On the global asymptotic stability of difference equations satisfying a Markus-Yamabe condition
Consider the \(n\)-dimensional reccurrence \[ x_{k+1} = F(x_k) \] with \(F:\mathbb R^n \to \mathbb R^n\) being a \(C^1\) map. The following result is proved. Theorem. Let \(F:\mathbb R^n \to \mathbb R^n\) be given by \[ F(x_1,\dots,x_n) = col(x_2,\dots,x_n,f(x_1,\dots, x_n)) \] with \(f:\mathbb R^n \to \mathbb R^n\) a \(C^1\) function. Then the following are equivalent (\(H_1\)) \(F\) is such that \(\rho(|DF(x)|)<1\), \(\forall x\in \mathbb R^n\); (\(H_2\)) \(\sum_1^n |D_i(f)(x)|<1\), \(\forall x\in \mathbb R^n\). If one of them holds and \(F\) has a fixed point then this fixed point is globally asymptotically stable. Here \(DF(x)=(D_j(F_i)(x))\) is the Jacobian matrix of \(F\) at \(x\).
scientific article; zbMATH DE number 6381929