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Existence of positive periodic solutions for higher-order ordinary differential equations. (English) Zbl 1231.34076

Summary: We consider the existence of positive periodic solutions for the \(n\)th-order ordinary differential equation \[ u^{n}(t)=f(t,u(t),u'(t)\dots,u^{n-1}(t)), \] where \(n\geq 2\), \(f\mathbb{R}\times [0,\infty)\times\mathbb{R}^{n-1}\to\mathbb{R}\) is a continuous function and is \(2\pi \)-periodic in \(t\). Some existence results of positive \(2\pi \)-periodic solutions are obtained assuming \(f\) satisfies some superlinear or sublinear growth conditions on \(x_0\dots,x_{n-1}\). The discussion is based on the fixed point index theory in cones.

MSC:

34C25 Periodic solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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