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

Volume 17, Issue 2, pp 301-307
Publication Date: June 15, 2017 Submission Date: May 31, 2016
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### 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 $\vartheta_j\in \textbf{AA}\textbf{P}_0^r(\mathbb{Z}_+,\Delta)$ .

### Share and Cite

##### ISRP Style

Shuhong Tang, Akbar Zada, Habiba Khalid, Tongxing Li, Asymptotic behavior of discrete semigroups of bounded linear operators over Banach spaces, Journal of Mathematics and Computer Science, 17 (2017), no. 2, 301-307

##### AMA Style

Tang Shuhong, Zada Akbar, Khalid Habiba, Li Tongxing, Asymptotic behavior of discrete semigroups of bounded linear operators over Banach spaces. J Math Comput SCI-JM. (2017); 17(2):301-307

##### Chicago/Turabian Style

Tang, Shuhong, Zada, Akbar, Khalid, Habiba, Li, Tongxing. "Asymptotic behavior of discrete semigroups of bounded linear operators over Banach spaces." Journal of Mathematics and Computer Science, 17, no. 2 (2017): 301-307

### Keywords

• Banach space
• difference equation
• uniform exponential stability
• almost periodic sequence
• relatively compact

•  39A10
•  39A30

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