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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, Taiwan30013, 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 City106,Taiwan (ROC) e-mail: d917205@oz.nthu.edu.tw
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Abstract

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
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

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