Existence of nontrivial solutions for a nonlinear second order periodic boundary value problem with derivative term (Q780424)

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.