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Stability Analysis of the First Order Non-linear Impulsive Time Varying Delay Dynamic System on Time Scales

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Abstract

In this paper, we study Hyers–Ulam stability and Hyers–Ulam–Rassias stability of first order non-linear impulsive time varying delay dynamic system on time scales, via a fixed point approach. We obtain some results of existence and uniqueness of solutions by using Picard operator. The main tools for our results are the Grönwall’s inequality on time scales, abstract Grönwall lemma and Banach contraction principle. In order to overcome difficulties arises in our considered model, we pose some conditions along with Lipchitz condition. At the end, an example is given that shows the validity of our main results.

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Acknowledgements

The authors express their sincere gratitude to the Editor and referees for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results.

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Authors and Affiliations

  1. Department of Mathematics, University of Peshawar, Peshawar, Pakistan

    Syed Omar Shah & Akbar Zada

  2. Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

    Alaa E. Hamza

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  1. Syed Omar Shah
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  2. Akbar Zada
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  3. Alaa E. Hamza
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Correspondence to Akbar Zada.

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All the authors contributed equally and significantly in writing this paper. All the authors read and approved the final manuscript.

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Shah, S.O., Zada, A. & Hamza, A.E. Stability Analysis of the First Order Non-linear Impulsive Time Varying Delay Dynamic System on Time Scales. Qual. Theory Dyn. Syst. 18, 825–840 (2019). https://doi.org/10.1007/s12346-019-00315-x

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