EXISTENCE AND EXPONENTIAL STABILITY OF POSITIVE PERIODIC SOLUTIONS FOR SECOND-ORDER DYNAMIC EQUATIONS

Faycal Bouchelaghem     (High School of Management Sciences Annaba, Annaba 23000, Algeria)
Abdelouaheb Ardjouni     (Department of Mathematics and Informatics, Souk Ahras University, P.O. Box 1553, Souk Ahras, 41000, Algeria)
Ahcene Djoudi     (Department of Mathematics, Annaba University, P.O. Box 12, Annaba 23000, Algeria)

Abstract


In this article, we establish the existence of positive periodic solutions for second-order dynamic equations on time scales. The main method used here is the Schauder fixed point theorem. The exponential stability of positive periodic solutions is also studied. The results obtained here extend some results in the literature. An example is also given to illustrate this work.


Keywords


Positive periodic solutions, Exponential stability, Schauder fixed point theorem, Dynamic equations, Time scales

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References


Adivar M., Raffoul Y.N. Existence of periodic solutions in totally nonlinear delay dynamic equations. Electron. J. Qual. Theory Differ. Equ., 2009. Vol. 2009, No. 1. P. 1–20. DOI: 10.14232/ejqtde.2009.4.1

Ardjouni A., Djoudi A. Existence of positive periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale. Malaya Journal of Matematik, 2013. Vol. 2, No. 1. P. 60–67.

Ardjouni A., Djoudi A. Existence of periodic solutions for nonlinear neutral dynamic equations with functional delay on a time scale. Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 2013. Vol. 52, No. 1. P. 5–19.

Ardjouni A., Djoudi A. Existence of periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale. Commun. Nonlinear Sci. Numer. Simul., 2012. Vol. 17. P. 3061–3069. DOI: 10.1016/j.cnsns.2011.11.026

Ardjouni A., Djoudi A. Periodic solutions in totally nonlinear dynamic equations with functional delay on a time scale. Rend. Sem. Mat. Univ. Politec. Torino, 2010. Vol. 68, No. 4. P. 349-359.

Bohner M., Peterson A. Dynamic Equations on Time Scales, An Introduction with Applications. Birkhäuser, Boston, 2001. 358 p. DOI: 10.1007/978-1-4612-0201-1

Bohner M., Peterson A. Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, 2003. 348 p. DOI: 10.1007/978-0-8176-8230-9

Bouchelaghem F., Ardjouni A., Djoudi A. Existence and stability of positive periodic solutions for delay nonlinear dynamic equations. Nonlinear Stud., 2018. Vol. 25, No. 1. P. 191–202.

Bouchelaghem F., Ardjouni A., Djoudi A. Existence of positive solutions of delay dynamic equations. Positivity, 2017. Vol. 21, No. 4. P. 1483–1493. DOI: 10.1007/s11117-017-0480-2

Bouchelaghem F., Ardjouni A., Djoudi A. Existence of positive periodic solutions for delay dynamic equations. Proyecciones (Antofagasta), 2017. Vol. 36, No. 3. P. 449–460. DOI: 10.4067/S0716-09172017000300449

Cid J.Á., Propst G., Tvrdý M. On the pumping effect in a pipe/tank flow configuration with friction. Phys. D: Nonlinear Phenomena, 2014. Vol. 273/274. P. 28–33. DOI: 10.1016/j.physd.2014.01.010

Culáková I., Hanuštiaková L’., Olach R. Existence for positive solutions of second-order neutral nonlinear differential equations. Appl. Math. Lett., 2009. Vol. 22, No. 7. P. 1007–1010. DOI: 10.1016/j.aml.2009.01.009

Dorociaková B., Michalková M., Olach R., Sága M. Existence and Stability of Periodic Solution Related to Valveless Pumping. Math. Probl. Eng., 2018. Vol. 2018. Art. ID 3982432. P. 1–8. DOI: 10.1155/2018/3982432

Gouasmia M., Ardjouni A., Djoudi A. Periodic and nonnegative periodic solutions of nonlinear neutral dynamic equations on a time scale. Int. J. Anal. Appl., 2018. Vol. 16, No. 2. P. 162–177. DOI: 10.28924/2291-8639-16-2018-162

Hilger S. Ein Masskettenkalkul mit Anwendung auf Zentrumsmanningfaltigkeiten. PhD thesis, Universitat Wurzburg, 1988.

Kaufmann E.R., Raffoul Y.N. Periodic solutions for a neutral nonlinear dynamical equation on a time scale. J. Math. Anal. Appl., 2006. Vol. 319. P. 315–325. DOI: 10.1016/j.jmaa.2006.01.063

Lakshmikantham V., Sivasundaram S., Kaymarkcalan B. Dynamic Systems on Measure Chains. Dordrecht: Kluwer Academic Publishers, 1996. 294 p. DOI: 10.1007/978-1-4757-2449-3

Smart D.R. Fixed Points Theorems. Cambridge, UK: Cambridge Univ. Press, 1980. 104 p.




DOI: http://dx.doi.org/10.15826/umj.2020.1.004

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