Nontrivial solutions for a second order periodic boundary value problem with the nonlinearity dependent on the derivative

Abstract

In this paper, we study the existence of nontrivial solutions to the nonlinear differential equation with derivative term u′′(t)+a(t)u(t)=f(t,u(t),u′(t)),t∈[0,ω],u(0)=u(ω),u′(0)=u′(ω), where a: [0,ω]→R+(R+=[0,+∞)) is a continuous function with a(t)⁄≡0, and f: [0,ω]×R×R→R is continuous and may be sign-changing and unbounded from below. Without making any nonnegative assumption on the nonlinearity, by 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 C1[0,ω] for the sublinear case. Moreover, two examples are given to demonstrate the validity of our main results.