Journal of Function Spaces and Applications

Volume 2013, Article ID 543839, 7 pages

http://dx.doi.org/10.1155/2013/543839

## The -Fractional Analogue for Gronwall-Type Inequality

^{1}Department of Mathematics and Computer Science, Çankaya University, Ögretmenler Caddesi 14, Balgat, 06530 Ankara, Turkey^{2}Department of Mathematics and Physical Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia

Received 8 May 2013; Accepted 7 July 2013

Academic Editor: Dashan Fan

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.

#### Abstract

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.

#### 1. Introduction

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 [12]. 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 [13], 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 [32]). *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 [14]). *If , then the Caputo left -fractional derivative of order of a function is defined by
where . In case , we may write .

Lemma 3 (see [14]). *Assume that and is defined in a suitable domain. Then
**
and if , then
*

For solving linear -fractional equations, the following identity is essential: where and . See, for instance, the recent papers [14, 15] for more information.

The -analogue of Mittag-Leffler function with double index is first introduced in [14]. Indeed, it was defined as follows.

*Definition 4 (see [14]). *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 [14]). *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
*

The proof of the previous statement is a straightforward implementation of Theorems 8 and 12 by setting .

In case , we deduce the following immediate consequence of Theorem 13 which can be considered as the well-known -Gronwall's Lemma; consult, for instance, the paper [42].

Corollary 14. *Let for all . If
**
then
**
where is the nabla -exponential function on the time scale . *

#### 4. Applications

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.

#### References

- S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives: Theory and Applications*, Gordon and Breach Science, Yverdon, Switzerland, 1993. View at Zentralblatt MATH · View at MathSciNet - I. Podlubny,
*Fractional Differential Equations*, vol. 198, Academic Press, San Diego, Calif, USA, 1999. View at Zentralblatt MATH · View at MathSciNet - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204 of*North-Holland Mathematics Studies*, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Finkelstein and E. Marcus, “Transformation theory of the $q$-oscillator,”
*Journal of Mathematical Physics*, vol. 36, no. 6, pp. 2652–2672, 1995. View at Publisher · View at Google Scholar · View at MathSciNet - R. J. Finkelstein, “The $q$-Coulomb problem,”
*Journal of Mathematical Physics*, vol. 37, no. 6, pp. 2628–2636, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - R. Floreanini and L. Vinet, “Automorphisms of the $q$-oscillator algebra and basic orthogonal polynomials,”
*Physics Letters A*, vol. 180, no. 6, pp. 393–401, 1993. View at Publisher · View at Google Scholar · View at MathSciNet - R. Floreanini and L. Vinet, “Symmetries of the $q$-difference heat equation,”
*Letters in Mathematical Physics*, vol. 32, no. 1, pp. 37–44, 1994. View at Publisher · View at Google Scholar · View at MathSciNet - R. Floreanini and L. Vinet, “Quantum symmetries of $q$-difference equations,”
*Journal of Mathematical Physics*, vol. 36, no. 6, pp. 3134–3156, 1995. View at Publisher · View at Google Scholar · View at MathSciNet - P. G. O. Freund and A. V. Zabrodin, “The spectral problem for the $q$-Knizhnik-Zamolodchikov equation and continuous $q$-Jacobi polynomials,”
*Communications in Mathematical Physics*, vol. 173, no. 1, pp. 17–42, 1995. View at Publisher · View at Google Scholar · View at MathSciNet - G.-N. Han and J. Zeng, “On a $q$-sequence that generalizes the median Genocchi numbers,”
*Annales des Sciences Mathématiques du Québec*, vol. 23, no. 1, pp. 63–72, 1999. View at Google Scholar · View at MathSciNet - J. O. Alzabut and T. Abdeljawad, “Perron's theorem for $q$-delay difference equations,”
*Applied Mathematics & Information Sciences*, vol. 5, no. 1, pp. 74–84, 2011. View at Google Scholar · View at MathSciNet - V. Kac and P. Cheung,
*Quantum Calculus*, Springer, New York, NY, USA, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - F. H. Jackson, “$q$-difference equations,”
*American Journal of Mathematics*, vol. 32, no. 4, pp. 305–314, 1910. View at Publisher · View at Google Scholar · View at MathSciNet - T. Abdeljawad and D. Baleanu, “Caputo $q$-fractional initial value problems and a $q$-analogue Mittag-Leffler function,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 12, pp. 4682–4688, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - T. Abdeljawad and D. Baleanu, “Fractional differences and integration by parts,”
*Journal of Computational Analysis and Applications*, vol. 13, no. 3, pp. 574–582, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Abdeljawad, B. Benli, and D. Baleanu, “A generalized $q$-Mittag-Leffler function by $q$-Captuo fractional linear equations,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 546062, 11 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - F. Jarad, T. Abdeljawad, and D. Baleanu, “Stability of $q$-fractional non-autonomous systems,”
*Nonlinear Analysis: Real World Applications*, vol. 14, no. 1, pp. 780–784, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - T. Abdeljawad, F. Jarad, and D. Baleanu, “A semigroup-like property for discrete Mittag-Leffler functions,”
*Advances in Difference Equations*, vol. 2012, article 72, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - N. R. O. Bastos, R. A. C. Ferreira, and D. F. M. Torres, “Necessary optimality conditions for fractional difference problems of the calculus of variations,”
*Discrete and Continuous Dynamical Systems A*, vol. 29, no. 2, pp. 417–437, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. R. O. Bastos, R. A. C. Ferreira, and D. F. M. Torres, “Discrete-time variational problems,”
*Signal Process*, vol. 91, no. 3, 2011. View at Google Scholar - F. M. Atıcı and S. Şengül, “Modeling with fractional difference equations,”
*Journal of Mathematical Analysis and Applications*, vol. 369, no. 1, pp. 1–9, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. M. Atici and P. W. Eloe, “A transform method in discrete fractional calculus,”
*International Journal of Difference Equations*, vol. 2, no. 2, pp. 165–176, 2007. View at Google Scholar · View at MathSciNet - F. M. Atici and P. W. Eloe, “Initial value problems in discrete fractional calculus,”
*Proceedings of the American Mathematical Society*, vol. 137, no. 3, pp. 981–989, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. S. Goodrich, “Continuity of solutions to discrete fractional initial value problems,”
*Computers & Mathematics with Applications*, vol. 59, no. 11, pp. 3489–3499, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. S. Goodrich, “Solutions to a discrete right-focal fractional boundary value problem,”
*International Journal of Difference Equations*, vol. 5, no. 2, pp. 195–216, 2010. View at Google Scholar · View at MathSciNet - G. A. Anastassiou, “Nabla discrete fractional calculus and nabla inequalities,”
*Mathematical and Computer Modelling*, vol. 51, no. 5-6, pp. 562–571, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-F. Cheng and Y.-M. Chu, “Fractional difference equations with real variable,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 918529, 24 pages, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. F. Cheng and G. C. Wu, “Solutions of fractional difference equations of order $(2,q)$,”
*Acta Mathematica Sinica*, vol. 55, no. 3, pp. 469–480, 2012. View at Google Scholar · View at MathSciNet - W. A. Al-Salam, “Some fractional $q$-integrals and $q$-derivatives,”
*Proceedings of the Edinburgh Mathematical Society II*, vol. 15, no. 2, pp. 135–140, 1966. View at Publisher · View at Google Scholar · View at MathSciNet - R. P. Agarwal, “Certain fractional $q$-integrals and $q$-derivatives,”
*Proceedings of the Cambridge Philosophical Society*, vol. 66, pp. 365–370, 1969. View at Google Scholar · View at MathSciNet - W. A. Al-Salam and A. Verma, “A fractional Leibniz $q$-formula,”
*Pacific Journal of Mathematics*, vol. 60, no. 2, pp. 1–9, 1975. View at Publisher · View at Google Scholar · View at MathSciNet - F. M. Atici and P. W. Eloe, “Fractional $q$-calculus on a time scale,”
*Journal of Nonlinear Mathematical Physics*, vol. 14, no. 3, pp. 333–344, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - P. M. Rajković, S. D. Marinković, and M. S. Stanković, “Fractional integrals and derivatives in $q$-calculus,”
*Applicable Analysis and Discrete Mathematics*, vol. 1, no. 1, pp. 311–323, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - Z. S. I. Mansour, “Linear sequential $q$-difference equations of fractional order,”
*Fractional Calculus & Applied Analysis*, vol. 12, no. 2, pp. 159–178, 2009. View at Google Scholar · View at MathSciNet - Y. Zhao, H. Chen, and Q. Zhang, “Existence results for fractional $q$-difference equations with nonlocal $q$-integral boundary conditions,”
*Advances in Difference Equations*, vol. 2013, article 48, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - M. H. Annaby and Z. S. Mansour,
*q-fractional Calculus and Equations*, vol. 2056 of*Lecture Notes in Mathematics*, Springer, Heidelberg, Germany, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - H. Ye, J. Gao, and Y. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation,”
*Journal of Mathematical Analysis and Applications*, vol. 328, no. 2, pp. 1075–1081, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q.-H. Ma and J. Pečarić, “Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations,”
*Journal of Mathematical Analysis and Applications*, vol. 341, no. 2, pp. 894–905, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. M. Furati and N.-E. Tatar, “Inequalities for fractional differential equations,”
*Mathematical Inequalities & Applications*, vol. 12, no. 2, pp. 279–293, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q.-X. Kong and X.-L. Ding, “A new fractional integral inequality with singularity and its application,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 937908, 12 pages, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. M. Atıcı and P. W. Eloe, “Gronwall's inequality on discrete fractional calculus,”
*Computers & Mathematics with Applications*, vol. 64, no. 10, pp. 3193–3200, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - W. N. Li and W. Sheng, “Some Gronwall type inequalities on time scales,”
*Journal of Mathematical Inequalities*, vol. 4, no. 1, pp. 67–76, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet