Exponential dichotomy and boundedness for retarded functional difference equations. (English) Zbl 1162.39002

The authors investigate the retarded linear functional equation
\[ x(n+1)=L(n,x_n),\,\,\,n\geq 0,\tag{E} \]
where \(\{L(n,\cdot)\}_{n\geq 0}\) is a uniformly bounded sequence of bounded linear operators mapping \({\mathcal B}_{\gamma}\) (\(\gamma>0\)) into \(\mathbb{C}^r\), with the phase space \({\mathcal B}_{\gamma}\) defined by \({\mathcal B}_{\gamma}=\{\varphi:\mathbb{Z}^-\to\mathbb{C}^r;\;\sup_{\theta\in\mathbb{Z}^-}|\varphi(\theta)|e^{\gamma\theta}<+\infty\},\) and \(x_n(s)=x(n+s)\) for \(s\in\mathbb{Z}^-\). They characterize the exponential dichotomy of equation (E) and give conditions for the exponential stability of the orbit of the solution operator of (E). The robustness of the exponential dichotomy and the boundedness for (E) under several type of perturbations are also studied. In the last section of the paper the authors apply the obtained results to analyze bounded solutions of a class of Volterra difference equations with infinite delay.


39A10 Additive difference equations
39A12 Discrete version of topics in analysis
39A30 Stability theory for difference equations
Full Text: DOI


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