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Zbl 1190.34049
Liu, Bingmei; Liu, Lishan; Wu, Yonghong
Existence of nontrivial periodic solutions for a nonlinear second order periodic boundary value problem. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 7-8, 3337-3345 (2010). ISSN 0362-546X

Summary: We study the existence of nontrivial periodic solutions to the following nonlinear differential equation $$\cases u''(t)+a(t)u(t)=f(t,u(t)),\quad t\in\Bbb R,\\ u(0)=u(\omega),\quad u'(0)=u'(\omega),\endcases$$ where $a:\Bbb R\to\Bbb R^+$ is an $\omega$-periodic continuous function with $a(t)\not\equiv 0$, $f:\Bbb R\times \Bbb R\to\Bbb R$ is continuous, may take negative values and can be sign-changing. Without making any nonnegative assumption on nonlinearity, by using the first eigenvalue corresponding to the relevant linear operator and the topological degree, the existence of nontrivial periodic solutions to the above periodic boundary value problem is established. Finally, three examples are given to demonstrate the validity of our main results.

MSC 2000:: *34C25 Periodic solutions of ODE
34B15 Nonlinear boundary value problems of ODE
47N20 Appl. of operator theory to differential and integral equations

Keywords: nontrivial periodic solutions; spectral radius; topological degree; fixed point

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