On Bohl-Perron type theorems for linear difference equations. (English) Zbl 1060.39002

The authors obtain a Bohl-Perron type theorem for linear difference equations. They consider the linear nonhomogeneous difference equation \[ x(n+1)-x(n)=\sum_{k=-d}^n B(n,k)x(k)+g(n),\quad n\geq 0\tag{\(*\)} \] where \(B(n,k)\) are \(m\times m\) matrices and \(x_n=\varphi (n),\;n\leq 0\) are the initial conditions. They prove that if a solution \(\{x(n)\}\) of Eq. (\(*\)) is bounded for any bounded sequence \(\{g(n)\}\), then Eq. (\(*\)) is exponentially stable. As an application, the authors obtain the following
Theorem: Suppose that the following conditions hold \[ \sum_{k=1}^N\;\sum_{n=0}^\infty a_k(n)=\infty\qquad \underset {n\rightarrow \infty} {\lim\sup}\sum_{k=1}^N\;\sum_{\ell=\min_k\{g_k(n)\}}^{n-1}\,a_k(\ell)<1. \] Then the scalar equation \(x(n+1)-x(n)=-\sum_{k=1}^N a_k(n) x(g_k(n))\) where \(a_k(n)\geq 0,\;0\leq n-g_k(n)\leq K\) is exponentially stable.
Reviewer: Fozi Dannan (Doha)


39A11 Stability of difference equations (MSC2000)