Notes on the existence and uniqueness of solutions of Stieltjes differential equations. (English) Zbl 07746479

Summary: We study some classical uniqueness and existence results, such as Peano’s or Osgood’s uniqueness criteria, in the context of Stieltjes differential equations. This type of equation is based on derivatives with respect to monotone functions, and enables the investigation of discrete and continuous problems from a common standpoint. We compare our results with previous work on the topic and illustrate the advantages of the theorems presented in this paper with an example. Finally, we make some remarks regarding analogous uniqueness results which can be derived for measure differential equations.
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26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34A36 Discontinuous ordinary differential equations
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[1] R. P.Agarwal and V.Lakshmikantham, Uniqueness and nonuniqueness criteria for ordinary differential equations, World Scientific, Singapore, 1993. · Zbl 0785.34003
[2] J.Ángel Cid and R.López Pouso, Integration by parts and by substitution unified, Green’s theorem and uniqueness for ODEs, arXiv: 1503.05699. · Zbl 1342.26033
[3] J.Ángel Cid and R.López Pouso, Does Lipschitz with respect to x imply uniqueness for the differential equation \(y^\prime = f ( x , y )\)?, Amer. Math. Monthly116 (2009), 61-66. · Zbl 1176.34006
[4] E.Bompiani, Un teorema di confronto ed un teorema di unicità per l’equazione differenziale \(y^\prime = f ( x , y )\), Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nature.1 (1925), 298-302 (Italian). · JFM 51.0331.06
[5] J.Diblík, C.Nowak, and S.Siegmund, A general Lipschitz uniqueness criterion for scalar ordinary differential equations, Electron. J. Qual. Theory Differ. Equ.34 (2014), 6 pp.
[6] V.Ene, Thomson’s variational measure and some classical theorems, Real Anal. Exchange25 (1999), 521-546. · Zbl 1010.26009
[7] M.Federson, J. G.Mesquita, and A.Slavík, Measure functional differential equations and functional dynamic equations on time scales, J. Differential Equations252 (2012), 3816-3847. · Zbl 1239.34076
[8] M.Frigon and R.López Pouso, Theory and applications of first‐order systems of Stieltjes differential equations, Adv. Nonlinear Anal.6 (2017), 13-36. · Zbl 1361.34010
[9] R.López Pouso and I.Márquez Albés, General existence principles for Stieltjes differential equations with applications to mathematical biology, J. Differential Equations264 (2018), 5388-5407. · Zbl 1386.34021
[10] R. LópezPouso, I.Márquez Albés, and G. A.Monteiro, Extremal solutions of systems of measure differential equations and applications in the study of Stieltjes differential problems, Electron. J. Qual. Theory Differ. Equ.38 (2018), 24 pp.
[11] R.López Pouso and A.Rodríguez, A new unification of continuous, discrete, and impulsive calculus through Stieltjes derivatives, Real Anal. Exchange40 (2015), 1-35.
[12] G. A.Monteiro and B.Satco, Distributional, differential and integral problems: Equivalence and existence results, Electron. J. Qual. Theory Differ. Equ.7 (2017), 26 pp. · Zbl 1413.34062
[13] G. A.Monteiro and A.Slavík, Extremal solutions of measure differential equations, J. Math. Anal. Appl.444 (2016), 568-597. · Zbl 1356.34094
[14] G. A.Monteiro, A.Slavík, and M.Tvrdý, Kurzweil-Stieltjes integral: theory and applications, World Scientific, Singapore, 2018.
[15] P.Montel, Sur l’intègrale supèrieure et l’intègrale infèrieure d’une èquation diffèrentielle, Bull. Sci. Math.50 (1926), 205-217 (French). · JFM 52.0437.03
[16] W. F.Osgood, Beweis der Existenz einer Lösung der Differentialgleichung \(d y / d x = f ( x , y )\) ohne Hinzunahme der Cauchy-Lipschitz’schen Bedingung, Monatsh. Math. Phys.9 (1898), 331-345 (German). · JFM 29.0260.03
[17] O.Perron, Über Ein‐und Mehrdeutigkeit des Integrals eines Systems von Differentialgleichungen, Math. Ann.95 (1926), 98-101 (German). · JFM 51.0331.07
[18] A.Slavík, Well‐posedness results for abstract generalized differential equations and measure functional differential equations, J. Differential Equations259 (2015), 666-707. · Zbl 1319.34116
[19] S.Schwabik, Generalized ordinary differential equations, World Scientific, Singapore, 1992. · Zbl 0781.34003
[20] L.Tonelli, Sull’ unicità della soluzione di un’equazione differenziale ordinaria, Rend. Accad. Naz. Lincei1 (1937), 144-152 (Italian).
[21] T.Yosie, Über die Unität der Lösung der gewöhnlichen Differentialgleichungen erster Ordnung, Japan. J. Math.2 (1926), 161-173 (German). · JFM 52.0438.02
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