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Asymptotic constancy for systems of delay differential equations. (English) Zbl 0959.34058

Results of the paper of K. L. Cooke and J. A. Yorke [Math. Biosci. 16, 75-101 (1973; Zbl 0251.92011)] and the author’s papers [Funkc. Ekvacioj., Ser. Int. 39, No. 3, 519-540 (1996; Zbl 0872.34052) and J. Math. Anal. Appl. 205, No. 2, 512-530 (1997; Zbl 0885.34060)] are extended to the \(n\)-dimensional system \[ x'(t)= A\bigl(x(t- \tau_1)-x(t-\tau _2)\bigr), \quad A\in \mathbb{R}^{n\times n},\;0<\tau_1<\tau_2. \] For an arbitrary \(2\times 2\)-matrix the author gives precise tests for the asymptotic behavior of the solutions depending on \(A\). The solutions may approach a constant or may approach a periodic orbit, or may be unbounded. For the \(n\)-dimensional systems conditions for the existence of an asymptotic equilibrium point are given.

MSC:

34K13 Periodic solutions to functional-differential equations
34K11 Oscillation theory of functional-differential equations
34K20 Stability theory of functional-differential equations
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References:

[1] Cooke, K.; Yorke, J., Some Equations Modelling Growth Processes and Gonorrhea Epidemics, Math. Biosci., 16, 75-101 (1973) · Zbl 0251.92011
[2] Diekmann, O.; Van Gils, S. A.; Lunel, S. M. Verduyn; Walther, H.-O., (Delay Equations: Functional-, Complex-, and Nonlinear Analysis (1995), Springer-Verlag: Springer-Verlag Warszawa) · Zbl 0826.34002
[3] Hale, J., (Theory of Functional Differential Equations (1977), Springer-Verlag) · Zbl 0352.34001
[4] Hale, J.; Lunel, S. M. Verduyn, (Introduction to Functional Differential Equations (1993), Springer-Verlag) · Zbl 0787.34002
[5] Kuang, Y., (Delay Differential Equations With Applications in Population Dynamics (1993), Academic Press) · Zbl 0777.34002
[6] Murakami, K., Asymptotic Constancy and Periodic Solutions for Linear Autonomous Delay Differential Equation, Funkcial. Ekvac., 39, 519-540 (1996) · Zbl 0872.34052
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