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On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis

Author

Listed:
  • Pshtiwan Othman Mohammed

    (Department of Mathematics, College of Education, University of Sulaimani, Sulaymaniyah 46001, Kurdistan Region, Iraq)

  • Thabet Abdeljawad

    (Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
    Department of Medical Research, China Medical University, Taichung 40402, Taiwan
    Department of Computer Science and Information Engineering, Asia University, Taichung 41354, Taiwan)

  • Faraidun Kadir Hamasalh

    (Department of Mathematics, College of Education, University of Sulaimani, Sulaymaniyah 46001, Kurdistan Region, Iraq)

Abstract

Monotonicity analysis of delta fractional sums and differences of order υ ∈ ( 0 , 1 ] on the time scale h Z are presented in this study. For this analysis, two models of discrete fractional calculus, Riemann–Liouville and Caputo, are considered. There is a relationship between the delta Riemann–Liouville fractional h -difference and delta Caputo fractional h -differences, which we find in this study. Therefore, after we solve one, we can apply the same method to the other one due to their correlation. We show that y ( z ) is υ -increasing on M a + υ h , h , where the delta Riemann–Liouville fractional h -difference of order υ of a function y ( z ) starting at a + υ h is greater or equal to zero, and then, we can show that y ( z ) is υ -increasing on M a + υ h , h , where the delta Caputo fractional h -difference of order υ of a function y ( z ) starting at a + υ h is greater or equal to − 1 Γ ( 1 − υ ) ( z − ( a + υ h ) ) h ( − υ ) y ( a + υ h ) for each z ∈ M a + h , h . Conversely, if y ( a + υ h ) is greater or equal to zero and y ( z ) is increasing on M a + υ h , h , we show that the delta Riemann–Liouville fractional h -difference of order υ of a function y ( z ) starting at a + υ h is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h -difference of order υ of a function y ( z ) starting at a + υ h is greater or equal to − 1 Γ ( 1 − υ ) ( z − ( a + υ h ) ) h ( − υ ) y ( a + υ h ) on M a , h . Furthermore, we consider some related results for strictly increasing, decreasing, and strictly decreasing cases. Finally, the fractional forward difference initial value problems and their solutions are investigated to test the mean value theorem on the time scale h Z utilizing the monotonicity results.

Suggested Citation

  • Pshtiwan Othman Mohammed & Thabet Abdeljawad & Faraidun Kadir Hamasalh, 2021. "On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis," Mathematics, MDPI, vol. 9(11), pages 1-17, June.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:11:p:1303-:d:570009
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    References listed on IDEAS

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    1. Thabet Abdeljawad, 2013. "On Delta and Nabla Caputo Fractional Differences and Dual Identities," Discrete Dynamics in Nature and Society, Hindawi, vol. 2013, pages 1-12, July.
    2. Thabet Abdeljawad & Ferhan M. Atici, 2012. "On the Definitions of Nabla Fractional Operators," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-13, October.
    3. Pshtiwan Othman Mohammed, 2019. "A Generalized Uncertain Fractional Forward Difference Equations of Riemann-Liouville Type," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 11(4), pages 43-50, August.
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    1. Almusawa, Musawa Yahya & Mohammed, Pshtiwan Othman, 2023. "Approximation of sequential fractional systems of Liouville–Caputo type by discrete delta difference operators," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).

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