In this paper we investigate the existence of the periodic solutions of a nonlinear differential equation with a general piecewise constant argument, in short DEPCAG, that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. In the process we use the Green’s function for periodic solutions and convert the given DEPCAG into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii’s fixed point theorem to show the existence of a periodic solution of this type of nonlinear differential equations. We also use the contraction mapping principle to show the existence of a unique periodic solution. Appropriate examples are given to show the feasibility of our results.

在本文中，我们研究具有一般分段常数参数（简称DEPCAG）的非线性微分方程周期解的存在性，即DEPCAG，即该参数是一般阶跃函数。当关联的线性齐次系统接受非平凡的周期解时，我们考虑临界情况。获得了这类方程的周期解的存在性准则。在此过程中，我们将格林函数用于周期解，并将给定的DEPCAG转换为等效的积分方程。然后，我们构造适当的映射，并使用Krasnoselskii不动点定理来证明这种类型的非线性微分方程的周期解的存在。我们还使用收缩映射原理来显示唯一周期解的存在。