Research Article | Open Access
Thabet Abdeljawad, Jehad O. Alzabut, "The -Fractional Analogue for Gronwall-Type Inequality", Journal of Function Spaces, vol. 2013, Article ID 543839, 7 pages, 2013. https://doi.org/10.1155/2013/543839
The -Fractional Analogue for Gronwall-Type Inequality
We utilize -fractional Caputo initial value problems of order to derive a -analogue for Gronwall-type inequality. Some particular cases are derived where -Mittag-Leffler functions and -exponential type functions are used. An example is given to illustrate the validity of the derived inequality.
The fractional differential equations have conspicuously received considerable attention in the last two decades. Many researchers have investigated these equations due to their significant applications in various fields of science and engineering such as in viscoelasticity, capacitor theory, electrical circuits, electroanalytical chemistry, neurology, diffusion, control theory, and statistics; see, for instance, the monographs [1–3]. The study of -difference equations, on the other hand, has gained intensive interest in the last years. It has been shown that these types of equations have numerous applications in diverse fields and thus have evolved into multidisciplinary subjects [4–11]. For more details on -calculus, we refer the reader to the reference . The corresponding fractional difference equations, however, have been comparably less considered. Indeed, the notions of fractional calculus and -calculus are tracked back to the works of Jackson , respectively. However, the idea of fractional difference equations is considered to be very recent; we suggest the new papers [14–28] whose authors have taken the lead to promote the theory of fractional difference equations.
The -fractional difference equations which serve as a bridge between fractional difference equations and -difference equations have become a main object of research in the last years. Recently, many papers have appeared which study the qualitative properties of solutions for -fractional differential equations [29–33], whereas few results exist for -fractional difference equations [34–36]. The integral inequalities which are considered as effective tools for studying solutions properties have been also under consideration. In particular, we are interested in Gronwall's inequality which has been a main target for many researchers. There are several versions for Gronwall's inequality in the literature; we list here those results which are concerned with fractional order equations [37–41]. To the best of authors' observation, however, the -fractional analogue for Gronwall-type inequality has not been investigated yet.
A primary purpose of this paper is to utilize the -fractional Caputo initial value problems of order to derive a -analogue for Gronwall-type inequality. Some particular cases are derived where -Mittag-Leffler functions and -exponential type functions are used. An example is given to illustrate the validity of the derived inequality.
2. Preliminary Assertions
Before stating and proving our main results, we introduce some definitions and notations that will be used throughout the paper. For , we define the time scale as follows: where is the set of integers. In general, if is a nonnegative real number then we define the time scale and thus we may write . For a function , the nabla -derivative of is given by The nabla -integral of is given by The -factorial function for is defined by In case is a nonpositive integer, the -factorial function is defined by In the following lemma, we present some properties of -factorial functions.
Lemma 1 (see ). For , one has the following. (I). (II). (III)The nabla -derivative of the -factorial function with respect to is (IV)The nabla -derivative of the -factorial function with respect to is
For a function , the left -fractional integral of order and starting at is defined by where One should note that the left -fractional integral maps functions defined on to functions defined on .
Definition 2 (see ). If , then the Caputo left -fractional derivative of order of a function is defined by where . In case , we may write .
Lemma 3 (see ). Assume that and is defined in a suitable domain. Then and if , then
The -analogue of Mittag-Leffler function with double index is first introduced in . Indeed, it was defined as follows.
Definition 4 (see ). For and , the -Mittag-Leffler function is defined by In case , we may use .
The following example clarifies how -Mittag-Leffler functions can be used to express the solutions of Caputo -fractional linear initial value problems.
Example 5 (see ). Let and consider the left Caputo -fractional difference equation Applying to (16) and using (13), we end up with To obtain an explicit form for the solution, we apply the method of successive approximation. Set and For , we have by the power formula (14) For , we also see that If we proceed inductively and let , we obtain the solution That is,
Remark 6. If instead we use the modified -Mittag-Leffler function then, the solution representation (17) becomes
Remark 7. If we set , , , and , we reach to the -exponential formula on the time scale , where with . It is known that , where is a special case of the basic hypergeometric series, given by where is the -Pochhammer symbol.
3. The Main Results
Throughout the remaining part of the paper, we assume that . Consider the following -fractional initial value problem: Applying to both sides of (26), we obtain Set where In the following, we present a comparison result for the fractional summation operator.
Theorem 8. Let and satisfy respectively. If , then for .
Proof. Set . We claim that for . Let us assume that is valid for , where . Then, for we have or It follows that Since and , (34) can be written in the form where is used. It follows that By (29), we conclude that .
Define the following operator The following lemmas are essential in the proof of the main theorem. We only state these statements as their proofs are straightforward.
Lemma 9. For any constant , one has
Lemma 10. For any constant , one has
Lemma 11. Let be such that for . Then
The next result together with Theorem 8 will give us the desired -fractional Gronwall-type inequality.
Theorem 12. Let for . Then, the -fractional integral equation for where , has a solution
Proof. The proof is achieved by utilizing the successive approximation method. Set
We observe that
Inductively, we end up with
Taking the limit as , we have
It remains to prove the convergence of the series in (46). The subsequent analyses are carried out for .
In virtue of (29), we obtain However, for we observe that For , it follows that For , we have or For , we get Therefore, (47) becomes Let . Then, We observe that Setting we deduce that Hence, convergence is guaranteed. In case , we can proceed in a similar way taking into account that as .
Theorem 13 (-fractional Gronwall's lemma). Let and be nonnegative real valued functions such that for all (in particular if ) and Then
Corollary 14. Let for all . If then where is the nabla -exponential function on the time scale .
In this section, we show, by the help of the -fractional Gronwall inequality proved in the previous section, that small changes in the initial conditions of Caputo -fractional initial value problems lead to small changes in the solution.
Let satisfy a Lipschitz condition with constant for all and .
Example 15. Consider the following -fractional initial value problems: It follows that Taking the absolute value, we obtain or By using Theorem 13, we get Consider the following -fractional initial value problem: where . If the solution of (67) is denoted by , then for all we have Hence as . This clearly verifies the dependence of solutions on the initial conditions.
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Copyright © 2013 Thabet Abdeljawad and Jehad O. Alzabut. 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.