Existence of positive periodic solutions to nonlinear second order differential equations

In this paper, we discuss the existence of positive periodic solutions to the nonlinear differential equation u″(t)+a(t)u(t)=f(t,u(t)),t∈R, where a:R→[0,+∞) is an ω-periodic continuous function with a(t)⁄≡0, f:R×[0,+∞)→[0,+∞) is continuous and f(⋅,u):R→[0,+∞) is also an ω-periodic function for each u∈[0,+∞). Using the fixed point index theory in a cone, we get an essential existence result because of its involving the first positive eigenvalue of the linear equation with regard to the above equation.