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A note on discrete maximal regularity for functional difference equations with infinite delay

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Abstract

Using exponential dichotomies, we get maximal regularity for retarded functional difference equations. Applications on Volterra difference equations with infinite delay are shown.

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References

  1. 1.

    Arendt W, Bu S: The operator-valued Marcinkiewicz multiplier theorem and maximal regularity. Mathematische Zeitschrift 2002,240(2):311–343. 10.1007/s002090100384

  2. 2.

    Beyn W-J, Lorenz J: Stability of traveling waves: dichotomies and eigenvalue conditions on finite intervals. Numerical Functional Analysis and Optimization 1999,20(3–4):201–244. 10.1080/01630569908816889

  3. 3.

    Beyn W-J, Lorenz J: Stability of viscous profiles: proofs via dichotomies. preprint, 2004

  4. 4.

    Blunck S: Analyticity and discrete maximal regularity on L p -spaces. Journal of Functional Analysis 2001,183(1):211–230. 10.1006/jfan.2001.3740

  5. 5.

    Blunck S: Maximal regularity of discrete and continuous time evolution equations. Studia Mathematica 2001,146(2):157–176. 10.4064/sm146-2-3

  6. 6.

    Cuevas C: Weighted convergent and bounded solutions of Volterra difference systems with infinite delay. Journal of Difference Equations and Applications 2000,6(4):461–480. 10.1080/10236190008808241

  7. 7.

    Cuevas C, Del Campo L: An asymptotic theory for retarded functional difference equations. Computers & Mathematics with Applications 2005,49(5–6):841–855. 10.1016/j.camwa.2004.06.032

  8. 8.

    Cuevas C, Pinto M: Asymptotic behavior in Volterra difference systems with unbounded delay. Journal of Computational and Applied Mathematics 2000,113(1–2):217–225. 10.1016/S0377-0427(99)00257-5

  9. 9.

    Cuevas C, Pinto M: Asymptotic properties of solutions to nonautonomous Volterra difference systems with infinite delay. Computers & Mathematics with Applications 2001,42(3–5):671–685.

  10. 10.

    Cuevas C, Pinto M: Convergent solutions of linear functional difference equations in phase space. Journal of Mathematical Analysis and Applications 2003,277(1):324–341. 10.1016/S0022-247X(02)00570-X

  11. 11.

    Cuevas C, Vidal C: Discrete dichotomies and asymptotic behavior for abstract retarded functional difference equations in phase space. Journal of Difference Equations and Applications 2002,8(7):603–640. 10.1080/10236190290032499

  12. 12.

    Elaydi S, Murakami S, Kamiyama E: Asymptotic equivalence for difference equations with infinite delay. Journal of Difference Equations and Applications 1999,5(1):1–23. 10.1080/10236199908808167

  13. 13.

    Hale JK, Kato J: Phase space for retarded equations with infinite delay. Funkcialaj Ekvacioj 1978,21(1):11–41.

  14. 14.

    Hino Y, Murakami S, Naito T: Functional-Differential Equations with Infinite Delay, Lecture Notes in Mathematics. Volume 1473. Springer, Berlin; 1991:x+317.

  15. 15.

    Kolmanovskii VB, Castellanos-Velasco E, Torres-Muñoz JA: A survey: stability and boundedness of Volterra difference equations. Nonlinear Analysis 2003,53(7–8):861–928. 10.1016/S0362-546X(03)00021-X

  16. 16.

    Matsunaga H, Murakami S: Some invariant manifolds for functional difference equations with infinite delay. Journal of Difference Equations and Applications 2004,10(7):661–689. 10.1080/10236190410001685021

  17. 17.

    Murakami S: Representation of solutions of linear functional difference equations in phase space. Nonlinear Analysis. Theory, Methods & Applications 1997,30(2):1153–1164. 10.1016/S0362-546X(97)00296-4

  18. 18.

    Murakami S: Some spectral properties of the solution operator for linear Volterra difference systems. In New Developments in Difference Equations and Applications (Taipei, 1997). Gordon and Breach, Amsterdam; 1999:301–311.

  19. 19.

    Weis L: A new approach to maximal L p -regularity. In Evolution Equations and Their Applications in Physical and Life Sciences (Bad Herrenalb, 1998), Lecture Notes in Pure and Appl. Math.. Volume 215. Marcel Dekker, New York; 2001:195–214.

  20. 20.

    Weis L: Operator-valued Fourier multiplier theorems and maximal L p -regularity. Mathematische Annalen 2001,319(4):735–758. 10.1007/PL00004457

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Correspondence to Claudio Cuevas.

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Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Functional Equation