Article ID: MTJPAM-D-21-00038

Title: Variational approach for a class of delay second-order differential equations


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-21-00038; Volume 5 / Issue 1 / Year 2023, Pages 71-77

Document Type: Research Paper

Author(s): Fatima Dib a , Mustapha Yebdri b , Naima Daoudi-Merzagui c

aDepartment of Sciences and Technology, University Center of Maghnia, 13300 Maghnia, Algeria – Laboratory of Dynamical Systems and Applications, University of Tlemcen

bLaboratory of Dynamical Systems and Applications, University of Tlemcen

cDepartment of Mathematics, University of Tlemcen, 13000 Tlemcen, Algeria

Received: 3 June 2021, Accepted: 1 June 2023, Published: 21 July 2023

Corresponding Author: Fatima Dib (Email address: fatimadib1967@yahoo.fr)

Full Text: PDF

Abstract

We consider a class of nonautonomous second-order delay differential equations. By means of variational approach, we prove the existence of at least one periodic solution.

Keywords: Differential equation, periodic solution, variational method

References:
  1. M. Ayachi and J. Blot, Variational methods for almost periodic solutions of a class of neutral delay equations, Abstr. Appl. Anal. 2008, 2008; Article ID: 153285.
  2. L. Bai, J. J. Nieto and J. M. Uzal, On a delayed epidemic model with non-instantaneous impulses, Commun. Pure Appl. Anal. 19 (4), 1915–1930, 2020.
  3. H. Brezis, Analyse fonctionnelle, théorie et applications, Masson, Paris, 1983.
  4. N. Daoudi-Merzagui and F. Dib, Positive periodic solutions to impulsive delay differential equations, Turkish J. Math. 41 (4), 969–982, 2017.
  5. F. Dib and N. Daoudi-Merzagui, Multiple periodic solutions for second order impulsive delay differential equations, Int. J. Dyn. Syst. Differ. Equ. 9 (3), 298–312, 2019.
  6. L. C. Evans, Partial differential equations, American Mathematical Society, Graduate Studies in Mathematics, 19, 1998.
  7. C. Guo and Z. Guo, Existence of multiple periodic solutions for a class of second-order delay differential equations, Nonlinear Anal., Real World Appl. 10 (5), 3285–3297, 2009.
  8. Z. Guo and J. Yu, Multiplicity results for periodic solutions to delay differential equations via critical point theory, J. Differential Equations 218 (1), 15–35, 2005.
  9. J. Hale and S. M. Verduyn Lunel, Introduction to functional differential equations, Springer-Verlag, New York, 1993.
  10. O. Kavian, Introduction à la théorie des points critiques et application aux problèmes elliptiques, Springer-Verlag, France, Paris, 1993.
  11. J. Mawhin and M. Willem, Critical point theory and Hamiltonian systems, Springer-Verlag, Berlin, 1989.
  12. J. J. Nieto and D. O’Regan, Variational approach to impulsive differential equations, Nonlinear Anal., Real World Appl. 10 (2), 680–690, 2009.
  13. J. J. Nieto and J. M. Uzal, Nonlinear second-order impulsive differential problems with dependence on the derivative via variational structure, J. Fixed Point Theory Appl. 22, 2020; Article ID: 19.
  14. X-B. Shu and Y-T. Xu, Multiple periodic solutions to class of second-order nonlinear mixed-type functional differential equations, Int. J. Math. Math. Sci. 2005 (17), 2689–2702, 2005; Article ID: 851685.
  15. M. Struwe, Variational methods applications to nonlinear partial differential equations and Hamiltonian systems, Springer-Verlag, Berlin Heidelberg, 2000.
  16. G. Q. Wang and J. R. Yan, Existence of periodic solutions for second-order nonlinear neutral delay equations, Acta Math. Sinica 47 (2), 379–384, 2004.