Research PapersNo Access

Comparison principle for difference equations with variable-time impulses

Chongqing Key Laboratory of Nonlinear Circuits and Intelligent Information Processing, College of Electronic and Information Engineering, Southwest University, Chongqing 400715, China

Search for more papers by this author

Chuandong Li

http://orcid.org/0000-0001-6155-4849

Chongqing Key Laboratory of Nonlinear Circuits and Intelligent Information Processing, College of Electronic and Information Engineering, Southwest University, Chongqing 400715, China

E-mail Address: licd@cqu.edu.cn

Corresponding author.

Search for more papers by this author

, and

Tingwen Huang

Department of Mathematics, Texas A&M University at Qatar, Qatar

Search for more papers by this author

https://doi.org/10.1142/S0217984918500136Cited by:2 (Source: Crossref)

Abstract

In this paper, a class of difference equations with variable-time impulses is considered. By applying comparison principle, we shall show that difference equations with variable-time impulse can be reduced to the corresponding difference equations with fixed-time impulses under well-selected conditions. Meanwhile, the fixed-time impulsive systems can be regarded as the comparison system of the difference equations with variable-time impulses. Furthermore, we use a series of sufficient criteria to illustrate the same stability properties between variable-time impulsive difference equations and the fixed-time ones. We then establish several sufficient conditions guaranteeing the global exponential stability of variable-time impulsive difference equations by comparison principle. As an application, global exponential stability of discrete-time neural networks with variable-time impulses is discussed. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed results.

Keywords:

References

1. L. Nie et al., Biosystems 98(2) (2009) 67–72. Crossref, Web of Science, Google Scholar
2. L. Nie et al., Nonlinear Anal.-Real 11(3) (2010) 1364–1373. Crossref, Web of Science, Google Scholar
3. L. Nie, Z. Teng and L. Hu, Math. Comput. Simulat. 79(7) (2009) 2122–2134. Crossref, Web of Science, Google Scholar
4. V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations (World Scientific, Singapore, 1989). Link, Google Scholar
5. M. Akhmet, Principles of Discontinuous Dynamical Systems (Springer, New York, 2010). Crossref, Google Scholar
6. I. Rachunková and J. Tomeček, State-Dependent Impulses: Boundary Value Problems on Compact Interval (Atlantis Press, Paris, 2015). Crossref, Google Scholar
7. W. Zhu, D. Xu and Z. Yang, Appl. Math. Comput. 181(1) (2006) 65–72. Web of Science, Google Scholar
8. J. Wang and X. Li, Appl. Math. Comput. 217(5) (2010) 1933–1938. Web of Science, Google Scholar
9. S. Zhong, C. Li and X. Liao, Neurocomputing 73(16–18) (2010) 3132–3138. Crossref, Web of Science, Google Scholar
10. S. Wu et al., Neurocomputing 82(4) (2012) 29–36. Crossref, Web of Science, Google Scholar
11. Y. Zhang, Math. Methods Appl. Sci. 35(11) (2012) 1287–1299. Crossref, Web of Science, Google Scholar
12. X. Zhao, Nonlinear Anal. 71(12) (2009) 2873–2878. Crossref, Web of Science, Google Scholar
13. Z. Li et al., Nonlinear Dynam. 82(4) (2015) 2081–2096. Crossref, Web of Science, Google Scholar
14. Y. Zhang, J. Franklin Inst. 350(4) (2013) 737–756. Crossref, Web of Science, Google Scholar
15. X. Wang et al., Nonlinear Dynam. 84(3) (2016) 1747–1757. Crossref, Web of Science, Google Scholar
16. S. Wu, K. Li and J. Zhang, Appl. Math. Comput. 218(12) (2012) 6972–6986. Crossref, Web of Science, Google Scholar
17. A. Abdurahman, H. Jiang and Z. Teng, Nonlinear Dynam. 78(4) (2014) 2627–2638. Crossref, Web of Science, Google Scholar
18. Y. Zhang, Math. Comput. Simulat. 132 (2017) 183–194. Crossref, Web of Science, Google Scholar
19. S. Zheng, Nonlinear Dynam. 67(4) (2012) 2621–2630. Crossref, Web of Science, Google Scholar
20. J. Wang, H. Jiang and C. Hu, Neurocomputing 142(1) (2014) 542–550. Crossref, Web of Science, Google Scholar
21. J. Zhou, Q. Wu and L. Xiang, Nonlinear Dynam. 69(3) (2012) 1393–1403. Crossref, Web of Science, Google Scholar
22. C. Liu et al., Neurocomputing 171 (2015) 1644–1654. Crossref, Web of Science, Google Scholar
23. M. Şaylı and E. Yılmaz, Ann. Oper. Res. (2016) 1–27. Google Scholar
24. M. Şaylı and E. Yılmaz, Neural Networks 68 (2015) 1–11. Crossref, Web of Science, Google Scholar
25. M. Şaylı and E. Yılmaz, Neural Networks 60 (2014) 67–73. Crossref, Web of Science, Google Scholar
26. M. Şaylı and E. Yılmaz, Neurocomputing 171 (2016) 1375–1386. Crossref, Web of Science, Google Scholar
27. X. Yang et al., Neurocomputing 207 (2016) 276–286. Crossref, Web of Science, Google Scholar
28. X. Liu and Q. Wang, Appl. Math. Comput. 174(1) (2006) 271–288. Web of Science, Google Scholar
29. X. Liu and G. Ballinger, Nonlinear Anal.-Theory 51(4) (2002) 633–647. Crossref, Web of Science, Google Scholar
30. C. L. Hwang, C. Jan and Y. H. Chen, IEEE T. Ind. Electron. 48(1) (2001) 47–59. Crossref, Web of Science, Google Scholar
31. W. Zhang et al., Neural Comput. Appl. 25(6) (2014) 1275–1283. Crossref, Web of Science, Google Scholar
32. Y. Wang, J. Cao and J. Hu, Neural Comput. Appl. 26(7) (2015) 3259–3272. Google Scholar