In this paper,~we study the existence of positive solutions for a class of second-order nonlinear Neumann problem \[ \begin{cases} y''+a(t)y=\lambda g(t)f(y),~~t\in[0,1],\y'(0)=y'(1)=0, \end{cases} \] where~$\lambda>0$~is a positive parameter,~$f$~is superlinear at infinity,allowed to change sign， and the Green's function associated with this problem may vanish at some points. The proof of the main result is based on the Krasnosel'skii fixed-point theorem.
