Smooth invariant manifolds and foliations for the differential equations with piecewise constant argument

In this work, we establish the theory of smooth invariant manifolds and smooth invariant foliations for the differential equations with piecewise constant argument of a generalized type (DEPCAGs). Suppose that the linear DEPCAGs admits a α-exponential dichotomy, we obtain the existence of Lipschitz stable (unstable) invariant manifolds and Lipschitz stable (unstable) invariant foliations, which are based on the Lyapunov-Perron integrals with piecewise constant argument and other non-trivial techniques (such as, dichotomy inequalities with piecewise constant argument). Furthermore, we formulate and prove the C¹-smoothness of these manifolds and foliations for DEPCAGs by means of the fiber contraction theorem.