Nonlinear periodic boundary value problems with sign-changing Green’s function

We study the existence and nonexistence of positive solutions of nonlinear periodic boundary value problems u″+a(t)u=λb(t)f(u),a.e.t∈[0,T],u(0)=u(T),u′(0)=u′(T), where b≻0b≻0, a≻0a≻0 and the Green’s function G(t,s)G(t,s) of the linear problem u″+a(t)u=0,a.e.t∈[0,T],u(0)=u(T),u′(0)=u′(T) may change sign on [0,T]×[0,T][0,T]×[0,T].