On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis

Pshtiwan Othman Mohammed, Thabet Abdeljawad and Faraidun Kadir Hamasalh
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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
Faraidun Kadir Hamasalh: Department of Mathematics, College of Education, University of Sulaimani, Sulaymaniyah 46001, Kurdistan Region, Iraq

Mathematics, 2021, vol. 9, issue 11, 1-17

Abstract: Monotonicity analysis of delta fractional sums and differences of order ? ? ( 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 ? -increasing on M a + ? h , h , where the delta Riemann–Liouville fractional h -difference of order ? of a function y ( z ) starting at a + ? h is greater or equal to zero, and then, we can show that y ( z ) is ? -increasing on M a + ? h , h , where the delta Caputo fractional h -difference of order ? of a function y ( z ) starting at a + ? h is greater or equal to ? 1 ? ( 1 ? ? ) ( z ? ( a + ? h ) ) h ( ? ? ) y ( a + ? h ) for each z ? M a + h , h . Conversely, if y ( a + ? h ) is greater or equal to zero and y ( z ) is increasing on M a + ? h , h , we show that the delta Riemann–Liouville fractional h -difference of order ? of a function y ( z ) starting at a + ? h is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h -difference of order ? of a function y ( z ) starting at a + ? h is greater or equal to ? 1 ? ( 1 ? ? ) ( z ? ( a + ? h ) ) h ( ? ? ) y ( a + ? 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.

Keywords: discrete fractional calculus; ? -monotonicity analysis; discrete delta fractional operators; mean value theorem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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