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Existence and Roughness of Nonuniform Exponential Dichotomies on Time Scales

  • Soniya Dhama [1] ; Samuel Castillo [4] ; Syed Abbas [2] ; Manuel Pinto [3]
    1. [1] Rajiv Gandhi Institute of Petroleum Technology

      Rajiv Gandhi Institute of Petroleum Technology

      India

    2. [2] Indian Institute of Technology Mandi

      Indian Institute of Technology Mandi

      India

    3. [3] Universidad de Chile

      Universidad de Chile

      Santiago, Chile

    4. [4] Universidad del Bio Bio
    Mostrar afiliaciones +
  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 2, 2024
  • Idioma: inglés
  • Enlaces
  • Resumen

    • In this manuscript, we discuss the roughness of nonuniform exponential dichotomies on a time scale in a Banach space and give the existence results for nonuniform and uniform exponential dichotomy for the time scale. We extend and unify the previous results by considering the equation on a time scale. For a given linear equation on a time scale, the existence of exponential dichotomy persists under an adequately small variable linear perturbation. To establish the results, we acquire related roughness results for the case of uniform exponential contractions. In the end, a suitable example is given for illustration.

  • Referencias bibliográficas
    • 1. Perron, O.: Die Stabilitatsfrage bei differentialgleichungen. Math. Z. 32, 703–728 (1930)
    • 2. Coppel, W.A.: Dichotomies in Stability Theory. In: Lecture Notes in Mathematics, vol. 629, Springer, Berlin (1978)
    • 3. Chow, S.N., Leiva, H.: Existence and roughness of the exponential dichotomy for skew-product semiflows in Banach spaces. J. Differ. Equ....
    • 4. Li, T.: Die Stabilitasfrage bei differenzengleichungen. Acta Math. 63, 99–141 (1934)
    • 5. Alonso, A.I., Hong, J.L., Obaya, R.: Exponential dichotomy and trichotomy for difference equations. Comput. Appl. Math. 38, 41–49 (1999)
    • 6. Aulbach, B., Van Minh, N.: The concept of spectral dichotomy for linear difference equations. I. J. Math. Anal. Appl. 185, 275–287 (1994)
    • 7. Aulbach, B., Van Minh, N.: The concept of spectral dichotomy for linear difference equations. II. J. Differ. Equ. Appl. 2, 251–262 (1996)
    • 8. Kurzweil, J.: Topological equivalence and structural stability for linear difference equations. J. Differ. Equ. 89, 89–94 (1991)
    • 9. Papaschinopoulos, G., Schinas, J.: Criteria for an exponential dichotomy of difference equations. Czechoslov. Math. J. 35, 295–299 (1985)
    • 10. Hilger, S.: Ein Makettenkalkaul mit Anwendung auf Zentrumsmannigfaltigkeiten (Ph.D. thesis) Universität Wäurzburg (1988)
    • 11. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. Birkhäuser, Basel (2001)
    • 12. Bohner, M., Peterson, A. (eds.): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)
    • 13. Bohner, M., Peterson, A.: A survey of exponential functions on time scales. Cubo Mat. Educ. 3(2), 285–301 (2001)
    • 14. Agarwal, R., Bohner, M., O’Regan, D., Peterson, A.: Dynamic equations on time scales: a survey. J. Comput. Appl. Math. 141(1–2), 1–26...
    • 15. Agarwal, R., Bohner, M.: Basic calculus on time scales and some of its applications. Resultate der Mathematik 35(1), 3–22 (1999)
    • 16. Castillo, S., Pinto, M.: Asymptotic behavior of functional dynamic equations in time scales. Dyn. Syst. Appl. 19, 165–168 (2010)
    • 17. Huy, N.: Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line. J. Funct. Anal. 235, 330–354...
    • 18. Sasu, B.: Uniform dichotomy and exponential dichotomy of evolution families on the half-line. J. Math. Anal. Appl. 323(2), 1465–1478 (2006)
    • 19. Mihail, M., Sasu, A.L., Sasu, B.: Theorems of Perron type for uniform exponential dichotomy of linear skew-product semiflows. Bull. Belgian...
    • 20. Naulin, R., Pinto, M.: Admissible perturbations of exponential dichotomy roughness. Nonlinear Anal. 31, 559–571 (1998)
    • 21. Pliss, V., Sell, G.: Robustness of exponential dichotomies in infinite-dimensional dynamical systems. J. Dyn. Differ. Equ. 11, 471–513...
    • 22. Popescu, L.: Exponential dichotomy roughness on Banach spaces. J. Math. Anal. Appl. 314, 436–454 (2006)
    • 23. Massera, J.L., Schaffer, J.J.: Linear differential equations and functional analysis, II. Equations with periodic coefficients. Ann. Math....
    • 24. Barreira, L., Claudia, V.: Robustness of nonuniform exponential dichotomies in Banach spaces. J. Differ. Equ 244(10), 2407–2447 (2008)
    • 25. Barreira, L., Valls, C.: Nonuniform exponential dichotomies and Lyapunov regularity. J. Dyn. Differ. Equ. 19, 215–241 (2007)
    • 26. Barreira, L., Valls, C.: A Grobman–Hartman theorem for nonuniformly hyperbolic dynamics. J. Differ. Equ. 228, 285–310 (2006)
    • 27. Zhang, J.M., Fan, M., Zhu, H.P.: Existence and roughness of exponential dichotomies of linear dynamic equations on time scales. Comput....
    • 28. Zhang, J.M., Fan, M., Zhu, H.P.: Necessary and sufficient criteria for the existence of exponential dichotomy on time scales. Comput....
    • 29. Zhang, J.M., Song, Y.J., Zhao, Z.T.: General exponential dichotomies on time scales and parameter dependence of roughness. In: Advances...
    • 30. Naulin, R., Pinto, M.: Roughness of (h, k)-dichotomies. J. Differ. Equ. 118, 20–35 (1995)
    • 31. Pinto, M.: Discrete dichotomies. Comput. Math. Appl. 28(1–3), 259–270 (1994)
    • 32. Naulin, R., Pinto, M.: Stability of discrete dichotomies for linear difference systems. J. Differ. Equ. Appl. 3(2), 101–123 (1997)
    • 33. Pinto, M., Sepúlveda, D.: H-asymptotic stability by fixed point in neutral nonlinear differential equations with delay. Nonlinear Anal....
    • 34. Dhama, S., Abbas, S., Pinto, M., Castillo, S., Tomar, S.: Weighted pseudo almost automorphic solution for abstract dynamic equations under...
    • 35. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. In: Applied Mathematical Sciences, vol. 44,...
    • 36. Palmer, K.J.: Exponential dichotomies and transversal homoclinic points. J. Diff. Equ. 55, 225–256 (1984)
    • 37. Barreira, L., Valls, C.: Smooth invariant manifolds in Banach spaces with nonuniform exponential dichotomy. J. Funct. Anal. 238, 118–148...
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