Stability and asymptotic behavior of difference equations
The author considers the the following system of difference equations \[ y(n+1)= A(n)y(n)+ f(n,y(n)),\tag{1} \] where \(y(n)\in\mathbb{R}^m\), \(f:N_0\times\mathbb{R}^m\to \mathbb{R}^m\), \(N_0=\{n_0,n_0+1,\dots\}\), \(n_0\) is a nonnegative integer, \(f(n,0)= 0\) for all \(n\in N_0\) and \(A(n)\) is a discrete \(m\times m\) matrix-function defined for all \(n\in N_0\). The system (1) is said to be \(h\)-stable if there exist a positive bounded function \(h:N_0\to \mathbb{R}\) and constants \(c\geq 1\), \(\delta>0\) such that \(|y(n,n_0,y_0)|\leq c|y_0|h(n)h^{-1}(n_0)\), \(n\geq n_0\) for \(|y_0|\leq\delta\), (\(y(n,n_0,y_0)\), \(y(n_0,n_0,y_0)=y_0\) is the solution of (1)). Sufficient conditions for \(h\)-stability of the system (1) are given. Some results about asymptotic behavior for perturbed systems under general hypotheses are obtained.
scientific article; zbMATH DE number 1034164
scientific article