• Nguyen, Sinh Bay (Department of Mathematics University of Commerce) ;
  • Le, Van Hien (Department of Mathematics Hanoi National University of Education) ;
  • Hieu, Trinh (School of Engineering Deakin University)
  • Received : 2016.02.22
  • Accepted : 2017.06.28
  • Published : 2017.10.31


The problems of global and local exponential stability analysis of a class of nonlinear non-autonomous difference equations in Banach spaces are studied in this paper. By a novel comparison technique, new explicit exponential stability conditions are derived. Numerical examples are given to illustrate the effectiveness of the obtained results.


  1. M. Adivar and E. A. Bohner, Halanay type inequalities on time scales with applications, Nonlinear Anal. 74 (2011), no. 18, 7519-7531.
  2. R. P. Agarwal, Y. H. Kim, and S. K. Sen, New discrete Halanay inequalities: stability of difference equations, Commun. Appl. Anal. 12 (2008), no. 1, 83-90.
  3. G. Akrivis, Stability of implicit-explicit backward difference formulas for nonlinear par- abolic equations, SIAM J. Numer. Anal. 53 (2015), no. 1, 464-484.
  4. S. Badia, R. Codina, and H. Espinoza, Stability, convergence, and accuracy of stabilized nite element methods for the wave equation in mixed form, SIAM J. Numer. Anal. 52 (2014), no. 4, 1729-1752.
  5. N. S. Bay, On the exponential stability of nonlinear difference equations in Banach spaces, Comm. Appl. Nonlinear Anal. 21 (2014), no. 4, 16-26.
  6. N. S. Bay and V. N. Phat, Stability analysis of nonlinear retarded difference equations in Banach spaces, Comput. Math. Appl. 45 (2003), no. 6-9, 951-960.
  7. A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Clarendon Press, Oxford, 2003.
  8. L. Berezansky and E. Braverman, Stability conditions for scalar delay differential equa- tions with a non-delay term, Appl. Math. Comput. 250 (2015), 157-164.
  9. T. Erneux, Applied Delay Differential Equations, Springer, New York, 2009.
  10. M. I. Gil', Difference Equations in Normed Spaces: Stability and Oscilations, Elservier, Amsterdam, 2007.
  11. I. Gyori and F. Hartung, Asymptotic behaviour of nonlinear difference equations, J. Difference Equ. Appl. 18 (2012), no. 9, 1485-1509.
  12. L. V. Hien, A novel approach to exponential stability of nonlinear non-autonomous difference equations with variable delays, Appl. Math. Lett. 38 (2014), 7-13.
  13. L. V. Hien, Global asymptotic behaviour of positive solutions to a non-autonomous Nichol- son's blowflies model with delays, J. Biol. Dyn. 8 (2014), no. 1, 135-144.
  14. L. V. Hien, V. N. Phat, and H. Trinh, New generalized Halanay inequalities with appli- cations to stability of nonlinear nonautonomous time-delay systems, Nonlinear Dynam. 82 (2015), no. 1-2, 563-575.
  15. A. Ivanov, E. Liz, and S. Trofikmchuk, Halanay inequality, Yorke 3/2 stability criterion, and differential equations with maxima, Tohoku Math. J. 54 (2002), no. 2, 277-295.
  16. B. Li and Q. Song, Asymptotic behaviors of non-autonomous impulsive difference equa- tion with delays, Appl. Math. Model. 35 (2011), no. 7, 3423-3433.
  17. E. Liz, Stability of non-autonomous difference equations: simple ideas leading to useful results, J. Difference Equ. Appl. 17 (2011), no. 2, 203-220.
  18. E. Liz and J. B. Ferreiro, A note on the global stability of generalized difference equations, Appl. Math. Lett. 15 (2002), no. 6, 655-659.
  19. E. Liz, A. Ivanov, and J. B. Ferreiro, Discrete Halanay-type inequalities and applications, Nonlinear Anal. 55 (2003), no. 6, 669-678.
  20. H. R. Marzban, Optimal control of linear multi-delay systems based on a multi-interval decomposition scheme, Optim. Control Appl. Meth. 37 (2016), no. 1, 190-211.
  21. R. Medina, Delay difference equations in infinite-dimensional spaces, J. Difference Equ. Appl. 12 (2006), no. 8, 799-809.
  22. S. Mohamad and K. Gopalsamy, Continuous and discrete Halanay-type inequalities, Bull. Austral. Math. Soc. 61 (2000), no. 3, 371-385.
  23. R. K. Mohanty and R. Kumar, A new fast algorithm based on half-step discretization for one space dimensional quasilinear hyperbolic equations, Appl. Math. Comput. 244 (2014), 624-641.
  24. S. Singh and P. Lin, High order variable mesh off-step discretization for the solution of 1-D non-linear hyperbolic equation, Appl. Math. Comput. 230 (2014), 629-638.
  25. H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, 2011.
  26. Y. Song, Y. Shen, and Q. Yin, New discrete Halanay-type inequalities and applications, Appl. Math. Lett. 26 (2013), no. 2, 258-263.
  27. S. Udpin and P. Niamsup, New discrete type inequalities and global stability of nonlinear difference equations, Appl. Math. Lett. 22 (2009), no. 6, 856-859.
  28. T. Vyhlidal, J. F. Lafay, and R. Sipahi, Delay Systems: From Theory to Numerics and Applications, Springer, Dordrecht, 2014.
  29. L. Wang and X. Ding, Dissipativity of ${\theta}$-methods for a class of nonlinear neutral delay integro-differential equations, Int. J. Comput. Math. 89 (2012), no. 15, 2029-2046.
  30. L. Xu, Generalized discrete Halanay inequalities and the asymptotic behavior of non- linear discrete systems, Bull. Korean Math. Soc. 50 (2013), no. 5, 1555-1565.