Existence of Positive Periodic Solutions for Scalar Delay Differential Equations with and without Impulses
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The paper is concerned with a broad family of scalar periodic delay differential equations with linear impulses, for which the existence of a positive periodic solution is established under very general conditions. The proofs rely on fixed point arguments, employing either the Schauder theorem or Krasnoselskii fixed point theorem in cones. The results are illustrated with applications to an impulsive hematopoiesis model or generalized Nicholson’s equations, among other selected examples from mathematical biology. The method presented here turns out to be very powerful, in the sense that the derived theorems largely generalize and improve other results in recent literature, even for the situation without impulses.
KeywordsDelay differential equation Impulses Positive periodic solution Fixed point theorems Hematopoiesis model Nicholson equation
Mathematics Subject Classification34K13 34K45 92D25
This work was partially supported by Fundação para a Ciência e a Tecnologia under Project UID/MAT/04561/2013 (Teresa Faria) and UID/MAT/00013/2013 (José J. Oliveira). The authors thank the referee, for bringing to their attention some relevant recent references. They also express their gratitude to the Editorial Board of this journal, for preparing a Special Issue in memory of Professor George Sell.
- 15.Krasnoselskii, M.A.: Positive Solutions of Operator Equations. P. Noordhoff Ltd., Groningen (1964)Google Scholar
- 23.Sacker, R.J., Sell, G.R.: Lifting Properties in Skew-Product Flows with Applications to Differential Equations, vol. 11, no. 190. Memoirs of the American Mathematical Society, Providence, RI (1977)Google Scholar