Stability of a linear difference equation and estimation of its fundamental solution
The authors analyze different types of stability for the following non-autonomous difference equation with delays \[ x(n+1)-x(n) + \sum_{k=0}^{N} A_{k}(n)x(n-\alpha_k(n))=f(n),\tag{1} \] where \(m\in\{0,1,\dots\}\), \(n\in\mathbb{N}_{m}:=\{m,m+1,\dots\}\), \(f:\mathbb{N}_0\rightarrow \mathbb{C}^{r}\), \(A_k: \mathbb{N}_0\rightarrow \mathbb{C}^{r\times r}\) are matrix functions and \(\alpha_k:\mathbb{N}_0\rightarrow \mathbb{N}_0\), \(k=0,\dots,N\). The authors impose the so-called \(V\)-condition, namely, the quantity \(V:=\sup_{n\in\mathbb{N}_0}\sum_{j=n-\alpha(n)}^{n} a(i)\) is finite (here \(\alpha\) denotes \(\max_{k}\{\alpha_k\}\) and \(a(n)=\sum_{k=0}^{N}|A_k(n)|\) if \(n\geq 0\) and \(a(n)=0\) otherwise). The main results of the paper are the following: 1. If \(f\equiv 0\) and \(m\geq 0\), assuming the \(V\)-condition they characterize different notions of stability (Lyapunov, uniform, asymptotical, uniformly asymptotical and exponential) in terms of the associate fundamental solution \(K(n,m)\), \(n\geq m\). 2. If \(m=0\) in \((1)\), and the zero initial function is considered, the stability properties of (1) are studied with respect to certain linear normed spaces of functions. To this end the authors consider an operator of (1) which is strongly linked with the Cauchy function appearing in a related functional differential equation. 3. Finally, they apply the results of the second point to establish a sufficient condition which ensures the exponential stability of \(x(n+1)-x(n) + A x(n-\alpha(n))=f(n)\), \(n\in\mathbb{N}_0\), where \(A\) is a constant \(\mathbb{C}^{r\times r}\) matrix.
non-autonomous difference equation with delays
scientific article; zbMATH DE number 6108756