Existence of nontrivial solutions for a nonlinear second order periodic boundary value problem with derivative term

Abstract

In this paper, we study the existence of nontrivial solutions to the following nonlinear differential equation with derivative term: $$\begin{aligned} {\left\{ \begin{array}{l}u''(t)+a(t)u(t)=f\big (t,u(t),u'(t)\big ),\quad t\in [0,\omega ],\\ u(0)=u(\omega ),\quad u'(0)=u'(\omega ),\end{array}\right. } \end{aligned}$$ where a: $$[0,\omega ]\rightarrow \mathbb {R}^{+}\big (\mathbb {R}^{+}=[0,+\infty )\big )$$ is a continuous function with $$a(t)\not \equiv 0$$ , f: $$[0,\omega ]\times \mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}$$ is continuous and may be sign-changing and unbounded from below. Without making any nonnegative assumption on nonlinearity, using the first eigenvalue corresponding to the relevant linear operator and the topological degree, the existence of nontrivial solutions to the above periodic boundary value problem is established in $$C^1[0,\omega ]$$ . Finally, an example is given to demonstrate the validity of our main result.