×

Exponential dichotomy and boundedness for retarded functional difference equations. (English) Zbl 1162.39002

The authors investigate the retarded linear functional equation
\[ x(n+1)=L(n,x_n),\,\,\,n\geq 0,\tag{E} \]
where \(\{L(n,\cdot)\}_{n\geq 0}\) is a uniformly bounded sequence of bounded linear operators mapping \({\mathcal B}_{\gamma}\) (\(\gamma>0\)) into \(\mathbb{C}^r\), with the phase space \({\mathcal B}_{\gamma}\) defined by \({\mathcal B}_{\gamma}=\{\varphi:\mathbb{Z}^-\to\mathbb{C}^r;\;\sup_{\theta\in\mathbb{Z}^-}|\varphi(\theta)|e^{\gamma\theta}<+\infty\},\) and \(x_n(s)=x(n+s)\) for \(s\in\mathbb{Z}^-\). They characterize the exponential dichotomy of equation (E) and give conditions for the exponential stability of the orbit of the solution operator of (E). The robustness of the exponential dichotomy and the boundedness for (E) under several type of perturbations are also studied. In the last section of the paper the authors apply the obtained results to analyze bounded solutions of a class of Volterra difference equations with infinite delay.

MSC:

39A10 Additive difference equations
39A12 Discrete version of topics in analysis
39A30 Stability theory for difference equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Agarwal R.P., Difference Equations and Inequalities (1992) · Zbl 0925.39001
[2] DOI: 10.1016/j.camwa.2004.06.032 · Zbl 1080.39007
[3] DOI: 10.1016/S0377-0427(99)00257-5 · Zbl 0940.39005
[4] Pinto M., J. Math. Anal. Appl. 227 pp 324– (2003)
[5] DOI: 10.1080/10236190290032499 · Zbl 1019.39008
[6] Vidal C., Adv. Difference Equ. 2006 pp 1– (2006)
[7] Elaydi S., Undergraduate Texts in Mathematics, 3. ed. (2005)
[8] Elaydi S., J. Difference Equ. Appl. 5 pp 1– (1999)
[9] Furumochi T., Fields Inst. Commun. AMS 42 pp 159– (2004)
[10] Nagabuchi Y., Japan. J. Math. 30 pp 387– (2004)
[11] DOI: 10.1080/1023619021000035836 · Zbl 1033.39009
[12] DOI: 10.1155/S0161171203206098 · Zbl 1029.34065
[13] Hino Y., Lecture Notes in Mathematics 1473, in: Functional Differential Equations with Infininte Delay (1991)
[14] DOI: 10.1155/ADE/2006/58453 · Zbl 1139.39305
[15] DOI: 10.1016/S0898-1221(01)00155-9 · Zbl 1016.39007
[16] DOI: 10.1016/S0362-546X(03)00021-X · Zbl 1031.39005
[17] DOI: 10.1080/10236190410001685021 · Zbl 1057.39015
[18] DOI: 10.1016/j.jmaa.2004.10.065 · Zbl 1074.39009
[19] S. Murakami, Some spectral properties of the solution operator for linear Volterra difference system, Proceedings of the third international conference of difference equations, Taipe, China (1997), pp. 301–311 · Zbl 0938.39017
[20] DOI: 10.1016/S0362-546X(97)00296-4 · Zbl 0889.39012
[21] Palmer K.J., Dyn. Reported 1 pp 265– (1998)
[22] Simon J., Ann. Mat. Pura Appl. pp 65– (1987)
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.