CONVERGENCE OF CERTAIN BOUNDED SEQUENCES

Let V be a nontrivial finite-dimensional real vector space, ordered by a cone K, and equipped with the standard norm topology. If K contains no affine line, then for each alpha(1), ..., alpha(p) is an element o f R the following statements are equivalent: (i) Every bounded sequenc e (x(n))(n=1)(infinity), in V satisfying x(n+p) greater than or equal to Sigma(j=1)(p) alpha(j)x(n+p-j), n = 1, 2, ... is convergent; (ii) T he polynomial P(t) = t(p) - alpha(1)t(p-1) - ... - alpha(p-1)t - alpha (p), has 1 as a zero and has no other complex zeroes of absolute value i. If alpha(j) greater than or equal to 0 for j = 1, ..., p, then (ii ) can be replaced by (ii) Sigma(j=1)(p) alpha(j) = 1, and the natural numbers j less than or equal to p satisfying alpha(j) > 0 are relativ ely prime. (C) 1998 Published by Elsevier Science Inc. All rights rese rved.