Bulletin des Sciences Mathématiques ( IF 1.241 ) Pub Date : 2021-04-30 , DOI: 10.1016/j.bulsci.2021.102991
M. Federson, J. Mawhin, C. Mesquita

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 Arzelà-Ascoli-type theorem for regulated functions.

J. Kurzweil在1957年提出的广义常微分方程（简称GODE）包含其他类型的方程。本文的第一个主要结果是将非自治ODE周期解的存在性扩展到GODEs的一些经典条件。通过脉冲微分方程（简称IDE）和GODE之间的对应关系，我们将结果转换为IDE。不需要经典的假设（即IDE右侧的功能是分段连续的），就足以要求它们在Lebesgue的意义上是可集成的，从而使这些功能具有许多不连续性。我们的第二个主要结果为根据参数的GODE周期边值问题的平凡解提供了一个分叉点存在的条件，并且再次，我们将这样的结果应用于IDE。获得主要结果的方法是拓扑度理论，紧凑算子理论的工具以及可调节函数的Arzelà-Ascoli型定理。

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