On the number and complete continuity of weighted eigenvalues of measure differential equations

Abstract

The classical eigenvalue theory for second-order ordinary differential equations (ODE) describes the spatial oscillation of strings whose distributions of masses are absolutely continuous. For general distributions of masses, including completely singular ones, the spatial oscillation can be explained using measure differential equations (MDE). In this paper we will study weighted eigenvalue problems for second-order MDE with general distributions or measures. It will be shown that the numbers of weighted eigenvalues depend on measures and may be finite. Furthermore, it will be proved that weighted eigenvalues and eigenfunctions are completely continuous in measures, i.e., when measures are convergent in the weak$^*$ topology, these eigen-pairs are strongly convergent. The present paper and the work of Meng and Zhang (Dependence of solutions and eigenvalues of measure differential equations on measures, J. Differential Equations, 254 (2013), 2196-2232, have given an extension of the classical Sturm-Liouville theory to the measure version of ODE.