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

Abstract

In this paper, we investigate sign-changing points of nontrivial real-valued solutions of homogeneous Sturm–Liouville differential equations of the form $-d(du/d\alpha )+ud\beta =0$, where $d\alpha$ is a positive Borel measure supported everywhere on $(a,b)$ and $d\beta$ 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 $d\beta$.

COMMUNICATED BY:

• B. P. Belinskiy

Keywords:

• Sturm separation theorem
• Sturm comparison theorem
• measure-valued coefficients

2010 Mathematics Subject Classifications:

• 47B25
• 47A06
• 34B24

Disclosure statement

No potential conflict of interest was reported by the author(s).