Oscillations in systems of differential equations with piecewise constant argument (Q804803)

The authors consider linear systems of differential equations with piecewise constant argument of the form \[ (*)\quad x'(t)=Ax(t)+Bx([t+]),\quad x(0)=c_ 0,\quad x'(t)=Ax(t)+Bx([t]),\quad x(0)=c_ 0 \] and the associated nonhomogeneous equations, where [. ] denotes the greatest integer function. They obtain sufficient conditions under which the above equations admit unique solutions on \([0,\infty)\). Asymptotic stability of the zero solution and oscillatory behaviour and periodicity of solutions of (*) are studied. These properties depend upon the eigenvalues of a certain matrix associated with (*).