Positive periodic solutions of a class of delay differential system with feedback control. (English) Zbl 1057.34093
The authors present several sufficient conditions for guaranteeing the global existence of positive periodic solutions for a delay differential system with feedback control by using the continuation theorem given by Gaines and Mawhin and coincidence degree theory. In addition, the obtained results can be applied to some special delay population models.
Reviewer: Jinde Cao (Nanjing)
MSC:
| 34K13 | Periodic solutions to functional-differential equations |
| 92D25 | Population dynamics (general) |
Keywords:
delay differential equation; feedback control; positive periodic solution; coincidence degreeReferences:
| [1] | H.F. Huo, W.T. Li, Periodic solution of a periodic two-species competition model with delays, Int. J. Appl. Math., in press; H.F. Huo, W.T. Li, Periodic solution of a periodic two-species competition model with delays, Int. J. Appl. Math., in press · Zbl 1043.34074 |
| [2] | Huo, H. F.; Li, W. T.; Cheng, S. S., Periodic solutions of two-species diffusion models with continuous time delays, Demonstratio Mathematica, XXXV, 2, 432-446 (2002) · Zbl 1013.92035 |
| [3] | Saker, S. H.; Agarwal, S., Oscillation and global attractivity in a nonlinear delay periodic model of resiratory dynamics, Comput. Math. Appl., 44, 5-6, 623-632 (2002) · Zbl 1041.34073 |
| [4] | Saker, S. H.; Agarwal, S., Oscillation and global attractivity in a periodic Nichlson’s blowflies model, Math. Comput. Model., 35, 7-8, 719-731 (2002) · Zbl 1012.34067 |
| [5] | Yan, J.; Feng, Q., Global attractivity and oscillation in a nonlinear delay equation, Nonlinear Anal., 43, 101-108 (2001) · Zbl 0987.34065 |
| [6] | Lasalle, J.; Lefschetz, S., Stability by Lyapunov’s Direct Method (1961), Academic Press: Academic Press New York · Zbl 0098.06102 |
| [7] | Lefschetz, S., Stability of Nonlinear Control System (1965), Academic Press: Academic Press New York · Zbl 0136.08801 |
| [8] | Kuang, Y., Delay Differential Equations with Application in Population Dynamics (1993), Academic Press: Academic Press New York · Zbl 0777.34002 |
| [9] | Gopalsamy, K.; Weng, P. X., Feedback regulation of Logistic growth, J. Math. Math. Sci., 1, 177-192 (1993) · Zbl 0765.34058 |
| [10] | 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. Diff. Eqs., 17, 4, 377-384 (2001) · Zbl 1004.34030 |
| [11] | Fan, M.; Wang, K., Periodicity in a delayed ratio-dependent predator-prey system, J. Math. Anal. Appl., 262, 179-190 (2001) · Zbl 0994.34058 |
| [12] | Li, Y. K., Periodic solutions of a periodic delay predator-prey system, Proc. Am. Math. Soc., 127, 1331-1335 (1999) · Zbl 0917.34057 |
| [13] | Li, Y. K.; Kuang, Y., Periodic solutions of periodic delay Lotka-Volterra equations and systems, J. Math. Anal. Appl., 255, 260-280 (2001) · Zbl 1024.34062 |
| [14] | Gaines, R. E.; Mawhin, J. L., Coincidence Degree and Nonlinear Differential Equations (1977), Springer: Springer Berlin · Zbl 0326.34021 |
| [15] | Li, Y. K., Existence and global attractivity of a positive periodic solution of a class of delay differential equation, Sci. China (Ser. A), 41, 3, 273-284 (1998) · Zbl 0955.34057 |
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