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

Academic Article


  • In this paper, we investigate sign-changing points of nontrivial real-valued solutions of homogeneous Sturm–Liouville differential equations of the form (Formula presented.), where (Formula presented.) is a positive Borel measure supported everywhere on (Formula presented.) and (Formula presented.) is a locally finite real Borel measure on (Formula presented.). 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 (Formula presented.). 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 (Formula presented.).
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  • Ghatasheh A; Weikard R