Nodal solutions of multi-point boundary value problems

We study the nonlinear boundary value problem consisting of the equation y″+w(t)f(y)=0 on [a,b]y″+w(t)f(y)=0 on [a,b] and a multi-point boundary condition. By relating it to the eigenvalues of a linear Sturm–Liouville problem with a two-point separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. We also discuss the changes in the existence question for different types of nodal solutions as the problem changes.