On an inverse spectral problem and a generalized Sturm's nodal theorem for nonlinear boundary value problems

TL;DR

This paper explores an inverse spectral problem for a Sturm-Liouville operator and establishes a generalized Sturm's nodal theorem for nonlinear boundary value problems, linking spectral data to solution properties.

Contribution

It introduces a novel connection between inverse spectral problems and nonlinear boundary value problems, extending Sturm's nodal theorem to nonlinear cases.

Findings

Proves the inverse spectral problem relates to solutions of nonlinear equations.

Establishes a generalized Sturm's nodal theorem for nonlinear boundary problems.

Provides theoretical foundation for nonlinear spectral analysis.

Abstract

We consider an inverse optimization spectral problem for the Sturm-Liouville operator $$\mathcal{L}[q] u:=-u''+q(x)u$$ subject to the separated boundary conditions. In the main result, we prove that this problem is related to the existence of solutions of boundary value problems for the nonlinear equations of the form $$-u"+q_0(x) u=\lambda u+\sigma u^3$$ with $\sigma=1$ or $\sigma=-1$. The key outcome of this relationship is a generalized Sturm's nodal theorem for the nonlinear boundary value problems.