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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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