Number of positive periodic solutions for first -order nonlinear difference equations with feedback

This paper presents a sufficient condition to determine the number of positive periodic solutions of scalar nonlinear difference equations with time delays. The difference equations considered here contain several terms that act as feedback dominated by time delays. The main result is proved using the Krasnosel'skii fixed point theorem. Our results also show the range in which positive periodic solutions exist. An example and its numerical simulations are provided to illustrate our results. Applying this example to our results guarantees the existence of at least four positive periodic solutions. The simulation shows that there are exactly four positive periodic solutions. Hence, it can be concluded that our results are satisfactory. Finally, applicability of our results is demonstrated using a Mackey-Glass-type discrete hematopoiesis model with a unimodal production function. (c) 2020 Elsevier Inc. All rights reserved.