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

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 .