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Zbl 1064.34052
Wang, Haiyan
Positive periodic solutions of functional differential equations.
(English)
[J] J. Differ. Equations 202, No. 2, 354-366 (2004). ISSN 0022-0396

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, $a,b\in C(\bbfR\to [0,\infty))$ 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 $\tau\in C(\bbfR\to \bbfR)$ 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.
[Jurang Yan (Taiyuan)]
MSC 2000:
*34K13 Periodic solutions of functional differential equations

Keywords: positive periodic solution; existence; multiplicity; fixed-point index

Cited in: Zbl 1166.34038 Zbl 1117.34067

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