## Existence of positive periodic solutions to nonlinear second order differential equations.(English)Zbl 1088.34038

Summary: We discuss the existence of positive periodic solutions to the nonlinear differential equation $u''(t)+a(t)u(t)=f\bigl(t,u(t)\bigr),\;t\in \mathbb{R},$ where $$a:\mathbb{R}\to[0,+\infty)$$ is an $$\omega$$-periodic continuous function with $$a(t)\not \equiv 0$$, $$f:\mathbb{R}\times[0,+\infty)\to[0,+\infty)$$ is continuous and f$$(\cdot,u):\mathbb{R} \to[0,+\infty)$$ is also an $$\omega$$-periodic function for each $$u\in[0,+ \infty)$$. Using the fixed-point index theory in a cone, we get an essential existence result because of its involving the first positive eigenvalue of the linear equation with regard to the above equation.

### MSC:

 34C25 Periodic solutions to ordinary differential equations 47H10 Fixed-point theorems
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### References:

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