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Periodic solutions and stability in a nonlinear neutral system of differential equations with infinite delay

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Abstract

In this paper, we study the existence of periodic solutions of the nonlinear neutral system of differential equations

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}x\left( t\right) =A\left( t\right) x\left( t\right) +\frac{\mathrm{d}}{\mathrm{d}t} Q\left( t,x\left( t-g\left( t\right) \right) \right) +\int _{-\infty }^{t}D\left( t,s\right) F\left( x\left( s\right) \right) \mathrm{d}s. \end{aligned}$$

Using Krasnoselskii’s fixed point theorem, we obtain the existence of periodic solution and by contraction mapping principle we obtain the uniqueness of periodic solution and stability of the zero solution. Our results extend some earlier publications.

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Acknowledgments

The authors gratefully acknowledge the reviewers for their helpful comments.

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Authors and Affiliations

  1. Applied Mathematics Lab, Department of Mathematics, Faculty of Sciences, University of Annaba, P.O. Box 12, 23000, Annaba, Algeria

    Mouataz Billah Mesmouli, Abdelouaheb Ardjouni & Ahcene Djoudi

  2. Department of Mathematics and Informatics, University of Souk Ahras, P.O. Box 1553, 41000, Souk Ahras, Algeria

    Abdelouaheb Ardjouni

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  1. Mouataz Billah Mesmouli
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  2. Abdelouaheb Ardjouni
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  3. Ahcene Djoudi
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Correspondence to Abdelouaheb Ardjouni.

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Mesmouli, M.B., Ardjouni, A. & Djoudi, A. Periodic solutions and stability in a nonlinear neutral system of differential equations with infinite delay. Bol. Soc. Mat. Mex. 24, 239–255 (2018). https://doi.org/10.1007/s40590-016-0155-1

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