Global stability for mixed monotone systems

We show that the embedding method described in J.-L. Gouze and P. Hadeler (Monotone flows and order intervals, Nonlinear World 1 (1994), pp. 23-34) and H.L. Smith (The discrete dynamics of monotonically decomposable maps, J. Math. Biol. 53 (2006), pp. 747-758) leads immediately to the global stability results in M. Kulenovic and O. Merino (A global attractivity result for maps with invariant boxes. Discrete Contin. Dyn. Syst. Series. B, 6 (2006), pp. 97-110). This allows the extension of some results on global stability for higher order difference equations due to Gerry Ladas and collaborators. Further, we provide a new result suggests that embedding into monotone systems may not be necessary for global stability results.