Global stability for a class of delay differential equations. (English) Zbl 1064.34061

Foryś, U.

Appl. Math. Lett. 17, No. 5, 581-584 (2004).

The paper is concerned with the study of the behavior of solutions of the delay differential equation \[ {\dot x}(t) = \alpha f(x(t-1)), x(t) = x^0(t) \quad\text{for }t \in [-1, 0] \] where \(\alpha \geq 0\) and \(f\) is a continuously differentiable unimodal mapping with \(f(0)=f(1)=0\). The author gives sufficient conditions for the nonnegativity of solutions of the equation and for the global stability of the solutions.

Reviewer: Bogdan Sasu (Timişoara)

MSC:

34K20

Stability theory of functional-differential equations

Keywords:

delay differential equation; stability; global stability

Software:

PSPManalysis

References:

[1]	Hutchinson, G. E., Circular casual systems in ecology, Ann. N.Y. Acad. Sci., 50, 221-246 (1948)
[2]	Schuster, R.; Schuster, H., Reconstruction models for the Ehrlich ascites tumor of the mouse, (Arino, O.; Axelrod, D.; Kimmel, M., Mathematical Population Dynamics, Volume 2 (1995), Wuertz: Wuertz Winnipeg)
[3]	Foryś, U.; Kolev, M., Time delays in proliferation and apoptosis for solid avascular tumour, (Mathematical Modelling of Population Dynamics, Volume 63 (2004), Banach Center Publications: Banach Center Publications Warsaw), Warsaw University, RW 02-10 (110), (Preprint, August 2002) · Zbl 1058.35107
[4]	Kirkilionis, M.; Diekman, O.; Lisser, B.; Nool, M.; Sommeijer, B.; de Roos, A., Numerical continuation of equilibria of physiologically structured population models. I. Theory, Math. Models Methods Appl. Sci., 6, 1101-1127 (2001) · Zbl 1013.92036
[5]	Hale, J., Theory of Functional Differential Equations (1997), Springer: Springer New York
[6]	Foryś, U.; Marciniak-Czochra, A., Delay logistic equation with diffusion, (Foryś, U.; Zió̷ko, M., Proceedings of VIII National Conference on Mathematics Applied to Biology and Medicine, Lajs 2002. Proceedings of VIII National Conference on Mathematics Applied to Biology and Medicine, Lajs 2002, Warsaw (2002))
[7]	Foryś, U.; Marciniak-Czochra, A., Logistic equation in tumour growth modelling, Int. J. Appl. Math. Comp. Sci., 13, 3, 317-325 (2003) · Zbl 1035.92017
[8]	Bodnar, M., The nonnegativity of solutions of delay differential equations, Appl. Math. Lett., 13, 6, 91-95 (2000) · Zbl 0958.34049
[9]	Foryś, U., On the Mikhailov criterion and stability of delay differential equations (November 2001), Warsaw University, RW 01-14 (97) Preprint
[10]	Foryś, U., Biological delay systems and the Mikhailov criterion of stability, J. Biol. Sys., 12, 1, 1-16 (2004)