Asymptotic behavior of solutions of functional difference equations
The authors consider a difference equation of the form \(x(n+1)=L(x_{n})+G(n)x_{n}\) where \(L\) is a bounded linear operator defined on a linear space and \(G(n)\) also is a bounded linear operator defined on a complex space. They investigate the asymptotic behavior of the solution under the condition \(\lim_{n\rightarrow \infty }\left\| G(n)\right\| =0\), where \(\left\| .\right\|\) is the operator norm of \(G(n).\) Some auxiliary lemmas that are used in the proof of the main results are stated and proved. The results are applied to different types of Volterra difference equations and the asymptotic behavior of their solutions are given. The results can be considered as the extension of the results given by \textit{M. Pituk} [J. Difference Equ. Appl. 8, No. 3, 201--216 (2002; Zbl 1002.39014)].
scientific article; zbMATH DE number 2162009