Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2013, Article ID 341706, 8 pages
http://dx.doi.org/10.1155/2013/341706
Research Article

Some New Gronwall-Bellman-Type Inequalities Based on the Modified Riemann-Liouville Fractional Derivative

School of Science, Shandong University of Technology, Zibo, Shandong 255049, China

Received 22 May 2013; Accepted 28 August 2013

Academic Editor: Francisco J. Marcellán

Copyright © 2013 Bin Zheng. 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

By use of the properties of the modified Riemann-Liouville fractional derivative, some new Gronwall-Bellman-type inequalities are researched. First, we derive some new explicit bounds for the unknown functions lying in these inequalities, which are of different forms from some existing bounds in the literature. Then, we apply the results established to research the boundedness, uniqueness, and continuous dependence on the initial value for the solution to a certain fractional differential equation.

1. Introduction

During the past decades, a lot of integral and difference inequalities have been discovered, which play an important role in the research of the theory of differential, integral, and difference equations. In these inequalities, the Gronwall-Bellman inequality and their generalizations have been paid much attention by many authors (e.g., see [122]), as such inequalities provide explicit bounds for the unknown functions concerned and can be used as a handy tool in the research of boundedness, global existence, uniqueness, stability, and continuous dependence of solutions to differential and integral equations as well as difference equations. However, most of the Gronwall-Bellman-type inequalities established so far can only be used in the research of differential equations of integer order, while in order to fulfill qualitative and quantitative analysis for solutions to some certain differential equations of fractional order, the earlier inequalities established are inadequate. So it is necessary to establish new inequalities so as to obtain the desired analysis for fractional differential equations.

In this paper, we establish some new Gronwall-Bellman-type inequalities. Based on some basic properties of the modified Riemann-Liouville fractional derivative, we derive explicit bounds for unknown functions concerned in these inequalities. The presented inequalities can be used as a handy tool in the qualitative as well as quantitative analysis of solutions to fractional differential equations.

The modified Riemann-Liouville fractional derivative, defined by Jumarie in [2326], has many excellent characters in handling many fractional calculus problems. Many authors have investigated various applications of the modified Riemann-Liouville fractional derivative (e.g., see [2733]). We now list the definition for it as follows.

Definition 1. The modified Riemann-Liouville derivative of order is defined by the following expression:

Definition 2. The Riemann-Liouville fractional integral of order on the interval is defined by
Some important properties for the modified Riemann-Liouville derivative and fractional integral are listed as follows (the interval concerned below is always defined by ):(a), (b),(c),(d),(e).

The next part of this paper is organized as follows. In Section 2, we present some new Gronwall-Bellman-type inequalities, and based on them we derive explicit bounds for unknown functions lying in these inequalities. In Section 3, for illustrating the validity of the established results, we apply them to research boundedness, uniqueness, and continuous dependence on initial data for the solution to a certain fractional differential equation, where the fractional derivative is defined in the sense of the modified Riemann-Liouville derivative. Finally, some summary comments are presented in Section 4.

2. Main Results

Theorem 3. Suppose that , , and   are nonnegative continuous functions defined on . If the following inequality satisfies: then, one has the following explicit estimate for :

Proof. Let . Then, we have Furthermore, as is continuous, there exists such that for , where . So, for , we have , which implies that . So, Then, by the properties (b) and (c), we get that Since by (a) it holds that , then furthermore, we have Substituting with and fulfilling fractional integral of order for (8) with respect to from to we deduce that which implies that Combining (5) and (10), we get (4).

Remark 4. In Theorem 3, we note that the bound for established in (4) is essentially different from that in [34, Theorem 1]. The bound in [34, Theorem 1] is expressed in a series, whose convergence has not been proved in fact, while the bound in (4) here is more explicit and easy to use. Moreover, the function here is not necessarily nondecreasing, which is a generalization of the condition in [34, Theorem 1]. Besides, the result established here has been derived based on the modified Riemann-Liouville derivative, which is also different from [34].

Theorem 5. Suppose that , , and are defined as in Theorem 3. If the following inequality satisfies: then, one has the following explicit estimate for :

Proof. Let . Then, we have Considering that and is continuous satisfying , , where is a positive constant, we obtain So, we have , and furthermore, Multiplying on both sides of (15), we get that Since , then furthermore, we have Using , substituting with , and fulfilling fractional integral of order for (17) with respect to from to , we deduce that which implies that Combining (13) and (19), we get (12).

Theorem 6. Suppose that , , and   are defined as in Theorem 3, and furthermore, assume that and are both nondecreasing. is nonnegative continuous on . If the following inequality satisfies: then, for , one has the following explicit estimate for :

Proof. Denote . Then, Since and are both nondecreasing, then is also nondecreasing, and so, we obtain that So, Notice that the structure of (24) is similar to that of inequality (3). So, a suitable application of Theorem 3 to (24) yields that Combining (23) and (25), we can obtain the desired result.

Theorem 7. Suppose that , , and are defined as in Theorem 3, and furthermore, assume that and are both nondecreasing. is nonnegative continuous on . If the following inequality satisfies: then, for and , one has the following explicit estimate for :

The proof of Theorem 7 is similar to that of Theorem 6, except that Theorem 5 instead of Theorem 3 is applied at the end of the proof.

3. Applications

In this section, we will show that the inequalities established above are useful in the research of boundedness, uniqueness, and continuous dependence on the initial value for solutions to fractional differential equations. Consider the following fractional differential equation: with the initial condition , where .

Theorem 8. Suppose that is a solution of (28). If , then one has the following estimate for :

Proof. Similar to [24, Equation ], we can obtain the equivalent integral form of (28) with the initial condition as follows: So, Then, a suitable application of Theorem 5 to (31) (with ) yields Since , then furthermore, we have which is the desired result.

Remark 9. In Theorem 8, if satisfies instead, then one can apply Theorem 3 instead of Theorem 5 in the proof and obtain the following estimate for : Furthermore, we have

Theorem 10. If , then (28) has a unique solution.

Proof. Suppose that (28) has two solutions and . Then, similar to Theorem 8, we can obtain Furthermore, which implies that Treating as one independent function and applying Theorem 5 to (38) (with ), we get that , which implies that . So, the proof is complete.

Now we research the continuous dependence on the initial value for the solution of (28).

Theorem 11. Under the conditions of Theorem 10, the solution of (28) depends continuously on the initial value of , where is arbitrarily large.

Proof. Let be the solution of (28), and let be the solution of the following fractional differential equation: with the initial condition .
Similar to Theorem 8, we get that So, we have which implies that Applying Theorem 5 to (42), after some basic computation, we can get that From (43), one can see that small change in the initial value leads to small change in the solution on the closed interval . So, the proof is complete.

4. Conclusions

We have presented some new Gronwall-Bellman-type inequalities, and based on them we derived explicit bounds for the unknown functions concerned, which are different from the existing bounds in the literature. As one can see, the results established are useful in fulfilling qualitative and quantitative analyses such as the boundedness, uniqueness, and continuous dependence on the initial value and parameter for solutions to certain fractional differential equations. Finally, we note that the method used in Theorems 3 and 5 is of generality and can be used to establish other Gronwall-Bellman-type inequalities with more complicated and general forms, which are expected to further research.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author would like to thank the reviewers very much for their valuable suggestions on improving this paper.

References

  1. T. H. Gronwall, “Note on the derivatives with respect to a parameter of the solutions of a system of differential equations,” Annals of Mathematics, vol. 20, no. 4, pp. 292–296, 1919. View at Publisher · View at Google Scholar · View at MathSciNet
  2. R. Bellman, “The stability of solutions of linear differential equations,” Duke Mathematical Journal, vol. 10, pp. 643–647, 1943. View at Publisher · View at Google Scholar · View at MathSciNet
  3. Q. Feng and B. Zheng, “Generalized Gronwall-Bellman-type delay dynamic inequalities on time scales and their applications,” Applied Mathematics and Computation, vol. 218, no. 15, pp. 7880–7892, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  4. Q. H. Ma, “Estimates on some power nonlinear Volterra-Fredholm type discrete inequalities and their applications,” Journal of Computational and Applied Mathematics, vol. 233, no. 9, pp. 2170–2180, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  5. W. S. Wang, “A class of retarded nonlinear integral inequalities and its application in nonlinear differential-integral equation,” Journal of Inequalities and Applications, vol. 2012, article 154, 10 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  6. S. H. Saker, “Some nonlinear dynamic inequalities on time scales,” Mathematical Inequalities & Applications, vol. 14, no. 3, pp. 633–645, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  7. R. A. C. Ferreira and D. F. M. Torres, “Generalized retarded integral inequalities,” Applied Mathematics Letters, vol. 22, no. 6, pp. 876–881, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  8. Q. Feng, F. Meng, and B. Zheng, “Gronwall-Bellman type nonlinear delay integral inequalities on time scales,” Journal of Mathematical Analysis and Applications, vol. 382, no. 2, pp. 772–784, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  9. O. Lipovan, “Integral inequalities for retarded Volterra equations,” Journal of Mathematical Analysis and Applications, vol. 322, no. 1, pp. 349–358, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  10. S. H. Saker, “Some nonlinear dynamic inequalities on time scales and applications,” Journal of Mathematical Inequalities, vol. 4, no. 4, pp. 561–579, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  11. B. Zheng, Q. Feng, F. Meng, and Y. Zhang, “Some new Gronwall-Bellman type nonlinear dynamic inequalities containing integration on infinite intervals on time scales,” Journal of Inequalities and Applications, vol. 2012, article 201, 20 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  12. B. G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press, New York, NY, USA, 1998. View at MathSciNet
  13. R. P. Agarwal, S. Deng, and W. Zhang, “Generalization of a retarded Gronwall-like inequality and its applications,” Applied Mathematics and Computation, vol. 165, no. 3, pp. 599–612, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  14. W. S. Cheung and J. Ren, “Discrete non-linear inequalities and applications to boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 319, no. 2, pp. 708–724, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  15. W. N. Li, “Some delay integral inequalities on time scales,” Computers & Mathematics with Applications, vol. 59, no. 6, pp. 1929–1936, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  16. W. S. Wang, “Some retarded nonlinear integral inequalities and their applications in retarded differential equations,” Journal of Inequalities and Applications, vol. 2012, article 75, 8 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  17. Y. H. Kim, “Gronwall, Bellman and Pachpatte type integral inequalities with applications,” Nonlinear Analysis A, vol. 71, no. 12, pp. e2641–e2656, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  18. W. N. Li, M. Han, and F. W. Meng, “Some new delay integral inequalities and their applications,” Journal of Computational and Applied Mathematics, vol. 180, no. 1, pp. 191–200, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  19. Q. Feng, F. Meng, Y. Zhang, B. Zheng, and J. Zhou, “Some nonlinear delay integral inequalities on time scales arising in the theory of dynamics equations,” Journal of Inequalities and Applications, vol. 2011, article 29, 14 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  20. Q. Feng, F. Meng, and Y. Zhang, “Generalized Gronwall-Bellman-type discrete inequalities and their applications,” Journal of Inequalities and Applications, vol. 2011, article 47, 12 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  21. Z. Meng, B. Zheng, and C. Wen, “Some new integral inequalities on time scales containing integration on infinite intervals,” Journal of Inequalities and Applications, vol. 2013, article 245, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  22. Y. Wang, Q. Feng, S. Zhou, and F. Xu, “Some new generalized Gronwall-Bellman-type integral inequalities in two independent variables on time scales,” Journal of Inequalities and Applications, vol. 2013, article 234, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  23. G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  24. G. Jumarie, “Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions,” Applied Mathematics Letters, vol. 22, no. 3, pp. 378–385, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  25. G. Jumarie, “Cauchy's integral formula via the modified Riemann-Liouville derivative for analytic functions of fractional order,” Applied Mathematics Letters, vol. 23, no. 12, pp. 1444–1450, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  26. G. Jumarie, “Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution,” Journal of Applied Mathematics & Computing, vol. 24, no. 1-2, pp. 31–48, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  27. S. Zhang and H. Q. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs,” Physics Letters A, vol. 375, no. 7, pp. 1069–1073, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  28. S. Guo, L. Mei, Y. Li, and Y. Sun, “The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics,” Physics Letters A, vol. 376, no. 4, pp. 407–411, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  29. B. Lu, “The first integral method for some time fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 395, no. 2, pp. 684–693, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  30. B. Zheng, “(G′/G)-expansion method for solving fractional partial differential equations in the theory of mathematical physics,” Communications in Theoretical Physics, vol. 58, no. 5, pp. 623–630, 2012. View at Publisher · View at Google Scholar
  31. Q. Feng, “Exact solutions for fractional differential-difference equations by an extended riccati sub-ODE method,” Communications in Theoretical Physics, vol. 59, no. 5, pp. 521–527, 2013. View at Publisher · View at Google Scholar
  32. G.- c. Wu and E. W. M. Lee, “Fractional variational iteration method and its application,” Physics Letters A, vol. 374, no. 25, pp. 2506–2509, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  33. B. Tang, Y. He, L. Wei, and X. Zhang, “A generalized fractional sub-equation method for fractional differential equations with variable coefficients,” Physics Letters A, vol. 376, no. 38-39, pp. 2588–2590, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  34. 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 MathSciNet