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Advanced Gronwall-Bellman-Type Integral Inequalities and Their Applications

Authors: Zixin Liu, Shu Lü, Shouming Zhong, Mao Ye


In this paper, some new nonlinear generalized Gronwall-Bellman-Type integral inequalities with mixed time delays are established. These inequalities can be used as handy tools to research stability problems of delayed differential and integral dynamic systems. As applications, based on these new established inequalities, some p-stable results of a integro-differential equation are also given. Two numerical examples are presented to illustrate the validity of the main results.

Keywords: Gronwall-Bellman-Type integral inequalities, integrodifferential equation, p-exponentially stable, mixed delays

Digital Object Identifier (DOI):

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[1] A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York, NY, USA, 1966.
[2] R. Agarwal, Y. Kim, and S. Sen, Advanced Discrete Halanay-Type Inequalities: Stability of Difference Equations, Volume 2009, Article ID 535849, 11 pages doi:10.1155/2009/535849.
[3] J. L, On Some New Impulsive Integral Inequalities, Journal of Inequalities and Applications Volume 2008, Article ID 312395, 8 pages doi:10.1155/2008/312395.
[4] D. Xu, Z. Yang, Impulsive delay diffrential inequality and stability of neural networks, Journal of Mathematical Analysis and Applications, 305 (2005) 107-120.
[5] Z. Ma, X. Wang, A new singular impulsive delay dierential inequality and its application, Journal of Inequalities and Applications, Accepted Article.
[6] W. Cheung, D. Zhao, Gronwall-Bellman-Type Integral Inequalities and applications to BVPs, Journal of Inequalities and Applications, Accepted Article.
[7] Y. Yang, J. Cao, Solving Quadratic Programming Problems by Delayed Projection Neural Network, IEEE Transactions on neural metworks. 17 (2006) 1630-1634.
[8] X. Hu, Applications of the general projection neural network in solving extended linear-quadratic programming problems with linear constraints, Neurocomputing (2008), doi:10.1016/j.neucom.2008.02.016
[9] Y. Yang, J. Cao, A feedback neural network for solving convex ..., Appl. Math. Comput. (2008), doi:10.1016/j.amc.2007.12.029
[10] Li P et al., Delay-dependent robust BIBO stabilization ..., Chaos, Solitons and Fractals (2007), doi:10.1016/j.chaos.2007.08.059
[11] O. Lipovan, A Retarded Gronwall-Like Inequality and Its Applications, Journal of Mathematical Analysis and Applications 252 (2000) 389-401 .
[12] R. Agarwal, S. Deng and W. Zhang, Generalization of a retarded Gronwall-like inequality and its applications, Applied Mathematics and Computation 165 (2005) 599-612.
[13] A. Gallo, A. Piccirillo, About new analogies of GronwallCBellmanCBihari type inequalities for discontinuous functions and estimated solutions for impulsive differential systems, Nonlinear Analysis 67 (2007) 1550- 1559.
[14] W. Wang, A generalized retarded Gronwall-like inequality in two variables and applications to BVP, Applied Mathematics and Computation 191 (2007) 144-154.
[15] W. Zhang, S. Deng, Projected GronwallCBellmans inequality for integral functions, Mathematical and Computer Modelling. 34 (2001) 393- 402.
[16] X. Mao, Stochastic Differential Equations and Applications, Horwood Publication, Chichester, 1997.
[17] Yonghui Sun,Jinde Cao, pth moment exponential stability of stochastic recurrent neural networks with time-varying delays, Nonlinear Analysis: RealWorld Applications, 8 (2007) 1171-1185.
[18] Y. Xia, Z. Huang, M. Han, Exponential p-stability of delayed Cohen- Grossberg-type BAM neural networks with impulses, Chaos, Solitons and Fractals, 38 (2008) 806-818.
[19] L. Sheng, H. Yang, Exponential synchronization of a class of neural networks with mixed time-varying delays and impulsive effects, Neurocomputing, 71 (2008) 3666-3674.
[20] H. Kopka and P. W. Daly, A Guide to LATEX, 3rd ed. Harlow, England: Addison-Wesley, 1999.