Abstract
In this work, the nabla discrete new Riemann–Liouville and Caputo fractional proportional differences of order on the time scale are formulated. The differences and summations of discrete fractional proportional are detected on , and the fractional proportional sums associated to with order are defined. The relation between nabla Riemann–Liouville and Caputo fractional proportional differences is derived. The monotonicity results for the nabla Caputo fractional proportional difference are proved; specifically, if then is increasing, and if is strictly increasing on and , then . As an application of our findings, a new version of the fractional proportional difference of the mean value theorem (MVT) on is proved.
1. Introduction
Many problems in science, engineering, and media can be formulated using continuous and discrete fractional calculus [1–14]. The fractional sums and differences and their monotonicity properties are deeply studied in [15–25]. 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 [27–29]. 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 . In [3], deep monotonicity analysis is done for nabla discrete fractional differences with a discrete Mittag–Leffler kernel in the time scale with and . The results of the research generalized those obtained in [22] where and . 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 on the time scale where are formulated, and the monotonicity results for the nabla 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 on the time scale . They also proved a new version of the fractional proportional difference of the mean value theorem (MVT) on .
The 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.
2. Definitions and Preliminary Results
Definition 1. The discrete proportional difference of order for the function is defined byand is an integer.
Definition 2. Let , , and , then .
Definition 3. For any real number , the rising function is , such that , , where is the gamma function.
Definition 4. (nabla fractional proportional sums).
For a function , , and , , the nabla left fractional proportional sum of starting at is defined by
For the function , the nabla right fractional proportional sum ending at is defined by
We notice that by setting , the given definitions of the fractional sums are generalizations of the Riemann fractional sums.
Lemma 1. Let be a function, , , and , then
Proof.
Lemma 2. Let , , , and , then
Proof. hence,Note that if , we get
Definition 5. (Riemann–Liouville (RL) fractional proportional differences)
For , , , and be a function defined on or on , then the left Riemann–Liouville fractional proportional difference starting at is defined byand the right Riemann–Liouville fractional proportional difference ending at is defined by
We notice that by setting , the given definitions of the fractional differences are generalizations of the Riemann fractional differences.
Definition 6. (Caputo fractional proportional differences)
For , , , and be a function defined on or on , then the left Caputo fractional proportional difference starting at is defined byand the right Caputo fractional proportional difference ending at is defined by
We notice that by setting , 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 , , and , the relation between nabla RL and Caputo fractional proportional differences is given as follows:(i)(ii)
Proof. Numerical calculations have been done in order to verify the first equation in Proposition 1. The values used are , , and . The results are illustrated in Figure 1.
In addition to that, the data are presented in Table 1.
Lemma 3. Let , , and , then .
Proof.
3. Monotonicity Results
The following two monotonicity definitions are given in [18].
Definition 7. Let be a function satisfying , . Then, is called an increasing function on if .
Definition 8. Let be a function satisfying , . Then, is called an decreasing function on if .
In the following, we report the new proportional monotonicity main results.
Theorem 1. Let be a function, and suppose that () for and . Then, .
Proof. LetThen,Hence, from the assumption, we have . That is,Therefore,Hence,Clearly, . So, we can start the induction from the next step. When , we get ; also, when , we haveNow for , replace by , then we getHence, is increasing which completes the proof.
Using Theorem 1 and Proposition 1 we can state the following Caputo fractional proportional difference monotonicity result.
Corollary 1. Let be a function, and suppose that for and . Suppose thatthen .
Proof. now, from the assumption we havehence,which means that
Now, from Theorem 1, we conclude that is .
Theorem 2. Let be a function which satisfies , and suppose that for and . If is increasing on , then we have
Proof. Sincewhen , we have from the assumption .
Clearly, . So, we can start the induction from the next step.
Assume that . We shall show that .
Since, from assumption, is increasing, it follows that :
Theorem 3. Let be a function which satisfies and be strictly increasing on , where and Then,
Proof. Sincewhen , we have . Clearly, , and so we can start the induction from the next step.
Assume that . We shall show that .
Since, from assumption, is increasing it follows that :
Theorem 4. Let be a function, and suppose that () for and Then,
Proof. Let be a function such that ; hence,Now by Theorem 1, we conclude that is .
Hence,which isthat is to say, is .
Theorem 5. Let a function be decreasing on such that . Then, for and , we have
Proof. The proof follows by applying Theorem 2 to .
Using Theorem 4.3 in [4] we can state the following.
Theorem 6. (see [4]). For any , , , and , the following equality holds:
4. Application: Mean Value Theorem (MVT)
First, for the sake of simplification, depending on Theorem 6, we shall writewhere
Theorem 7. (the fractional proportional difference MVT)
Let and be functions defined on , where . Assume that is strictly increasing, , and and . Then, there exist such that
Proof. First we need to show that . Since is strictly increasing, then by Theorem 3 we haveApplying the fractional sum operator associated to on both sides of the inequality, by means of (40), we getor we haveFor , we getTo prove the theorem, we use the proof by contradiction. Assume (42) is not true, then eitherorAgain, since is strictly increasing, then by Theorem 3 we conclude thatHence, (47) becomesApplying the fractional sum operator on both sides of the inequality at and by making use of (43), we see thatand hence, , which is a contradiction. In a similar way, (48) leads to contradiction.
5. Conclusions
The conclusions of this article are summarized as follows:(1)The summation and difference of a discrete fractional proportional have been detected.(2)The nabla discrete new Riemann–Liouville and Caputo fractional proportional differences of order on the time scale have been formulated.(3)The fractional proportional sums associated to with order have been defined.(4)The relation between nabla Riemann–Liouville and Caputo fractional proportional differences has been detected.(5)The monotonicity results for the nabla Caputo fractional proportional difference which are if , then is increasing; if is strictly increasing on and , then has been proved as well.(6)A new version of the fractional proportional difference of the mean value theorem on has been proved as an application.
Data Availability
No data were used to support this study.
Conflicts of Interest
The 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.
Acknowledgments
The corresponding author would like to thank Prince Sultan University for funding this work through the research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM), group number RG-DES-2017-01-17.