New global stability conditions for a class of difference equations

Abstract

In this paper we consider some classes of difference equations, including the well-known Clark model, and study the stability of their solutions. In order to do that we introduce a property, namely semicontractivity, and study relations between ‘semi-contractive’ functions and sufficient conditions for the solution of the difference equation to be globally asymptotically stable. Moreover, we establish new sufficient conditions for the solution to be globally asymptotically stable, and we improve the ‘3/2 criteria’ type stability conditions.

Keywords

Clark model / global asymptotic stability / semi-contractive function / nonlinear difference equation

References

[1.] Elaydi S. An Introduction to difference Equations, 2005 3rd ed. New York: Springer-Verlag.

[2.] El-Morshedy H. A. The global attractivity of difference equations of nonincreasing nonlinearities with applications. Comput Math Appl, 2003, 45: 749-758.

[3.] El-Morshedy H. A., Liz E. Globally attracting fixed points in higher order discrete population models. J Math Biol, 2006, 53: 365-384.

[4.] El-Morshedy H. A., López V. J., Liz E. Periodic points and stability in Clark’s delayed recruitment model. Nonlinear Analysis: RWA, 2008, 9: 776-790.

[5.] Györi I., Hartung F. Stability in delay perturbed differential and difference equations. Fields Inst Commun, 2001, 29: 181-194.

[6.] Györi I., Pituk M. Asymptotic stability in a linear delay differential equation. Proceedings of SICDEA, Hungary, 1997, Langhorne: Gorden and Breach Science.

[7.] Karakostas G., Philos C. G., Sficas Y. G. The dynamics of some discrete population models. Nonlinear Analysis TMA, 1991, 17: 1069-1084.

[8.] Liz E. A sharp global stability result for a discrete population model. J Math Anal Appl, 2007, 330: 740-743.

[9.] Muroya Y. A global stability criterion in scalar delay differential equations. J Math Anal Appl, 2007, 236: 209-227.

[10.] Muroya Y, Ishiwata E. Global stability for nonlinear difference equations with variable delay. In: Proceedings of the Third International Conference on Nonlinear Analysis and Convex Analysis, Tokyo, Japan, 2003. 2004, 347–358

[11.] Muroya Y., Ishiwata E., Guglielmi N. Global stability for nonlinear difference equations with variable coefficients. J Math Anal Appl, 2007, 334: 232-247.

[12.] So J. W. -H., Yu J. S. Global stability in a logistic equation with piecewise constant arguments. Hokkaido Math J, 1995, 24: 269-286.

[13.] Tkachenko V., Trofimchuk S. Global stability in difference equations satisfying the generalized Yorke condition. J Math Anal Appl, 2005, 303: 173-187.

[14.] Tkachenko V., Trofimchuk S. A global attractivity criterion for nonlinear nonautonomous difference equations. J Math Anal Appl, 2006, 322: 901-912.

[15.] Uesugi K., Muroya Y., Ishiwata E. On the global attractivity for a logistic equation with piecewise constant arguments. J Math Anal Appl, 2004, 294: 560-580.

[16.] Wazewska-Czyzewska M., Lasota A. Mathematical problems of the dynamics of the red-blood cells systems. Ann Polish Math Soc Series III, Appl Math, 1988, 17: 23-40.

[17.] Yu J. S. Asymptotic stability for a linear difference equation with variable delay. Comput Math Appl, 1998, 36: 203-210.