The purpose of this work is to establish an extension of a Bai-Ge type multiple fixed point theorem for a sum of two operators. The arguments are based upon recent fixed point index theory in cones of Banach spaces for a k-set contraction perturbed by an expansive operator. As an illustration, our approach is applied to prove the existence of at least three nontrivial nonnegative solutions for a class of eigenvalue three-point boundary value problems (BVPs) for a class of fourth-order ordinary differential equations (ODEs for short).