Embedding and global stability in periodic 2-dimensional maps of mixed monotonicity

In this paper, we consider nonautonomous second order difference equations of the form x_n+1=F(n,x_n,x_n−1), where F is p-periodic in its first component, non-decreasing in its second component and non-increasing in its third component. The map F is referred to as periodic of mixed monotonicity, which broadens the notion of maps of mixed monotonicity. We introduce the concept of artificial cycles, and we develop the embedding technique to tackle periodicity and globally attracting cycles in periodic 2-dimensional maps of mixed monotonicity. We present a result on globally attracting cycles and illustrate its application to periodic systems. The first application is a periodic rational difference equation of second order, and the second application is a population model with periodic stocking. In both cases, we prove the existence of a globally attracting cycle.