Skip to main content
SpringerLink
Log in

Summability of Formal Solutions for Some Generalized Moment Partial Differential Equations

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

The concept of moment differentiation is extended to the class of moment summable functions, giving rise to moment differential properties. The main result leans on accurate upper estimates for the integral representation of the moment derivatives of functions under exponential-like growth at infinity, and appropriate deformation of the integration paths. The theory is applied to obtain summability results of certain family of generalized linear moment partial differential equations with variable coefficients.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Balser, W.: Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations, Universitext. Springer, New York (2000)

    MATH  Google Scholar 

  2. Balser, W., Yoshino, M.: Gevrey order of formal power series solutions of inhomogeneous partial differential equations with constant coefficients. Funkcial. Ekvac. 53, 411–434 (2010)

    Article  MathSciNet  Google Scholar 

  3. Chaumat, J., Chollet, A.M.: Surjectivité de l’application restriction à un compact dans des classes de fonctions ultradifférentiables. Math. Ann. 298(1), 7–40 (1994)

    Article  MathSciNet  Google Scholar 

  4. Dreyfus, T., Lastra, A., Malek, S.: Multiple-scale analysis for some linear partial q-difference and differential equations with holomorphic coefficients. Adv. Differ. Equ. 2019, 326 (2019)

    Article  MathSciNet  Google Scholar 

  5. Goldberg, A.A., Ostrovskii, I.V.: Value Distribution of Meromorphic Functions, Translation of Mathematical Monograph, vol. 236. American Mathematical Society, Providence (2008)

    Book  Google Scholar 

  6. Ichinobe, K., Adachi, S.: On \(k\)-summability of formal solutions to the Cauchy problems for some linear \(q\)-difference-differential equations, complex differential and difference equations. In: De Gruyter Proceedings in Mathematics, pp. 447–464 (2020)

  7. Immink, G.K.: Exact asymptotics of nonlinear difference equations with levels 1 and 1+. Ann. Fac. Sci. Toulouse 2, 309–356 (2008)

    Article  MathSciNet  Google Scholar 

  8. Immink, G.K.: Accelero-summation of the formal solutions of nonlinear difference equations. Ann. Inst. Fourier (Grenoble) 61(1), 1–51 (2011)

    Article  MathSciNet  Google Scholar 

  9. Jiménez-Garrido, J., Kamimoto, S., Lastra, A., Sanz, J.: Multisummability in Carleman ultraholomorphic classes by means of nonzero proximate orders. J. Math. Anal. Appl. 472(1), 627–686 (2019)

    Article  MathSciNet  Google Scholar 

  10. Jiménez-Garrido, J., Sanz, J., Schindl, G.: Log-convex sequences and nonzero proximate orders. J. Math. Anal. Appl. 448(2), 1572–1599 (2017)

    Article  MathSciNet  Google Scholar 

  11. Jiménez-Garrido, J., Sanz, J., Schindl, G.: Injectivity and surjectivity of the asymptotic Borel map in Carleman ultraholomorphic classes. J. Math. Anal. Appl. 469, 136–168 (2019)

    Article  MathSciNet  Google Scholar 

  12. Lastra, A., Malek, S.: On multiscale Gevrey and Gevrey asymptotics for some linear difference differential initial value Cauchy problems. J. Differ. Equ. Appl. 23(8), 1397–1457 (2017)

    Article  MathSciNet  Google Scholar 

  13. Lastra, A., Malek, S., Sanz, J.: Summability in general Carleman ultraholomorphic classes. J. Math. Anal. Appl. 430, 1175–1206 (2015)

    Article  MathSciNet  Google Scholar 

  14. Lastra, A., Malek, S., Sanz, J.: Strongly regular multi-level solutions of singularly perturbed linear partial differential equations. Results. Math. 70, 581–614 (2016)

    Article  MathSciNet  Google Scholar 

  15. Lastra, A., Michalik, S., Suwińska, M.: Estimates of formal solutions for some generalized moment partial differential equations (2020) (Submitted). arXiv:1911.11998

  16. Michalik, S.: Summability and fractional linear partial differential equations. J. Dyn. Control Syst. 16(4), 557–584 (2010)

    Article  MathSciNet  Google Scholar 

  17. Michalik, S.: Analytic solutions of moment partial differential equations with constant coefficients. Funkcial. Ekvac. 56, 19–50 (2013)

    Article  MathSciNet  Google Scholar 

  18. Michalik, S.: Summability of formal solutions of linear partial differential equations with divergent initial data. J. Math. Anal. Appl. 406, 243–260 (2013)

    Article  MathSciNet  Google Scholar 

  19. Michalik, S.: Analytic summable solutions of inhomogeneous moment partial differential equations. Funkcial. Ekvac. 60, 325–351 (2017)

    Article  MathSciNet  Google Scholar 

  20. Michalik, S., Suwińska, M.: Gevrey estimates for certain moment partial differential equations, complex differential and difference equations. In: De Gruyter Proceedings in Mathematics, pp. 391–408 (2020)

  21. Michalik, S., Tkacz, B.: The Stokes phenomenon for some moment partial differential equations. J. Dyn. Control Syst. 25, 573–598 (2019)

    Article  MathSciNet  Google Scholar 

  22. Petzsche, H.-J.: On E. Borel’s theorem. Math. Ann. 282(2), 299–313 (1988)

    Article  MathSciNet  Google Scholar 

  23. Remy, P.: Gevrey order and summability of formal series solutions of some classes of inhomogeneous linear partial differential equations with variable coefficients. J. Dyn. Control Syst. 22, 693–711 (2016)

    Article  MathSciNet  Google Scholar 

  24. Sanz, J.: Flat functions in Carleman ultraholomorphic classes via proximate orders. J. Math. Anal. Appl. 415(2), 623–643 (2014)

    Article  MathSciNet  Google Scholar 

  25. Sanz, J.: Asymptotic Analysis and Summability of Formal Power Series, Analytic, Algebraic and Geometric Aspects of Differential Equations, Trends Mathematics, pp. 199–262. Springer, Cham (2017)

    Google Scholar 

  26. Suwińska, M.: Gevrey estimates of formal solutions for certain moment partial differential equations with variable coefficients (2020) Submitted. arXiv:2002.02845

  27. Thilliez, V.: Division by flat ultradifferentiable functions and sectorial extensions. Results Math. 44, 169–188 (2003)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the anonymous referee for the suggestions made, and the constructive remarks on the promising future directions to be considered.

Author information

Authors and Affiliations

  1. Departamento de Física y Matemáticas, University of Alcalá, Ap. de Correos 20, E-28871 Alcalá de Henares, Madrid, Spain

    Alberto Lastra

  2. Faculty of Mathematics and Natural Sciences, College of Science, Cardinal Stefan Wyszyński University, Wóycickiego 1/3, 01-938, Warszawa, Poland

    Sławomir Michalik & Maria Suwińska

Authors
  1. Alberto Lastra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sławomir Michalik
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Maria Suwińska
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alberto Lastra.

Ethics declarations

Conflict of interest

The authors declare no conflicts of interest.

Funding

A. Lastra is partially supported by the project PID2019-105621GB-I00 of Ministerio de Ciencia e Innovación, Spain, and by Dirección General de Investigación e Innovación, Consejería de Educación e Investigación of the Comunidad de Madrid (Spain), and Universidad de Alcalá under grant CM/JIN/2019-010, Proyectos de I+D para Jóvenes Investigadores de la Universidad de Alcalá 2019.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lastra, A., Michalik, S. & Suwińska, M. Summability of Formal Solutions for Some Generalized Moment Partial Differential Equations. Results Math 76, 22 (2021). https://doi.org/10.1007/s00025-020-01324-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-020-01324-y

Keywords

Mathematics Subject Classification

Search

Navigation