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Zbl 1073.34082
Wan, Aying; Jiang, Daqing; Xu, Xiaojie
A new existence theory for positive periodic solutions to functional differential equations.
(English)
[J] Comput. Math. Appl. 47, No. 8-9, 1257-1262 (2004). ISSN 0898-1221

The aim of the paper is to obtain the existence of positive periodic solutions for the nonlinear functional-differential equation $$y'(t)=-a(t)y(t)+g(t,y(t-\tau(t))), \tag 1$$ where $a\in C(\bbfR,(0,\infty))$, $\tau\in C(\bbfR,\bbfR)$, $g\in C(\bbfR\times[0,\infty),[0,\infty))$ and $a,\tau, g(t,\cdot)$ are all $\omega$-periodic functions, $\omega>0$ is a constant. By using a fixed-point theorem in a cone, the authors establish the following result: equation (1) has at least one positive $\omega$-periodic solution provided one of the following conditions holds: $$\liminf_{u\downarrow 0}\min_{0\leq t\leq\omega}\frac{g(t,u)}{a(t)u}>1\quad \text {and} \quad \limsup_{u\uparrow\infty}\max_{0\leq t\leq\omega}\frac{g(t,u)}{a(t)u}<1;\tag i$$ $$\limsup_{u\downarrow 0}\max_{0\leq t\leq\omega}\frac{g(t,u)}{a(t)u}<1\quad \text {and} \quad \liminf_{u\uparrow\infty}\min_{0\leq t\leq\omega}\frac{g(t,u)}{a(t)u}=\infty.\tag ii$$
[Jurang Yan (Taiyuan)]
MSC 2000:
*34K13 Periodic solutions of functional differential equations

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

Cited in: Zbl 1156.34053

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