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Existence and Roughness of Nonuniform Exponential Dichotomies on Time Scales

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Abstract

In this manuscript, we discuss the roughness of nonuniform exponential dichotomies on a time scale in a Banach space and give the existence results for nonuniform and uniform exponential dichotomy for the time scale. We extend and unify the previous results by considering the equation on a time scale. For a given linear equation on a time scale, the existence of exponential dichotomy persists under an adequately small variable linear perturbation. To establish the results, we acquire related roughness results for the case of uniform exponential contractions. In the end, a suitable example is given for illustration.

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Acknowledgements

We are thankful to the reviewers for their constructive comments and suggestions which helped us to improve the manuscript. Also Syed Abbas, thanks for the support of Matrices SERB project No. IITM/SERB/SB/284. Samuel Castillo, thanks for the support of DIUBB 164408 3/R. Manuel Pinto, thanks for the support of Fondecyt project 1170466.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Rajiv Gandhi Institute of Petroleum Technology Jais, Amethi, UP, 229305, India

    Soniya Dhama

  2. GISDA Grupo de Investigacion en Sistemas Dinamicos y Aplicaciones, Universidad del Bio Bio, Concepcion, Chile

    Samuel Castillo

  3. School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Mandi, HP, 175005, India

    Syed Abbas

  4. Departamento de Matematicas, Facultad de Ciencias, Universidad de Chile, Santiago, Chile

    Manuel Pinto

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  1. Soniya Dhama
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  2. Samuel Castillo
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  3. Syed Abbas
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  4. Manuel Pinto
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All authors contributed equally.

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Correspondence to Syed Abbas.

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Dhama, S., Castillo, S., Abbas, S. et al. Existence and Roughness of Nonuniform Exponential Dichotomies on Time Scales. Qual. Theory Dyn. Syst. 23, 90 (2024). https://doi.org/10.1007/s12346-023-00949-y

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  • DOI: https://doi.org/10.1007/s12346-023-00949-y

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