Positive solution of a singular nonlinear third-order periodic boundary value problem

The singular nonlinear third-order periodic boundary value problem u″′+ρ3u=f(t,u), 0⩽t⩽2π, with u(i)(0)=u(i)(2π), i=0,1,2, is studied, where ρ>0 and f is singular at u=0. Under suitable conditions on f, it is proved by employing a priori estimates, perturbation technique and Schauder fixed point theorem that the problem has at least one positive solution if ρ∈(0,13).