Perturbations of nonlinear system of difference equations (Q1331236)

The author considers the initial value problems for the difference system (i) \(x_{n + 1} = f(n,x_ n)\), \(x_{n_ 0} = x_ 0\), \(n \geq n_{n_ 0}\), and its perturbed system (ii) \(y_{n + 1} = f(n,y_ n) + F(n,y_ n)\), \(y_{n_ 0} = x_ 0\), \(n \geq n_{n_ 0}\), where \(f,F : \mathbb{N}_{n_ 0} \times \mathbb{R}^ s \to \mathbb{R}^ s\), and \(\partial f/ \partial x\) exists and be continuous and invertible on \(\mathbb{N}_{n_ 0} \times \mathbb{R}^ s\). Under some conditions, to complicated to be presented here, he proves that the zero solution of (ii) is exponentially asymptotically stable, resp. it is uniformly Lipschitz stable, resp. it is slowly growing.