Monotonicity Analysis of Fractional Proportional Differences
In this work, the nabla discrete new Riemann–Liouville and Caputo fractional proportional differences of order 0<ε<1 on the time scale ℤ are formulated. The differences and summations of discrete fractional proportional are detected on ℤ, and the fractional proportional sums associated to ∇cRχ...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2020-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2020/4867927 |
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| Summary: | In this work, the nabla discrete new Riemann–Liouville and Caputo fractional proportional differences of order 0<ε<1 on the time scale ℤ are formulated. The differences and summations of discrete fractional proportional are detected on ℤ, and the fractional proportional sums associated to ∇cRχε,ρz with order 0<ε<1 are defined. The relation between nabla Riemann–Liouville and Caputo fractional proportional differences is derived. The monotonicity results for the nabla Caputo fractional proportional difference are proved; specifically, if ∇c−1Rχε,ρz>0 then χz is ερ −increasing, and if χz is strictly increasing on ℕc and χc>0, then ∇c−1Rχε,ρz>0. As an application of our findings, a new version of the fractional proportional difference of the mean value theorem (MVT) on ℤ is proved. |
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| ISSN: | 1026-0226 1607-887X |