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

In this paper, we study the existence of nontrivial periodic solutions to the following nonlinear differential equation {u″(t)+a(t)u(t)=f(t,u(t)),t∈R,u(0)=u(ω),u′(0)=u′(ω), where a:R→R+a:R→R+ is an ωω-periodic continuous function with a(t)≢0,f:R×R→Ra(t)≢0,f:R×R→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.