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Abstract and Applied Analysis
Volume 2012, Article ID 860716, 35 pages
http://dx.doi.org/10.1155/2012/860716
Research Article

On -Gevrey Asymptotics for Singularly Perturbed -Difference-Differential Problems with an Irregular Singularity

1Facultad de Ciencias, Universidad de Valladolid, Calle del Doctor Mergelina s/n, 47011 Valladolid, Spain
2UFR de Mathématiques, Université Lille 1, Cité Scientifique M2, 59655 Villeneuve d'Ascq Cedex, France

Received 30 September 2011; Revised 28 November 2011; Accepted 28 November 2011

Academic Editor: Allan C. Peterson

Copyright © 2012 Alberto Lastra and Stéphane Malek. 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.

Linked References

  1. S. Malek, “On complex singularity analysis for linear q-difference-differential equations,” Journal of Dynamical and Control Systems, vol. 15, no. 1, pp. 83–98, 2009. View at Publisher · View at Google Scholar
  2. S. Malek, “On functional linear partial differential equations in Gevrey spaces of holomorphic functions,” Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série 6, vol. 16, no. 2, pp. 285–302, 2007. View at Google Scholar
  3. A. Lastra, S. Malek, and J. Sanz, “On q-asymptotics for q-difference-differential equations with Fuchsian and irregular singularities,” preprint.
  4. S. Malek, “On singularly perturbed q-difference-differential equations with irregular singularity,” Journal of Dynamical and Control Systems, vol. 17, no. 2, pp. 243–271, 2011. View at Publisher · View at Google Scholar
  5. M. Canalis-Durand, J. Mozo-Fernández, and R. Schäfke, “Monomial summability and doubly singular differential equations,” Journal of Differential Equations, vol. 233, no. 2, pp. 485–511, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. S. Malek, “On the summability of formal solutions for nonlinear doubly singular partial differential equations,” Journal of Dynamical and Control Systems, vol. 18, no. 1, 2012. View at Publisher · View at Google Scholar
  7. C. Zhang, “Transformations de q-Borel-Laplace au moyen de la fonction thêta de Jacobi,” Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, vol. 331, no. 1, pp. 31–34, 2000. View at Publisher · View at Google Scholar
  8. W. Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations, Springer, Berlin, Germany, 2000. View at Zentralblatt MATH
  9. O. Costin, Asymptotics and Borel Summability, vol. 141 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, CRC Press, Boca Raton, Fla, USA, 2009.
  10. L. Di Vizio, J.-P. Ramis, J. Sauloy, and C. Zhang, “Équations aux q-différences,” Gazette des Mathématiciens, no. 96, pp. 20–49, 2003. View at Google Scholar
  11. L. Di Vizio and C. Zhang, “On q-summation and confluence,” Université de Grenoble. Annales de l'Institut Fourier, vol. 59, no. 1, pp. 347–392, 2009. View at Publisher · View at Google Scholar
  12. J. Chaumat and A.-M. Chollet, “Surjectivité de l'application restriction à un compact dans des classes de fonctions ultradifférentiables,” Mathematische Annalen, vol. 298, no. 1, pp. 7–40, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. J. Bonet, R. W. Braun, R. Meise, and B. A. Taylor, “Whitney's extension theorem for nonquasianalytic classes of ultradifferentiable functions,” Studia Mathematica, vol. 99, no. 2, pp. 155–184, 1991. View at Google Scholar · View at Zentralblatt MATH
  14. J. Sanz, “Linear continuous extension operators for Gevrey classes on polysectors,” Glasgow Mathematical Journal, vol. 45, no. 2, pp. 199–216, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. V. Thilliez, “Division by flat ultradifferentiable functions and sectorial extensions,” Results in Mathematics, vol. 44, no. 1-2, pp. 169–188, 2003. View at Google Scholar · View at Zentralblatt MATH
  16. A. Lastra and J. Sanz, “Extension operators in Carleman ultraholomorphic classes,” Journal of Mathematical Analysis and Applications, vol. 372, no. 1, pp. 287–305, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. J.-P. Ramis, J. Sauloy, and C. Zhang, “Local analytic classification of q-difference equations,” preprint.
  18. B. Malgrange, “Travaux d'Écalle et de Martinet-Ramis sur les systèmes dynamiques,” in Bourbaki N. Seminar, vol. 92, pp. 59–73, Société Mathématique de France, Paris, France, 1982. View at Google Scholar · View at Zentralblatt MATH
  19. J.-M. Kantor, “Classes non-quasi analytiques et décomposition des supports des ultradistributions,” Anais da Academia Brasileira de Ciências, vol. 44, pp. 171–180, 1972. View at Google Scholar · View at Zentralblatt MATH
  20. B. Malgrange, Ideals of Differentiable Functions, vol. 3 of Tata Institute of Fundamental Research Studies in Mathematics, Oxford University Press, London, UK, 1966.
  21. R. Narasimhan and Y. Nievergelt, Complex Analysis in One Variable, Birkhäuser Boston, Boston, Mass, USA, 2nd edition, 2001.