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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.

34K13Periodic solutions of functional differential equations
34K40Neutral functional-differential equations
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
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