Advanced Gronwall-Bellman-Type Integral Inequalities and Their Applications

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.

Authors:

• Zixin Liu
• Shu Lü
• Shouming Zhong
• Mao Ye

Keywords:

• Gronwall-Bellman-Type integral inequalities
• integrodifferential equation
• p-exponentially stable
• mixed delays.

References:

[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.