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Zbl 1088.34038
Li, Fuyi; Liang, Zhanping
Existence of positive periodic solutions to nonlinear second order differential equations.
(English)
[J] Appl. Math. Lett. 18, No. 11, 1256-1264 (2005). ISSN 0893-9659

Summary: We discuss the existence of positive periodic solutions to the nonlinear differential equation $$u''(t)+a(t)u(t)=f\bigl(t,u(t)\bigr),\ t\in \bbfR,$$ where $a:\bbfR\to[0,+\infty)$ is an $\omega$-periodic continuous function with $a(t)\not \equiv 0$, $f:\bbfR\times[0,+\infty)\to[0,+\infty)$ is continuous and f$(\cdot,u):\bbfR \to[0,+\infty)$ 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 2000:
*34C25 Periodic solutions of ODE
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: positive periodic solution; fixed-point index; Krein-Rutman theorem; the first positive eigenvalue

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