A new stability functional is introduced for analyzing the stability and consistency of linear multistep methods. Using it and the general theory of [t] we prove that a linear multistep method of design order q ~p ~ t which satisfies the weak stability root condition, applied to the differential equation y" (t) = / (t, y (t)) where f is Lipschitz continuous in its second argument, will exhibit actual convergence of order o(h p-l) if y has a (p--t)th derivative y(p-1) that is a Riemann integral and order o(h t') if y(p-1} is the integral of a function of bounded variation. This result applies for a function y taking on values in any real vector space, finite or infinite dimensional.
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