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Periodicity in a logistic type system with several delays. (English) Zbl 1061.34050
Summary: With the help of a continuation theorem based on Gaines and Mawhin’s coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of the following logistic-type system with several delays \[ \frac{du(t)}{dt}=u(t)\left[a(t)-\sum^m_{j=1}b_j(t) \biggl(u\bigl(t-\tau_j(t)\bigr)\biggr)^{\theta_j} \right], \] where \(a(t)\), \(b_j(t)\) are positive periodic continuous functions with periodic \(\omega>0\), \(\tau_j(t)\) are nonnegative continuous periodic functions with periodic \(\omega>0\). After that, by constructing a suitable Lyapunov functional, some sufficient conditions which guarantee the stability of the positive periodic solutions are obtained.

MSC:
34K13 Periodic solutions to functional-differential equations
34K20 Stability theory of functional-differential equations
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[1] Gopalsamy, K., ()
[2] Gyori, I.; Ladas, G., Oscillation theory of delay differential equations, (1991), Oxford Science Publications Dordrecht · Zbl 0780.34048
[3] Lenhart, S.M.; Travis, C.C., Global stability of a biological model with time delay, (), 75-78 · Zbl 0602.34044
[4] Kuang, Y., ()
[5] Yu, J.S., Global attractivity of zero solution for a class of functional equations and its applications, Science in China, series A, 26, 23-33, (1996), (in Chinese)
[6] Cao, Y.L.; Thomas, C.G., Ultimate bounds and global asymptotic stability for differential delay equations, Rocky mountain journal of mathematics, 25, 119-131, (1995) · Zbl 0829.34066
[7] Li, Y.K., Global attractivity in a periodic delay single species model, System science and complexity, 13, 1-6, (2000) · Zbl 0969.92022
[8] Li, Y.K., On a periodic logistic equation with several delay, Advances in mathematics, 28, 135-142, (1999) · Zbl 1054.34513
[9] Chen, Y.M., Periodic solution of a delayed periodic logistic equation, Appl. math. lett., 16, 7, 1047-1051, (2003) · Zbl 1118.34327
[10] 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
[11] Yan, J.R.; Feng, Q.X., Global existence and oscillation in a nonlinear delay equation, Nonlinear analysis, 43, 101-108, (2001) · Zbl 0987.34065
[12] Gaines, R.E.; Mawhin, J.L., Coincidence degree and nonlinear differential equations, (1977), Springer-Verlag Boston, MA · Zbl 0326.34021
[13] Gui, Z.; Chen, L., Persistence and periodic solutions of a periodic logistic equation with time delays, Journal of mathematical research and exposition, 23, 1, 109-114, (2003), (in Chinese) · Zbl 1038.34080
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