Nodal solutions for second-order m-point boundary value problems with nonlinearities across several eigenvalues

We study the existence of nodal solutions of the m-point boundary value problem u″+f(u)=0,00αi>0 and 0<∑i=1m-2αi⩽1. We give conditions on the ratio f(s)/sf(s)/s at infinity and zero that guarantee the existence of nodal solutions. The proofs of our main results are based upon bifurcation techniques.