EnIn this note we observe a decreasing property of ​Fn​G​2 along the numerical solution of the autonomous differential system y′=f(x) which satisfies a monotonicity condition;such a solution is obtained by means of a class of linear k-step A-stable methods and we have set Fn=(fT(yn),fT(yn+1​),...,fT(yn+k−1​))T and G is a symmetric positive definite matrix of order k. We study also a particular subclass of linear multistep G-stable methods of maximum order,in which the matrix G is actually constructed.The associated Lyapunov function ensures the stability of the set of equilibrium points