Extension, embedding and global stability in two dimensional monotone maps

We consider the general second order difference equation x_n+1 = F(xn, x_n-1) in which F is continuous and of mixed monotonicity in its arguments. In equations with negative terms, a persistent set can be a proper subset of the positive orthant, which motivates studying global stability with respect to compact invariant domains. In this paper, we assume that F has a semi-convex compact invariant domain, then make an extension of F on a rectangular domain that contains the invariant domain. The extension preserves the continuity and monotonicity of F. Then we use the embedding technique to embed the dynamical system generated by the extended map into a higher dimensional dynamical system, which we use to characterize the asymptotic dynamics of the original system. Some illustrative examples are given at the end.