Existence of almost periodic solutions for neutral delay difference systems

Abstract

In this paper, the existence of almost periodic solutions is studied via the Lyapunov function. Razumikhin type theorems are established on the existence, uniqueness and uniformly asymptotic stability of almost periodic solutions. Two examples are given to explain our results.

Keywords

Neutral delay difference system / almost periodic solution / Razumikhin technique / asymptotic stability

References

[1] Agarwal R P. Difference Equations and Inequalities. 2nd Ed. New York: Dekker, 2000

[2] Cooke K L. Stability analysis for a vector disease model. Rocky Mountain J Math, 1977, 9: 31-42

[3] Corduneanu C. Almost Periodic Functions. New York: Chelsea, 1989

[4] Fink A M, Seifert G. Liapunov functions and almost periodic solutions for almost periodic systems. J Diff Eqns, 2969, 5: 307-313

[5] Gopalsamy K, Mohamad S. Canonical solutions and almost periodicity in a discrete logistic equation. Appl Math Comput, 2000, 113(2-3): 305-323

[6] Hale J K. Periodic and almost periodic solution of functional differential equations. Arch Rational Mech Anal, 1964, 15: 289-309

[7] Hale J K. Theory of Functional Differential Equations. Berlin: Springer, 1977

[8] Hamaya Y. Existence of an almost periodic solution in a difference equation by Liapunov functions. Nonlinear Studies, 2001, 8(3): 373-379

[9] Hino Y. Almost periodic solutions of functional differential equations with infinite retardation. II. Tohoku Math J, 1980, 32: 525-530

[10] Lakshmikantham V, Trigiante D. Theory of Difference Equations. New York: Academic Press, INC, 1988

[11] Meisters G H. On almost periodic solutions of a class of differential equations. Proc Amer Math Soc, 1959, 10: 113-119

[12] Nakajima F. Existence of almost periodic solutions by Liapunov functions. Tohoku Math J, 1975, 27: 69-74

[13] Sawano K. Exponential asymptotic stability for functional-differential equations with infinite retardations. Tohoku Math J, 1979, 31: 363-382

[14] Yang X, Yuan R. On the module and spectral of almost periodic sequence in Banach space. J Beijing Normal University, 2005, 41(5): 455-459

[15] Yoshizawa T. Extreme stability and almost periodic solutions of functionaldi fferential equations. Arch Rational Mech Anal, 1964, 17: 148-170

[16] Yoshizawa T. Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions. Applied Mathematical Sciences, Vol 14. Berlin: Springer-Verlag, 1975

[17] Yuan R. Existence of almost periodic solution of functional differential equations. Ann of Diff Eqs, 1991, 7(2): 234-242

[18] Yuan R. Existence of almost periodic solutions of functional differential equations of neutral type. J Math Anal Appl, 1992, 165(2): 524-538

[19] Yuan R. Existence of almost periodic solutions of neutral functional-differential equations via Liapunov-Razumikhin function. Z Angew Math Phys, 1998, 49: 113-136

[20] Yuan R. The existence of almost periodic solutions of retarded differential equations with piecewise constant argument. Nonlinear Anal, 2002, 48: 1013-1032

[21] Yuan R, Hong J L. The existence of almost periodic solutions for a class of differential equations with piecewise constant argument. Nonlinear Analysis TMA, 1997, 28(8): 1439-1450

[22] Zhang S. Existence of almost periodic solutions for differences systems. Ann of Diff Eqs, 2000, 16(2): 184-206

[23] Zhang S, Liu P, Gopalsamy K. Almost periodic solutions of nonautonomous linear difference equations. Appl Anal, 2002, 81(2): 281-301

[24] Zheng G, Zhang S. Existence of almost periodic solutions of neutral delay difference systems. Dynamics of Continuous, Discrete and Impulsive Systems, Ser A: Math Anal, 2002, 9: 523-540

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg