Asymptotic behavior of discrete semigroups of bounded linear operators over Banach spaces


Authors

Shuhong Tang - School of Information and Control Engineering, Weifang University, Weifang, Shandong 261061, P. R. China.
Akbar Zada - Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan.
Habiba Khalid - Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan.
Tongxing Li - LinDa Institute of Shandong Provincial Key Laboratory of Network Based Intelligent Computing, Linyi University, Linyi, Shandong 276005, P. R. China.


Abstract

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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