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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References:

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