Monotonicity Analysis of Fractional Proportional Differences

Department of Mathematics and Statistics, Arab American University, P.O. Box 240 Jenin 13, Zababdeh, State of Palestine Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia Department of Medical Research, China Medical University, Taichung 40402, Taiwan Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan


Introduction
Many problems in science, engineering, and media can be formulated using continuous and discrete fractional calculus [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. e fractional sums and differences and their monotonicity properties are deeply studied in [15][16][17][18][19][20][21][22][23][24][25]. In [26], Atangana and Baleanu solved the fractional heat transfer model using new fractional derivatives with exponential kernels, and they presented many applications of the new notations of fractional derivatives. Applications of discrete fractional calculus are successfully discussed by many researchers in the last decade, for example, in [27][28][29]. Recently, studying the monotonicity for fractional difference operators with nonsingular discrete kernels is under focus [30,31]. Monotonicity results for fractional difference operators with discrete exponential kernels were studied in [32] when the time step h � 1. In [3], deep monotonicity analysis is done for nabla h− discrete fractional differences with a discrete Mittag-Leffler kernel in the time scale hZ with 0 < ε < 1 and 0 < h ≤ 1. e results of the research generalized those obtained in [22] where 0 < ε < 0.5 and h � 1. After that, monotonicity analysis of fractional proportional differences is studied and then the results are prettified by formulating a new version of mean value theorem as an application. In [33], the nabla fractional sums and differences of order 0 < ε < 1 on the time scale hZ where 0 < h ≤ 1 are formulated, and the monotonicity results for the nabla h− Caputo fractional difference operator were concluded. In this paper, the authors formulated the nabla discrete new Riemann-Liouville (RL) and Caputo fractional proportional differences of order 0 < ε < 1 on the time scale Z. ey also proved a new version of the fractional proportional difference of the mean value theorem (MVT) on Z.
e article is organized as follows: Section 2 presents the main definitions and needed preliminaries. In Section 3, the monotonicity results for fractional proportional differences are classified. In Section 4, we formulate a new version of the mean value theorem as an application. Finally, we provide the conclusions in Section 5.
Definition 3. For any real number α, the α rising function is Definition 4 (nabla fractional proportional sums).
For a function χ: N c ⟶ R, ρ > 0, and ε ∈ C, 0 < Re(ε) < 1, the nabla left fractional proportional sum of χ starting at c is defined by We notice that by setting ρ � 1, the given definitions of the fractional sums are generalizations of the Riemann fractional sums. Proof.
□ 2 Discrete Dynamics in Nature and Society Lemma 2. Let χ: N c ⟶ R, p � (ρ − 1/ρ), 0 < ε < 1, and 0 < ρ ≤ 1, then Proof. hence, Note that if ρ � 1, we get For 0 < ρ ≤ 1, ε ∈ C, 0 < Re(ε) < 1, and χ be a function defined on N c or on d N , then the left Riemann-Liouville fractional proportional difference starting at c is defined by and the right Riemann-Liouville fractional proportional difference ending at d is defined by We notice that by setting ρ � 1, the given definitions of the fractional differences are generalizations of the Riemann fractional differences.

Definition 6 (Caputo fractional proportional differences)
For 0 < ρ ≤ 1, ε ∈ C, 0 < Re(ε) < 1, and χ be a function defined on N c or on d N , then the left Caputo fractional proportional difference starting at c is defined by and the right Caputo fractional proportional difference ending at d is defined by We notice that by setting ρ � 1, the given definitions of the fractional differences are generalizations of the Caputo fractional differences.

Proposition 1 (the relation between nabla RL and Caputo fractional proportional differences)
For any ε ∈ C, 0 < Re(ε) < 1, and 0 < ρ ≤ 1, the relation between nabla RL and Caputo fractional proportional differences is given as follows: Discrete Dynamics in Nature and Society Proof.
Discrete Dynamics in Nature and Society Discrete Dynamics in Nature and Society 5 Numerical calculations have been done in order to verify the first equation in Proposition 1.
In addition to that, the data are presented in Table 1. Proof.

Monotonicity Results
e following two monotonicity definitions are given in [18].
en, y(t) is called an α− decreasing function on N a if y(t + 1) ≤ αy(t) ∀t ∈ N a .
Using eorem 4.3 in [4] we can state the following.

Application: Mean Value Theorem (MVT)
First, for the sake of simplification, depending on eorem 6, we shall write where S(z, c) � e p (z, c) Theorem 7 (the fractional proportional difference MVT)

Let Θ and θ be functions defined on
Assume that θ is strictly increasing, θ(c) > 0, and 0 < ε< 1 and 0 < ρ≤ 1. en, there exist s 1 , s 2 ∈ N c ∩ d N such that Since θ is strictly increasing, then by eorem 3 we have Applying the fractional sum operator associated to ( R c ∇ ε,ρ θ)(z) on both sides of the inequality, by means of (40), we get or we have For z � d , we get To prove the theorem, we use the proof by contradiction. Assume (42) is not true, then either or Again, since θ is strictly increasing, then by eorem 3 we conclude that Hence, (47) becomes Discrete Dynamics in Nature and Society ∀z ∈ N c ∩ d N. (50) Applying the fractional sum operator on both sides of the inequality at z � d and by making use of (43), we see that and hence, Θ(d) < Θ(d), which is a contradiction. In a similar way, (48) leads to contradiction.

Conclusions
e conclusions of this article are summarized as follows: (1) e summation and difference of a discrete fractional proportional have been detected. (2) e nabla discrete new Riemann-Liouville and Caputo fractional proportional differences of order 0 < ε < 1 on the time scale Z have been formulated. (3) e fractional proportional sums associated to ( R c ∇ ε,ρ χ)(z) with order 0 < ε < 1 have been defined. (4) e relation between nabla Riemann-Liouville and Caputo fractional proportional differences has been detected. (5) e monotonicity results for the nabla Caputo fractional proportional difference which are if ( R c− 1 ∇ ε,ρ χ)(z) > 0, then χ(z) is ερ − increasing; if χ(z) is strictly increasing on N c and χ(c) > 0, then ( R c− 1 ∇ ε,ρ χ)(z) > 0 has been proved as well. (6) A new version of the fractional proportional difference of the mean value theorem on Z has been proved as an application.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
All the authors participated in obtaining the main results of this manuscript and drafted the manuscript. All authors read and approved the final manuscript.