Difference monotonicity analysis on discrete fractional operators with discrete generalized Mittag-Leffler kernels

In this paper, we present the monotonicity analysis for the nabla fractional differences with discrete generalized Mittag-Leffler kernels (ABR a–1∇ y)(η) of order 0 < δ < 0.5, β = 1, 0 < γ ≤ 1 starting at a – 1. If (ABR a–1∇ y)(η)≥ 0, then we deduce that y(η) is δ2γ -increasing. That is, y(η + 1) ≥ δ2γ y(η) for each η ∈Na := {a,a + 1, . . .}. Conversely, if y(η) is increasing with y(a) ≥ 0, then we deduce that (ABR a–1∇ y)(η)≥ 0. Furthermore, the monotonicity properties of the Caputo and right fractional differences are concluded to. Finally, we find a fractional difference version of the mean value theorem as an application of our results. One can see that our results cover some existing results in the literature.


Introduction
In the past two decades, fractional calculus and its applications have been applied into various fields due to its accurate describing in many scientific fields, such as fractional stochastic noise [1], fraction order memristive chaotic circuits [2], fractional order financial models [3], and fractional order relaxation-oscillation models [4]. Also, it has wide application in various research areas, which one can find in the references [5][6][7][8][9][10][11][12][13].
Along the years, fractional calculus has attracted more and more researchers' attention and has found applications in several fields of engineering and the applied sciences (see [5][6][7][8]). Recently, many fractional models were proposed showing the significance of fractional calculus. Discrete fractional calculus can be seen as the most recent model of fractional calculus which has been widely used.
Recently, discrete fractional calculus gains a great deal of interest by many researchers. In [14,15], the authors introduced the discrete fractional sums and differences which produced directly from the Riemann-Liouville (RL) fractional integrals and derivatives, respectively. To review the history of discrete fractional operators, their properties and information related to discrete fractional calculus applications one can refer to [16][17][18][19][20][21][22] and the references therein. Nowadays, being new fractional integral and derivative operators make the researchers attempt to introduce a new definition of discrete fractional sum and difference operators corresponding to them. Those models are receiving the attention of many researchers (see [23][24][25][26]).
Monotonocity analysis has become very important in discrete fractional calculus which was firstly applied for the discrete fractional operators of RL version by Atici and Uyanik in [27]. In [28], the authors found the monotonicity analysis for the Caputo-Fabrizo (CF) version of discrete fractional operators. In [29], the monotonicity analysis for the Atangana-Baleanu (AB) version of discrete fractional operators with discrete Mittag-Leffler (ML) kernels was done. In addition, the monotonicity analysis has been established for the hdiscrete fractional operators in [30,31] (see also [32]).
However, to the best of our knowledge so far, the monotonicity results have not been considered for the discrete fractional operators with discrete generalized ML kernels [33]. Therefore, our aim in this article is to establish the monotonicity analysis for the above model of discrete fractional operators that can cover the monotonicity results in [29].
The structure of the article is designed as follows: Sect. 2.1 deals with recalling the RLfractional sums and generalized discrete ML functions. Section 2.2 deals with recalling the generalized discrete AB fractional operators with their equivalent formulas and definition of δ-monotonicity. In Sect. 3 we discuss the monotonicity analysis for the 2-parameter fractional difference operators involving the discrete generalized ML kernels. Section 4 deals with the application of our findings on the mean value theorem, and in Sect. 5 we conclude the article.

Preliminaries
This section deals with some basic concepts on discrete fractional operators and discrete ML functions. ([24, 25, 33]) The increasing factorial function of η is given by

RL-fractional sums and generalized ML function
Generally, the increasing factorial function is given by for η ∈ R \ {. . . , -2, -1, 0}, where R denotes the set of real numbers. Then, for any function f : N a → R, the nabla left fractional sum of order δ > 0 in the sense of RL is defined by Also, for any function f . .} → R, the nabla right fractional sum of order δ > 0 in the sense of RL is defined by Lemma 2.1 ([24, 25, 33]) For any a, b ∈ R and δ 1 , δ 2 > 0, we have Lemma 2.2 ([24, 25, 33]) For any δ 1 , δ 2 ∈ R and any f defined on N a , we have

Lemma 2.3 ([25])
Let f be defined on N a , then, for any 0 < δ < 1, we have The nabla discrete ML functions are important; they are recalled now.
Proof In proving (i), we need the following identity:

Generalized discrete ABR and ABC and monotonicity definitions
The discrete ABR and ABC fractional differences and sums were introduced in [24] using the one parameter discrete ML function. After that, the generalized discrete ABR and ABC fractional differences and sums were introduced by Abdeljawad in [33] using the generalized discrete ML function: 1-δ and 0 < δ < 1/2. Then, for γ ∈ R and Re(β) > 0, the left generalized discrete ABR fractional difference is defined by 8) and the right generalized discrete ABR fractional difference is defined by and the right generalized discrete ABC fractional difference is defined by where B(δ) is a multiplier and it satisfies B(0) = B(1) = 1.
In this article, we consider a specific case where 0 < γ ≤ 1 and β = 1. Then we can rewrite the above definitions as follows. 12) and the right 2-parameter discrete ABR fractional difference is defined by and the right 2-parameter discrete ABC fractional difference is defined by ) Let y be defined on N a with b ≡ a (mod 1), then, for any λ = -δ 1-δ , 0 < δ < 1/2, γ ∈ R and 0 < Re(β) < 1, we have the following relationships between the discrete ABC and discrete ABR fractional differences: ABC a ∇ δ,β,γ y (η) = ABR a ∇ δ,β,γ y (η) - in the left-side sense and in the right-side sense.

Definition 2.7
Let y : N a → R be a function satisfying y(a) ≥ 0. Then y is called a δincreasing function on N a , if Observe that, if y(η) is increasing on N a , then y(η + 1) ≥ y(η) for all η ∈ N a , and thus y(η) is δ-increasing on N a . Definition 2.8 Let y : N a → R be a function satisfying y(a) ≤ 0. Then y is called a δdecreasing function on N a , if Observe that, if y(η) is decreasing on N a , then y(η + 1) ≤ y(η) for all η ∈ N a , and thus y(η) is δ-decreasing on N a . Remark 2.2 Note that, if δ = 1 in Definition 2.7, then the increasing and δ-increasing concepts coincide, and if δ = 1 in Definition 2.8, then the decreasing and δ-decreasing concepts coincide.

Corollary 3.1 Let y : N a-1 → R be a function. Suppose that, for
Proof The proof follows directly from Theorem 3.1 and Theorem 2.1 with β = 1.
Then, from (3.1), we have Since (1δ) γ > 0 and y(η) ≥ y(η -1), we have Then, by using this in (3.2), we get which we can rearrange to get the desired result.
The following theorems are similar to Theorem 3.3.

MVT application
This section deals with the application of our results to the mean value theorem (MVT). First, we need the following lemmas.
On the other hand, from the definition of discrete nabla fractional sum, we have For By taking ∇ to both sides of (4.3), we obtain By using (4.4) in (4.2), we obtain which completes the proof.
we can obtain the following relationship between the combination formula and the Pochhammer symbol: This is a useful tool in the proof of the next theorem. Now, from [33], we see that AB a ∇ -(δ,γ )ABR a ∇ δ,γ y (η) = y(η). (4.5) One can note that Eq. (4.5) does not contain y(a). However, the next result contains an initial value y(a) which will be a great tool to prove our fractional difference MVT.