Analytical and Numerical Monotonicity Analyses for Discrete Delta Fractional Operators

Abstract

In this paper, first, we intend to determine the relationship between the sign of $\left({{}_{c_0}\Delta }^{\beta }y\right)(c_0+1),$ for $1<\beta <2$, and $\left(\Delta y\right)(c_0+1)>0$, in the case we assume that $\left({{}_{c_0}\Delta }^{\beta }y\right)(c_0+1)$ is negative. After that, by considering the set ${\mathcal{D}}_{\ell +1,\theta }\subseteq {\mathcal{D}}_{\ell ,\theta }$, which are subsets of $(1,2)$, we will extend our previous result to make the relationship between the sign of $\left({{}_{c_0}\Delta }^{\beta }y\right)\left(z\right)$ and $\left(\Delta y\right)\left(z\right)>0$ (the monotonicity of y), where $\left({{}_{c_0}\Delta }^{\beta }y\right)\left(z\right)$ will be assumed to be negative for each $z\in N_{c_0}^T:=\{c_0,c_0+1,c_0+2,\cdots ,T\}$ and some $T\in N_{c_0}:=\{c_0,c_0+1,c_0+2,\cdots \}$. The last part of this work is devoted to see the possibility of information reduction regarding the monotonicity of y despite the non-positivity of $\left({{}_{c_0}\Delta }^{\beta }y\right)\left(z\right)$ by means of numerical simulation.

1. Introduction

In the last few years, much attention was devoted to the study of discrete fractional models displaying a discrete transition between various discrete fractional orders to describe highly heterogeneous system of equations. Such scenarios motivated the development of discrete fractional operators with new defined kernels, for which the kernels themselves become functions of space and time. These efforts therefore led to the formulation of several inequivalent definitions of fractional differences and sums; see, for example, Refs. [1,2,3,4,5,6,7,8].

In addition to the existence and uniqueness issue coming with fractional order discrete operators, there is also the problem of providing a monotonicity analysis for each of these discrete operators. Contemporary research into discrete fractional models for monotonicity analyses are mostly focused on the development of discrete fractional calculus. For this reason, monotonicity analyses for various types of discrete fractional difference and sum operators with Riemann–Liouville, Caputo, Caputo–Fabrizio (exponential in kernel), Attangana–Baleanu differences, and other generalized difference operators have proven to be important tools for the mathematical analysis and modeling of various phenomena in applied science and engineering, see for example Refs. [9,10,11,12,13].

On the other hand, by using variational techniques, a number of important results of the monotonicity analysis defined using different types of discrete fractional operators have been established with the function defined on the time scale ${\mathrm{N}}_{c_0}\{c_0,c_0+1,\dots \}$ satisfying various conditions. For example, in [14], Atici and Uyanik established some $\beta -$monotonicity results on ${\mathrm{N}}_0$ for a function $\mathtt{y}$ satisfying an initial condition and $\left({{}_{c_0}\Delta }^{\beta }\right)\left(\mathtt{z}\right)\ge 0$. This work were generalized for other types of difference operators, for example, in [15], Abdeljawad and Baleanu established some ${\beta }^2-$monotonicity results on ${\mathrm{N}}_{c_0}$ for a function $\mathtt{y}$ satisfied the Atangana–Baleanu difference operator condition defined using Mittag-Liffler kernel $\left({{}_{c_0-1}{}^{ABR}\nabla }^{\beta }\right)\left(\mathtt{z}\right)\ge 0$. Besides, the work extended on the time scale ${\mathrm{N}}_{c_0}^h$ by Suwan et al. in [16]. In [17], the authors have demonstrated a strong connection between the positivity of the $\beta$th order Riemann–Liouville fractional difference and the convexity of the function. Moreover, in [18], the authors have established the ${\beta }^2$—monotonicity results for fractional Attangana–Baleanu difference operators. More relevant results and recent developments with different classes of conditions, we refer the readers to [17,19,20,21,22,23,24] and their references therein.

Motivated by the aforementioned research articles, we dedicate the main contributions of our work as follows:

• We consider the delta fractional difference $\left({{}_{c_0}\Delta }^{\beta }\mathtt{y}\right)(c_0+1)>-\theta \mathtt{y}\left(c_0\right)$ together with two conditions as stated in Theorem 1, to get one time step positivity $\left(\Delta \mathtt{y}\right)(c_0+1)\ge 0$.
• We introduce a set ${\mathcal{D}}_{\ell ,\theta }$ as defined in (5) and then we show that this set is decreasing for each values $\ell \ge 0$ and $\theta \ge 0$.
• The decreasing of this set makes the first theorem (Theorem 1) to be correct for each value of t in ${\mathrm{N}}_{c_0}$. This result will be proved in Corollary 1.
• Finally, the possibility of the negative lower bound will be demonstrated by numerical simulation of the sets ${\mathcal{D}}_{\ell ,\theta }$.
• Our results tell us that $\left({{}_{c_0}\Delta }^{\beta }\mathtt{y}\right)(c_0+1)$ can be negative and yet $\mathtt{y}$ can be increasing monotonically for finite time steps, compared to the results established in [10,17,18,25], when $\left({{}_{c_0}\nabla }^{\beta }\mathtt{y}\right)(c_0+1)$ and $\left({{}_{c_0}\Delta }^{\beta }\mathtt{y}\right)(c_0+1)$ needed to be positive for $\mathtt{y}$ to be monotone increasing.

The outline of the study is as follows. In Section 2, we introduce the pertinent properties and notations of discrete fractional operators of delta Riemann–Lioville type. In Section 3, we study the delta positivity and monotonicity results of our proposed difference operator. Furthermore, we define and compact the set ${\mathcal{D}}_{\ell ,\theta }$ which comes from the assumption of our first result in Theorem 1. In addition, we will generalize the first result by using this defined set. In Section 4, we deal with and example and four illustrative figures in which the time steps will be applied on the set ${\mathcal{D}}_{\ell ,\theta }$ as an application. Finally, in Section 5, we have our conclusion and future directions.

2. Basic Definitions

In this section we introduce tools for establishing the results in subsequent sections. The formulation of the definitions of delta fractional difference and sum adopted in this work, and additional information regarding the discrete fractional calculus can be found in the monograph [1]. Furthermore, the interested reader can visit the references [5,7,10] for further backgrounds on the discrete fractional operators.

Definition 1

 (see [26]). Let $\mathcal{D}$ be a set and $\mathcal{D}=P\left(\mathcal{D}\right)$ be the power set of $\mathcal{D}$. If ${\left\{{\mathcal{D}}_{\ell }\right\}}_{\ell =0}^{\infty }$ be a nested sequence of subsets of $\mathcal{D}$ such that ${\mathcal{D}}_{\ell +1}\subseteq {\mathcal{D}}_{\ell }$ for each $\ell \ge 0$. Then, ${\left\{{\mathcal{D}}_{\ell }\right\}}_{\ell =0}^{\infty }$ is a decreasing sequence of sets (in $\mathcal{D}$).

Definition 2

 (see Definition 2.24 in [1]). The β-th order delta fractional Taylor monomial and the falling factorial function are given by

$$\begin{array}{c}\hfill h_{\beta }(\mathtt{z},c_0):={\displaystyle \frac{{(\mathtt{z}-c_0)}^{\left(\beta \right)}}{\mathrm{\Gamma }(\beta +1)}}\mathit{and}{\mathtt{z}}^{\left(r\right)}:={\displaystyle \frac{\mathrm{\Gamma }(\mathtt{z}+1)}{\mathrm{\Gamma }(\mathtt{z}+1-r)}}\end{array}$$

respectively. These are valid for$\mathtt{z},r\in \mathcal{R}$provided that neither$\mathtt{z}+1$nor$\mathtt{z}+1-r$is a pole of the Gamma function. We use the convention that${\mathtt{z}}^{\left(r\right)}:=0$such that$\mathtt{z}+1-r$is a pole of the Gamma function and$\mathtt{z}+1$is not a pole of the Gamma function.

Definition 3

 (see Definition 2.25 in [1]). Let $\beta >0$ be the order, $c_0\in \mathcal{R}$ be a starting point and $\mathtt{y}:{\mathrm{N}}_{c_0+1}\to \mathcal{R}$ be a given function. Then, the β-th order delta fractional sum is defined for $\mathtt{z}\in {\mathrm{N}}_{c_0+\beta }:$

$$\begin{array}{c}\hfill {\displaystyle \left({{}_{c_0}\Delta }^{-\beta }\mathtt{y}\right)\left(\mathtt{z}\right):=\sum _{r=c_0}^{\mathtt{z}-\beta }{h}_{\beta -1}\left(\mathtt{z},r+1\right)\mathtt{y}\left(r\right).}\end{array}$$

Definition 4

 (see Theorem 2.33 in [1]). Suppose that $\mathtt{y}:{\mathrm{N}}_{c_0}\to \mathcal{R}$, $\beta >0$, and $P-1<\beta <P$ for $P\in {\mathrm{N}}_0$. Then, the β-th order delta fractional difference is defined for $\mathtt{z}\in {\mathrm{N}}_{c_0+P-\beta }:$

$$\begin{array}{c}\hfill {\displaystyle \left({{}_{c_0}\Delta }^{\beta }\mathtt{y}\right)\left(\mathtt{z}\right):=\sum _{r=c_0}^{\mathtt{z}+\beta }{h}_{-\beta -1}\left(\mathtt{z},r+1\right)\mathtt{y}\left(r\right).}\end{array}$$

3. Negative Lower Bound Results

As a first step, we wish to establish our first result in which $\left({\Delta }_{c_0}^{\beta }\mathtt{y}\right(c_0+1)$ can be negative, but it still remains $\mathtt{y}$ monotone increasing for a time step $c_0+1$. Recall that $\mathtt{y}$ is increasing at t if $\left(\Delta \mathtt{y}\right)\left(\mathtt{z}\right)\ge 0$ or $\mathtt{y}(\mathtt{z}+1)\ge \mathtt{y}\left(\mathtt{z}\right)$ for each $\mathtt{z}\in {\mathrm{N}}_{c_0}$. Having the relationship between the sign of $\left({\Delta }_{c_0}^{\beta }\mathtt{y}\right(c_0+1)$ and $(\Delta \mathtt{y})(c_0+1)$ can be relevant for computation purposes. This is due to the fact that $(\Delta \mathtt{y})(c_0+1)$ can be taken from $\left({\Delta }_{c_0}^{\beta }\mathtt{y}\right(c_0+1)$. One can understand this analysis further in the following theorem.

Theorem 1.

Let the function$\mathtt{y}$be defined on${\mathrm{N}}_{c_0+1}$, and let$1<\beta <2$and$\theta >0$. Assume that

$$\begin{array}{c}\hfill \left({{}_{c_0}\Delta }^{\beta }\mathtt{y}\right)(c_0+1)>-\theta \mathtt{y}\left(c_0\right).\end{array}$$

(1)

If$\mathtt{y}(c_0+1)\ge \mathtt{y}\left(c_0\right)\ge 0$and${\displaystyle \frac{\mathrm{\Gamma }(3-\beta )}{2\mathrm{\Gamma }(1-\beta )}}<-\theta ,$then$\left(\Delta \mathtt{y}\right)(c_0+1)\ge 0.$

Proof. 

By making use of Theorem 3.2 in [25] and the condition (1), we have

$$\begin{array}{c}\left(\Delta \mathtt{y}\right)(c_0+\ell +1)\ge -\mathtt{y}\left(c_0\right)\left[{\displaystyle \frac{\mathrm{\Gamma }(\ell +3-\beta )}{\mathrm{\Gamma }(1-\beta )\mathrm{\Gamma }(\ell +3)}}+\theta \right]\hfill \\ -{\displaystyle \sum _{r=c_0}^{c_0+\ell }}{h}_{-\beta }(c_0+\ell +2-\beta ,r+1)\left(\Delta \mathtt{y}\right)\left(r\right),\hfill \end{array}$$

(2)

for each $\ell \ge 0$, where $\ell =\mathtt{z}+\beta -c_0-2$ for $\mathtt{z}\in {\mathrm{N}}_{c_0+2-\beta }$. Since $\left(\Delta \mathtt{y}\right)\left(c_0\right)\ge 0$ by assumption, we have

$$\begin{array}{c}\hfill -\sum _{r=c_0}^{c_0}{h}_{-\beta }(c_0+2-\beta ,r+1)\left(\Delta \mathtt{y}\right)\left(r\right)=-\underset{<0}{\underbrace{(1-\beta )}}\underset{\ge 0}{\underbrace{(\Delta \mathtt{y})\left(c_0\right)}}\ge 0.\end{array}$$

(3)

Again, since $\mathtt{y}\left(c_0\right)>0,$ then by making use of inequalities (2) and (3) with $\ell =0$ we find that $\left(\Delta \mathtt{y}\right)(c_0+1)\ge 0$ as required such that

$$\begin{array}{c}\hfill \frac{\mathrm{\Gamma }(3-\beta )}{\mathrm{\Gamma }(1-\beta )\mathrm{\Gamma }\left(3\right)}+\theta <0,\end{array}$$

(4)

but this inequality is true according to the assumption. Thus, the proof is completed. □

Remark 1.

The applicability of Theorem 1 needs the allowable range of θ as follows:

$$\begin{array}{c}\hfill \theta \in \left(0,-{\displaystyle \frac{1}{2}}(2-\beta )(1-\beta )\right).\end{array}$$

for a fixed$\beta \in (1,2)$. Furthermore, Figure 1 shows the graph of$\beta \mapsto \frac{\mathrm{\Gamma }(3-\beta )}{2\mathrm{\Gamma }(1-\beta )}$for$\beta \in (1,2)$.

Let us define the set ${\mathcal{D}}_{\ell ,\theta }$ as follows

$$\begin{array}{c}\hfill {\mathcal{D}}_{\ell ,\theta }:=\left\{\beta \in (1,2):\frac{\mathrm{\Gamma }(\ell +3-\beta )}{\mathrm{\Gamma }(1-\beta )\mathrm{\Gamma }(\ell +3)}<-\theta \right\}\subseteq (1,2),\ell \ge 0.\end{array}$$

(5)

The following lemma shows that ${\left\{{\mathcal{D}}_{\ell ,\theta }\right\}}_{\ell =c_0+1}^{\infty }$ is the collection of decreasing sets in a nested form for each $\theta >0$. Further, we can see that this lemma is almost similar to Lemma 3.2 in [11].

Lemma 1.

Let$1<\beta <2$. Then, we have that${\mathcal{D}}_{\ell +1,\theta }\subseteq {\mathcal{D}}_{\ell ,\theta }$for each$\theta >0$and$\ell \ge 0$. Additionally, we have that${\displaystyle \bigcap _{\ell =0}^{\infty }}{\mathcal{D}}_{\ell ,\theta }=⌀$.

Proof. 

Let $\beta \in {\mathcal{D}}_{\ell +1,\theta }$ for some fixed $\ell \ge 0$ and arbitrary $\theta >0$. Then, we have

$$\begin{array}{cc}\hfill & \frac{\mathrm{\Gamma }(\ell +4-\beta )}{\mathrm{\Gamma }(1-\beta )\mathrm{\Gamma }(\ell +4)}=\frac{(\ell +3-\beta )\mathrm{\Gamma }(\ell +3-\beta )}{(\ell +3)\mathrm{\Gamma }(1-\beta )\mathrm{\Gamma }(\ell +3)}<-\theta .\hfill \end{array}$$

Considering $1<\beta <2$ and $\ell \ge 0$, we have $0<{\displaystyle \frac{\ell +3-\beta }{\ell +3}}<1$. Therefore, we have

$$\begin{array}{c}\hfill \frac{\mathrm{\Gamma }(\ell +3-\beta )}{\mathrm{\Gamma }(1-\beta )\mathrm{\Gamma }(\ell +3)}<-\theta .\underset{>1}{\underbrace{\frac{\ell +3}{\ell +3-\beta }}}<-\theta .\end{array}$$

This implies that $\beta \in {\mathcal{D}}_{\ell ,\theta }$, and thus ${\mathcal{D}}_{\ell +1,\theta }\subseteq {\mathcal{D}}_{\ell ,\theta }$, as required.

For the second part of the theorem, we use the fact that

$$\begin{array}{c}\hfill \frac{\mathrm{\Gamma }(\ell +3-\beta )}{\mathrm{\Gamma }(1-\beta )\mathrm{\Gamma }(\ell +3)}\equiv \frac{\mathrm{\Gamma }(n-c_0-\beta )}{\mathrm{\Gamma }(1-\beta )\mathrm{\Gamma }(n-c_0)},\end{array}$$

where we use the change of the variable $\ell :=n-c_0-3$ for each $n\in {\mathrm{N}}_{c_0+3}$. Then, by using the same sort of calculation as in Equation (3.25) in [11], we can deduce:

$$\begin{array}{cc}\hfill \underset{\ell \to \infty }{lim}\frac{\mathrm{\Gamma }(\ell +3-\beta )}{\mathrm{\Gamma }(1-\beta )\mathrm{\Gamma }(\ell +3)}& =\underset{n\to \infty }{lim}\frac{\mathrm{\Gamma }(n-c_0-\beta )}{\mathrm{\Gamma }(1-\beta )\mathrm{\Gamma }(n-c_0)}\hfill \\ \hfill & =\underset{n\to \infty }{lim}\left[\frac{\mathrm{\Gamma }(n-c_0-\beta )}{{(n-c_0)}^{-\beta }\mathrm{\Gamma }(n-c_0)}·\frac{{(n-c_0)}^{-\beta }}{\mathrm{\Gamma }(1-\beta )}\right]=0.\hfill \end{array}$$

Considering $\theta >0$, we see that

$$\begin{array}{c}\hfill {\ell }_0:=sup\left\{\ell \ge 0:\frac{\mathrm{\Gamma }(\ell +3-\beta )}{\mathrm{\Gamma }(1-\beta )\mathrm{\Gamma }(\ell +3)}<-\theta \right\}<+\infty ,\end{array}$$

which implies that ${\displaystyle \bigcap _{\ell =0}^{\infty }}{\mathcal{D}}_{\ell ,\theta }=⌀$. Hence the result. □

In conclusion, Lemma 1 gives us a chance to extend Theorem 1 to be valid for all elements in ${\mathrm{N}}_{c_0}^T$ for some $T\in {\mathrm{N}}_{c_0}$. The following corollary is dedicated for that result.

Corollary 1.

Assume that$1<\beta <2$and the function$\mathtt{y}:{\mathrm{N}}_{c_0+1}\to \mathcal{R}$satisfies

$$\begin{array}{c}\hfill \left({{}_{c_0}\Delta }^{\beta }\mathtt{y}\right)(c_0+\ell +1)>-\theta \mathtt{y}\left(c_0\right),\end{array}$$

(6)

for each$\ell \in \mathcal{K}:=\left\{0,1,\dots ,n\right\}$and some$n\ge 0$. If$\mathtt{y}(c_0+1)\ge \mathtt{y}\left(c_0\right)\ge 0$and$\beta \in {\mathcal{D}}_{n,\theta }$, then$\left(\Delta \mathtt{y}\right)(c_0+\ell +1)\ge 0$for each$\ell \in \mathcal{K}\cup \{-1\}$.

Proof. 

By using the assumption $\beta \in {\mathcal{D}}_{n,\theta }$ and according to Lemma 1, we see that $\beta \in {\mathcal{D}}_{n,\theta }={\mathcal{D}}_{n,\theta }\cap {\displaystyle \bigcap _{\ell =0}^{n-1}}{\mathcal{D}}_{\ell ,\theta }$. This leads to

$$\begin{array}{c}\hfill \frac{\mathrm{\Gamma }(\ell +3-\beta )}{\mathrm{\Gamma }(1-\beta )\mathrm{\Gamma }(\ell +3)}<-\theta ,\end{array}$$

(7)

for each $\ell \in \mathcal{K}$.

To end the proof, we must proceed by induction process: For $\ell =0$, we can directly obtain $\left(\Delta \mathtt{y}\right)(c_0+1)\ge 0$ as in Theorem 1 by using inequalities (6) and (7) and the given assumption that $(\Delta \mathtt{y})\left(c_0\right)\ge 0$. Therefore, by iterating the inequality (2) inductively, we can obtain

$$\begin{array}{c}\hfill \left(\Delta \mathtt{y}\right)(c_0+\ell +1)\ge 0,\end{array}$$

for each $\ell \in \mathcal{K}\cup \{-1\}$ as required. □

4. Numerical Performances

In the last part of the study, we will perform numerical analysis of ${\mathcal{D}}_{\ell ,\theta }$. The computations in this section were performed with MATLAB software. According to Theorem 1 and Corollary 1, we see that these sets are the main door to understanding the number of time steps which are applicable on the corollary for arbitrary selection of $\beta$ and $\theta$. For this reason, we dedicate an example together with four numerical figures later.

Example 1.

Considering Definition 4 for $\mathtt{z}:=c_0+\ell +2-\beta$:

$$\begin{array}{cc}\hfill \left({{}_{c_0}\Delta }^{\beta }\mathtt{y}\right)(c_0+\ell +2-\beta )& =\frac{1}{\mathrm{\Gamma }(-\beta )}\sum _{r=c_0}^{(c_0+\ell +2-\beta )+\beta }{\left(c_0+\ell +2-\beta -r-1\right)}^{(-\beta -1)}\mathtt{y}\left(r\right)\hfill \\ \hfill & =\frac{1}{\mathrm{\Gamma }(-\beta )}\sum _{r=c_0}^{c_0+\ell +2}{\left(c_0+\ell +1-\beta -r\right)}^{(-\beta -1)}\mathtt{y}\left(r\right).\hfill \end{array}$$

For $\ell =c_0=0$, it follows that

$$\begin{array}{cc}\hfill \left({{}_0\Delta }^{\beta }\mathtt{y}\right)(2-\beta )& =\frac{1}{\mathrm{\Gamma }(-\beta )}\sum _{r=0}^2{\left(1-\beta -r\right)}^{(-\beta -1)}\mathtt{y}\left(r\right)\hfill \\ \hfill & =\frac{1}{\mathrm{\Gamma }(-\beta )}\left\{{(1-\beta )}^{(-\beta -1)}\mathtt{y}\left(0\right)+{(-\beta )}^{(-\beta -1)}\mathtt{y}\left(1\right)+{(1-\beta )}^{(-\beta -1)}\mathtt{y}\left(2\right)\right\}\hfill \\ \hfill & =\frac{1}{\mathrm{\Gamma }(-\beta )}\left\{\mathtt{y}\left(0\right)\frac{(1-\beta )(-\beta )\mathrm{\Gamma }(-\beta )}{\mathrm{\Gamma }\left(3\right)}+\mathtt{y}\left(1\right)\frac{(-\beta )\mathrm{\Gamma }(-\beta )}{\mathrm{\Gamma }\left(2\right)}+\mathtt{y}\left(2\right)\frac{\mathrm{\Gamma }(-\beta )}{\mathrm{\Gamma }\left(1\right)}\right\}\hfill \\ \hfill & =\frac{1}{2}(-\beta )(1-\beta )\mathtt{y}\left(0\right)+(-\beta )\mathtt{y}\left(1\right)+\mathtt{y}\left(2\right).\hfill \end{array}$$

If we take $\beta =1.99,\mathtt{y}\left(0\right)=1,\mathtt{y}\left(1\right)=1,\mathtt{y}\left(2\right)=1.001$, and $\theta =0.004$, we have

$$\begin{array}{cc}\hfill \left({{}_0\Delta }^{1.99}\mathtt{y}\right)\left(0.01\right)& =\frac{1}{2}(-1.99)(-0.99)\left(1\right)+(-1.99)\left(1\right)+\left(1.001\right)\hfill \\ \hfill & =-0.00395>-0.004=-\theta \mathtt{y}\left(1\right).\hfill \end{array}$$

We can see that $\left({{}_0\Delta }^{1.99}\mathtt{y}\right)\left(0.01\right)<0$. Moreover, we found that $(\Delta \mathtt{y})\left(1\right)>0$ as Theorem 1 correctly predicted. Therefore, the collection of functions is fruitful when Theorem 1 is verified.

In Figure 2, the sets ${\mathcal{D}}_{\ell ,0.004}$ and ${\mathcal{D}}_{\ell ,0.0002}$ are demonstrated for different values of ℓ, respectively, in Figure 2a,b. These mean the range of $\beta \in {\mathcal{D}}_{\ell ,\theta }$ with $\theta =0.004$ and $\theta =0.0002$, respectively. It can be noted that ${\mathcal{D}}_{\ell ,0.004}$ and ${\mathcal{D}}_{\ell ,0.0002}$ decrease by increasing the values of ℓ. In addition, the set ${\mathcal{D}}_{\ell ,0.004}$ is empty for $\ell \ge 25$ (see Figure 2a), while the set ${\mathcal{D}}_{\ell ,0.0002}$ is non-empty for many larger values of ℓ up to 170 (see Figure 2b). Surprising, the measure of ${\mathcal{D}}_{\ell ,0.004}$ is not asymmetrically distributed when k increases as in Figure 2a, while the same asymmetrical distribution can be observed for ${\mathcal{D}}_{\ell ,0.0002}$ when ℓ increases as in Figure 2b. There is still no idea about the observation of this symmetric behavior. In fact, it is not clear why the discrete delta operator ${{}_{c_0}\Delta }^{\beta }$ seems to encode monotonically when $\beta \to 1$ rather than for $\beta \to 2$, specifically, it gives a maximal information when $\beta \approx 1.2$. Note that ${\mathrm{N}}_{c_0}^b:=\{c_0,c_0+1,c_0+2,\cdots ,b\}$.

On the other hand, we consider two heat maps in Figure 3a,b, these represent the cardinality of the set $\{\ell :\beta \in {\mathcal{D}}_{\ell ,\theta }\}$. In these figures, the warm colors are such as red ones and the cool colors are such as blue ones. In addition, $\beta$ is on the vertical axis and $\theta$ is on the horizontal axis.

In Figure 3a, we chose the values of $\theta$ between 0 and $0.0012$ (in the x-axis), but in the Figure 3b, we chose the values of $\theta$ between $0.0012$ and $0.004$. Again, a symmetry concentration can be observed in Figure 3a,b. The color shifts slightly closer to one rather than closer to two (look at the right side in Figure 3b). In addition, we can see that the warmest colors move strongly towards the lower values of $\beta$ for small enough value of $\theta$ (look at the left side in Figure 3a). In fact, when as $\beta$ increases to up to $1.65$, it drops sharply from magenta to cyan, which implies a sharp decrease in the cardinality of $\{\ell :\beta \in {\mathcal{D}}_{\ell ,\theta }\}$ when a slight increase happens in $\theta$. Moreover, we do not have a clear idea about the sensitivity of the set $\left|\{\ell :\beta \in {\mathcal{D}}_{\ell ,\theta }\}\right|$ when slight increasing in $\theta$ is observed for $\beta$ close to 2 compares with $\beta$ close to 1. However, our numerical data strongly recommend this correlation.

In general, the numerical results can be summarized as follows: For a smaller value of $\theta$, the set ${\mathcal{D}}_{\ell ,\theta }$ approaches to remain non-empty, unlikely for a larger value of $\theta$. Furthermore, for every $\theta$ and $\beta$ close to 1, the set ${\mathcal{D}}_{\ell ,\theta }$ seems to be concentrated strongly toward those values of $\beta$, but less concentrated for those values of $\beta$ close to 2. Specifically, the set ${\mathcal{D}}_{\ell ,\theta }$ is most concentrated for $\beta \approx 1.2$ as illustrated in both Figure 2a,b and Figure 3a,b. All in all, we conclude that Corollary 1 may be applicable for the largest number of time steps when $\beta$ approaches to $1.2$ and $0<\theta \ll 1$.

5. Conclusions

Based on the conditions $\left({{}_{c_0}\Delta }^{\beta }\mathtt{y}\right)(c_0+1)>-\theta \mathtt{y}\left(c_0\right)$, $\mathtt{y}(c_0+1)\ge \mathtt{y}\left(c_0\right)\ge 0$ and ${\displaystyle \frac{\mathrm{\Gamma }(3-\beta )}{2\mathrm{\Gamma }(1-\beta )}}<-\theta$, we have found that $\mathtt{y}$ is convex at $c_0+1$, that is, $\left(\Delta \mathtt{y}\right)(c_0+1)\ge 0$, in Theorem 1.

By defining the set ${\mathcal{D}}_{\ell ,\theta }$ in (5), we have proved that ${\mathcal{D}}_{\ell +1,\theta }\subseteq {\mathcal{D}}_{\ell ,\theta }$ in Lemma 1 for each $1<\beta <2$, $\theta >0$ and $\ell \ge 0$. Based on this lemma, we have extended the result in Theorem 1 to obtain the monotonicity of $\mathtt{y}$ at $\mathtt{z}\in \{c_0,c_0+1,c_0+2,\cdots ,T\}$, for some $T\in {\mathrm{N}}_{c_0}$ in Corollary 1.

On the other hand, the last section of this work is devoted to see the possibility of information reduction regarding the monotonicity of $\mathtt{y}$ despite the non-positivity of $\left({\Delta }_{c_0}^{\beta }\mathtt{y}\right)(c_0+1)$ by means of numerical simulation as demonstrated in Figure 2 and Figure 3.

Due to the large employment of discrete fractional operators in a wide variety of mathematical analysis and discrete fractional calculus, the presented methodology is suitable not only for discrete Riemann–Liouville fractional operators with standard kernel, but also for many other discrete fractional operators including the exponential or Mittag-Leffler kernels, as there are some recently published papers including them, see for example [3,5,6,9].