The paper establishes sufficient conditions for the existence of solutions of Neumann’s problem for the differential equation $\mu y"+ky=f(t,y)$ which tend to the solution of the reduced problem $ky=f(t,y)$ on $[0,1]$ as $\mu \to 0.$

We consider multipoint and two-point BVPs for second order ordinary differential equations with a Carathéodory right hand side. We prove the existence of solutions provided there exist upper and lower solutions of the BVP and the upper solution is less than the lower one.