C^1,1-smooth center manifolds for the differential equations with piecewise constant arguments of a generalized type

In this paper, we first formulate and prove the existence of smooth center manifolds for the differential equations with piecewise constant arguments of a generalized type (DEPCAGs) by supposing that the linear DEPCAGs admit an exponential trichotomy. Furthermore, we emphasize that we use a result from ElBialy (2000) to establish smooth center manifolds outside the continuous and non-differentiable points. This result allows us to establish the existence and C^1,1smoothness of the center manifolds simultaneously using a fixed point.