International Scientific Research PublicationsJournal of Mathematics and Computer Science(JMCS) ISSN 2008-949X17220170615Asymptotic behavior of discrete semigroups of bounded linear operators over Banach spaces301307http://dx.doi.org/10.22436/jmcs.017.02.12ENShuhongTangSchool of Information and Control Engineering, Weifang University, Weifang, Shandong 261061, P. R. China.AkbarZadaDepartment of Mathematics, University of Peshawar, Peshawar 25000, Pakistan.HabibaKhalidDepartment of Mathematics, University of Peshawar, Peshawar 25000, Pakistan.TongxingLiLinDa Institute of Shandong Provincial Key Laboratory of Network Based Intelligent Computing, Linyi University, Linyi, Shandong 276005, P. R. China. Assume that \(\vartheta_j\) is the solution of the nonhomogeneous Cauchy problem
\[\vartheta_{j+1}=\rho(1)\vartheta_j+f(j+1),\quad \vartheta_0=0,\]
where \(\rho(1)\) is the algebraic generator of the discrete semigroup \(\textbf{T}=\{\rho(j): j\in \mathbb{Z}_+\}\) acting on a complex Banach space \(\Delta\). Suppose
further that \(\textbf{AA}\textbf{P}_0^r(\mathbb{Z}_+,\Delta)\) is the space of asymptotically almost periodic sequences with relatively compact ranges. We prove
that the system
\[u_{j+1}=\rho(1)u_j\]
is uniformly exponentially stable if and only if for each \(f\in \textbf{AA}\textbf{P}_0^r(\mathbb{Z}_+,\Delta)\) the solution \(\vartheta_j\in \textbf{AA}\textbf{P}_0^r(\mathbb{Z}_+,\Delta)\) .
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