PERIODIC SOLUTIONS OF A CLASS OF SECOND ORDER NEUTRAL DIFFERENTIAL EQUATIONS WITH MULTIPLE DIFFERENT DELAYS

Rabah Khemis     (Laboratory of Applied Mathematics and History and Didactics of Mathematics (LAMAHIS), University of 20 August 1955, B.P.26 route d’El-Hadaiek, Skikda, 21000, Algeria)
Abdelouaheb Ardjouni     (Department of Mathematics and Informatics, Souk Ahras University, Souk Ahras, 41000, Algeria)
Ahlème Bouakkaz     (Laboratory of Applied Mathematics and History and Didactics of Mathematics (LAMAHIS), University of 20 August 1955, B.P.26 route d’El-Hadaiek, Skikda, 21000, Algeria)

Abstract


The present work mainly probes into the existence and uniqueness of periodic solutions for a class of second-order neutral differential equations with multiple delays. Our approach is based on using Banach and  Krasnoselskii's fixed point theorems as well as the Green's function method. Besides, two examples are exhibited to validate the effectiveness of our findings which complement and extend some relevant ones in the literature.


Keywords


Fixed point theorem, Green's function, Neutral differential equation, Periodic solutions

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References


REFERENCES

  1. Ardjouni A., Djoudi A. Existence of positive periodic solutions for a second-order nonlinear neutral differential equation with variable delay. Adv. Nonlinear Anal., 2013. Vol. 2, No. 2. P. 151–161. DOI: 10.1515/anona-2012-0024
  2. Ardjouni A., Djoudi A. Periodic solutions for a second-order nonlinear neutral differential equation with variable delay. Electron. J. Differ. Equ., 2011. Vol. 2011, No. 128. P. 1–7. URL: https://ejde.math.txstate.edu/Volumes/2011/128/ardjouni.pdf
  3. Bouakkaz A., Ardjouni A., Djoudi A. Existence of positive periodic solutions for a second-order nonlinear neutral differential equation by the Krasnoselskii’s fixed point theorem. Nonlinear Dyn. Syst. Theory, 2017. Vol. 17, No. 3. P. 230–238. URL: https://e-ndst.kiev.ua/v17n3/2(60).pdf
  4. Bouakkaz A., Ardjouni A., Djoudi A. Periodic solutions for a second order nonlinear functional differential equation with iterative terms by Schauder’s fixed point theorem. Acta Math. Univ. Comen., 2018. Vol. 87, No. 2. P. 223–235. URL: http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/671/621
  5. Bouakkaz A., Khemis R. Positive periodic solutions for a class of second-order differential equations with state-dependent delays. Turkish J. Math., 2020. Vol. 44, No. 4. P. 1412–1426. DOI: 10.3906/mat-2004-52
  6. Burton T.A. Stability by Fixed Point Theory for Functional Differential Equations. Dover Publications, New York, 2006. 368 p.
  7. Guerfi A., Ardjouni A. Periodic solutions for second order totally nonlinear iterative differential equations. J. Anal., 2022. Vol. 30. P. 353–367. DOI: 10.1007/s41478-021-00347-0
  8. Kaufmann E.R. A nonlinear neutral periodic differential equation. Electron. J. Differ. Equ., 2010. Vol. 2010, No. 88. P. 1–8. URL: https://ejde.math.txstate.edu/Volumes/2010/88/kaufmann.pdf
  9. Khemis R., Ardjouni A., Djoudi A. Existence of periodic solutions for a second-order nonlinear neutral differential equation by the Krasnoselskii’s fixed point technique. Matematiche, 2017. Vol. 72, No. 1. P. 145–156. DOI: 10.4418/2017.72.1.11
  10. Liu Y., Ge W. Positive periodic solutions of nonlinear Duffing equations with delay and variable coefficients. Tamsui Oxf. J. Math. Sci., 2004. Vol. 20, No. 2. P. 235–255. 
  11. Smart D.S. Fixed Point Theorems. Cambridge, UK: Cambridge Univ. Press, 1980. 104 p.
  12. Wang Y., Lian H., Ge W. Periodic solutions for a second order nonlinear functional differential equation. Appl. Math. Lett., 2007. Vol. 20, No. 1. P. 110–115. DOI: 10.1016/j.aml.2006.02.028
  13. Yankson E. Positive periodic solutions for second-order neutral differential equations with functional delay. Electron. J. Differ. Equ., 2012. Vol. 2012, No. 14. P. 1–6. URL: http://ejde.math.txstate.edu/Volumes/2012/14/yankson.pdf
  14. Zeidler E. Applied Functional Analysis. Springer-Verlag, New York, 1995. 481 p. DOI: 10.1007/978-1-4612-0815-0




DOI: http://dx.doi.org/10.15826/umj.2022.2.006

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