A Liapunov functional for a matrix neutral difference-differential equation with one delay

For the matrix neutral difference-differential equation $\text{x}\text{?}\text{(t) + A}\text{x}\text{?}\text{(t ? τ)  Bx(t) + Cx(t ? τ)}$ we construct a quadratic Liapunov functional which gives necessary and sufficient conditions for the asymptotic stability of the solutions of that equation. We consider a difference equation approximation of the difference-differential equation, and for this difference equation we construct a Liapunov function from which we obtain the desired Liapunov functional by an appropriate limiting process. The Liapunov functional thus obtained gives the best possible estimate for the rates of growth or decay of the solutions of the matrix neutral difference-differential equation. The results obtained are natural generalizations of previous results obtained for a matrix retarded difference-differential equation with one delay.