Stability analysis of a class of higher order difference equations. (English) Zbl 1470.39001

Liu, Yuanyuan; Meng, Fanwei

Abstr. Appl. Anal. 2014, Article ID 434621, 7 p. (2014).

Summary: We consider the sufficient conditions for asymptotic stability and instability of certain higher order nonlinear difference equations with infinite delays in finite-dimensional spaces. With the aid of the general comparison condition on the right-hand side function \(f_k(\cdot)\), we generalize the stability and instability result.

MSC:

39A05

General theory of difference equations

References:

[1]	Agarwal, R. P., Difference Equations and Inequalities. Theory, Methods, and Applications. Difference Equations and Inequalities. Theory, Methods, and Applications, Monographs and Textbooks in Pure and Applied Mathematics, 228 (2000), New York, NY, USA: Marcel Dekker, New York, NY, USA · Zbl 0952.39001
[2]	Elaydi, S., An Introduction to Difference Equations (1995), New York, NY, USA: Springer, New York, NY, USA · Zbl 0855.39003
[3]	Kocić, V. L.; Ladas, G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and Its Applications, 256 (1993), Dordrecht, The Netherlands: Kluwer Academic Publishers Group, Dordrecht, The Netherlands · Zbl 0787.39001 · doi:10.1007/978-94-017-1703-8
[4]	Leung, A. Y. T.; Xu, J. N.; Tsui, W. S., Nonlinear delay difference equations for housing dynamics assuming heterogeneous backward-looking expectations, Applied Mathematics and Mechanics, 28, 6, 785-798 (2007) · Zbl 1231.39005 · doi:10.1007/s10483-007-0609-z
[5]	Cull, P.; Flahive, M.; Robson, R., Difference Equations. From Rabbits to Chaos (2005), New York, NY, USA: Springer, New York, NY, USA · Zbl 1085.39002
[6]	Phat, V. N., Constrained Control Problems of Discrete Processes (1996), Singapore: World Scientific, Singapore · Zbl 0917.93002 · doi:10.1142/9789812812889
[7]	Kipnis, M. M.; Komissarova, D. A., A note on explicit stability conditions for autonomous higher order difference equations, Journal of Difference Equations and Applications, 13, 5, 457-461 (2007) · Zbl 1119.39007 · doi:10.1080/10236190601132933
[8]	Mohamad, S.; Gopalsamy, K., Exponential stability of continuous-time and discrete-time cellular neural networks with delays, Applied Mathematics and Computation, 135, 1, 17-38 (2003) · Zbl 1030.34072 · doi:10.1016/S0096-3003(01)00299-5
[9]	Yao, J.; Meng, F., Asymptotic behavior of solutions of certain higher order nonlinear difference equation, Journal of Computational and Applied Mathematics, 205, 1, 640-650 (2007) · Zbl 1127.39027 · doi:10.1016/j.cam.2006.05.039
[10]	Bay, N. S.; Phat, V. N., Stability analysis of nonlinear retarded difference equations in Banach spaces, Computers & Mathematics with Applications, 45, 6-9, 951-960 (2003) · Zbl 1053.39004 · doi:10.1016/S0898-1221(03)00068-3
[11]	LaSalle, J. P., The Stability and Control of Discrete Processes. The Stability and Control of Discrete Processes, Applied Mathematical Sciences, 62 (1986), New York, NY, USA: Springer, New York, NY, USA · Zbl 0606.93001 · doi:10.1007/978-1-4612-1076-4
[12]	Naulin, R.; Pinto, M., Stability of discrete dichotomies for linear difference systems, Journal of Difference Equations and Applications, 3, 2, 101-123 (1997) · Zbl 0884.39002 · doi:10.1080/10236199708808090
[13]	Naulin, R.; Vanegas, C., Instability of discrete-time systems, Electronic Journal of Differential Equations, 33, 1-11 (1998) · Zbl 0912.39005
[14]	Xiao, H. M.; Liu, Y. Q., The stability of linear time-varying discrete systems with time-delay, Journal of Mathematical Analysis and Applications, 188, 1, 66-77 (1994) · Zbl 0817.93054 · doi:10.1006/jmaa.1994.1411
[15]	Yang, X. S.; Stepanenko, Y., A stability criterion for discrete nonlinear systems with time delayed feedback, IEEE Transactions on Automatic Control, 39, 3, 585-589 (1994) · Zbl 0814.93057 · doi:10.1109/9.280765