SpringerLink
Log in

LebesguE Regularity for Nonlocal Time-Discrete Equations with Delays

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

In this work we provide a new and effective characterization for the existence and uniqueness of solutions for nonlocal time-discrete equations with delays, in the setting of vector-valued Lebesgue spaces of sequences. This characterization is given solely in terms of the R-boundedness of the data of the problem, and in the context of the class of UMD Banach spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. Abadias, C. Lizama, Almost automorphic mild solutions to fractional partial difference-differential equations. Appl. Analysis 95, No 6 (2016), 1347–1369.

    MathSciNet  MATH  Google Scholar 

  2. L. Abadias, C. Lizama, P.J. Miana, M.P. Velasco, Cesáro sums and algebra homomorphisms of bounded operators. Israel J. Math. 216, No 1 (2016), 471–505.

    MathSciNet  MATH  Google Scholar 

  3. R.P. Agarwal, C. Cuevas, C. Lizama, Regularity of Difference Equations on Banach Spaces. Springer-Verlag, Cham (2014).

    MATH  Google Scholar 

  4. G. Akrivis, B. Li, C. Lubich, Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations. Math. Comp. 86, No 306 (2017), 1527–1552.

    MathSciNet  MATH  Google Scholar 

  5. H. Amann, Linear and Quasilinear Parabolic Problems. Ser. Monographs in Mathematics # 89, Birkhäuser, Basel (1995).

    Google Scholar 

  6. H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications. Math. Nachr. 186 (1997), 5–56.

    MathSciNet  MATH  Google Scholar 

  7. F.M. Atici, P.W. Eloe, A transform method in discrete fractional calculus. Int. J. Difference Equ. 2, No 2 (2007), 165–176.

    MathSciNet  Google Scholar 

  8. F.M. Atici, P.W. Eloe, Initial value problems in discrete fractional calculus. Proc. Amer. Math. Soc. 137, No 3 (2009), 981–989.

    MathSciNet  MATH  Google Scholar 

  9. S. Blünck, Maximal regularity of discrete and continuous time evolution equations. Studia Math. 146, No 2 (2001), 157–176.

    MathSciNet  MATH  Google Scholar 

  10. F. Borrelli, M. Baotic, A. Bemporad, M. Morari, Dynamic programming for constrained optimal control of discrete-time linear hybrid systems. Automatica 41 (2005), 1709–1721.

    MathSciNet  MATH  Google Scholar 

  11. D. Bothe, R. Denk, M. Hieber, R. Schnaubelt, G. Simonett, M. Wilke, R. Zacher, Special issue: Parabolic evolution equations, maximal regularity, and applications dedicated to Jan Prüss. J. Evol. Equ. 17, No 1 (2017), 1–15.

    MathSciNet  MATH  Google Scholar 

  12. S. Bu, Mild well-posedness of equations with fractional derivative. Math. Nachr. 285, No 2/3 (2012), 202–209.

    MathSciNet  MATH  Google Scholar 

  13. S. Bu, Well-posedness of fractional differential equations on vector-vauled function spaces. Integral Equations Operator Theory 71, No 2 (2001), 259–274.

    Google Scholar 

  14. J. Cermák, I. Gyóri, L. Nechvátal, L. On explicit stability conditions for a linear fractional difference system. Fract. Calc. Appl. Anal. 18, No 3 (2015), 651–672; 10.1515/fca-2015-0040; https://www.degruyter.com/view/j/fca.2015.18.issue-3/issue-files/fca.2015.18.issue-3.xml.

    MathSciNet  MATH  Google Scholar 

  15. R. Chill, S. Srivastava, Lq-maximal regularity for second order Cauchy problems. Math. Z. 251, No 4 (2005), 751–781.

    MathSciNet  MATH  Google Scholar 

  16. W. Chojnacki, On operator-valued cosine sequences on UMD spaces. Studia Math. 199, No 3 (2010), 267–278.

    MathSciNet  MATH  Google Scholar 

  17. C.W. Clark, A delayed-recruitment model of population dynamics, with an application to baleen whale populations. J. Math. Biol. 3 (1976), 381–391.

    MathSciNet  MATH  Google Scholar 

  18. S. Elaydi, An Introduction to Difference Equations. Springer, New York (2005).

    MATH  Google Scholar 

  19. X. Fu, M. Li, Spatial dynamics for lattice difference equations with a shifting habitat. J. Differential Equations 259 (2015), 1957–1989.

    Google Scholar 

  20. B. Kovács, B. Li, C. Lubich, A-stable time discretizations preserve maximal parabolic regularity. SIAM J. Numer. Anal. 54, No 6 (2016), 3600–3624.

    MathSciNet  MATH  Google Scholar 

  21. I. Gyóri, G. Ladas, Oscillation Theory of Delay Differential Equations. Clarendon, Oxford (1991).

    MATH  Google Scholar 

  22. T. Kemmochi, Discrete maximal regularity for abstract Cauchy problems. Studia Math. 234, No 3 (2016), 241–263.

    MathSciNet  MATH  Google Scholar 

  23. D. Leykekhman, B. Vexler, Discrete maximal parabolic regularity for Galerkin finite element methods. Numer. Math. 135, No 3 (2017), 923–952.

    MathSciNet  MATH  Google Scholar 

  24. B. Li, W. Sun, Maximal regularity of fully discrete finite element solutions of parabolic equations. SIAM J. Numer. Anal. 55, No 2 (2017), 521–542.

    MathSciNet  MATH  Google Scholar 

  25. Z.-M. Li, X.-H. Chang, K. Mathiyalagan, J. **ang, Robust energy-to-peak filtering for discrete-time nonlinear systems with measurement quantization. Signal Processing 139, No 1 (2017), 1339–1351.

    Google Scholar 

  26. C. Lizama, ℓp-maximal regularity for fractional difference equations on UMD spaces. Math. Nach. 288, No 17/18 (2015), 2079–2092.

    MathSciNet  MATH  Google Scholar 

  27. C. Lizama, M. Murillo-Arcila, ℓp-maximal regularity for a class of fractional difference equations on UMD spaces: The case 1 < α ≤ 2. Banach J. Math. Anal. 11, No 1 (2017), 188–206.

    MathSciNet  MATH  Google Scholar 

  28. C. Lizama, M. Murillo-Arcila, Maximal regularity in ℓp spaces for discrete time fractional shifted equations. J. Differential Equations 263, No 6 (2017), 3175–3196.

    MathSciNet  MATH  Google Scholar 

  29. C. Lizama, M. Murillo-Arcila, Well posedness for semidiscrete abstract fractional Cauchy problems with finite delay. J. Comput. Appl. Math. 339 (2018), 356–366.

    MathSciNet  MATH  Google Scholar 

  30. P. Portal, Discrete time analytic semigroups and the geometry of Banach spaces. Semigroup Forum 67, No 1 (2003), 125–144.

    MathSciNet  MATH  Google Scholar 

  31. P. Portal, Maximal regularity of evolution equations on discrete time scales. J. Math. Anal. Appl. 304, No 1 (2005), 1–12.

    MathSciNet  MATH  Google Scholar 

  32. W.J. Rugh, Linear System Theory, 2nd Ed. Prentice Hall, New York (1996).

    MATH  Google Scholar 

  33. P.A. Samuelson, Conditions that the roots of a polynomial be less than unity in absolute value. Ann. Math. Stat. 12, No 3 (1941), 360–364.

    MathSciNet  MATH  Google Scholar 

  34. E.H. Simsek, E. Gulec, H. Kavustu, Application of Kalman filter to determination of coal liquefaction mechanisms using discrete time models. Fuel 207, No 1 (2017), 814–820.

    Google Scholar 

  35. V.E. Tarasov, Fractional-order difference equations for physical lattices and some applications. J. Math. Phys. 56, No 10 (2015), 1–19.

    MathSciNet  MATH  Google Scholar 

  36. V.E. Tarasov, Fractional Liouville equation on lattice phase-space. Phys. A 421 (2015), 330–342.

    MathSciNet  MATH  Google Scholar 

  37. V.V. Tarasova, V.E. Tarasov, Logistic map with memory from economic model. Chaos Solitons Fractals 95 (2017), 84–91.

    MathSciNet  MATH  Google Scholar 

  38. L. Weis, Operator-valued Fourier multiplier theorems and maximal Lp-regularity. Math. Ann. 319 (2001), 735–758.

    MathSciNet  MATH  Google Scholar 

  39. G.C. Wu, D. Baleanu, Discrete chaos in fractional delayed logistic maps. Nonlinear Dynamics 80, No 4 (2016), 1697–1703.

    MathSciNet  Google Scholar 

  40. G.C. Wu, D. Baleanu, H-P. **e, Riesz Riemann-Liouville difference on discrete domains. Chaos 26, No 8 (2016), # 084308, 5pp.

    Google Scholar 

  41. G.C. Wu, D. Baleanu, Z.G. Deng, S.D. Zeng, Lattice fractional diffusion equation in terms of a Riesz-Caputo difference. Phys. A 438 (2015), 335–339.

    MathSciNet  MATH  Google Scholar 

  42. Z.X. Yu, Uniqueness of critical travelling waves for nonlocal lattice equations with delays. Proc. Amer. Math. Soc. 140, No 11 (2012), 3853–3859.

    MathSciNet  MATH  Google Scholar 

  43. A. Zygmund, Trigonometric Series, 2nd Ed., Vols. I, II. Cambridge University Press, New York (1959).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 173, Estación Central, Santiago, Chile

    Claudio Leal & Carlos Lizama

  2. Institut de Màtematiques i Aplicacions de Castelló (IMAC), Universitat Jaume I, Campus del Riu s/n, 12701, Castelló, Spain

    Marina Murillo-Arcila

Authors
  1. Claudio Leal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carlos Lizama
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marina Murillo-Arcila
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Claudio Leal.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Leal, C., Lizama, C. & Murillo-Arcila, M. LebesguE Regularity for Nonlocal Time-Discrete Equations with Delays. FCAA 21, 696–715 (2018). https://doi.org/10.1515/fca-2018-0037

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1515/fca-2018-0037

MSC 2010

Key Words and Phrases

Search

Navigation