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Existence of Positive Periodic Solutions for Scalar Delay Differential Equations with and without Impulses

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Abstract

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.

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Acknowledgements

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.

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Authors and Affiliations

  1. Departamento de Matemática and CMAF-CIO, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016, Lisbon, Portugal

    Teresa Faria

  2. CMAT and Departamento de Matemática e Aplicações, Escola de Ciências, Universidade do Minho, Campus de Gualtar, 4710-057, Braga, Portugal

    José J. Oliveira

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  1. Teresa Faria
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  2. José J. Oliveira
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Correspondence to Teresa Faria.

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To the memory of Professor George R. Sell

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Faria, T., Oliveira, J.J. Existence of Positive Periodic Solutions for Scalar Delay Differential Equations with and without Impulses. J Dyn Diff Equat 31, 1223–1245 (2019). https://doi.org/10.1007/s10884-017-9616-0

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  • DOI: https://doi.org/10.1007/s10884-017-9616-0

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