Bifurcation of periodic solutions to nonlinear measure differential equations

Maria Carolina Mesquita; Milan Tvrdý

Czechoslovak Mathematical Journal (2025)

  • Issue: 1, page 357-395
  • ISSN: 0011-4642

Abstract

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The paper is devoted to the periodic bifurcation problems for generalizations of ordinary differential systems. The bifurcation is understood in the static sense of Krasnoselski and Zabreko. First, the conditions necessary for the given point to be bifurcation point for non autonomous generalized ordinary differential equations (based on the Kurzweil gauge type generalized integral) are proved. Then, as the main contribution, analogous results are obtained also for the nonlinear non autonomous measure differential equations considered in the sense of distributions. To this aim their relationship to Kurzweil’s generalized differential equations is disclosed. Although the measure differential equations turned out to be special cases of those Kurzweil’s equations, the proofs of the main results of the paper are by no means the straightforward consequences of the analogous results for generalized differential equations. Essentially they rely on the theory of the Kurzweil-Stieltjes integration. It is worth noting that as the systems studied in the paper encompass many types of equations such as impulsive differential equations, ordinary differential equations, dynamic equations on time scales etc., the results of the paper offer applications to rather wide scale of practical problems. Two illustrating examples are included, as well.

How to cite

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Mesquita, Maria Carolina, and Tvrdý, Milan. "Bifurcation of periodic solutions to nonlinear measure differential equations." Czechoslovak Mathematical Journal (2025): 357-395. <http://eudml.org/doc/299902>.

@article{Mesquita2025,
abstract = {The paper is devoted to the periodic bifurcation problems for generalizations of ordinary differential systems. The bifurcation is understood in the static sense of Krasnoselski and Zabreko. First, the conditions necessary for the given point to be bifurcation point for non autonomous generalized ordinary differential equations (based on the Kurzweil gauge type generalized integral) are proved. Then, as the main contribution, analogous results are obtained also for the nonlinear non autonomous measure differential equations considered in the sense of distributions. To this aim their relationship to Kurzweil’s generalized differential equations is disclosed. Although the measure differential equations turned out to be special cases of those Kurzweil’s equations, the proofs of the main results of the paper are by no means the straightforward consequences of the analogous results for generalized differential equations. Essentially they rely on the theory of the Kurzweil-Stieltjes integration. It is worth noting that as the systems studied in the paper encompass many types of equations such as impulsive differential equations, ordinary differential equations, dynamic equations on time scales etc., the results of the paper offer applications to rather wide scale of practical problems. Two illustrating examples are included, as well.},
author = {Mesquita, Maria Carolina, Tvrdý, Milan},
journal = {Czechoslovak Mathematical Journal},
keywords = {periodic solution; bifurcation; Kurzweil integral; Kurzweil-Stieltjes integral; generalized differential equation; measure differential equation; distributional differential equation},
language = {eng},
number = {1},
pages = {357-395},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Bifurcation of periodic solutions to nonlinear measure differential equations},
url = {http://eudml.org/doc/299902},
year = {2025},
}

TY - JOUR
AU - Mesquita, Maria Carolina
AU - Tvrdý, Milan
TI - Bifurcation of periodic solutions to nonlinear measure differential equations
JO - Czechoslovak Mathematical Journal
PY - 2025
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 357
EP - 395
AB - The paper is devoted to the periodic bifurcation problems for generalizations of ordinary differential systems. The bifurcation is understood in the static sense of Krasnoselski and Zabreko. First, the conditions necessary for the given point to be bifurcation point for non autonomous generalized ordinary differential equations (based on the Kurzweil gauge type generalized integral) are proved. Then, as the main contribution, analogous results are obtained also for the nonlinear non autonomous measure differential equations considered in the sense of distributions. To this aim their relationship to Kurzweil’s generalized differential equations is disclosed. Although the measure differential equations turned out to be special cases of those Kurzweil’s equations, the proofs of the main results of the paper are by no means the straightforward consequences of the analogous results for generalized differential equations. Essentially they rely on the theory of the Kurzweil-Stieltjes integration. It is worth noting that as the systems studied in the paper encompass many types of equations such as impulsive differential equations, ordinary differential equations, dynamic equations on time scales etc., the results of the paper offer applications to rather wide scale of practical problems. Two illustrating examples are included, as well.
LA - eng
KW - periodic solution; bifurcation; Kurzweil integral; Kurzweil-Stieltjes integral; generalized differential equation; measure differential equation; distributional differential equation
UR - http://eudml.org/doc/299902
ER -

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