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Zbl 1149.34040
Luo, Yan; Wang, Weibing; Shen, Jianhua
Existence of positive periodic solutions for two kinds of neutral functional differential equations.
(English)
[J] Appl. Math. Lett. 21, No. 6, 581-587 (2008). ISSN 0893-9659

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.
[Satoshi Tanaka (Okayama)]
MSC 2000:
*34K13 Periodic solutions of functional differential equations
34K40 Neutral equations
47N20 Appl. of operator theory to differential and integral equations

Keywords: neutral functional differential equation; existence; positive periodic solution; fixed-point theorem

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