Upper and lower solutions for a second-order three-point singular boundary-value problem

We study the singular boundary-value problem $$displaylines{ u''+ q(t)g(t,u)=0,quad t in (0,1),; eta in (0,1),;gamma >0cr u(0)=0, quad u(1)=gamma u(eta),. }$$ The singularity may appear at $ t=0$ and the function $g$ may be superlinear at infinity and may change sign. The existence of solutions is obtained via an upper and lower solutions method.