Attractors in k-Dimensional Discrete Systems of Mixed Monotonicity

We consider k-dimensional discrete-time systems of the form \(x_{n+1}=F(x_n,\ldots ,x_{n-k+1})\) $x_{n+1}=F(x_n,\dots ,x_{n-k+1})$ in which the map F is continuous and monotonic in each one of its arguments. We define a partial order on \({\mathbb {R}}^{2k}_+\) ${\mathbb{R}}_{+}^{2k}$ , compatible with the monotonicity of F, and then use it to embed the k-dimensional system into a 2k-dimensional system that is monotonic with respect to this poset structure. An analogous construction is given for periodic systems. Using the characteristics of the higher-dimensional monotonic system, global stability results are obtained for the original system. Our results apply to a large class of difference equations that are pertinent in a variety of contexts. As an application of the developed theory, we provide two examples that cover a wide class of difference equations, and in a subsequent paper, we provide additional applications of general interest.