On the global asymptotic stability of nonlinear discrete time systems depending on several past steps (Q1178462)

The global asymptotic stability of discrete systems of the form \(x^{m+1}=g(x^ m,x^{m-1},\dots,x^{m-k})\) is considered using comparison and invariance principles coupled with the theory of \(M\)- functions. Basically the requirement is that the function \(p_ h(x)=h(x,\dots,x)\) is a (weak) \(M\)-function, where \[ h(x^ m,x^{m- 1},\dots,x^{m-k})=x^ m-g(x^ m,x^{m-1},\dots,x^{m-k}). \] The idea of \(M\)-functions generalizes the notion of \(M\)-matrix to nonlinear functions.