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

Volume 17, Issue 2, pp 301-307 http://dx.doi.org/10.22436/jmcs.017.02.12

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 \(\vartheta_j\in \textbf{AA}\textbf{P}_0^r(\mathbb{Z}_+,\Delta)\) .

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Keywords

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

MSC

•  39A10
•  39A30

References

• [1] N. Ahmad, H. Khalid, A. Zada, Uniform exponential stability of discrete semigroup and space of asymptotically almost periodic sequences, Z. Anal. Anwend., 34 (2015), 477–484.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] S. Balint, On the Perron–Bellman theorem for systems with constant coefficients, An. Univ. Timişoara Ser. Ştiinţ. Mat., 21 (1983), 3–8.
  • MathSciNet
  • MATH


• [3] A. S. Besicovitch, Almost periodic functions, Dover Publications, Inc., New York (1955)


• [4] C. Buşe, On the Perron–Bellman theorem for evolutionary processes with exponential growth in Banach spaces, New Zealand J. Math., 27 (1998), 183–190.
  • MathSciNet
  • MATH


• [5] C. Buşe, D. Barbu, Some remarks about the Perron condition for \(C_0\)-semigroups, An. Univ. Timişoara Ser. Mat.-Inform., 35 (1997), 3–8.
  • MathSciNet
  • Google Scholar
  • MATH


• [6] C. Buşe, A. Khan, G. Rahmat, A. Tabassum, Uniform exponential stability for discrete non-autonomous systems via discrete evolution semigroups, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 57 (105) (2014), 193–205.
  • MathSciNet
  • MATH


• [7] C. Buşe, A. Zada, Dichotomy and boundedness of solutions for some discrete Cauchy problems, Topics in operator theory, Volume 2, Systems and mathematical physics, Oper. Theory Adv. Appl., Birkhäuser Verlag, Basel , 203 (2010), 165– 174.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] C. Corduneanu, Almost periodic oscillations and waves, Springer, New York (2009)
  • View Article
  • MathSciNet


• [9] Ju. L. Daleckiĭ, M. G. Kreĭn, Stability of solutions of differential equations in Banach space, Translated from the Russian by S. Smith, Translations of Mathematical Monographs, American Mathematical Society, Providence, R.I. (1974)


• [10] G. Greiner, J. Voigt, M. Wolff, On the spectral bound of the generator of semigroups of positive operators, J. Operator Theory, 5 (1981), 245–256.
  • MathSciNet


• [11] A. Khan, G. Rahmat, A. Zada, On uniform exponential stability and exact admissibility of discrete semigroups, Int. J. Differ. Equ., 2013 (2013 ), 4 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] M. G. Kreĭn, On some questions related to the ideas of Lyapunov in the theory of stability, (Russian) Uspehi Matem. Nauk (N. S.), 3 (1948), 166–169.
  • MathSciNet
  • Google Scholar


• [13] Vu Quoc Phong, On stability of \(C_0\)-semigroups, Proc. Amer. Math. Soc., 129 (2001), 2871–2879.


• [14] M. Reghiş, C. Buşe, On the Perron–Bellman theorem for \(C_0\)-semigroups and periodic evolutionary processes in Banach spaces, Ital. J. Pure Appl. Math., 4 (1998), 155–166.
  • MathSciNet
  • MATH


• [15] J. M. A. M. van Neerven, Individual stability of \(C_0\)-semigroups with uniformly bounded local resolvent, Semigroup Forum, 53 (1996), 155–161.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] J. van Neerven, The asymptotic behaviour of semigroups of linear operators, Operator Theory: Advances and Applications, Birkhäuser Verlag, Basel (1996)
  • View Article
  • MathSciNet


• [17] Y.-F. Wang, A. Zada, N. Ahmad, D. Lassoued, T.-X. Li, Uniform exponential stability of discrete evolution families on space of p-periodic sequences, Abstr. Appl. Anal., 2014 (2014 ), 4 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [18] A. Zada, A characterization of dichotomy in terms of boundedness of solutions for some Cauchy problems, Electron. J. Differential Equations, 2008 (2008 ), 5 pages.
  • MathSciNet
  • Google Scholar
  • MATH


• [19] A. Zada, N. Ahmad, I. U. Khan, F. M. Khan, On the exponential stability of discrete semigroups, Qual. Theory Dyn. Syst., 14 (2015), 149–155.
  • View Article
  • MathSciNet
  • Google Scholar