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Asymptotic behavior of solutions of fractional differential equations with Hadamard fractional derivatives

  • Mohammed D. Kassim EMAIL logo and Nasser-eddine Tatar
Published/Copyright: May 9, 2021

Abstract

The asymptotic behaviour of solutions in an appropriate space is discussed for a fractional problem involving Hadamard left-sided fractional derivatives of different orders. Reasonable sufficient conditions are determined ensuring that solutions of fractional differential equations with nonlinear right hand sides approach a logarithmic function as time goes to infinity. This generalizes and extends earlier results on integer order differential equations to the fractional case. Our approach is based on appropriate desingularization techniques and generalized versions of Gronwall-Bellman inequality. It relies also on a kind of Hadamard fractional version of l'Hopital’s rule which we prove here.

Acknowledgements

The first author wants to thank Imam Abdulrahman Bin Faisal University for the support and facilities. The second author is grateful for the financial support and the facilities provided by King Fahd University of Petroleum and Minerals through Project number IN181008.

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Received: 2019-01-04
Revised: 2020-11-27
Published Online: 2021-05-09
Published in Print: 2021-04-27

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