Abstract
Monotonicity analysis of delta fractional sums and differences of order upsilon is an element of (0,1] on the time scale hZ 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 upsilon-increasing on M-a+upsilon h,M-h, where the delta Riemann-Liouville fractional h-difference of order upsilon of a function y(z) starting at a + upsilon h is greater or equal to zero, and then, we can show that y(z) is upsilon-increasing on M-a+upsilon h,M-h, where the delta Caputo fractional h-difference of order upsilon of a function y(z) starting at a + upsilon h is greater or equal to -1/Gamma(1-upsilon)(z-(a + upsilon h))(h)((-upsilon))y(a + upsilon h) for each z is an element of M-a+h,M-h. Conversely, if y(a + upsilon h) is greater or equal to zero and y(z) is increasing on M-a+upsilon h,M-h, we show that the delta Riemann-Liouville fractional h-difference of order upsilon of a function y(z) starting at a + upsilon h is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h-difference of order upsilon of a function y(z) starting at a + upsilon h is greater or equal to -1/Gamma(1-upsilon)(z-(a + upsilon h))(h)((-upsilon))y(a + upsilon h) on M-a,M-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 hZ utilizing the monotonicity results.