The asymptotic stability of difference equations
The paper discusses the stability properties of the linear delay difference equation \[ x_{n+1}-x_n=-a_nx_{n-k} , \tag{1} \] where \(a_n\geq 0\) and \(k\geq 1\). As the main result it is proved that if \[ \sum_{n=0}^{\infty}a_n=\infty \tag{"(C1)"} \] and \[ \underset{n\to\infty}\limsup \sum_{j=n-2k}^{n-1}a_j<\frac{7}{4} , \tag{"(C2)"} \] then every solution \(\{x_n\}\) of (1) tends to zero as \(n\to\infty\). Moreover, the presented example demonstrates that the constant in (C2) is sharp enough. The proof technique is also generalized to the case of nonlinear difference equation \[ x_{n+1}-x_n=f_n(x_{n-k}), \tag{2} \] where \(f_n(x)/x\leq 0\) for \(x\neq 0\) and \(f_n(0)=0\). The sufficient conditions for the asymptotic stability of (2) are simple modifications of (C1), (C2).
scientific article; zbMATH DE number 1429481