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Periodic solutions of a class of third-order functional differential equations with iterative source terms

  • Ahlème Bouakkaz
  • Abdelouaheb ArdjouniEmail author
  • Rabah Khemis
  • Ahcene Djoudi
Original Article

Abstract

This work deals with a class of third-order differential equations with iterative source terms. Some new results on the existence, uniqueness and continuous dependence of periodic solution for this class are established by virtue of Krasnoselskii’s and Banach’s fixed point theorems and some useful properties of Green’s function. Finally, we present an example to illustrate the effectiveness of our results.

Keywords

Periodic solutions Iterative differential equations Fixed point theorem Green’s function 

Mathematics Subject Classification

Primary 34K13 34A34 Secondary 34K30 34L30 

Notes

Acknowledgements

The authors would like to thank the anonymous referee for his/her valuable comments and good advice.

References

  1. 1.
    Abel, N.H.: Oeuvres compétes. Christiana I I, 36–39 (1881)Google Scholar
  2. 2.
    Andrzej, P.: On some iterative-differential equation I, Zeszyty Naukowe UJ. Prace Mat. 12, 53–56 (1968)MathSciNetGoogle Scholar
  3. 3.
    Andrzej, P.: On some iterative-differential equations II, Zeszyty Naukowe UJ. Prace Mat. 13, 49–51 (1969)Google Scholar
  4. 4.
    Andrzej, P.: On some iterative-differential equations III, Zeszyty Naukowe UJ. Prace Mat. 15, 125–1303 (1971)Google Scholar
  5. 5.
    Ardjouni, A., Djoudi, A.: Existence of positive periodic solutions for a nonlinear neutral differential equations with variable delay. Appl. Math. E-Notes 12, 94–101 (2012)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Babbage, C.: An essay towards the calculus of functions. Philos. Trans. R. Soc. Lond. 105, 389–432 (1815)CrossRefGoogle Scholar
  7. 7.
    Bouakkaz, A., Ardjouni, A., Djoudi, A.: Periodic solutions for a nonlinear iterative functional differential equation. Electron. J. Math. Anal. Appl. 7(1), 156–166 (2019)zbMATHGoogle Scholar
  8. 8.
    Bouakkaz, A., Ardjouni, A., Djoudi, A.: Periodic solutions for a second order nonlinear functional differential equation with iterative terms by Schauder fixed point theorem. Acta Math. Univ. Comen. LXXXVII(2), 223–235 (2018)Google Scholar
  9. 9.
    Chen, Y., Ren, J., Siegmund, S.: Green’s function for third-order differential equations. Rocky Mt. J. Math. 41(5), 1417–1448 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cheng, Z., Ren, J.: Existence of positive periodic solution for variable-coefficient third order differential equation with singularity. Math. Methods Appl. Sci. 37, 2281–2289 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Egri, E., Rus, I.A.: First order iterative functional-differential equation with parameter. Stud. Univ. Babes-Bolyai Math. 52(4), 67–80 (2007)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Fite, W.B.: Properties of the solutions of certain functional fifferential equations. Trans. Am. Math. Soc. 22, 311–319 (1921)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kaufmann, E.R.: Existence and uniqueness of solutions for a second-order iterative boundary-value problem functional differential equation. Electron. J. Differ. Equ. 2018(150), 1–6 (2018)Google Scholar
  14. 14.
    Liu, Y., Ge, W.: Positive periodic solutions of nonlinear Duffing equations with delay and variable coefficients. Tamsui Oxf. J. Math. Sci. 20, 235–255 (2004)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ren, J., Siegmund, S., Chen, Y.: Positive periodic solutions for third-order nonlinear differential equations. Electron. J. Differ. Equ. 2011(66), 1–19 (2011)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Schröder, E.: Über iterate funktionen. Math. Ann. 3, 295–322 (1871)Google Scholar
  17. 17.
    Smart, D.S.: Fixed point theorems. In: Cambridge Tracts in Mathematics, No. \(66\). Cambridge University Press, London (1974)Google Scholar
  18. 18.
    Wang, Y., Lian, H., Ge, W.: Periodic solutions for a second order nonlinear functional differential equation. Appl. Math. Lett. 20, 110–115 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zhang, P.: Analytic solutions for iterative functional-differential equations. Electron. J. Differ. Equ. 2012(180), 1–7 (2012)MathSciNetGoogle Scholar
  20. 20.
    Zhao, H.Y., Fečkan, M.: Periodic solutions for a class of differential equations with delays depending on state. Math. Commun. 22, 1–14 (2017)MathSciNetGoogle Scholar
  21. 21.
    Zhao, H.Y., Liu, J.: Periodic solutions of an iterative functional differential equation with variable coefficients. Math. Methods Appl. Sci. 40, 286–292 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Sociedad Matemática Mexicana 2019

Authors and Affiliations

  • Ahlème Bouakkaz
    • 1
  • Abdelouaheb Ardjouni
    • 2
    • 3
    Email author
  • Rabah Khemis
    • 1
  • Ahcene Djoudi
    • 3
  1. 1.Laboratoire LAMAHIS, Département de MathématiquesUniversité 20 Août 1955SkikdaAlgeria
  2. 2.Faculty of Sciences and Technology, Department of Mathematics and InformaticsUniversity of Souk AhrasSouk AhrasAlgeria
  3. 3.Applied Mathematics Lab, Faculty of Sciences, Department of MathematicsUniversity of AnnabaAnnabaAlgeria

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