Pseudo almost periodic solutions for equation with piecewise constant argument (Q990875)

The main objective of this paper is to study the existence and uniqueness of pseudo almost periodic solutions of the differential equation with piecewise constant argument \[ \text{(EPCA)}\quad x'(t)=A(t)x(t)+\sum_{j=0}^{r}A_{j}(t)x(\lfloor t-j\rfloor)+g(t,x(\lfloor t \rfloor),\dots,x(\lfloor t-r\rfloor )) \] where \(\lfloor \cdot \rfloor \) denotes the greatest integer function and \(A, A_j :\mathbb R \to M_{q}(\mathbb R)\) are almost periodic, \(g:\mathbb R\times\mathbb R^q \times\dots \times\mathbb R^q\) is pseudo almost periodic satisfying some Lipschitz conditions. The authors first remind some facts about discontinuous almost periodic functions and exponential dichotomy of discrete linear equations of the form \(x_{n+1}=A(n)x_n+h_n\). The main results are obtained by means of the contraction mapping principle.