Sign-changing points of solutions of homogeneous Sturm–Liouville equations with measure-valued coefficients.

In this paper, we investigate sign-changing points of nontrivial real-valued solutions of homogeneous Sturm–Liouville differential equations of the form − d (d u / d α) + u d β = 0 , where d α is a positive Borel measure supported everywhere on (a , b) and d β is a locally finite real Borel measure on (a , b). Since solutions for such equations are functions of locally bounded variation, sign-changing points are the natural generalization of zeros. We prove that sign-changing points for each nontrivial real-valued solution are isolated in (a , b). We also prove a Sturm-type separation theorem for two nontrivial linearly independent solutions and conclude the paper by proving a Sturm-type comparison theorem for two differential equations with distinct potentials d β. [ABSTRACT FROM AUTHOR]