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

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.