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Keywords:
periodic solution; bifurcation; Kurzweil integral; Kurzweil-Stieltjes integral; generalized differential equation; measure differential equation; distributional differential equation
periodic solution; bifurcation; Kurzweil integral; Kurzweil-Stieltjes integral; generalized differential equation; measure differential equation; distributional differential equation
Summary:
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\u ı and Zabre\u ıko. 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.
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