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Zbl 1231.34076
Li, Yongxiang; Fan, Hongxia
Existence of positive periodic solutions for higher-order ordinary differential equations.
(English)
[J] Comput. Math. Appl. 62, No. 4, 1715-1722 (2011). ISSN 0898-1221

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 $n\ge 2$, $f\bbfR\times [0,\infty)\times\bbfR^{n-1}\to\bbfR$ 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 2000:
*34C25 Periodic solutions of ODE
34B15 Nonlinear boundary value problems of ODE

Keywords: $n$th-order differential equation; positive periodic solution; cone; fixed point theorem in cones

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