Existence of a periodic solution for a state-dependent delay differential equation with a singular perturbation

Authors

  • Mohamed Nehari

DOI:

https://doi.org/10.54021/seesv5n1-078

Keywords:

state-dependent delay differential equation, singular perturbation, slowly oxillating periodic solution, fixed point theorem

Abstract

The idea of this article came from the works dones by Khung and Smith on the one hand in [2] for  and Arino and Al in [1] for  on the other hand, to proves the existence of a slowly oxillating preriodic solution by the fixed point theorem of Picard. I proposed the following question: Do the result remain valid if  1 ? by the same method with using the homotopy well defined link between the application of Khung and Smith and the application of Arino and  and indeed, I came to prove existence of a slowly oxillating preriodic solution by the fixed point theorem of Picard, therefore the result obtained in this work is generalization of two special cases. A State-dependent Delay Differential equation play important role in recent years to modeling of several phenomenon, in dynamics of populations, in biology and even in the social life for example: When we take our driving test, we learn that the reaction time of our nervous system when driving is of the order of a few seconds, and that we must take care to maintain a sufficient distance between two cars following each other, from this example, we notice that the delay can be useful. Although this work has added a new addition to the scientific field which is a generalization and linking between two special cases, it remains somewhat limited because it takes special case of Arino and Al. Therefore in the futur, we are thinking about a new work that takes into account the general case of Arino and .

References

Arino, O.; Hadeler, K.P.; and Hbib, M.L. Existence of periodic solutions for delay diffrential equations with state dependent delay.v J. Diff. Equ. vol. 144, N^∘ 2,p:263-301,1998.

Khuang, Y.; and Smith, H.L. Slowly oscillating periodic solutions of autonomous state-dependent delay equations, Nonl. Ana, Vol. 19, N9, 855-872, 1992.

Walther, H.O. A priodic solution of a differential equation with statedependent delay, J. Diff. Equ. 244 (2008).

Yebdri, M. Special symmetric slowly oscillating periodic solution of state dependent delay differential equation. Proceding Equadiff 99, International conference on differential equations, vol.2, page (1396,1400), World Scientific (1999).

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Published

2024-04-29

How to Cite

Nehari, M. (2024). Existence of a periodic solution for a state-dependent delay differential equation with a singular perturbation. STUDIES IN ENGINEERING AND EXACT SCIENCES, 5(1), 1506–1529. https://doi.org/10.54021/seesv5n1-078