## Positive periodic solutions of functional differential equations.(English)Zbl 1064.34052

The author considers the existence, multiplicity and nonexistence of positive $$\omega$$-periodic solutions for the periodic equation $x'(t)=a(t)g(x(t))x(t)-\lambda b(t)f(x(t-\tau(t))),$ where $$\lambda>0$$ is a positive parameter, $$a,b\in C(\mathbb{R}\to [0,\infty))$$ are $$\omega$$-periodic, $$\int_0^\omega a(t)\,dt>0$$, $$\int_0^\omega b(t)\,dt>0$$, $$f,g\in C([0,\infty),[0,\infty))$$, and $$f(u)>0$$ for $$u>0$$, $$g(x)$$ is bounded and $$\tau\in C(\mathbb{R}\to \mathbb{R})$$ is an $$\omega$$-periodic function. Define $f_0:=\lim_{u\to 0+}\frac{f(u)}{u}, \qquad f_\infty:=\lim_{u\to\infty}\frac{f(u)}{u},$ $$i_0:=$$ number of zeros in the set $$\{f_0,f_\infty\}$$ and $$i_\infty=$$ number of infinities in the set $$\{f_0,f_\infty\}$$. The author shows that the equation has $$i_0$$ or $$i_\infty$$ positive $$\omega$$-periodic solutions for sufficiently large or small $$\lambda>0$$, respectively. The proof is based on the fixed-point index theorem.

### MSC:

 34K13 Periodic solutions to functional-differential equations
Full Text:

### References:

 [1] Cheng, S; Zhang, G, Existence of positive periodic solutions for non-autonomous functional differential equations, Electron. J. differential equations, 59, 1-8, (2001) [2] Chow, S.-N, Existence of periodic solutions of autonomous functional differential equations, J. differential equations, 15, 350-378, (1974) · Zbl 0295.34055 [3] Deimling, K, Nonlinear functional analysis, (1985), Springer Berlin [4] L. Erbe, H. Wang, Existence and nonexistence of positive solutions for elliptic equations in an annulus, Inequalities and Applications, World Science Series in Application Analysis, Vol. 3, World Science Publishing, River Edge, NJ, 1994, pp. 207-217. · Zbl 0900.35144 [5] Freedman, H.I; Wu, J, Periodic solutions of single-species models with periodic delay, SIAM J. math. anal., 23, 689-701, (1992) · Zbl 0764.92016 [6] Guo, D; Lakshmikantham, V, Nonlinear problems in abstract cones, (1988), Academic Press Orlando, FL · Zbl 0661.47045 [7] Gurney, W.S; Blythe, S.P; Nisbet, R.N, Nicholson’s blowflies revisited, Nature, 287, 17-21, (1980) [8] Hadeler, K.P; Tomiuk, J, Periodic solutions of difference differential equations, Arch. rational mech. anal., 65, 87-95, (1977) · Zbl 0426.34058 [9] Jiang, D; Wei, J, Existence of positive periodic solutions of nonautonomous functional differential equations, Chinese ann. math. A, 20, 6, 715-720, (1999), (in Chinese) · Zbl 0948.34046 [10] Jiang, D; Wei, J; Zhang, B, Positive periodic solutions of functional differential equations and population models, Electron. J. differential equations, 71, 1-13, (2002) [11] Krasnoselskii, M, Positive solutions of operator equations, (1964), Noordhoff Groningen [12] Kuang, Y, Delay differential equations with application in population dynamics, (1993), Academic Press New York [13] Kuang, Y, Global attractivity and periodic solutions in delay-differential equations related to models in physiology and population biology, Jpn J. ind. appl. math., 9, 205-238, (1992) · Zbl 0758.34065 [14] Kuang, Y; Smith, H.L, Periodic solutions of differential delay equations with threshold-type delays, oscillations and dynamics in delay equations, Contemp. math., 129, 153-176, (1992) · Zbl 0762.34044 [15] Mackey, M.C; Glass, L, Oscillations and chaos in physiological control systems, Science, 197, 287-289, (1997) · Zbl 1383.92036 [16] Mallet-Paret, J; Nussbaum, R, D. global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation, Ann. mat. pura appl., 145, 4, 33-128, (1986) · Zbl 0617.34071 [17] Wan, A; Jiang, D, Existence of positive periodic solutions for functional differential equations, Kyushu J. math., 56, 193-202, (2002) · Zbl 1012.34068 [18] Wang, H, On the existence of positive solutions for semilinear elliptic equations in the annulus, J. differential equations, 109, 1-7, (1994) · Zbl 0798.34030 [19] Wang, H, On the number of positive solutions of nonlinear systems, J. math. anal. appl., 281, 287-306, (2003) · Zbl 1036.34032 [20] H. Wang, Y. Kuang, M. Fen, Periodic solutions of systems of delay differential equations, submitted for publication. [21] Wazewska-Czyzewska, M; Lasota, A, Mathematical problems of the dynamics of a system of red blood cells, Mat. stos., 6, 23-40, (1976), (in Polish)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.