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Stability of logarithmic type for a Hadamard fractional differential problem

Abstract

We study the long-time behavior of solutions for a general class of nonlinear fractional differential equations. These equations involve Hadamard fractional derivatives of different orders. We determine sufficient conditions on the nonlinear terms which guarantee that solutions exist globally and decay to zero as a logarithmic function. For this purpose, we combine and generalize some versions of Gronwall–Bellman inequality, appropriate regularization techniques and several properties of the Hadamard fractional derivative. Our findings are illustrated by examples.

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Acknowledgements

The authors would like to acknowledge the support provided by King Abdulaziz City of Science and Technology (KACST) under the National Science, Technology and Innovation Plan (NSTIP), Project No. 15-OIL4884-0124.

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Correspondence to M. D. Kassim.

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Kassim, M.D., Tatar, N. Stability of logarithmic type for a Hadamard fractional differential problem. J. Pseudo-Differ. Oper. Appl. 11, 447–466 (2020). https://doi.org/10.1007/s11868-019-00285-3

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Keywords

  • Asymptotic behavior
  • Boundedness
  • Fractional differential equation
  • Hadamard fractional derivative

Mathematics Subject Classification

  • 34E10
  • 34A08
  • 26A33