Admissibility in the Strong and Weak Senses

• Barreira, Luis ^[1] ; Valls, Claudia ^[1]
  1. [1] Universidade de Lisboa

     Universidade de Lisboa

     Socorro, Portugal

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00359-4
• Enlaces
  • Texto completo
• Resumen

  • For a dynamics with continuous time determined by a nonautonomous linear equation, we characterize completely the notion of an exponential dichotomy with respect to a family of norms in terms of an admissibility property. Moreover, we consider both strong and weak admissibility properties, which are expressed, respectively, in terms of classical and mild solutions. As a nontrivial application, we establish the robustness of the notions of an exponential dichotomy with respect to a family of norms and of a nonuniform exponential dichotomy in a very simple manner.

• Referencias bibliográficas
  • 1. Barreira, L., Pesin, Ya.: Nonuniform Hyperbolicity. Encyclopedia of Mathematics and its Applications, vol. 115. Cambridge University Press,...
  • 2. Barreira, L., Valls, C.: Robustness of nonuniform exponential dichotomies in Banach spaces. J. Differ. Equ. 244, 2407–2447 (2008)
  • 3. Barreira, L., Valls, C.: Stability of Nonautonomous Differential Equations. Lecture Notes in Mathematics, vol. 1926. Springer, Berlin (2008)
  • 4. Barreira, L., Valls, C.: Admissibility for nonuniform exponential contractions. J. Differ. Equ. 249, 2889–2904 (2010)
  • 5. Barreira, L., Valls, C.: Nonuniform exponential dichotomies and admissibility. Discrete Contin. Dyn. Syst. 30, 39–53 (2011)
  • 6. Chicone, C., Latushkin, Yu.: Evolution Semigroups in Dynamical Systems and Differential Equations. Mathematical Surveys and Monographs,...
  • 7. Chow, S.-N., Leiva, H.: Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces. J. Differ. Equ....
  • 8. Coffman, C., Schäffer, J.: Dichotomies for linear difference equations. Math. Ann. 172, 139–166 (1967)
  • 9. Coppel, W.: Dichotomies and reducibility. J. Differ. Equ. 3, 500–521 (1967)
  • 10. Coppel, W.: Dichotomies in Stability Theory. Lecture Notes in Mathematics, vol. 629. Springer, Berlin (1981)
  • 11. Dalec’ki˘ı, Ju, Kre˘ın, M.: Stability of Solutions of Differential Equations in Banach Space. Translations of Mathematical Monographs,...
  • 12. Folland, G.: Real Analysis. Wiley, Hoboken (1999)
  • 13. Huy, N.: Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line. J. Funct. Anal. 235, 330–354...
  • 14. Levitan, B., Zhikov, V.: Almost Periodic Functions and Differential Equations. Cambridge University Press, Cambridge (1982)
  • 15. Li, T.: Die Stabilitätsfrage bei Differenzengleichungen. Acta Math. 63, 99–141 (1934)
  • 16. Massera, J., Schäffer, J.: Linear differential equations and functional analysis. I. Ann. Math. 67, 517– 573 (1958)
  • 17. Massera, J., Schäffer, J.: Linear Differential Equations and Function Spaces, Pure and Applied Mathematics, vol. 21. Academic Press, Cambridge...
  • 18. Megan, M., Sasu, B., Sasu, A.: On nonuniform exponential dichotomy of evolution operators in Banach spaces. Integral Equ. Oper. Theory...
  • 19. Minh, N., Huy, N.: Characterizations of dichotomies of evolution equations on the half-line. J. Math. Anal. Appl. 261, 28–44 (2001)
  • 20. Naulin, R., Pinto, M.: Admissible perturbations of exponential dichotomy roughness. Nonlinear Anal. 31, 559–571 (1998)
  • 21. Perron, O.: Die Stabilitätsfrage bei Differentialgleichungen. Math. Z. 32, 703–728 (1930)
  • 22. Pliss, V., Sell, G.: Robustness of exponential dichotomies in infinite-dimensional dynamical systems. J. Dyn. Differ. Equ. 11, 471–513...
  • 23. Preda, P., Megan, M.: Nonuniform dichotomy of evolutionary processes in Banach spaces. Bull. Aust. Math. Soc. 27, 31–52 (1983)
  • 24. Preda, P., Pogan, A., Preda, C.: (L p, Lq )-admissibility and exponential dichotomy of evolutionary processes on the half-line. Integral...
  • 25. Preda, P., Pogan, A., Preda, C.: Schäffer spaces and exponential dichotomy for evolutionary processes. J. Differ. Equ. 230, 378–391 (2006)
  • 26. Van Minh, N., Räbiger, F., Schnaubelt, R.: Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations...