Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2006, Article ID 31641, 16 pages
http://dx.doi.org/10.1155/AAA/2006/31641

Exponential dichotomy for evolution families on the real line

Department of Mathematics, Faculty of Mathematics and Computer Science, West University of Timişoara, Boulevard Vasile Pârvan No. 4, Timişoara 300223, Romania

Received 20 October 2004; Accepted 26 September 2005

Copyright © 2006 Adina Luminiţa Sasu. 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. A. Ben-Artzi and I. Gohberg, “Dichotomy of systems and invertibility of linear ordinary differential operators,” Operator Theory: Advances and Applications, vol. 56, pp. 90–119, 1992. View at Google Scholar
  2. A. Ben-Artzi, I. Gohberg, and M. A. Kaashoek, “Invertibility and dichotomy of differential operators on a half-line,” Journal of Dynamics and Differential Equations, vol. 5, no. 1, pp. 1–36, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. C. Chicone and Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, vol. 70 of Mathematical Surveys and Monographs, American Mathematical Society, Rhode Island, 1999. View at Zentralblatt MATH · View at MathSciNet
  4. 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
  5. J. L. Dalec'kiĭ and M. G. Kreĭn, Stability of Differential Equations in Banach Space, vol. 43 of Translations of Mathematical Monographs, American Mathematical Society, Rhode Island, 1974. View at Zentralblatt MATH · View at MathSciNet
  6. G. Gühring, F. Räbiger, and W. M. Ruess, “Linearized stability for semilinear non-autonomous evolution equations with applications to retarded differential equations,” Differential and Integral Equations, vol. 13, no. 4–6, pp. 503–527, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. Y. Latushkin, T. Randolph, and R. Schnaubelt, “Exponential dichotomy and mild solutions of nonautonomous equations in Banach spaces,” Journal of Dynamics and Differential Equations, vol. 10, no. 3, pp. 489–510, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. M. Megan, B. Sasu, and A. L. Sasu, “On nonuniform exponential dichotomy of evolution operators in Banach spaces,” Integral Equations and Operator Theory, vol. 44, no. 1, pp. 71–78, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. 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 Zentralblatt MATH · View at MathSciNet
  10. M. Megan, A. L. Sasu, and B. Sasu, “Perron conditions for pointwise and global exponential dichotomy of linear skew-product flows,” Integral Equations and Operator Theory, vol. 50, no. 4, pp. 489–504, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, 1983. View at Zentralblatt MATH · View at MathSciNet
  12. V. A. Pliss and G. R. Sell, “Robustness of the exponential dichotomy in infinite-dimensional dynamical systems,” Journal of Dynamics and Differential Equations, vol. 11, no. 3, pp. 471–513, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. W. M. Ruess, “Existence and stability of solutions to partial functional-differential equations with delay,” Advances in Differential Equations, vol. 4, no. 6, pp. 843–876, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. W. M. Ruess, “Linearized stability for nonlinear evolution equations,” Journal of Evolution Equations, vol. 3, no. 2, pp. 361–373, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. R. J. Sacker and G. R. Sell, “Dichotomies for linear evolutionary equations in Banach spaces,” Journal of Differential Equations, vol. 113, no. 1, pp. 17–67, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A. L. Sasu, “Integral characterizations for stability of linear skew-product semiflows,” Mathematical Inequalities & Application, vol. 7, no. 4, pp. 535–541, 2004. View at Google Scholar · View at MathSciNet
  17. A. L. Sasu and B. Sasu, “A lower bound for the stability radius of time-varying systems,” Proceedings of the American Mathematical Society, vol. 132, no. 12, pp. 3653–3659, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. B. Sasu and A. L. Sasu, “Stability and stabilizability for linear systems of difference equations,” Journal of Difference Equations and Applications, vol. 10, no. 12, pp. 1085–1105, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. B. Sasu and A. L. Sasu, “Exponential dichotomy and (p,q)-admissibility on the half-line,” Journal of Mathematical Analysis and Applications, vol. 316, no. 2, pp. 397–408, 2006. View at Publisher · View at Google Scholar
  20. A. L. Sasu and B. Sasu, “Exponential dichotomy on the real line and admissibility of function spaces,” Integral Equations and Operator Theory, vol. 54, no. 1, pp. 113–130, 2006. View at Publisher · View at Google Scholar
  21. B. Sasu and A. L. Sasu, “Exponential trichotomy and p-admissibility for evolution families on the real line,” to appear in Mathematische Zeitschrift.
  22. S. Siegmund, “Dichotomy spectrum for non-autonomous differential equations,” Journal of Dynamics and Differential Equations, vol. 14, no. 1, pp. 243–258, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  23. N. Van Minh and N. T. Huy, “Characterizations of dichotomies of evolution equations on the half-line,” Journal of Mathematical Analysis and Applications, vol. 261, no. 1, pp. 28–44, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. N. Van Minh, F. Räbiger, and R. Schnaubelt, “Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line,” Integral Equations and Operator Theory, vol. 32, no. 3, pp. 332–353, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. W. N. Zhang, “The Fredholm alternative and exponential dichotomies for parabolic equations,” Journal of Mathematical Analysis and Applications, vol. 191, no. 1, pp. 180–201, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet