurzweil integral Mathematics - Classical Analysis and ODEs symbols kurzweil-stieltjes integral dushnik integral young integral |
| Zdroj: | Mathematica Bohemica, Vol 144, Iss 4, Pp 357-372 (2019) |
| Popis: | Integral equations of the form $$ x(t)=x(t_0)+\int_{t_0}^t d[A]\,x=f(t)-f(t_0)$$ are natural generalizations of systems of linear differential equations. Their main goal is that they admit solutions which need not be absolutely continuous. Up to now such equations have been considered by several authors starting with J. Kurzweil and T.H. Hildebrandt. These authors worked with several different concepts of the Stieltjes type integral like Young's (Hildebrandt), Kurzweil's (Kurzweil, Schwabik and Tvrd\'{y}), Dushnik's (H\"{o}nig) or Lebesgue's (Ashordia, Meng and Zhang). Thus an interesting question arises: what are the relationships between all these concepts? Our aim is to give an answer to this question. In addition, we present also convergence results that are new for the Young and Dushnik integrals. Let us emphasize that the proofs of all the assertions presented in this paper are based on rather elementary tools. |
| Databáze: | OpenAIRE |
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