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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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
Keywords
- Time scales
- Dichotomic inequalities
- Roughness
- Uniform and nonuniform exponential dichotomy
- Nonuniform exponential contraction