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AsymptotIc Behavior of Solutions of Fractional Differential Equations with Hadamard Fractional Derivatives

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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.

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Authors and Affiliations

  1. Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, 34151, Saudi Arabia

    Mohammed D. Kassim

  2. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Nasser-eddine Tatar

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  1. Mohammed D. Kassim
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  2. Nasser-eddine Tatar
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Correspondence to Mohammed D. Kassim.

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Kassim, M.D., Tatar, Ne. AsymptotIc Behavior of Solutions of Fractional Differential Equations with Hadamard Fractional Derivatives. Fract Calc Appl Anal 24, 483–508 (2021). https://doi.org/10.1515/fca-2021-0021

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  • DOI: https://doi.org/10.1515/fca-2021-0021

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