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Existence of nontrivial solutions for a nonlinear second order periodic boundary value problem with derivative term. (English) Zbl 1448.34053

This paper considers the existence of nontrivial solutions to the following nonlinear second order differential equation with derivative term of the type \[ \left\{ \begin{array}{l} u''(t)+a(t)u(t)=f(t,u(t),u'(t)), \quad t\in [0,\omega], \\ u(0)=u(\omega),\quad u'(0)=u'(\omega), \end{array} \right. \] where \(a: [0,\omega]\rightarrow \mathbb{R}^{+}:=[0,+\infty)\) is a continuous function with \(a(t)\not\equiv 0\) on \([0,\omega]\), \(f: [0,\omega] \times \mathbb{R}^2\rightarrow \mathbb{R}\) is continuous and may be sign-changing and unbounded from below.
Using the first eigenvalue corresponding to the relevant linear operator and the topological degree, the authors establish the existence of nontrivial solutions to the above periodic boundary value problem without making any nonnegative assumption on the nonlinearity. In addition, an example is given to demonstrate the validity of the main result.
Reviewer: Minghe Pei (Jilin)

MSC:

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