Almost periodic solutions of delay difference systems. (English) Zbl 1029.39011

Consider the system of delay difference equations \[ x(n+ 1)= f(n, x(n)),\qquad n\in\mathbb{Z}, \] where \(f(n,\varphi)\) is almost periodic in \(n\) uniformly for \(\|\varphi\|\leq B\). Under some assumptions, to complicate to be presented here, the authors show that if the system is stable with respect to so-called disturbance on \(f\), then the solution to the system is almost periodic, resp. asymptotically almost periodic etc.
The existence and uniqueness of such solutions is proved using some properties of Lyapunov functionals.


39A11 Stability of difference equations (MSC2000)
Full Text: DOI


[1] Hale, J. K., Periodic and almost periodic solution of functional differential equations, Arch. Rational Mech., 15, 289-309 (1964) · Zbl 0129.06006
[2] Yoshizawa, T., (Stability Properties in Almost Periodic Systems of Functional Differential Equations, Lecture Notes in Math, vol. 799 (1979), Springer: Springer New York), 385-409 · Zbl 0432.34049
[3] Yoshizawa, T., Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions (1975), Springer: Springer New York · Zbl 0304.34051
[4] Yuan, R., Existence of almost periodic solutions of functional differential equations of neutral type, J. Math. Anal. Appl., 165, 524-538 (1992) · Zbl 0754.34074
[5] Zhang, S., Almost periodic solutions of difference systems, Chinese Sci. Bull., 43, 2041-2047 (1998) · Zbl 1433.81116
[6] Zhang, S., Existence of almost periodic solution for difference systems, Ann. Differential Equations, 16, 2, 184-206 (2000) · Zbl 0981.39003
[7] Elaydi, S.; Zhang, S., Stability and periodicity of difference equations with finite delay, Funkcial. Ekvac., 37, 401-413 (1994) · Zbl 0819.39006
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.