Discrete time monotone systems: Criteria for global asymptotic stability and applications

For two classes of monotone maps on the n-dimensional positive orthant we show that for a discrete dynamical system induced by a map the origin of Rn+ is globally asymptotically stable, if and only if the map Γ is such that for any point in s ∈ R+, s 6= 0, the image-vector Γ(s) is such that at least one component is strictly less than the corresponding component of s. One class is the set of n × n matrices of class K∞ functions; these induce monotone operators on R+. Maps of the other class satisfy some geometric property for an invariant set. Keywords— monotone maps, spectral radius, one component decrease condition, global asymptotic stability