Abstract

The exponential stability concept for nonlinear non-instantaneous impulsive difference equations with a single delay is studied and some criteria are derived. These results are also applied for a neural networks with switching topology at certain moments and long time lasting impulses. It is considered the general case of time varying connection weights. The equilibrium is defined and exponential stability is studied. The obtained results are illustrated on examples.

Citation details of the article

Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 33
Issue: 2
Year: 2020

DOI: 10.12732/ijam.v33i2.1

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References

1. [1] R.P. Agarwal, Difference Equations and Inequalities. Theory, Methods and Applications, Monographs and Textbooks in Pure and Applied Mathematics, 2nd Ed., Marcel Dekker, Inc., New York, 2000.
2. [2] R. Agarwal, S. Hristova, R. O’Regan, Non-instantaneous impulses in Caputo fractional differential equations, Fract. Calc. Appl. Anal., 20, No 3 (2017), 595-622; DOI: 10.1515/fca-2017-0032.
3. [3] R. Agarwal, D. O’Regan, S. Hristova, Monotone iterative technique for the initial value problem for differential equations with non-instantaneous impulses, Appl. Math. Comput., 298, No 1 (2017), 45-56.
4. [4] M. Alshammari, Y. Raffoul, Exponential stability and instability in multiple delays difference equations, Khayyam J. Math., 1, No 2 (2015), 174- 184.
5. [5] D. Bainov, S. Hristova, The method of for the periodic boundary value problem for systems of impulsive differential equations, Appl. Math. Com- put., 117, No 1 (2001), 73-85.
6. [6] L. Barreira, C. Valls, Stability in delay difference equations with nonuniform exponential behavior, J. Diff. Eq., 238, No 2 (2007), 470-490.
7. [7] J. Cermak, Stability conditions for linear delay difference equations: Surveys and perspectives, Tatra Mt. Math. Publ., 63, No 1 (2015), 1-29.
8. [8] M. Danca, M. Feckan, M. Pospsil, Difference equations with impulses, Opuscula Math., 39, No 1 (2019), 5-22.
9. [9] J. Diblk, D. Khusainov, J. Bastinec, A. Sirenko, Exponential stability of linear discrete systems with constant coefficients and single delay, Appl. Math. Lett., 51 (2016), 68-73.
10. [10] S. Elaydi, An Introduction to Difference Equations, Springer, New York, 3rd Ed., 2005.
11. [11] S. Hristova, Integral stability in terms of two measures for impulsive functional differential equations, Math. Comput. Modell., 51, No 1-2 (2010), 100-108.
12. [12] V. Laksmikantham, D. Trigiante, Theory of Difference Equations (Numerical Methods and Applications), 2nd Ed., Marcel Dekker, 2002.
13. [13] H. Li, C. Li, T. Huang, Comparison principle for difference equations with variable-time impulses, Modern Physics Letters B, 32, No 2 (2018), DOI: 10.1142/S0217984918500136.
14. [14] D. Li, S. Long, X. Wang, Difference inequality for stability of impulsive difference equations with distributed delays, J. Ineq. Appl., 2011 (2011), Art. 8.
15. [15] R. Medina, Stability for linear Volterra difference equations in Banach spaces, Abst. Appl. Anal., 2018 (2018), Art. ID 3905632, 6 p.
16. [16] G. Wu, D. Baleanu, Stability analysis of impulsive fractional difference equations, Fract. Calc. Appl. Anal., 21, No 2 (2018), 354-375, DOI: 10.1515/fca-2018-0021.
17. [17] H. Xu, Y. Chen, K. Teo, Global exponential stability of impulsive discretetime neural networks with time-varying delays, Appl. Math. Comput., 217 (2010), 537-544.