We develop an approach which we used to deduce conditions of a new type for the existence of periodic solutions of ordinary differential equations and functional differential equations of point type. These conditions are based on the use of asymptotic properties of solutions of differential equations which can be observed on shifts of solutions and stated in terms of averages over the period on a distinguished sphere in the phase space. The development of this approach enables us to obtain conditions for the existence of bounded solutions for the same classes of functional differential equations.

中文翻译：

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点型泛函微分方程有界解存在性问题的一种新方法。

我们开发了一种方法，用于推导一种新类型的条件，该条件用于存在常微分方程和点型泛函微分方程的周期解。这些条件是基于对微分方程解的渐近性质的使用，这种微分方程的解可以在解的位移上观察到，并且可以用相空间中特定球面上的周期内的平均值表示。这种方法的发展使我们能够为相同类的泛函微分方程获得有界解的存在条件。