Difference monotonicity analysis on discrete fractional operators with discrete generalized Mittag-Leffler kernels
نویسندگان
چکیده
Abstract In this paper, we present the monotonicity analysis for nabla fractional differences with discrete generalized Mittag-Leffler kernels $( {}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y )(\eta )$ (a−1ABR∇δ,γy)(η) of order $0<\delta <0.5$ xmlns:mml="http://www.w3.org/1998/Math/MathML">0<δ<0.5 , $\beta =1$ xmlns:mml="http://www.w3.org/1998/Math/MathML">β=1 $0<\gamma \leq 1$ xmlns:mml="http://www.w3.org/1998/Math/MathML">0<γ≤1 starting at $a-1$ xmlns:mml="http://www.w3.org/1998/Math/MathML">a−1 . If $({}^{ABR}_{a-1}{\nabla ) ( \eta )\geq 0$ />a−1ABR∇δ,γy)(η)≥0 then deduce that $y(\eta xmlns:mml="http://www.w3.org/1998/Math/MathML">y(η) is $\delta ^{2}\gamma $ xmlns:mml="http://www.w3.org/1998/Math/MathML">δ2γ -increasing. That is, +1)\geq \delta ^{2} \gamma y(\eta xmlns:mml="http://www.w3.org/1998/Math/MathML">y(η+1)≥δ2γy(η) each $\eta \in \mathcal{N}_{a}:=\{a,a+1,\ldots\}$ xmlns:mml="http://www.w3.org/1998/Math/MathML">η∈Na:={a,a+1,…} Conversely, if increasing $y(a)\geq xmlns:mml="http://www.w3.org/1998/Math/MathML">y(a)≥0 \geq Furthermore, properties Caputo and right are concluded to. Finally, find a difference version mean value theorem as an application our results. One can see results cover some existing in literature.
منابع مشابه
Discrete Mittag-Leffler Functions in Linear Fractional Difference Equations
and Applied Analysis 3 and define recursively a∇−nf t ∫ t a a∇−n 1f τ ∇τ 2.4 for n 2, 3, . . .. Then we have the following. Proposition 2.1 Nabla Cauchy formula . Let n ∈ Z , a, b ∈ T and let f : T → R be ∇-integrable on a, b ∩ T. If t ∈ T, a ≤ t ≤ b, then a∇−nf t ∫ t a ̂ hn−1 ( t, ρ τ ) f τ ∇τ . 2.5 Proof. This assertion can be proved by induction. If n 1, then 2.5 obviously holds. Let n ≥ 2 an...
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ژورنال
عنوان ژورنال: Advances in Difference Equations
سال: 2021
ISSN: ['1687-1839', '1687-1847']
DOI: https://doi.org/10.1186/s13662-021-03372-2