Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-09T00:50:47.433Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE EIGENVALUES AND THE NODAL POINTS OF THE EIGENFUNCTIONS OF SOME EIGENVALUE PROBLEMS WITH EIGENPARAMETER-DEPENDENT BOUNDARY CONDITIONS

Part of: Boundary value problems

Published online by Cambridge University Press:  07 April 2020

CHI-HUA CHAN  and
PO-CHUN HUANG Open the ORCID record for PO-CHUN HUANG [Opens in a new window]
CHI-HUA CHAN
Affiliation:
Department of Mathematics, National Tsing Hua University, No. 101, Section 2, Kuang-Fu Road, Hsinchu, Taiwan 30013, R.O.C. e-mail: evolution6423@yahoo.com.tw
PO-CHUN HUANG
Affiliation:
Department of Mathematics and Information Education, National Taipei University of Education, No. 134, Section 2, Heping East Road, Da-an District, Taipei City 106,Taiwan (ROC) e-mail: d917205@oz.nthu.edu.tw
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Consider the following two eigenvalue problems: (0.1)

\begin{cases}\label{eqn:1abs}y"(x)+[\lambda^2-q(x)]y(x)=0, 0 \leq x \leq \pi,\\[3pt] y(0)=0, ay'(\pi)+\lambda y(\pi)=0, \end{cases}
and (0.2)
\begin{cases} z"(x)+[\mu^2-q(x)]z(x)=0, 0 \leq x \leq \pi,\\[3pt] z'(0)=0, az'(\pi)+\mu z(\pi)=0, \end{cases}

where $q(x)$ is real-valued and integrable on [0, $\pi$]. Let $\{\lambda_n\}_{n\in \mathbb{Z}\setminus \{0\}}$ and $\{\mu_n\}_{n\in \mathbb{Z}\setminus \{0\}}$ denote the eigenvalues of equations (0.1) and (0.2), respectively. Then

\[\cdots\lt\mu_{-3}\lt\lambda_{-2}\lt\mu_{-2}\lt\lambda_{-1}\lt\mu_{-1}\lt\mu_1\lt\lambda_1\lt\mu_2\lt\lambda_2\lt\mu_3\lt\cdots.\]
Moreover, the number of zeros of the eigenfunctions of (0.1) ((0.2), respectively) corresponding to $\lambda_n$ ($\mu_n$, respectively) in (0, $\pi$) is equal to $|n|-1$.

MSC classification

Primary: 34B09: Boundary eigenvalue problems

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 1 , January 2021 , pp. 158 - 178
Copyright
© The Author(s) 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

References

REFERENCES

Atkinson, F. V., Discrete and continuous boundary problems (Academic Press, New York, 1964).Google Scholar
Binding, P. A., Browne, P. J. and Seddighi, K., Sturm-Liouville problems with eigenparameter dependent boundary conditions, Proc. Edinburgh Math. Soc. 37 (1993), 57–72.CrossRefGoogle Scholar
Allahverdiev, B. P., Bairamov, E. and Ugurlu, E., Eigenparameter dependent Sturm-Liouville problems in boundary conditions with transmission conditions, J. Math. Anal. Appl. 401(1) (2013), 388–396.CrossRefGoogle Scholar
Browne, P. J. and Sleeman, B. D., Inverse nodal problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions, Inverse Prob. 12 (1996), 377–381.CrossRefGoogle Scholar
Chan, C.-H., Some eigenvalue problems for vectorial Sturm-Liouville equations with eigenparameter dependent boundary conditions, Trans. AMS 364 (2012), 119–136.CrossRefGoogle Scholar
Gheorghiu, C. I., On the numerical treatment of the eigenparameter dependent boundary conditions, Numer Algorithms 77(1) (2018), 77–93.CrossRefGoogle Scholar
Eastham, M. S. P., Kong, Q., Wu, H. and Zettl, A., Inequalities among eigenvalues of Sturm-Liouville problems, J. Inequalities Appl. 3 (1999), 25–43.Google Scholar
Gesztesy, F. and Simon, B., A new approach to the inverse spectral theory, II. General real potentials and the connection to the spectral measure, Ann. Math. 152 (2000), 593–643.Google Scholar
Hochstadt, H., On the inverse problems associated with second-order differential operators, Acta Math. 119 (1967), 173–192.CrossRefGoogle Scholar
Aydemir, K. and Mukhtarov, O. S., Variational principles for spectral analysis of one Sturm-Liouville problem with transmission conditions, Adv. Differ. Equ. 76 (2016), 1–14.Google Scholar
Pöschel, J. and Trubowitz, E., Inverse spectral theory (Academic Press, New York, 1987).Google Scholar
Binding, P. A., Browne, P. J. and Watson, B. A., Inverse spectral problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions, J. London Math. Soc. 62(1) (2000), 161–182.CrossRefGoogle Scholar
Shen, C.-L., Some inverse spectral problems for vectorial Sturm-Liouville equations, Inverse Prob. 17 (2001), 1253–1294.CrossRefGoogle Scholar
Simon, B., A new approach to the inverse spectral theory, I. Fundamental formalism, Ann. Math. 150 (1999), 1029–1057.CrossRefGoogle Scholar
Zheng, Z., Cai, J., Li, K. and Zhang, M., A discontinuous Sturm-Liouville problem with boundary conditions rationally dependent on the eigenparameter, Boundary Value Prob. 103 (2018).CrossRefGoogle Scholar
Zhang, M. and Wang, Y., Dependence of eigenvalues of Sturm-Liouville problems with interface conditions, Appl. Math. Comput. 265 (2015), 31–39.Google Scholar