In this paper, we develop some new multiple fixed point theorems for the sum of two operators $T+S$ where $I-T$ is Lipschitz invertible and $S$ is a $k$-set contraction on translate of a cone in a Banach space. New existence criteria for multiple positive solutions of a singular generalized Sturm-Liouville multipoint boundary value problem are established. The article ends with an illustrative example.