# Dependence of Solutions and Eigenvalues of Third Order Linear Measure Differential Equations on Measures

## TL;DR

This paper studies how solutions and eigenvalues of a third order linear measure differential equation depend on measure coefficients, proving continuity and differentiability of eigenvalues with respect to these coefficients.

## Contribution

It establishes the continuity and Fréchet differentiability of eigenvalues of the measure differential equation in relation to the coefficients p and q.

## Key findings

- Eigenvalues are continuous in p, q under total variation and weak* topologies.
- Eigenvalues are Fréchet differentiable in p, q with the total variation norm.
- Solutions depend continuously on coefficients p, q under various topologies.

## Abstract

This paper deals with a complex third order linear measure differential equation \begin{equation*} i\mathrm{d}\left( y^{\prime }\right) ^{\bullet }+2iq\left( x\right) y^{\prime }\mathrm{d}x+y\left( i\mathrm{d}q\left( x\right) +\mathrm{d}p\left( x\right) \right) = \lambda y\mathrm{d}x \end{equation*} on a bounded interval with boundary conditions presenting a mixed aspect of the Dirichlet and the periodic problems. The dependence of eigenvalues on the coefficients $p$, $q$ is investigated. We prove that the $n$-th eigenvalue is continuous in $p$, $q$ when the norm topology of total variation and the weak$^*$ topology are considered. Moreover, the Fr\'{e}chet differentiability of the $n$-th eigenvalue in $p$, $q$ with the norm topology of total variation is also considered. To deduce these conclusions, we investigate the dependence of solutions of the above equation on the coefficients $p$, $q$ with different topologies and establish the counting lemma of eigenvalues according to the estimates of solutions.