Pseudo almost-periodic solutions to systems of differential equations with piecewise constant argument \([t]\) (Q1609715)

The author considers the system of differential equations with piecewise constant argument \[ \dot x(t)= Ax(t)+ Bx([t])+ f(t),\quad t\in\mathbb{R},\tag{1} \] and the nonlinear system with piecewise constant argument \[ \dot x(t)= Ax(t)+ Bx([t])+ f(t)+ \mu F(t,x),\quad t\in\mathbb{R},\tag{2} \] where \(A\), \(B\) are constant matrices and \(A\) is nonsingular (\([\cdot]\) denotes the greatest-integer function). The author introduces a generalization of almost-periodic vector sequences (he calls them pseudo almost-periodic vector sequences). Thus, by constructing pseudo almost-periodic sequence solutions for a system of difference equations, he investigates the existence of pseudo almost-periodic solutions to systems (1) and (2). He also proves that under certain conditions the solution is unique.