Skip to main content
SpringerLink
Log in

Periodicity and continuous dependence in iterative differential equations

  • Published:
Rendiconti del Circolo Matematico di Palermo Series 2 Aims and scope Submit manuscript

Abstract

In this work, we study the existence, uniqueness and continuous dependence of periodic solutions of the iterative differential equation

$$\begin{aligned} x^{\prime }(t)=\sum \limits _{m=1}^{N}\sum \limits _{l=1}^{\infty }C_{l,m}(t)\left( x^{[m]}(t)\right) ^{l}+\frac{d}{dt}g\left( t,x^{[1]}(t),x^{[2]} (t),\ldots ,x^{[N]}(t)\right) +h(t). \end{aligned}$$

Using Schauder’s fixed point theorem, we obtain the existence of periodic solution and by the contraction mapping principle we obtain the uniqueness. An example is given to illustrate this work. The results obtained here extend the work of Zhao and Feckan [14].

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.

Similar content being viewed by others

References

  1. Cabada, A.: Green’s Functions in the Theory of Ordinary Differential Equations. Springer, New York (2014)

    Book  Google Scholar 

  2. Eder, E.: The functional differential equation \(x^{\prime }(t)=x(x(t))\). J. Differ. Equ. 54, 390–400 (1984)

    Article  Google Scholar 

  3. Feckan, M.: On certain type of functional differential equations. Math. Slovaca 43, 39–43 (1993)

    MathSciNet  MATH  Google Scholar 

  4. Kendre, S.D., Kharat, V.V., Narute, R.: On existence of solution for iterative integrodifferential equations. Nonlinear Anal. Differ. Equ. 3, 123–131 (2015)

    Article  Google Scholar 

  5. Liua, B., Tunc, C.: Pseudo almost periodic solutions for a class of first order differential iterative equations. Appl. Math. Lett. 40, 29–34 (2015)

    Article  MathSciNet  Google Scholar 

  6. Si, J.G., Wang, X.P.: Analytic solutions of a second-order iterative functional differential equation. Comput. Math. Appl. 43, 81–90 (2002)

    Article  MathSciNet  Google Scholar 

  7. Smart, D.R.: Fixed Point Theorems. Cambridge University Press, Cambridge (1980)

    MATH  Google Scholar 

  8. Stanek, S.: Global properties of solutions of the functional differential equations \(x(t)x^{\prime }(t)=Kx(x(t))\),\( 0<\left|K\right|<1\). Funct. Differ. Equ. 9, 527–550 (2002)

    MathSciNet  MATH  Google Scholar 

  9. Tunc, C.: On the existence of periodic solutions of functional differential equations of the third order. Appl. Comput. Math. 15(2), 189–199 (2016)

    MathSciNet  MATH  Google Scholar 

  10. Tunc, C., Erdur, S.: New qualitative results for solutions of functional differential equations of second order. Discrete Dyn. Nat. Soc. 2018, 1–13 (2018)

    Article  MathSciNet  Google Scholar 

  11. Wang, K.: On the equation \(x^{\prime }(t)=f(x(x(t)))\). Funkc. Ekvacioj 33, 405–425 (1990)

    MATH  Google Scholar 

  12. Wang, X.P., Si, J.G.: Analytic solutions of an iterative functional differential equation. J. Math. Anal. Appl. 262, 490–498 (2001)

    Article  MathSciNet  Google Scholar 

  13. Yang, D., Zhang, W.: Solutions of equivariance for iterative differential equations. Appl. Math. Lett. 17, 759–765 (2004)

    Article  MathSciNet  Google Scholar 

  14. Zhao, H.Y., Feckan, M.: Periodic solutions for a class of differential equations with delays depending on state. Math. Commun. 23, 29–42 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Zhao, H.Y., Liu, J.: Periodic solutions of an iterative functional differential equation with variable coefficients. Math. Methods Appl. Sci 40, 286–292 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referee for his valuable comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, University of Annaba, P.O. Box 12, 23000, Annaba, Algeria

    Bouzid Mansouri

  2. Department of Mathematics and Informatics, Faculty of Sciences and Technology, University of Souk Ahras, P.O. Box 1553, 41000, Souk Ahras, Algeria

    Abdelouaheb Ardjouni

  3. Applied Mathematics Lab, Department of Mathematics, Faculty of Sciences, University of Annaba, P.O. Box 12, 23000, Annaba, Algeria

    Abdelouaheb Ardjouni & Ahcene Djoudi

Authors
  1. Bouzid Mansouri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abdelouaheb Ardjouni
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ahcene Djoudi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abdelouaheb Ardjouni.

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

Mansouri, B., Ardjouni, A. & Djoudi, A. Periodicity and continuous dependence in iterative differential equations. Rend. Circ. Mat. Palermo, II. Ser 69, 561–576 (2020). https://doi.org/10.1007/s12215-019-00420-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-019-00420-5

Keywords

Mathematics Subject Classification

Search

Navigation