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A New Gronwall–Bellman Inequality in Frame of Generalized Proportional Fractional Derivative

Author

Listed:
  • Jehad Alzabut

    (Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia)

  • Weerawat Sudsutad

    (Department of General Education, Faculty of Science and Health Technology, Navamindradhiraj University, Bangkok 10300, Thailand)

  • Zeynep Kayar

    (Department of Mathematics, Van Yuzuncu Yil University, 65080 Van, Turkey)

  • Hamid Baghani

    (Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran)

Abstract

New versions of a Gronwall–Bellman inequality in the frame of the generalized (Riemann–Liouville and Caputo) proportional fractional derivative are provided. Before proceeding to the main results, we define the generalized Riemann–Liouville and Caputo proportional fractional derivatives and integrals and expose some of their features. We prove our main result in light of some efficient comparison analyses. The Gronwall–Bellman inequality in the case of weighted function is also obtained. By the help of the new proposed inequalities, examples of Riemann–Liouville and Caputo proportional fractional initial value problems are presented to emphasize the solution dependence on the initial data and on the right-hand side.

Suggested Citation

  • Jehad Alzabut & Weerawat Sudsutad & Zeynep Kayar & Hamid Baghani, 2019. "A New Gronwall–Bellman Inequality in Frame of Generalized Proportional Fractional Derivative," Mathematics, MDPI, vol. 7(8), pages 1-15, August.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:8:p:747-:d:258017
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    References listed on IDEAS

    as
    1. Xueru Lin, 2014. "A Note on Gronwall’s Inequality on Time Scales," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-4, July.
    2. Lokenath Debnath, 2003. "Recent applications of fractional calculus to science and engineering," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2003, pages 1-30, January.
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    Cited by:

    1. Weerawat Sudsutad & Nantapat Jarasthitikulchai & Chatthai Thaiprayoon & Jutarat Kongson & Jehad Alzabut, 2022. "Novel Generalized Proportional Fractional Integral Inequalities on Probabilistic Random Variables and Their Applications," Mathematics, MDPI, vol. 10(4), pages 1-21, February.
    2. Jehad Alzabut & James Viji & Velu Muthulakshmi & Weerawat Sudsutad, 2020. "Oscillatory Behavior of a Type of Generalized Proportional Fractional Differential Equations with Forcing and Damping Terms," Mathematics, MDPI, vol. 8(6), pages 1-18, June.

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