×

On exponential dichotomy, Bohl–Perron type theorems and stability of difference equations. (English) Zbl 1068.39004

The authors consider an extension of the Bohl-Perron theorem for a linear difference system in a Banach space. New explicit conditions for exponential stability of a scalar nonautonomous delay difference equation are obtained.

MSC:

39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
34K20 Stability theory of functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Daleckiiˇ, Yu. L.; Krein, M. G., Stability of Solutions of Differential Equations in Banach Spaces (1974), American Mathematical Society: American Mathematical Society Providence, RI
[2] Halanay, A., Differential Equations: Stability, Oscillations, Time Lag (1966), Academic Press: Academic Press New York
[3] Tyshkevich, V. A., A perturbation-accumulation problem for linear differential equations with time-lag, Differential Equations, 14, 177-186 (1978) · Zbl 0409.34073
[4] Azbelev, N. V.; Berezansky, L. M.; Simonov, P. M.; Chistyakov, A. V., Stability of linear systems with time-lag, Differential Equations. Differential Equations, Differential Equations. Differential Equations. Differential Equations, Differential Equations, Differential Equations, 29, 153-160 (1993), 1165-1172
[5] Kurbatov, V. G., Stability of functional differential equations, Differential Equations, 17, 611-618 (1981) · Zbl 0479.34026
[6] Anokhin, A.; Berezansky, L.; Braverman, E., Exponential stability of linear delay impulsive differential equations, J. Math. Anal. Appl., 193, 923-941 (1995) · Zbl 0837.34076
[7] Aulbach, B.; Van Minh, N., The concept of spectral dichotomy for linear difference equations, II, J. Differ. Equations Appl., 2, 251-262 (1996) · Zbl 0880.39009
[8] Pituk, M., A criterion for the exponential stability of linear difference equations, Appl. Math. Lett., 17, 779-783 (2004) · Zbl 1068.39019
[9] Elaydi, S., Periodicity and stability of linear Volterra difference systems, J. Math. Anal. Appl., 181, 483-492 (1994) · Zbl 0796.39004
[10] Elaydi, S.; Zhang, S., Stability and periodicity of difference equations with finite delay, Funkcial. Ekvac., 37, 401-413 (1994) · Zbl 0819.39006
[11] Kolmanovskii, V.; Shaikhet, L., Some conditions for boundedness of solutions of difference Volterra equations, Appl. Math. Lett., 16, 857-862 (2003) · Zbl 1107.39301
[12] Györi, I.; Pituk, M., Asymptotic stability in a linear delay difference equation, (Proceedings of SICDEA, Veszprem, Hungary, August 6-11, 1995 (1997), Gordon and Breach Science: Gordon and Breach Science Langhorne, PA) · Zbl 0846.39003
[13] Cooke, K. L.; Györi, I., Numerical approximation of solutions of delay differential equations on an infinite interval using piecewise constant arguments. Advances in difference equations, Comput. Math. Appl., 28, 81-92 (1994) · Zbl 0809.65074
[14] Györi, I.; Hartung, F., Stability in delay perturbed differential and difference equations, Fields Inst. Commun., 20, 181-194 (2001) · Zbl 0990.34066
[15] Levin, S. A.; May, R. M., A note on difference-delay equations, Theoret. Popul. Biol., 9, 178-187 (1976) · Zbl 0338.92021
[16] Zhang, B. G.; Tian, C. J.; Wong, P. J.Y., Global attractivity of difference equations with variable delay, Dynam. Contin. Discrete Impuls. Systems, 6, 307-317 (1999) · Zbl 0938.39016
[17] Erbe, L. H.; Xia, H.; Yu, J. S., Global stability of a linear nonautonomous delay difference equation, J. Differ. Equations Appl., 1, 151-161 (1995) · Zbl 0855.39007
[18] Berezansky, L.; Braverman, E., On Bohl-Perron type theorems for linear difference equations, Functional Differential Equations, 11, 19-29 (2004) · Zbl 1060.39002
[19] Kocić, V. L.; Ladas, G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Math. Appl., vol. 256 (1993), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0787.39001
[20] Kuruklis, S. A., The asymptotic stability of \(x_{n + 1} - a x_n + b x_{n - k} = 0\), J. Math. Anal. Appl., 188, 719-731 (1994) · Zbl 0842.39004
[21] Ladas, G.; Qian, C.; Vlahos, P. N.; Yan, J., Stability of solutions of linear nonautonomous difference equations, Appl. Anal., 41, 183-191 (1991) · Zbl 0701.39001
[22] Driver, R. D.; Ladas, G.; Vlahos, P. N., Asymptotic behavior of a linear delay difference equation, Proc. Amer. Math. Soc., 115, 105-112 (1992) · Zbl 0751.39001
[23] Kovácsvölgyi, I., The asymptotic stability of difference equations, Appl. Math. Lett., 13, 1-6 (2000) · Zbl 0964.39013
[24] Yu, J. S., Asymptotic stability for a linear difference equation with variable delay. Advances in difference equations, II, Comput. Math. Appl., 36, 203-210 (1998) · Zbl 0933.39009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.