Open Access

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

Advances in Difference Equations20062006:097614

https://doi.org/10.1155/ADE/2006/97614

Received: 4 October 2005

Accepted: 1 November 2005

Published: 5 March 2006

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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Authors’ Affiliations

(1)
Departamento de Matemática, Universidade Federal de Pernambuco

References

  1. Arendt W, Bu S: The operator-valued Marcinkiewicz multiplier theorem and maximal regularity. Mathematische Zeitschrift 2002,240(2):311–343. 10.1007/s002090100384MathSciNetView ArticleMATHGoogle Scholar
  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. 10.1080/01630569908816889MathSciNetView ArticleMATHGoogle Scholar
  3. Beyn W-J, Lorenz J: Stability of viscous profiles: proofs via dichotomies. preprint, 2004Google Scholar
  4. Blunck S: Analyticity and discrete maximal regularity on L p -spaces. Journal of Functional Analysis 2001,183(1):211–230. 10.1006/jfan.2001.3740MathSciNetView ArticleMATHGoogle Scholar
  5. Blunck S: Maximal regularity of discrete and continuous time evolution equations. Studia Mathematica 2001,146(2):157–176. 10.4064/sm146-2-3MathSciNetView ArticleMATHGoogle Scholar
  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. 10.1080/10236190008808241MathSciNetView ArticleMATHGoogle Scholar
  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.2004.06.032MathSciNetView ArticleMATHGoogle Scholar
  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)00257-5MathSciNetView ArticleMATHGoogle Scholar
  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.MathSciNetView ArticleMATHGoogle Scholar
  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-XMathSciNetView ArticleMATHGoogle Scholar
  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. 10.1080/10236190290032499MathSciNetView ArticleMATHGoogle Scholar
  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. 10.1080/10236199908808167MathSciNetView ArticleMATHGoogle Scholar
  13. Hale JK, Kato J: Phase space for retarded equations with infinite delay. Funkcialaj Ekvacioj 1978,21(1):11–41.MathSciNetMATHGoogle Scholar
  14. Hino Y, Murakami S, Naito T: Functional-Differential Equations with Infinite Delay, Lecture Notes in Mathematics. Volume 1473. Springer, Berlin; 1991:x+317.MATHGoogle Scholar
  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-XMathSciNetView ArticleMATHGoogle Scholar
  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. 10.1080/10236190410001685021MathSciNetView ArticleMATHGoogle Scholar
  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)00296-4MathSciNetView ArticleMATHGoogle Scholar
  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.Google Scholar
  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.Google Scholar
  20. Weis L: Operator-valued Fourier multiplier theorems and maximal L p -regularity. Mathematische Annalen 2001,319(4):735–758. 10.1007/PL00004457MathSciNetView ArticleMATHGoogle Scholar

Copyright

© C. Cuevas and C. Vidal 2006

This article is published under license to BioMed Central Ltd. 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.