On Riemann–Liouville and Caputo Fractional Forward Difference Monotonicity Analysis

Monotonicity analysis of delta fractional sums and differences of order υ ∈ (0, 1] on the time scale h Z 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 υ-increasing on Ma+υh,h, where the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a + υh is greater or equal to zero, and then, we can show that y(z) is υ-increasing on Ma+υh,h, where the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+ υh is greater or equal to− 1 Γ(1−υ) (z− (a+ υh)) (−υ) h y(a+ υh) for each z ∈ Ma+h,h. Conversely, if y(a + υ h) is greater or equal to zero and y(z) is increasing on Ma+υh,h, we show that the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a + υh is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h-difference of order υ of a function y(z) starting at a + υh is greater or equal to − 1 Γ(1−υ) (z− (a + υh)) (−υ) h y(a + υh) on Ma,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 h Z utilizing the monotonicity results.


Introduction
Fractional differentiation and integration have opened many new doors for researchers in recent decades due to their wide and novel applicability in many fields of science including mathematical analysis, technology, and engineering (see [1][2][3][4][5][6][7]). Many techniques are used to deal with these new differential and integral operators; for instance, some researchers used analytical techniques including Laplace transform, spline interpolation, Green function, Crank-Nicolson approximation method, method of separation of variable, and many others to derive exact solutions to linear differential or integral equations (see [8][9][10][11][12][13][14]). Using the fixed-point technique, some researchers provided the conditions under which differential and integral equations have unique solutions. Some others provided numerical schemes that could be used to solve numerically differential and integral equations with fractional order. Very recently, fractional differentiation and integration found application in image processing, where the fractional kernel is used to remove noise in a given image.
and the delta right Caputo fractional h-difference of order υ is defined by In the following lemma, we show that z

Proof. From Definition 1 and Equation (3), we have
Since υ, h > 0, it follows that h , and this completes the proof.
The following theorem can be seen as an equivalence definition to the delta RL fractional h-differences. Theorem 1. Let 0 < υ < 1. Then, the delta left and delta right RL fractional h-differences of order υ defined on M a+(1−υ)h,h and b−(1−υ)h,h M, respectively, are defined by Proof. From Definitions 1 and 3, we have Then by using Lemma 1, we get which is the required Equation (4). In the same manner as Equation (4), we can prove Equation (5), and thus, the proof is completed.
A relationship between the delta RL fractional and delta Caputo fractional h-differences are presented in the following proposition.
Proof. From Definition 4 and the fact that z − σ h (rh) By using Lemma 1 and Theorem 1, we get which is the required Equation (6). Using the same technique used for Equation (6), we can prove Equation (7), and thus, the proof is completed.
In the following lemma, we prove and modify a power rule that appeared in ( [22], Lemma 4). We state and prove the modified result in a simpler way as follows: for z ∈ M a+(µ+υ)h,h .
Proof. Following ( [4], Lemma 1), we have for z ∈ M a+(µ+υ)h,h . Calculating both sides of Equation (9), we get Substituting the LHS and RHS results into Equation (9), we get Multiplying by a positive constant h υ+µ on both sides of the equality, we get the desired Equation (8).

The Monotonicity Results
This section illustrates the monotonicity of a discrete function. Monotonicity analysis of discrete functions defined on M 0 1 was originally introduced in [27], and there is extensive literature on monotonicity analysis techniques and its extensions on M a,h ; for example, see [22,23,26].
Definition 5 (see [22,23,26]). Let 0 < h, υ ≤ 1, and y : M a,h → R be a function satisfying y(a) ≥ 0. Then, y is called an υ-increasing function on M a,h if Observe that, if y(z) is increasing on M a,h , then y(z + h) ≥ y(z) for all z ∈ M a,h , and thus, y(z) is υ-increasing on M a,h . Definition 6 (see [22,23,26]). Let 0 < h, υ ≤ 1, and y : M a,h → R be a function satisfying y(a) ≥ 0. Then, y is called an υ-decreasing function on M a,h if Observe that, if y(z) is decreasing on M a,h , then y(z + h) ≤ y(z) for all z ∈ M a,h , and thus, y(z) is υ-decreasing on M a,h .

Remark 1.
Note that, if υ = 1 in Definition 5, then the increasing and υ-increasing concepts coincide and that, if υ = 1 in Definition 6, then the decreasing and υ-decreasing concepts coincide.
To provide motivation for the above monotonicity definitions, we prove a few fundamental results of the discrete RL and Caputo fractional operators.
Proof. From the assumption and proof of Theorem 1, we have For z = a, we see that and, thus, y(a + υh) ≥ 0. For z = a + h, we see that and, thus, y(a + υh + h) ≥ υy(a + υh) since h −υ > 0 and y(a + υh) ≥ 0. Now, inductively, we show that Replace z by z + h in Equation (10) to get It follows that This implies that y(z + υ h + h) ≥ y(z + υ h), and this completes the proof.
Proof. The proof follows from Proposition 1 and Theorem 2.

Continue this process to get
which completes the proof.

Fractional Forward Difference Initial Value Problem and Mean Value Theorem
In this section, we move on from monotonicity analysis to the MVT. Having established the monotonicity analysis for the discrete RL and Caputo fractional operators, we now obtain the MVT using those discrete monotonicity results.

Establishing the Riemann-Liouville case
The following is the main results to start off the MVT here.
Theorem 8. y is a solution of RL − FFDE in Equation (15) with the initial condition in Equation (16) if and only if it has the representation Proof. This proof follows from Theorem 7.
According to Theorem 7, we can write where It is worthwhile to mention that analyzing the monotonicity property of such fractional difference operators is useful for better understanding the qualitative properties of solutions of different discrete fractional dynamic equations. The monotonicity properties are part of the basics of the discrete fractional calculus, where for example, we used them in this article to prove a discrete fractional version of MVT. Then, it is of interest to obtain the following MVT for RL − FFDE in Equation (17).
Then, by applying Lemma 2 and the identity (3), it follows that Hence, the proof is complete.
Theorem 10. Let y : M a,h → R and 0 < h ≤ 1, 0 < υ < 1; then, Proof. We use Proposition 1 in order to proceed Taking a+h ∆ −υ h on both sides of Equation (24) and then using Theorem 7 and Lemma 5, we get and after some simplifications, the result is obtained.
Consider the following Caputo−FFDE: with the initial condition where a ∈ R, h, υ ∈ (0, 1), and c 2 is a constant.
Theorem 11. y is a solution of Caputo−FFDE in Equation (25) with the initial condition in Equation (26) if and only if it has the representation y(z) = c 2 + a+h ∆ −υ h f (z, y(z)).
Proof. The proof follows from Theorem 10 directly.

Remark 2.
In the case of Caputo, MVT for Caputo−FFDE in Equation (23), does not hold, where the assumptions of Theorem 9 are supposed to be given. The reason for this is that we do not know whether C a+υh ∆ υ h g (z) > 0 when, by assumption and Corollary 3, we have since the three quantities 1 Γ(1−υ) , z − (a + υh) , and g(a + υh) are all positive for υ ∈ (0, 1) and z ∈ I.

Conclusions
In this paper, a discrete υ-monotonicity analysis for discrete functions defined on M a,h in the framework of the discrete RL fractional sums, and RL and Caputo fractional differences on the time scale h Z were successfully studied. The relation between delta RL and delta Caputo fractional h-differences was established. The RL − FFDE and Caputo−FFDE were considered, and their solutions were discussed. Utilizing the monotonicity results and RL − FFDE solution, the MVT was presented. However, MVT for Caputo−FFDE was not valid using its corresponding monotonicity results and Caputo−FFDE.
A further extension of this study includes improving our findings to study other classes of discrete fractional sums and differences, including those of exponential kernel CFR a ∆ υ h y (z) or Mittag-Liffler kernel ABR a ∆ υ h y (z), defined in [22]. Fortunately, some research is in progress in this area.