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Existence of nontrivial periodic solutions for a nonlinear second order periodic boundary value problem. (English) Zbl 1190.34049

Summary: We study the existence of nontrivial periodic solutions to the following nonlinear differential equation
\[ \begin{cases} u''(t)+a(t)u(t)=f(t,u(t)),\quad t\in\mathbb R,\\ u(0)=u(\omega),\quad u'(0)=u'(\omega),\end{cases} \]
where \(a:\mathbb R\to\mathbb R^+\) is an \(\omega\)-periodic continuous function with \(a(t)\not\equiv 0\), \(f:\mathbb R\times \mathbb R\to\mathbb R\) 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:

34C25 Periodic solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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