# Two-Parameter Uniformly Elliptic Sturm–Liouville Problems With Eigenparameter-Dependent Boundary Conditions

Bhattacharyya, Bhattacharyya T and Mohandas, Mohandas JP (2005) Two-Parameter Uniformly Elliptic Sturm–Liouville Problems With Eigenparameter-Dependent Boundary Conditions. In: Proceedings of the Edinburgh Mathematical Society (Series 2), 48 . pp. 531-547.

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## Abstract

We consider the two-parameter Sturm–Liouville system $$-y_1''+q_1y_1=(\lambda r_{11}+\mu r_{12})y_1\quad\text{on }[0,1],$$ with the boundary conditions $$\frac{y_1'(0)}{y_1(0)}=\cot\alpha_1\quad\text{and}\quad\frac{y_1'(1)}{y_1(1)}=\frac{a_1\lambda+b_1}{c_1\lambda+d_1},$$ and $$-y_2''+q_2y_2=(\lambda r_{21}+\mu r_{22})y_2\quad\text{on }[0,1],$$ with the boundary conditions $$\frac{y_2'(0)}{y_2(0)} =\cot\alpha_2\quad\text{and}\quad\frac{y_2'(1)}{y_2(1)}=\frac{a_2\mu+b_2}{c_2\mu+d_2},$$ subject to the uniform-left-definite and uniform-ellipticity conditions; where $q_{i}$ and $r_{ij}$ are continuous real valued functions on $[0,1]$, the angle $\alpha_{i}$ is in $[0,\pi)$ and $a_{i}$, $b_{i}$, $c_{i}$, $d_{i}$ are real numbers with $\delta_{i}=a_{i}d_{i}-b_{i}c_{i}>0$ and $c_{i}\neq0$ for $i,j=1,2$. Results are given on asymptotics, oscillation of eigenfunctions and location of eigenvalues.

Item Type: Journal Article Copyright of this article belongs to Cambridge University Press. Division of Physical & Mathematical Sciences > Mathematics 19 Aug 2011 10:13 19 Aug 2011 10:13 http://eprints.iisc.ernet.in/id/eprint/39980