## Journal of Applied Mathematics

### The Modified Parseval Equality of Sturm-Liouville Problems with Transmission Conditions

#### Abstract

We consider the Sturm-Liouville (S-L) problems with very general transmission conditions on a finite interval. Firstly, we obtain the sufficient and necessary condition for $\lambda$ being an eigenvalue of the S-L problems by constructing the fundamental solutions of the problems and prove that the eigenvalues of the S-L problems are bounded below and are countably infinite. Furthermore, the asymptotic formulas of the eigenvalues and eigenfunctions of the S-L problems are obtained. Finally, we derive the eigenfunction expansion for Green's function of the S-L problems with transmission conditions and establish the modified Parseval equality in the associated Hilbert space.

#### Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 619358, 11 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808207

Digital Object Identifier
doi:10.1155/2013/619358

Mathematical Reviews number (MathSciNet)
MR3127464

Zentralblatt MATH identifier
06950781

#### Citation

Bai, Mudan; Sun, Jiong; Yao, Siqin. The Modified Parseval Equality of Sturm-Liouville Problems with Transmission Conditions. J. Appl. Math. 2013 (2013), Article ID 619358, 11 pages. doi:10.1155/2013/619358. https://projecteuclid.org/euclid.jam/1394808207

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