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For a family of difference equations x nþ1 ¼ ax n þ f ðx n2k Þ; n ¼ 0; 1; . . .; where a [ ð0; 1Þ; k [ {1; 2; . . .}; and f : ½0; 1Þ ! ð0; 1Þ is continuous and decreasing, we find sufficient conditions for the convergence of all solutions to the unique positive equilibrium. In particular, we improve, up to our knowledge, all previous results on the global asymptotic stability of the equilibrium in the particular cases of the discrete MackeyGlass and Lasota-Wazewska models in blood-cells production.