September/October 2018 On the number and complete continuity of weighted eigenvalues of measure differential equations
Meirong Zhang, Zhiyuan Wen, Gang Meng, Jiangang Qi, Bing Xie
Differential Integral Equations 31(9/10): 761-784 (September/October 2018). DOI: 10.57262/die/1528855439


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.


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Meirong Zhang. Zhiyuan Wen. Gang Meng. Jiangang Qi. Bing Xie. "On the number and complete continuity of weighted eigenvalues of measure differential equations." Differential Integral Equations 31 (9/10) 761 - 784, September/October 2018.


Published: September/October 2018
First available in Project Euclid: 13 June 2018

zbMATH: 06945781
MathSciNet: MR3814566
Digital Object Identifier: 10.57262/die/1528855439

Primary: 34A30 , 34L40 , 58C07

Rights: Copyright © 2018 Khayyam Publishing, Inc.


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Vol.31 • No. 9/10 • September/October 2018
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