Attractors in \(k\)-dimensional discrete systems of mixed monotonicity (Q6608690)

Given a system \(x_{n+1}=F\left(x_n,\ldots,x_{n-k+1}\right)\), where the \(k\)-dimensional map \(F:V^k\to V\) is monotonically increasing with respect to a specified partial ordering \(\tau\) on its domain \(V^k\), the authors firstly introduce the construction of a \(2k\)-dimensional map \(G_\tau : V^k \times V^k\to V^k\times V^k\) which is monotonically increasing with respect to a certain partial ordering on \(V^k \times V^k\) explicitly associated to \(\tau\). Exploiting the result, the authors subsequently establish sufficient conditions for which the above system admits a globally asymptotically stable fixed point in the case of \(V\) being a closed interval. Additionally, the authors generalise the above construction to the case of the periodic system \(x_{n+1}=F_n\left(x_n,\ldots,x_{n-k+1}\right)\). As illustrative examples, the authors apply their results to address the global asymptotic stability problem of the \(k\)-dimensional Ricker model, and of a family of rational difference equations.