Abstract
In this paper, we present some new Gronwall-type inequalities. Explicit bounds for the unknown functions concerned are derived based on these inequalities and the properties of the modified Riemann-Liouville fractional derivative. The inequalities established are of new forms compared with the existing results so far in the literature. For illustrating the validity of the inequalities established, we apply them to research the boundedness, quantitative property, and continuous dependence on the initial value for the solution to a certain fractional integral equation.
MSC:26D10.
Similar content being viewed by others
1 Introduction
Recently, with the development of the theory of differential equations, many authors have researched various inequalities and investigated the boundedness, global existence, uniqueness, stability, and continuous dependence on the initial value and parameters of solutions to differential equations, integral equations as well as difference equations. The Gronwall-Bellman inequality [1, 2] is widely used in the qualitative and quantitative analysis of differential equations, as it can provide explicit bound for an unknown function lying in the inequality. In the last few decades, many authors have researched various generalizations of the Gronwall-Bellman inequality; for example, we refer the reader to [3–28] and the references therein. These Gronwall-type inequalities established can be used as a handy tool in the research of the theory of differential and integral equations as well as difference equations. However, we notice that the existing results in the literature are inadequate for researching the qualitative and quantitative properties of solutions to some fractional integral equations, for example, the following fractional integral equation:
where , is a constant, denotes the Riemann-Liouville fractional integral of order α.
So it is necessary to establish some new Gronwall-type inequalities in order to fulfill the desired analysis result.
The modified Riemann-Liouville fractional derivative, presented by Jumarie in [29, 30], is defined by the following expression.
Definition 1 The modified Riemann-Liouville derivative of order α is defined by the following expression:
Definition 2 The Riemann-Liouville fractional integral of order α on the interval is defined by
Some important properties for the modified Riemann-Liouville derivative and fractional integral are listed as follows (see [31, 32] and the interval concerned below is always defined by ):
-
(a)
.
-
(b)
.
-
(c)
.
-
(d)
.
-
(e)
.
-
(f)
, where C is a constant.
The modified Riemann-Liouville derivative has many excellent characters in handling many fractional calculus problems. Many authors have investigated various applications of the modified Riemann-Liouville fractional derivative. For example, in [32, 33], the authors sought exact solutions for some types of fractional differential equations based on the modified Riemann-Liouville fractional derivative, and in [34], the modified Riemann-Liouville fractional derivative was used in fractional calculus of variations, where a fractional basic problem of the calculus of variations with free boundary conditions as well as problems with isoperimetric and holonomic constraints were considered. In [35], Khan et al. presented a fractional homotopy perturbation method (FHPM) for solving fractional differential equations of any fractional order based on the modified Riemann-Liouville fractional derivative. In [36–38], the fractional variational iteration method based on the modified Riemann-Liouville fractional derivative was concerned. In [39], a fractional variational homotopy perturbation iteration method was proposed.
Based on the analysis above, in Section 2, we present some new Gronwall-type inequalities, based on which and some basic properties of the modified Riemann-Liouville fractional derivative, we derive explicit bounds for the unknown functions concerned in these inequalities. In Section 3, we apply the results established in Section 2 to research boundedness, quantitative property, and continuous dependence on the initial data for the solution to a certain fractional integral equation.
2 Main results
Lemma 1 Suppose , f is a continuous function, then .
Proof Since f is continuous, then there exists a constant M such that for , where . So, for , we have . Then one can see . Therefore,
Letting , we obtain that
where denotes the beta function. The proof is complete. □
Theorem 2 Suppose , the functions u, g are nonnegative continuous functions defined on , is a constant. If the following inequality is satisfied
then we have the following explicit estimate for :
provided that .
Proof Denote the right-hand side of (1) by . Then we have
and, by use of Lemma 1 and the property , we obtain
Furthermore, by the properties (a), (b), (c), we have:
Substituting t with τ, fulfilling a fractional integral of order α for (3) with respect to τ from 0 to t, we deduce that
which implies
On the other hand, we have
which is followed by
Combining (3), (5), (6), we can get the desired result. □
Now we study the inequality of the following form:
where , the functions u, g, h are nonnegative continuous functions defined on , and is a constant, p, q are constants with .
The following lemma is useful in deriving explicit bound for the function in (7).
Lemma 3 [24]
Assume that , , and , then for any ,
Theorem 4 The inequality admits the following explicit estimate for :
provided that , where , and
Proof Denote the right-hand side of (7) by . Then we have
and considering , it follows that
Let . Then
which implies that
So
Using Lemma 3, we get that
where is defined as above, and
Let . Then
and
By the properties (a), (b), and (c), we get that
Substituting t with τ, fulfilling a fractional integral of order α for (13) with respect to τ from 0 to t, and using , we deduce that
which implies
Combining (11), (12), and (14), we get that
which implies
under the condition .
The desired result can be obtained by the combination of (11), (12), (14), and (15). □
Theorem 5 Suppose , the function u is a nonnegative continuous function defined on , p, T are constants with , , satisfying for , , where is a constant. If the following inequality is satisfied
then we have the following explicit estimate for :
provided that .
Proof Denote the right-hand side of (16) by . Then we have
and
Then a suitable application of Theorem 2 to (19) yields the desired result. □
3 Applications
In this section, we present one example for the results established above, in which the boundedness, quantitative property, and continuous dependence on the initial value for the solutions to one certain fractional integral equation are researched.
Example Consider the following fractional integral equation:
where , , is a constant, denotes the Riemann-Liouville fractional integral of order α on the interval as defined in Definition 2.
Theorem 6 For Eq. (20), if , where , then under the condition , we have the following estimate:
Proof By Eq. (20) we in fact have
So,
Then a suitable application of Theorem 2 to (22) yields the desired result. □
Remark 1 The result of Theorem 6 shows that the trivial solution to Eq. (20) is uniformly stable on the initial value.
Theorem 7 If the function f satisfies the Lipschitz condition with , where A is the Lipschitz constant, then under the condition of the same initial value, Eq. (20) has at most one solution.
Proof Suppose that Eq. (20) has two solutions , with the same initial value . Then we have
Furthermore,
which implies
After a suitable application of Theorem 2 to (26) (with being treated as one independent function), we obtain that , which implies . So the proof is complete. □
Theorem 8 Let be the solution of Eq. (20), and let be the solution of the following fractional integral equation:
If f satisfies the Lipschitz condition with A being the Lipschitz constant, then we have the following estimate:
Proof By Eq. (27) we have
So, we have
Furthermore,
Applying Theorem 2 to (31), after some basic computation, we can get the desired result. □
Remark 2 The result of Theorem 8 shows that the solution to Eq. (20) depends continuously on the initial value.
4 Conclusions
In this paper, we have derived new explicit bounds for the unknown functions concerned in some new Gronwall-type inequalities. In the proof for the main results, we have used the properties of the modified Riemann-Liouville fractional derivative. As for applications, we have presented one example, in which the boundedness, uniqueness, and continuous dependence on the initial value for the solution to a certain fractional integral equation are investigated. Finally, we note that these inequalities can be generalized to more general forms, as well as be generalized to 2D cases.
References
Gronwall TH: Note on the derivatives with respect to a parameter of solutions of a system of differential equations. Ann. Math. 1919, 20: 292-296. 10.2307/1967124
Bellman R: The stability of solutions of linear differential equations. Duke Math. J. 1943, 10: 643-647. 10.1215/S0012-7094-43-01059-2
Ma QH: Estimates on some power nonlinear Volterra-Fredholm type discrete inequalities and their applications. J. Comput. Appl. Math. 2010, 233: 2170-2180. 10.1016/j.cam.2009.10.002
Pachpatte BG: Inequalities for Differential and Integral Equations. Academic Press, New York; 1998.
Sun YG: On retarded integral inequalities and their applications. J. Math. Anal. Appl. 2005, 301: 265-275. 10.1016/j.jmaa.2004.07.020
Agarwal RP, Deng SF, Zhang WN: Generalization of a retarded Gronwall-like inequality and its applications. Appl. Math. Comput. 2005, 165: 599-612. 10.1016/j.amc.2004.04.067
Li LZ, Meng FW, Ju PJ: Some new integral inequalities and their applications in studying the stability of nonlinear integro-differential equations with time delay. J. Math. Anal. Appl. 2010, 377: 853-862.
Gallo A, Piccirillo AM: About some new generalizations of Bellman-Bihari results for integro-functional inequalities with discontinuous functions and applications. Nonlinear Anal. 2009, 71: e2276-e2287. 10.1016/j.na.2009.05.019
Ma QH, Pečarić J: The bounds on the solutions of certain two-dimensional delay dynamic systems on time scales. Comput. Math. Appl. 2011, 61: 2158-2163. 10.1016/j.camwa.2010.09.001
Lipovan O: Integral inequalities for retarded Volterra equations. J. Math. Anal. Appl. 2006, 322: 349-358. 10.1016/j.jmaa.2005.08.097
Feng QH, Zheng B: Generalized Gronwall-Bellman-type delay dynamic inequalities on time scales and their applications. Appl. Math. Comput. 2012, 218: 7880-7892. 10.1016/j.amc.2012.02.006
Kim YH: Gronwall, Bellman and Pachpatte type integral inequalities with applications. Nonlinear Anal. 2009, 71: e2641-e2656. 10.1016/j.na.2009.06.009
Pachpatte BG: Explicit bounds on certain integral inequalities. J. Math. Anal. Appl. 2002, 267: 48-61. 10.1006/jmaa.2001.7743
Agarwal RP, Bohner M, Peterson A: Inequalities on time scales: a survey. Math. Inequal. Appl. 2001,4(4):535-557.
Wang WS: Some retarded nonlinear integral inequalities and their applications in retarded differential equations. J. Inequal. Appl. 2012,2012(75):1-8.
Li WN: Some delay integral inequalities on time scales. Comput. Math. Appl. 2010, 59: 1929-1936. 10.1016/j.camwa.2009.11.006
Saker SH: Some nonlinear dynamic inequalities on time scales. Math. Inequal. Appl. 2011, 14: 633-645.
Feng QH, Meng FW, Zhang YM: Generalized Gronwall-Bellman-type discrete inequalities and their applications. J. Inequal. Appl. 2011,2011(47):1-21.
Feng QH, Meng FW, Zheng B: Gronwall-Bellman type nonlinear delay integral inequalities on time scales. J. Math. Anal. Appl. 2011, 382: 772-784. 10.1016/j.jmaa.2011.04.077
Wang WS: A class of retarded nonlinear integral inequalities and its application in nonlinear differential-integral equation. J. Inequal. Appl. 2012,2012(154):1-10.
Saker SH: Some nonlinear dynamic inequalities on time scales and applications. J. Math. Inequal. 2010, 4: 561-579.
Zheng B, Feng QH, Meng FW, Zhang YM: Some new Gronwall-Bellman type nonlinear dynamic inequalities containing integration on infinite intervals on time scales. J. Inequal. Appl. 2012,2012(201):1-20.
Li WN, Han MA, Meng FW: Some new delay integral inequalities and their applications. J. Comput. Appl. Math. 2005, 180: 191-200. 10.1016/j.cam.2004.10.011
Jiang FC, Meng FW: Explicit bounds on some new nonlinear integral inequality with delay. J. Comput. Appl. Math. 2007, 205: 479-486. 10.1016/j.cam.2006.05.038
Feng QH, Meng FW, Zhang YM, Zheng B, Zhou JC: Some nonlinear delay integral inequalities on time scales arising in the theory of dynamics equations. J. Inequal. Appl. 2011,2011(29):1-14.
Ferreira RAC, Torres DFM: Generalized retarded integral inequalities. Appl. Math. Lett. 2009, 22: 876-881. 10.1016/j.aml.2008.08.022
Cheung WS, Ren JL: Discrete non-linear inequalities and applications to boundary value problems. J. Math. Anal. Appl. 2006, 319: 708-724. 10.1016/j.jmaa.2005.06.064
Ye HP, Gao JM, Ding YS: A generalized Gronwall inequality and ins application to a fractional differential equation. J. Math. Anal. Appl. 2007, 328: 1075-1081. 10.1016/j.jmaa.2006.05.061
Jumarie G: Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput. Math. Appl. 2006, 51: 1367-1376. 10.1016/j.camwa.2006.02.001
Jumarie G: Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions. Appl. Math. Lett. 2009, 22: 378-385. 10.1016/j.aml.2008.06.003
Wu GC, Lee EWM: Fractional variational iteration method and its application. Phys. Lett. A 2010, 374: 2506-2509. 10.1016/j.physleta.2010.04.034
Zheng B: -expansion method for solving fractional partial differential equations in the theory of mathematical physics. Commun. Theor. Phys. 2012, 58: 623-630. 10.1088/0253-6102/58/5/02
Feng QH: Exact solutions for fractional differential-difference equations by an extended Riccati Sub-ODE method. Commun. Theor. Phys. 2013, 59: 521-527. 10.1088/0253-6102/59/5/01
Almeida R, Torres DFM: Fractional variational calculus for nondifferentiable functions. Comput. Math. Appl. 2011, 61: 3097-3104. 10.1016/j.camwa.2011.03.098
Khan Y, Wu Q, Faraz N, Yildirim A, Madani M: A new fractional analytical approach via a modified Riemann-Liouville derivative. Appl. Math. Lett. 2012, 25: 1340-1346. 10.1016/j.aml.2011.11.041
Faraz N, Khan Y, Jafari H, Yildirim A, Madani M: Fractional variational iteration method via modified Riemann-Liouville derivative. J. King Saud Univ., Sci. 2011, 23: 413-417. 10.1016/j.jksus.2010.07.025
Khana Y, Faraz N, Yildirim A, Wu Q: Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science. Comput. Math. Appl. 2011, 62: 2273-2278. 10.1016/j.camwa.2011.07.014
Merdan M: Analytical approximate solutions of fractional convection-diffusion equation with modified Riemann-Liouville derivative by means of fractional variational iteration method. Iran. J. Sci. Technol., Trans. A, Sci. 2013,37(1):83-92.
Guo S, Mei L, Li Y: Fractional variational homotopy perturbation iteration method and its application to a fractional diffusion equation. Appl. Math. Comput. 2013, 219: 5909-5917. 10.1016/j.amc.2012.12.003
Acknowledgements
The authors would thank the referees very much for their valuable suggestions on improving this paper. This work was partially supported by the Natural Science Foundation of Shandong Province (in China) (grant No. ZR2013AQ009), and Doctoral Initializing Foundation of Shandong University of Technology (in China) (grant No. 4041-413030).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Author’s contributions
BZ carried out the main part of this article. The author read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Zheng, B. Explicit bounds derived by some new inequalities and applications in fractional integral equations. J Inequal Appl 2014, 4 (2014). https://doi.org/10.1186/1029-242X-2014-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-4