## Journal of Applied Mathematics

### Some New Gronwall-Bellman-Type Inequalities on Time Scales and Their Applications

#### Abstract

We establish some new Gronwall-Bellman-type inequalities on time scales. These inequalities are of new forms compared with other Gronwall-Bellman-type inequalities established so far in the literature. Based on them, new bounds for unknown functions are derived. For illustrating the validity of the inequalities established, we present some applications for them, in which the boundedness for solutions for some certain dynamic equations on time scales is researched.

#### Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 625063, 9 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394808224

Digital Object Identifier
doi:10.1155/2013/625063

Mathematical Reviews number (MathSciNet)
MR3130996

Zentralblatt MATH identifier
06950784

#### Citation

Zheng, Bin; Feng, Qinghua; Meng, Fanwei. Some New Gronwall-Bellman-Type Inequalities on Time Scales and Their Applications. J. Appl. Math. 2013 (2013), Article ID 625063, 9 pages. doi:10.1155/2013/625063. https://projecteuclid.org/euclid.jam/1394808224

#### References

• 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.
• R. Bellman, “The stability of solutions of linear differential equations,” Duke Mathematical Journal, vol. 10, pp. 643–647, 1943.
• Ou Yang-Liang, “The boundedness of solutions of linear differential equations $y''+A(t)y=0$,” Advances in Mathematics, vol. 3, pp. 409–415, 1957.
• F. Jiang and F. Meng, “Explicit bounds on some new nonlinear integral inequalities with delay,” Journal of Computational and Applied Mathematics, vol. 205, no. 1, pp. 479–486, 2007.
• A. Gallo and A. M. Piccirillo, “About some new generalizations of Bellman-Bihari results for integro-functional inequalities with discontinuous functions and applications,” Nonlinear Analysis A, vol. 71, no. 12, pp. e2276–e2287, 2009.
• 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.
• 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.
• O. Lipovan, “A retarded integral inequality and its applications,” Journal of Mathematical Analysis and Applications, vol. 285, no. 2, pp. 436–443, 2003.
• Q. H. Ma and E. H. Yang, “Some new Gronwall-Bellman-Bihari type integral inequalities with delay,” Periodica Mathematica Hungarica, vol. 44, no. 2, pp. 225–238, 2002.
• W. S. Wang, “Some generalized nonlinear retarded integral inequalities with applications,” Journal of Inequalities and Applications, vol. 2012, article 31, 14 pages, 2012.
• 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.
• L. Li, F. Meng, and L. He, “Some generalized integral inequalities and their applications,” Journal of Mathematical Analysis and Applications, vol. 372, no. 1, pp. 339–349, 2010.
• O. Lipovan, “Integral inequalities for retarded Volterra equations,” Journal of Mathematical Analysis and Applications, vol. 322, no. 1, pp. 349–358, 2006.
• B. G. Pachpatte, “Explicit bounds on certain integral inequalities,” Journal of Mathematical Analysis and Applications, vol. 267, no. 1, pp. 48–61, 2002.
• 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.
• 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.
• R. A. C. Ferreira and D. F. M. Torres, “Generalized retarded integral inequalities,” Applied Mathematics Letters, vol. 22, no. 6, pp. 876–881, 2009.
• R. Xu and Y. G. Sun, “On retarded integral inequalities in two independent variables and their applications,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1260–1266, 2006.
• S. Hilger, “Analysis on measure chains–-a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990.
• W. N. Li, “Some new dynamic inequalities on time scales,” Journal of Mathematical Analysis and Applications, vol. 319, no. 2, pp. 802–814, 2006.
• F. H. Wong, C. C. Yeh, and W. C. Lian, “An extension of Jensen's inequality on time scales,” Advances in Dynamical Systems and Applications, vol. 1, no. 1, pp. 113–120, 2006.
• M. Z. Sarikaya, “On weighted Iyengar type inequalities on time scales,” Applied Mathematics Letters, vol. 22, no. 9, pp. 1340–1344, 2009.
• 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.
• S. H. Saker, “Some nonlinear dynamic inequalities on time scales,” Mathematical Inequalities & Applications, vol. 14, no. 3, pp. 633–645, 2011.
• M. Bohner and T. Matthews, “The Grüss inequality on time scales,” Communications in Mathematical Analysis, vol. 3, no. 1, pp. 1–8, 2007.
• Q. A. Ngô, “Some mean value theorems for integrals on time scales,” Applied Mathematics and Computation, vol. 213, no. 2, pp. 322–328, 2009.
• W. Liu and Q. A. Ngô, “Some Iyengar-type inequalities on time scales for functions whose second derivatives are bounded,” Applied Mathematics and Computation, vol. 216, no. 11, pp. 3244–3251, 2010.
• S. H. Saker, “Some nonlinear dynamic inequalities on time scales and applications,” Journal of Mathematical Inequalities, vol. 4, pp. 561–579, 2010.
• R. Agarwal, M. Bohner, and A. Peterson, “Inequalities on time scales: a survey,” Mathematical Inequalities and Applications, vol. 4, no. 4, pp. 535–557, 2001.
• W. S. Cheung and Q. H. Ma, “On certain new Gronwall-Ou-Iang type integral inequalities in two variables and their applications,” Journal of Inequalities and Applications, vol. 2005, no. 4, pp. 347–361, 2005.
• C. J. Chen, W. S. Cheung, and D. Zhao, “Gronwall-bellman-type integral inequalities and applications to BVPs,” Journal of Inequalities and Applications, vol. 2009, Article ID 258569, 15 pages, 2009.
• O. Lipovan, “A retarded Gronwall-like inequality and its applications,” Journal of Mathematical Analysis and Applications, vol. 252, no. 1, pp. 389–401, 2000.
• M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001.