## Positive periodic solutions of functional differential equations.(English)Zbl 1064.34052

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, $$a,b\in C(\mathbb{R}\to [0,\infty))$$ 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 $$\tau\in C(\mathbb{R}\to \mathbb{R})$$ 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.

### MSC:

 34K13 Periodic solutions to functional-differential equations
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### References:

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