Dissipativity and stability of a class of nonlinear multiple time delay integro-differential equations
- College of Computational Science, Zhongkai University of Agriculture and Engineering, Guangzhou, 510225, P. R. China.
- Systems Engineering Institute, South China University of Technology, Guangzhou, 510640, P. R. China.
- School of Applied Mathematics, Guangdong University of Technology, Guangzhou, 510006, P. R. China.
- School of computer and electronic information, Guangdong University of Petrochemical Technology, Maoming 525000, P. R. China.
This paper is concerned with the dissipativity and stability of the theoretical
solutions of a class of nonlinear multiple time
delay integro-differential equations. At the first, we give a generalized Halanay inequality which plays an important role
in the study of dissipativity and stability of integro-differential equations. Then, we apply the
generalized Halanay inequality to the dissipativity and the
stability the theoretical solution of delay integro-differential equations
(or by small \(\epsilon\) perturbed) and some interesting results are obtained. Our results generalize a few previous known results.
Finally, two examples are provided to demonstrated the effectiveness and advantage of the theoretical results.
- Delay integro-differential equations
- dynamical systems
- Halanay inequality
S. Arik, On the global dissipativity of dynamical neural networks with time delays, Phys. Lett. A, 326 (2004), 126–132.
C. T. H. Baker, Development and application of Halanay-type theory: Evolutionary differential and difference equations with time lag, J. Comput. Appl. Math., 234 (2010), 2663–2682
C. T. H. Baker, A. Tang, Generalized Halanay inequalities for Volterra functional differential equations and discreted versions, in: Volterra Centennial Meeting, 2000 (2000), 39–55.
Z. W. Cai, L. H. Huang, Functional differential inclusions and dynamic behaviors for memristor-based BAM neural networks with time-varying delays, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1279–1300.
M. El-Gebeily, D. O’Regan, Existence and boundary behavior for singular nonlinear differential equations with arbitrary boundary conditions, J. Math. Anal. Appl., 334 (2007), 140–156.
A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York-London (1966)
D. W. C. Ho, J. L. Liang, J. Lam, Global exponential stability of impulsive high-order BAM neural networks with timevarying delay, Neural Networks, 19 (2006), 1581–1590.
C. X. Huang, Y. G. He, H. Wang, Mean square exponential stability of stochastic recurrent neural networks with timevarying delays, Comput. Math. Appl., 56 (2008), 1773–1778.
X. D. Li, Existence and global exponential stability of periodic solution for impulsive CohenCGrossberg-type BAM neural networks with continuously distributed delays, Appl. Math. Comput., 215 (2009), 292–307.
X. D. Li, M. Bohner, Exponential synchronization of chaotic neural networks with mixed delays and impulsive effects via output coupling with delay feedback, Math. Comput. Modelling, 52 (2010), 643–653.
X. D. Li, M. Bohner, An impulsive delay differential inequality and applications, Comput. Math. Appl., 64 (2012), 1875–1881.
B. Li, D. Xu, Mean square asymptotic behavior of stochastic neural networks with infinitely distributed delays, Neurocomputing, 72 (2009), 3311–3317.
X. Liao, J. Wang, Global dissipativity of continuous-time recurrent neural networks with time delay, Phys. Rev. E, 68 (2003), 7 pages.
X. Y. Liu, T. P. Chen, J. D. Cao, W. L. Lu, Dissipativity and quasi-synchronization for neural networks with discontinuous activations and parameter mismatches, Neural Networks, 24 (2011), 1013–1021.
B. Liu, W. L. Lu, T. P. Chen, Generalized Halanay inequalities and their applications to neural networks with unbounded time-varying delays, IEEE Trans. Neural Networks, 22 (2011), 1508–1513.
X. Z. Liu, K. L. Teo, B. G. Xu, Exponential stability of impulsive high-order Hopfield-type neural networks with timevarying delay, IEEE Trans. Neural Networks, 16 (2005), 1329–1339.
Y. Liu, Z. Wang, X. Liu, Global asymptotic stability of generalized bi-directional as sociative memory networks with discrete and distributed delays, Chaos Solitons Fractals, 28 (2006), 793–803.
K. Y. Liu, H. Q. Zhang, An improved global exponential stability criterion for delayed neural networks, Nonlinear Anal. Real World Appl., 10 (2009), 2613–2619.
X. Y. Lou, B. T. Cui , Global robust dissipativity for integro-differential systems modeling neural networks with delays, Chaos Solitons Fractals, 36 (2008), 469–478.
S. Mohamad, K. Gopalsamy, Continuous and discrete Halanay-type inequalities, Bull. Austral. Math. Soc., 61 (2000), 371–385.
S. Mohamad, K. Gopalsamy, H. Akca, Exponential stability of artificial neural networks with distributed delays and large impulses, Nonlinear Anal. Real World Appl., 9 (2008), 872–888.
D. O’Regan, M. El-Gebeily, Existence, upper and lower solutions and quasilinearization for singular differential equations, IMA J. Appl. Math., 73 (2008), 323–344.
J.-L. Shao, T.-Z. Huang, X.-P. Wang , Further analysis on global robust exponential stability of neural networks with time-varying delays, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1117–1124.
J.-L. Shao, T.-Z. Huang, S. Zhou, Some improved criteria for global robust exponential stability of neural networks with time varying delays, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 3782–3794.
H. J. Tian, The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag, J. Math. Anal. Appl., 270 (2002), 143–149.
Z. W. Tu, J. G. Jian, K. Wang, Global exponential stability in Lagrange sense for recurrent neural networks with both time-varying delays and general activation functions via LMI approach, Nonlinear Anal. Real World Appl., 12 (2011), 2174–2182
Z. W. Tu, L. W. Wang, Z. W. Zha, J. Jian, Global dissipativity of a class of BAM neural networks with time-varying and unbound delays, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2562–2570.
L. S. Wang, D. Xu, Global exponential stability of Hopfield reaction-diffusion neural networks with time-varying delays, Sci. China Ser. F, 46 (2003), 466–474.
L. S. Wang, L. Zhang, X. H. Ding, Global dissipativity of a class of BAM neural networks with both time-varying and continuously distributed delays, Neurocomputing, 152 (2015), 250–260.
L. P. Wen, W. S. Wang, Y. X. Yu, Dissipativity and asymptotic stability of nonlinear neutral delay integro-differential equations, Nonlinear Anal., 72 (2010), 1746–1754.
L. P. Wen, Y. X. Yu, W. S. Wang, Genernalized Halanay inequalities for disspativity of Volterra functional differential equations, J. Math. Anal. Appl., 347 (2008), 169–178.
D. Xu, Z. C. yang , Impulsive delay differential inequality and stability of neural networks, J. Math. Anal. Appl., 305 (2005), 107–120.
X. S. Yang, Z. C. Yang, Synchronization of TS fuzzy complex dynamical networks with time-varying impulsive delays and stochastic effects, Fuzzy Sets and Systems, 235 (2014), 25–43.
D. Yue, S. F. Xu, Y. Q. Liu, A differential inequality with delay and impulse and its applications to the design of robust controllers, Control Theory Appl., 16 (1999), 519–524.
Y. Zhang, J. T. Sun , Stability of impulsive functional differential equations, Nonlinear Anal., 68 (2008), 3665–3678.
Y. Zhang, J. T. Sun, Stability of impulsive linear hybrid systems with time delay, J. Syst. Sci. Complex., 23 (2010), 738–747.
X. Y. Zhao, F. Q. Deng, Moment stability of nonlinear discrete stochastic systems with time-delays based on Hrepresentation technique, Automatica J. IFAC, 50 (2014), 530–536.
C. Zheng, N. L. JindeCao, Matrix measure based stability criteria for high-order neural networks with proportional delay, Neurocomputing, 149 (2015), 1149–1154.