In this note we observe a decreasing property of {\big \Vert {Fn} \big \Vert} _{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(Error rendering LaTeX formula) 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.

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