Positive solutions for singular non-linear third-order periodic boundary value problems

In this paper, we are concerned with the problem of the existence of positive solutions for non-linear third-order periodic boundary value problemu‴+ρ3u=f(t,u),0⩽t⩽2π,u(i)(0)=u(i)(2π),i=0,1,2.Here, ρ∈(0,13) is a positive constant and our non-linearity f(t,u)f(t,u) may be singular at u=0u=0. The proof relies on a non-linear alternative of Leray–Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones.