Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2016, Article ID 4375069, 10 pages
http://dx.doi.org/10.1155/2016/4375069
Research Article

Approaching the Discrete Dynamical Systems by means of Skew-Evolution Semiflows

Department of Mathematics and Computer Science, Aurel Vlaicu University of Arad, 2 Elena Drăgoi Street, 310330 Arad, Romania

Received 10 April 2016; Accepted 9 June 2016

Academic Editor: Allan C. Peterson

Copyright © 2016 Codruţa Stoica. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. B. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1981. View at MathSciNet
  2. O. Perron, “Die Stabilitätsfrage bei Differentialgleichungen,” Mathematische Zeitschrift, vol. 32, no. 1, pp. 703–728, 1930. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. S.-N. Chow and H. Leiva, “Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces,” Journal of Differential Equations, vol. 120, no. 2, pp. 429–477, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  4. Y. Latushkin and R. Schnaubelt, “Evolution semigroups, translation algebras, and exponential dichotomy of cocycles,” Journal of Differential Equations, vol. 159, no. 2, pp. 321–369, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. M. Megan, A. L. Sasu, and B. Sasu, “Discrete admissibility and exponential dichotomy for evolution families,” Discrete and Continuous Dynamical Systems, vol. 9, no. 2, pp. 383–397, 2003. View at Google Scholar · View at MathSciNet
  6. P. H. Anh Ngoc and T. Naito, “New characterizations of exponential dichotomy and exponential stability of linear difference equations,” Journal of Difference Equations and Applications, vol. 11, no. 10, pp. 909–918, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. A. L. Sasu, “New criteria for exponential stability of variational difference equations,” Applied Mathematics Letters, vol. 19, no. 10, pp. 1090–1094, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. A. L. Sasu, “Discrete methods and exponential dichotomy of semigroups,” Acta Mathematica Universitatis Comenianae, vol. 73, no. 2, pp. 197–205, 2004. View at Google Scholar
  9. B. Sasu, “Uniform dichotomy and exponential dichotomy of evolution families on the half-line,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1465–1478, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. B. Sasu, “On exponential dichotomy of variational difference equations,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 324273, 18 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. B. Sasu, “On dichotomous behavior of variational difference equations and applications,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 140369, 16 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. R. J. Sacker and G. R. Sell, “Existence of dichotomies and invariant splittings for linear differential systems III,” Journal of Differential Equations, vol. 22, no. 2, pp. 497–522, 1976. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. S. Elaydi and O. Hajek, “Exponential trichotomy of differential systems,” Journal of Mathematical Analysis and Applications, vol. 129, no. 2, pp. 362–374, 1988. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. J. Zhang, “Lyapunov function and exponential trichotomy on time scales,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 958381, 22 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. L. Barreira and C. Valls, “Stability in delay difference equations with nonuniform exponential behavior,” Journal of Differential Equations, vol. 238, no. 2, pp. 470–490, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. S. Elaydi and K. Janglajew, “Dichotomy and trichotomy of difference equations,” Journal of Difference Equations and Applications, vol. 3, no. 5-6, pp. 417–448, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  17. G. Papaschinopoulos and G. Stefanidou, “Trichotomy of a system of two difference equations,” Journal of Mathematical Analysis and Applications, vol. 289, no. 1, pp. 216–230, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  18. C.-L. Mihiţ, M. Megan, and T. Ceauşu, “The equivalence of Datko and Lyapunov properties for (h,k)-trichotomic linear discrete-time systems,” Discrete Dynamics in Nature and Society, vol. 2016, Article ID 3760262, 8 pages, 2016. View at Publisher · View at Google Scholar
  19. X.-q. Song, T. Yue, and D.-q. Li, “Nonuniform exponential trichotomy for linear discrete-time systems in Banach spaces,” Journal of Function Spaces and Applications, vol. 2013, Article ID 645250, 6 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  20. A. L. Sasu and B. Sasu, “Discrete admissibility and exponential trichotomy of dynamical systems,” Discrete and Continuous Dynamical Systems, vol. 34, no. 7, pp. 2929–2962, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. B. Sasu and A. L. Sasu, “On the dichotomic behavior of discrete dynamical systems on the half-line,” Discrete and Continuous Dynamical Systems—Series A, vol. 33, no. 7, pp. 3057–3084, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. S. Matucci, “The lp trichotomy for difference systems and applications,” Archivum Mathematicum, vol. 36, pp. 519–529, 2000. View at Google Scholar
  23. M. Megan and C. Stoica, “Exponential instability of skew-evolution semiflows in Banach spaces,” Studia Universitatis Babeş-Bolyai Mathematica, vol. 53, no. 1, pp. 17–24, 2008. View at Google Scholar
  24. A. J. G. Bento and C. M. Silva, “Nonuniform dichotomic behavior: Lipschitz invariant manifolds for ODEs,” Bulletin des Sciences Mathmatiques, vol. 138, no. 1, pp. 89–109, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. P. Viet Hai, “Continuous and discrete characterizations for the uniform exponential stability of linear skew-evolution semiflows,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 12, pp. 4390–4396, 2010. View at Publisher · View at Google Scholar · View at Scopus
  26. P. Viet Hai, “Discrete and continuous versions of Barbashin-type theorem of linear skew-evolution semiflows,” Applicable Analysis, vol. 90, no. 12, pp. 1897–1907, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. P. Viet Hai, “A generalization for theorems of Datko and Barbashin type,” Journal of Function Spaces, vol. 2015, Article ID 517348, 5 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. T. Yue, X.-Q. Song, and D.-Q. Li, “On weak exponential expansiveness of skew-evolution semiflows in Banach spaces,” Journal of Inequalities and Applications, vol. 2014, no. 1, article 165, pp. 1–11, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. C. Stoica and M. Megan, “On uniform exponential stability for skew-evolution semiflows on Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1305–1313, 2010. View at Publisher · View at Google Scholar · View at Scopus
  30. C. Stoica, “Trichotomy for dynamical systems in Banach spaces,” The Scientific World Journal, vol. 2013, Article ID 793813, 8 pages, 2013. View at Publisher · View at Google Scholar · View at Scopus
  31. M. Megan and C. Stoica, “On exponential trichotomy for discrete time skew-evolution semiows in Banach spaces,” Scientific Bulletin of the Politehnica University of Timisoara, Mathematics-Physics Series, vol. 51, no. 2, pp. 25–32, 2006. View at Google Scholar
  32. M. Megan and C. Stoica, “Trichotomy for discrete skew-evolution semiows in Banach spaces,” Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity, vol. 5, pp. 3–9, 2007. View at Google Scholar
  33. C. Stoica and M. Megan, “Discrete asymptotic behaviors for skew-evolution semiflows on Banach spaces,” Carpathian Journal of Mathematics, vol. 24, no. 3, pp. 348–355, 2008. View at Google Scholar · View at Scopus