Matrix representations of Atkinson-type Sturm-Liouville problems with coupled eigenparameter-dependent conditions

Jinming Cai; Shuang Li; Kun Li; Jinming Cai; Shuang Li; Kun Li

doi:10.3934/math.20241235

AIMS Mathematics

2024, Volume 9, Issue 9: 25297-25318. doi: 10.3934/math.20241235

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Research article

Matrix representations of Atkinson-type Sturm-Liouville problems with coupled eigenparameter-dependent conditions

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

Received: 15 March 2024 Revised: 16 July 2024 Accepted: 29 July 2024 Published: 29 August 2024
MSC : 34B05, 34L30, 47E05

We investigate the Sturm-Liouville (S-L) operator with boundary and transfer conditions dependent on the eigen-parameter. By utilizing interval partitioning and factorization techniques of characteristic function, it is proven that this problem has a finite number of eigenvalues when the coefficients of the equation meet certain conditions, and some conditions for determining the number of eigenvalues are provided. The results indicate that the number of eigenvalues in this problem varies when the transfer conditions depend on the eigen-parameter. Furthermore, the equivalence between this problem and matrix eigenvalue problems is studied, and an equivalent matrix representation of the S-L problem is presented.
- Atkinson-type S-L problems,
- finite spectrum,
- eigenparameter-dependent boundary condition,
- transfer condition,
- matrix representation
Citation: Jinming Cai, Shuang Li, Kun Li. Matrix representations of Atkinson-type Sturm-Liouville problems with coupled eigenparameter-dependent conditions[J]. AIMS Mathematics, 2024, 9(9): 25297-25318. doi: 10.3934/math.20241235

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Abstract

We investigate the Sturm-Liouville (S-L) operator with boundary and transfer conditions dependent on the eigen-parameter. By utilizing interval partitioning and factorization techniques of characteristic function, it is proven that this problem has a finite number of eigenvalues when the coefficients of the equation meet certain conditions, and some conditions for determining the number of eigenvalues are provided. The results indicate that the number of eigenvalues in this problem varies when the transfer conditions depend on the eigen-parameter. Furthermore, the equivalence between this problem and matrix eigenvalue problems is studied, and an equivalent matrix representation of the S-L problem is presented.

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