非线性中立型变延迟微分方程的长时间稳定性

Abstract

Abstract: The long-time behaviour of nonlinear neutral delay differential equations (NDDEs) with variable delay is analysed. A main result guaranteeing uniform ultimate boundedness for a variable delay nonlinear system has been established. The sufficient conditions for the uniform ultimate boundedness for two typical examples of NDDEs, the constant delay differential equations and the proportional delay differential equations, are derived from the main result. Some examples are given to illustrate the applications of these results.

Key words: Neutral delay differential equations, Variable delay, Long-time stability, Asymptotic behaviour, Uniform ultimate boundedness, Dissipativity

CLC Number:

O177

Wang Wansheng, Zhong Peng, Zhao Xinyang. Long-Time Stability of Nonlinear Neutral Differential Equations with Variable Delay[J].Acta mathematica scientia,Series A, 2018, 38(1): 96-109.

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References

[1] Baker C T H, Bocharov G A, Rihan F A. A Report on the Use of Delay Differential Equations in Numerical Modelling in the Biosciences. Manchester:Manchester Univ, 1999
[2] Bocharov G A, Rihan F A. Numerical modelling in biosciences using delay differential equations. J Comput Appl Math, 2000, 125:183-199
[3] Bocharov K G, Hadeler P. Structured population models, conservation laws, and delay equations. J Differ Equ, 2000, 168:212-237
[4] Burton T A. Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Orlando:Academic Press, 1985
[5] Chukwu E N, Simpson H C. The solution operator in W_p⁽¹⁾ for systems of neutral type. Nonlinear Anal TMA, 1997, 29:1173-1195
[6] Gan S Q. Dissipativity of θ-methods for nonlinear Volterra delay-integro-differential equations. J Comput Appl Math, 2007, 206:898-907
[7] Gan S Q. Dissipativity of θ-methods for nonlinear delay differential equations of neutral type. Appl Numer Math, 2009, 59:1354-1365
[8] Hale J K. Theory of Functional Differential Equations. New York:Springer-Verlag, 1977
[9] Hale J K. Asymptotic Behavior of Dissipative Systems. Providence:American Mathematical Society, 1988
[10] Hill A T. Global dissipativity for A-stable methods. SIAM J Numer Anal, 1997, 34:119-142
[11] Hill A T. Dissipativity of Runge-Kutta methods in Hilbert spaces. BIT, 1997, 37:37-42
[12] Huang C H. Dissipativity of Runge-Kutta methods for dynamical systems with delays. IMA J Numer Anal, 2000, 20:153-166
[13] Humphries A R, Stuart A M. Runge-Kutta methods for dissipative and gradient dynamical systems. SIAM J Numer Anal, 1994, 31:1452-1485
[14] Li M Y, Wang C C, Wei J J. Global Hopf bifurcation analysis of a Nicholson's blowflies equation of neutral type. J Dyn Diff Equat, 2014, 26:165-179
[15] Li S F. Stability analysis of solutions to nonlinear stiff Volterra functional differential equations in Banach spaces. Sci China Math, 2005, 48:372-387
[16] Makay G. Uniform boundedness and uniform ultimate boundedness for functional differential equations. Funkcialaj Ekwacioj, 1995, 38:283-296
[17] Mackey M C, Glass L. Oscillations and chaos in physiological control systems. Science, 1977, 197:287-289
[18] Stuart A M, Humphries A R. Model problems in numerical stability theory for initial value problems. SIAM Review, 1994, 36:226-257
[19] Stuart A M, Humphries A R. Dynamical Systems and Numerical Analysis. Cambridge:Cambridge Unversity Press, 1996
[20] Tian H J. Numerical and analytic dissipativity of the θ-method for delay differential equation with a bounded variable lag. International J Bifurcation and Chaos, 2004, 14:1839-1845
[21] Tian H J, Guo N, Shen A L. Dissipativity of delay functional differential equations with bounded lag. J Math Anal Appl, 2009, 355:778-782
[22] Wang W S, Li S F. Dissipativity of Runge-Kutta methods for neutral delay differential equations with piecewise constant delay. Appl Math Lett, 2008, 21:983-991
[23] Wang W S, Zhang C J. Preserving stability implicit Euler method for nonlinear Volterra and neutral functional differential equations in Banach space. Numer Math, 2010, 115:451-474
[24] Wang W S. A generalized Halanay inequality for stability of nonlinear neutral functional differential equations. J Inequal Appl, 2010, DOI:10.1155/2010/475019
[25] Wang W S, Zhang C J. Analtical and numerical dissipativity for nonlinear generalized pantograph equations. Discrete Contin Dyn Syst, 2011, 29:1245-1260
[26] Wang W S, Li S F, Yang Y S. Contractivity and exponential stability of solutions to nonlinear neutral functional differential equations in Banach spaces. Acta Math Appl Sin Engl Ser, 2012, 28:289-304
[27] Wang W S, Fan Q, Zhang Y, Li S F. Asymptotic stability of solution to nonlinear neutral and Volterra functional differential equations in Banach spaces. Appl Math Comput, 2014, 237:217-226
[28] Wang W S. Uniform ultimate boundedness of numerical solutions to nonlinear neutral delay differential equations. J Comput Appl Math, 2017, 309:132-144
[29] Wen L P, Li S F. Dissipativity of Volterra functional differential equations. J Math Anal Appl, 2006, 324:696-706
[30] Wen L P, Yu Y X, Wang W S. Generalized Halanay inequalities for dissipativity of Volterra functional differential equations. J Math Anal Appl, 2008, 347:169-178
[31] Wen L P, Wang W S, Yu Y X. Dissipativity and asymptotic stability of nonlinear neutral delay integro-differential equations. Nonlinear Anal, 2010, 72:1746-1754
[32] Zhang C J, Li S F. Dissipativity and exponentially asymptotic stability of the solutions for nonlinear neutral functional-differential equations. Appl Math Comput, 2001, 119:109-115
[33] Zhang C J, Vandewalle S. Stability criteria for exact and discrete solutions of neutral multidelay-integro-differential equations. Adv Comput Math, 2008, 28:383-399

Long-Time Stability of Nonlinear Neutral Differential Equations with Variable Delay

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