Existence of periodic solutions and bifurcation points for generalized ordinary differential equations

Abstract The generalized ordinary differential equations (shortly GODEs), introduced by J. Kurzweil in 1957, encompass other types of equations. The first main result of this paper extends to GODEs some classical conditions on the existence of a periodic solution of a nonautonomous ODE. By means of the correspondence between impulse differential equations (shortly IDEs) and GODEs, we translate the result to IDEs. Instead of the classical hypotheses that the functions on the righthand side of an IDE are piecewise continuous, it is enough to require that they are integrable in the sense of Lebesgue, allowing such functions to have many discontinuities. Our second main result provides conditions for the existence of a bifurcation point with respect to the trivial solution of a periodic boundary value problem for a GODE depending upon a parameter, and, again, we apply such result to IDEs. The machinery employed to obtain the main results are the topological degree theory, tools from the theory of compact operators and an Arzela-Ascoli-type theorem for regulated functions.

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