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Non-Trivial Periodic Solutions for a Class of Second Order Differential Equations with Large Delay

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Abstract

We provide a result on the existence of a positive periodic solution for the following class of delay equations

$$ \theta ''(t)-\theta (t)+f(\theta (t-r))=0. $$

In particular, we find an infinite family of disjoint intervals having the following property: if the delay is within one of these intervals, then the equation admits a non-trivial and even \(2r\)-periodic solution. Furthermore, the length of these intervals is constant and depends on the size of the term \(|f'(\eta )|\), where \(\eta \) is the unique positive equilibrium point of the equation. Consequently, we can find periodic solutions for arbitrarily large delays.

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Acknowledgements

The authors would like to thank the editor and the anonymous referee for their feedback and comments, which improve the quality of the manuscript.

Funding

The work of M. Zamora was supported by Agencia Estatal de Investigación. Spain, project MCIU-22-PID2021-128418NA-I00.

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Authors and Affiliations

  1. Departamento de Matemática, Grupo de investigación en Sistemas Dinámicos y Aplicaciones (GISDA), Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile

    Adrian Gomez

  2. Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Alonso de Ribera 2850, Concepción, Chile

    Nolbert Morales

  3. Departament of Mathematics, University of Oviedo, Polytechnic School of Engineering (EPI), Campus of Viesques, Gijón, Spain

    Manuel Zamora

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  1. Adrian Gomez
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  2. Nolbert Morales
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Correspondence to Manuel Zamora.

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Gomez, A., Morales, N. & Zamora, M. Non-Trivial Periodic Solutions for a Class of Second Order Differential Equations with Large Delay. Acta Appl Math 188, 3 (2023). https://doi.org/10.1007/s10440-023-00613-2

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