Advanced Theoretical and Applied Studies of Fractional Differential EquationsView this Special Issue
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On the Definitions of Nabla Fractional Operators
We show that two recent definitions of discrete nabla fractional sum operators are related. Obtaining such a relation between two operators allows one to prove basic properties of the one operator by using the known properties of the other. We illustrate this idea with proving power rule and commutative property of discrete fractional sum operators. We also introduce and prove summation by parts formulas for the right and left fractional sum and difference operators, where we employ the Riemann-Liouville definition of the fractional difference. We formalize initial value problems for nonlinear fractional difference equations as an application of our findings. An alternative definition for the nabla right fractional difference operator is also introduced.
Do these two definitions lead the development of the theory of the nabla fractional difference equations in two directions?
In order to answer this question, we first obtain a relation between the operators in (1.1) and (1.2). Then we illustrate how such a relation helps one to prove basic properties of the one operator if similar properties of the other are already known.
In recent years, discrete fractional calculus gains a great deal of interest by several mathematicians. First Miller and Ross  and then Gray and Zhang  introduced discrete versions of the Riemann-Liouville left fractional integrals and derivatives, called the fractional sums and differences with the delta and nabla operators, respectively. For recent developments of the theory, we refer the reader to the papers [2, 4–19]. For further reading in this area, we refer the reader to the books on fractional differential equations [20–23].
The paper is organized as follows. In Section 2, we summarize some of basic notations and definitions in discrete nabla calculus. We employ the Riemann-Liouville definition of the fractional difference. In Section 3, we obtain two relations between the operators and . So by the use of these relations we prove some properties for -operator. Section 4 is devoted to summation by parts formulas. In Section 5, we formalize initial value problems and obtain corresponding summation equation with -operator. This section can be considered as an application of the results in Section 3. Finally, in Section 6, a definition of the nabla right fractional difference resembling the nabla right fractional sum is formulated. This definition can be used to prove continuity of the nabla right fractional differences with respect to the order .
2. Notations and Basic Definitions
Definition 2.1. (i) For a natural number , the rising (ascending) factorial of is defined by
(ii) For any real number , the rising function is defined by
Throughout this paper, we will use the following notations.(i) For real numbers and , we denote and .(ii) For , we define
Definition 2.2. Let be the backward jump operator. Then(i)the (nabla) left fractional sum of order (starting from ) is defined by
(ii)the (nabla) right fractional sum of order (ending at ) is defined by
We want to point out that the nabla left fractional sum operator has the following characteristics.(i) maps functions defined on to functions defined on .(ii) satisfies the th order discrete initial value problem (iii)The Cauchy function satisfies .
In the same manner, it is worth noting that the nabla right fractional sum operator has the following characteristics.(i) maps functions defined on to functions defined on .(ii) satisfies the th order discrete initial value problem (iii)The Cauchy function satisfies .
Definition 2.3. (i)The (nabla) left fractional difference of order is defined by
(ii)The (nabla) right fractional difference of order is defined by
Here and throughout the paper , where is the greatest integer less than or equal .
Regarding the domains of the fractional difference operators we observe the following.(i)The nabla left fractional difference maps functions defined on to functions defined on (on if we think before ).(ii)The nabla right fractional difference maps functions defined on to functions defined on (on if we think after ).
3. A Relation between the Operators and
In this section we illustrate how two operators, and are related.
Lemma 3.1. The following holds:(i), (ii).
Lemma 3.2. The following holds:
Lemma 3.3. Let and . Then for , the following equality holds
Proof. It follows from Theorem 2.1 in 
Lemma 3.4. Let be a real-valued function defined on , and let . Then
Remark 3.5. Let and . Then, by Lemma 3.2 we have
Then, using the identity
we verified that (3.2) is valid for any real .
By using Lemma 3.1, Remark 3.5, and the identity , we arrive inductively at the following generalization.
Theorem 3.6. For any real number and any positive integer , the following equality holds: where is defined on .
Lemma 3.7. For any , the following equality holds:
Proof. By using of the following summation by parts formula
On the other hand, where the identity and the convention that are used.
Remark 3.8. Let and . Then, by the help of Lemma 3.7 we have
Then, using the identity
we verified that (3.12) is valid for any real .
By using Lemma 3.7, Remark 3.8, and the identity , we arrive inductively at the following generalization.
Theorem 3.9. For any real number and any positive integer , the following equality holds: where is defined on .
We finish this section by stating the commutative property for the right fractional sum operators without giving its proof.
Lemma 3.10. Let be a real valued function defined on , and let , . Then
4. Summation by Parts Formulas for Fractional Sums and Differences
We first state summation by parts formula for nabla fractional sum operators.
Theorem 4.1. For , , , defined on and defined on , the following equality holds
Proof. By the definition of the nabla left fractional sum we have If we interchange the order of summation we reach (4.1).
Theorem 4.2. For , and defined in a suitable domain , the following are valid
We recall that , where is the Riemann-Liouville fractional integral, is valid for sufficiently smooth functions such as continuous functions. As a result of this it is possible to obtain integration by parts formula for a certain class of functions (see  page 76, and for more details see ). Since discrete functions are continuous we see that the term , for disappears in (4.4), with the application of the convention that .
Theorem 4.3. For , and defined in a suitable domain , we have
Theorem 4.4. Let be noninteger. If is defined on and is defined on , then
5. Initial Value Problems
Let us consider the following initial value problem for a nonlinear fractional difference equation where and is any real number.
Apply the operator to each side of (5.1) to obtain
Then using the definition of the fractional difference and sum operators we obtain
It follows from Lemma 3.2 that
Hence we obtain which follows from the power rule in Lemma 3.3.
Thus, we have proved the following lemma.
Remark 5.2. A similar result has been obtained in the paper  with the operator . And the initial value problem has been defined in the following form where and is any real number. The subscript of the term on the left hand side of (5.10) indicates directly that the solution has a domain starts at . The nature of this notation helps us to use the nabla transform easily as one can see in the papers [2, 12]. On the other hand, the subscript in (5.1) indicates that the solution has a domain starts at .
6. An Alternative Definition of Nabla Fractional Differences
Recently, the authors in , by the help of a nabla Leibniz’s Rule, have rewritten the nabla left fractional difference in a form similar to the definition of the nabla left fractional sum. In this section, we do this for the nabla right fractional differences.
The following delta Leibniz’s Rule will be used:
Using the following identity and the definition of the nabla right fractional difference (ii) of Definition 2.3, for , we have
By applying the Leibniz’s Rule (6.1), number of times we get
In the above, it is to be insisted that is required due the fact that the term is undefined for negative integers. Therefore we can proceed and unify the definitions of nabla right fractional sums and differences similar to Definition 5.3 in . Also, the alternative formula (6.4) can be employed, similar to Theorem 5.4 in , to show that the nabla right fractional difference is continuous with respect to .
In fractional calculus there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and derivatives. The second approach is by iterating the derivative and then defining a fractional order by making use of the binomial theorem to obtain Grünwald-Letnikov fractional derivatives. In this paper we followed the discrete form of the first approach via the nabla difference operator. However, we noticed that in the right fractional difference case we used both the nabla and delta difference operators. This setting enables us to obtain reasonable summation by parts formulas for nabla fractional sums and differences in Section 4 and to obtain an alternative definition for nabla right fractional differences through the delta Leibniz’s Rule in Section 6.
While following the discrete form of the first approach, two types of fractional sums and hence fractional differences appeared; one type by starting from and the other type, which obeys the general theory of nabla time scale calculus, by starting from in the left case and ending at in the right case. Section 3 discussed the relation between the two types of operators, where certain properties of one operator are obtained by using the second operator.
An initial value problem discussed in Section 5 is an important application exposing the derived properties of the two types of operators discussed throughout the paper, where the solution representation was obtained explicitly for order . Regarding this example we remark the following. In fractional calculus, Initial value problems usually make sense for functions not necessarily continuous at (left case) so that the initial conditions are given by means of . Since sequences are nice continuous functions then in Theorem 4.2, which is the tool in solving our example, the identity (4.4) appears without any initial condition. To create an initial condition in our example we shifted the fractional difference operator so that it started at .
The second author is grateful to the Kentucky Science and Engineering Foundation for financial support through research Grant KSEF-2488-RDE-014.
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Copyright © 2012 Thabet Abdeljawad and Ferhan M. Atici. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.