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Existence of positive periodic solutions for two kinds of neutral functional differential equations. (English) Zbl 1149.34040
The following two classes of neutral functional differential equations $$ \frac{d}{dt} [ x(t) - c\,x(t-\tau) ] = -a(t) x(t) + f(t,x(t-\tau(t))) $$ and $$\frac{d}{dt} \left[ x(t) - c \int_{-\infty}^0 Q(r) x(t+r) dr \right] = -a(t) x(t) + b(t) \int_{-\infty}^0 Q(r) f(t,x(t+r)) dr$$ are considered, where $a$, $b \in C({\Bbb R},(0,\infty))$, $\tau \in C({\Bbb R},{\Bbb R})$, $f \in C({\Bbb R} \times {\Bbb R},{\Bbb R})$, and $a(t)$, $b(t)$, $\tau(t)$, $f(t,\cdot)$ are $\omega$-periodic functions, $\omega>0$ and $\vert c\vert <1$. Sufficient conditions for the existence of a positive $\omega$-periodic solution are obtained. The proof is based on Krasnoselskii’s fixed point theorem. The results obtained here are applied to various mathematical models.

MSC:
34K13Periodic solutions of functional differential equations
34K40Neutral functional-differential equations
47N20Applications of operator theory to differential and integral equations
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References:
[1] Gopalsamy, K.: Stability and oscillation in delay differential equations of population dynamics. (1992) · Zbl 0752.34039
[2] Gurney, W. S. C.; Blythe, S. P.; Nisbet, R. M.: Nicholson’s blowflies revisited. Nature 287, 17-20 (1980)
[3] Gil, M. I.: Existence and stability of periodic solutions of semilinear neutral type systems. Dyn. discrete contin. Syst. 7, 809-820 (2001) · Zbl 1021.34056
[4] Joseph, W.; So, H.; Yu, J.: Global attractivity and uniform persistence in Nicholson’s blowflies. Differential equations dynam. Systems 1, 11-18 (1994) · Zbl 0869.34056
[5] Jiang, D.; Wei, J.: Existence of positive periodic solutions for Volterra integro-differential equations. Acta math. Sci. 21B4, 553-560 (2002) · Zbl 1035.45003
[6] Li, Y.: Existence and global attractivity of a positive periodic solution of a class of delay differential equation. Sci. China 41A3, 273-284 (1998) · Zbl 0955.34057
[7] Luo, J.; Yu, J.: Global asymptotic stability of nonautonomous mathematical ecological equations with distributed deviating arguments. Acta math. Sinica 41, 1273-1282 (1998) · Zbl 1027.34088
[8] Li, Z.; Wang, X.: Existence of positive periodic solutions for neutral functional differential equations. Electron. J. Differential equations 34, 1-8 (2006) · Zbl 1099.34063
[9] Weng, P.: Existence and global attractivity of periodic solution of integrodifferential equation in population dynamics. Acta appl. Math. 4, 427-434 (1996) · Zbl 0886.45005
[10] Weng, P.; Liang, M.: The existence and behavior of periodic solution of hematopoiesis model. Math. appl. 4, 434-439 (1995) · Zbl 0949.34517
[11] Wan, A.; Jiang, D.: Existence of positive periodic solutions for functional differential equations. Kyushu J. Math. 1, 193-202 (2002) · Zbl 1012.34068
[12] Wan, A.; Jiang, D.; Xu, X.: A new existence theory for positive periodic solutions to functional differential equations. Comput. math. Appl. 47, 1257-1262 (2004) · Zbl 1073.34082
[13] You, B.: Ordinary differential equation complementary curriculum. (1982)