Existence and global attractivity of positive periodic solutions of functional differential equations with feedback control. (English) Zbl 1069.34100
Summary: Sufficient conditions are obtained for the existence and global attractivity of positive periodic solutions of the delay differential system with feedback control
\[
\begin{aligned} {dx\over dt} &= -b(t)x(t)+ F(t, x(t-\tau_1(t)),\dots, x(t-\tau_n(t)), u(t-\delta(t))),\\ {du\over dt} &= -\eta(t) u(t)+ a(t) x(t-\sigma(t)).\end{aligned}
\]
The method involves the application of Krasnoselskii’s fixed-point theorem and estimates on uniform upper and lower bounds of solutions. When these results are applied to some special delay population models with multiple delays, some new results are obtained and some known results are generalized.
MSC:
| 34K13 | Periodic solutions to functional-differential equations |
| 34K20 | Stability theory of functional-differential equations |
| 34K35 | Control problems for functional-differential equations |
Keywords:
Delay differential equations; Feedback control; Positive periodic solutions; Global attractivityReferences:
| [1] | Barbalat, I., Systems d’equations differentielle d’oscillations nonlineaires, Rev. Roumaine Math. Pures Appl., 4, 267-270 (1959) · Zbl 0090.06601 |
| [2] | Deimling, K., Nonlinear Functional Analysis (1985), Springer: Springer New York · Zbl 0559.47040 |
| [3] | Gaines, R. E.; Mawhin, J. L., Coincidence Degree and Nonlinear Differential Equations (1977), Springer: Springer Berlin · Zbl 0326.34021 |
| [4] | Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics (1992), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0752.34039 |
| [5] | Gopalsamy, K.; Kulenović, M. R.S.; Ladas, G., Environmental periodicity and time delays in a “food-limited” population model, J. Math. Anal. Appl., 147, 545-555 (1990) · Zbl 0701.92021 |
| [6] | Gopalsamy, K.; Lalli, B. S., Oscillatory and asymptotic behavior of a multiplicative delay logistic equation, Dynamic Stability Systems, 7, 35-42 (1992) · Zbl 0764.34049 |
| [7] | Gopalsamy, K.; Weng, P. X., Feedback regulation of logistic growth, Internat. J. Math. Sci., 1, 177-192 (1993) · Zbl 0765.34058 |
| [8] | Grace, S. R.; Györi, I.; Lalli, B. S., Necessary and sufficient conditions for the oscillations of a multiplicative delay logistic equation, Quart. Appl. Math., 53, 69-79 (1995) · Zbl 0837.34073 |
| [9] | Greaf, J. R.; Qian, C.; Spikes, P. W., Oscillation and global attractivity in a periodic delay equation, Canad. Math. Bull., 38, 275-283 (1996) · Zbl 0870.34073 |
| [10] | Greaf, J. R.; Qian, C.; Zhang, B., Asymptotic behavior of solutions of differential equations with variable delays, Proc. London Math. Soc., 81, 72-92 (2000) · Zbl 1030.34075 |
| [11] | Györi, I.; Ladas, G., Oscillation Theorem of Delay Differential Equations with Applications (1991), Clarendon Press: Clarendon Press Oxford · Zbl 0780.34048 |
| [12] | Huo, H. F.; Li, W. T., Positive periodic solutions of delay differential systems with feedback control, Appl. Math. Comput., 148, 35-46 (2004) · Zbl 1057.34093 |
| [13] | Jiang, D. Q.; Wei, J.; Zhang, B., Positive periodic solutions of functional differential equations and population models, Electron. J. Differential Equations, 71, 1-13 (2002) · Zbl 1010.34065 |
| [14] | Krasnoselskii, M. A., Positive Solutions of Operator Equations (1964), Gorninggen: Gorninggen Noordhoff · Zbl 0121.10604 |
| [15] | Kuang, Y., Delay Differential Equations with Application in Population Dynamics (1993), Academic Press: Academic Press New York · Zbl 0777.34002 |
| [16] | Ladde, G. S.; Lakshmikantham, V.; Zhang, B. Z., Oscillation Theory of Differential Equations with Deviating Arguments (1987), Marcel Dekker: Marcel Dekker New York · Zbl 0622.34071 |
| [17] | Lalli, B. S.; Yu, J. S.; Chen, M. P., Feedback regulation of a logistic growth, Dynam. Systems Appl., 5, 117-124 (1996) · Zbl 0848.34060 |
| [18] | Lalli, B. S.; Zhang, B. G., On a periodic delay population model, Quart. Appl. Math., 52, 35-42 (1994) · Zbl 0788.92022 |
| [19] | LaSalle, J.; Lefschetz, S., Stability by Lyapunov’s Direct Method (1961), Academic Press: Academic Press New York · Zbl 0098.06102 |
| [20] | Lefschetz, S., Stability of Nonlinear Control Systems (1965), Academic Press: Academic Press New York · Zbl 0136.08801 |
| [21] | Li, Y. K., Existence and global attractivity of a positive periodic solution of a class of delay differential equation, Sci. China Ser. A, 41, 273-284 (1998) · Zbl 0955.34057 |
| [22] | Liao, L., Feedback regulation of a logistic growth with variable coefficients, J. Math. Anal. Appl., 259, 489-500 (2001) · Zbl 1003.34069 |
| [23] | Mackey, M. C.; Glass, L., Oscillation and chaos in physiological control system, Science, 197, 287-289 (1977) · Zbl 1383.92036 |
| [24] | Nicholson, A. J., The balance of animal population, J. Animal Ecol., 2, 132-178 (1933) |
| [25] | Nisbet, R. M.; Gurney, W. S.C., Population dynamics in a periodically varying environment, J. Theoret. Biol., 56, 459-475 (1976) |
| [26] | Pianka, E. R., Evolutionary Ecology (1974), Harper and Row: Harper and Row New York |
| [27] | Qian, C., Global attractivity in nonlinear delay differential equations, J. Math. Anal. Appl., 197, 529-547 (1996) · Zbl 0851.34075 |
| [28] | Rosen, G., Time delays produced by essential nonlinearity in population growth models, Bull. Math. Biol., 28, 253-256 (1987) · Zbl 0614.92015 |
| [29] | Saker, S. H., Oscillation and global attractivity in a periodic delay hematopoiesis model, Appl. Math. Comput., 136, 241-250 (2003) · Zbl 1026.34082 |
| [30] | Saker, S. H.; Agarwal, S., Oscillation and global attractivity in a periodic Nichlson’s blowflies model, Math. Comput. Modelling, 35, 719-731 (2002) · Zbl 1012.34067 |
| [31] | Saker, S. H.; Agarwal, S., Oscillation and global attractivity in a nonlinear delay periodic model of population dynamics, Appl. Anal., 81, 787-799 (2002) · Zbl 1041.34061 |
| [32] | S.H. Saker, S. Agarwal, Oscillation and global attractivity in a nonlinear delay periodic model of respiratory dynamics, Comput. Math. Appl. 44 (2002) 5-6, 623-632.; S.H. Saker, S. Agarwal, Oscillation and global attractivity in a nonlinear delay periodic model of respiratory dynamics, Comput. Math. Appl. 44 (2002) 5-6, 623-632. · Zbl 1041.34073 |
| [33] | S.H. Saker, S. Agarwal, Oscillation and global attractivity of a periodic survival red blood cells model, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, in press.; S.H. Saker, S. Agarwal, Oscillation and global attractivity of a periodic survival red blood cells model, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, in press. · Zbl 1078.34062 |
| [34] | Schley, D.; Gourley, S. A., Linear stability criteria for population models with periodic perturbed delays, J. Math. Biol., 40, 500-524 (2000) · Zbl 0961.92026 |
| [35] | Tang, X.; Zou, X., A \(3/2\) stability result for a regulated logistic growth model, Discrete Contin. Dynam. Systems Ser. B, 2, 265-278 (2002) · Zbl 1009.34072 |
| [36] | Yan, J.; Feng, Q., Global attractivity and oscillation in a nonlinear delay equation, Nonlinear Analysis-TMA, 43, 101-108 (2001) · Zbl 0987.34065 |
| [37] | Yang, F.; Jiang, D. Q., Existence and global attractivity of positive periodic solution of a logistic growth system with feedback control and deviating arguments, Ann. Differential Equations, 17, 377-384 (2001) · Zbl 1004.34030 |
| [38] | Zhang, B. G.; Gopalsamy, K., Global attractivity and oscillations in a periodic delay-logistic equation, J. Math. Anal. Appl., 150, 274-283 (1990) · Zbl 0711.34090 |
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