# Two parameter uniformly elliptic Sturm-Liouville problems with eigenparameter dependent boundary conditions

Bhattacharyya, Tirthankar and Mohandas, JP (2004) Two parameter uniformly elliptic Sturm-Liouville problems with eigenparameter dependent boundary conditions. [Preprint]

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We Consider the two parameter Sturm Liouville system (1) $-y_{1}^{''} + q_{1} y_{1} = ( \lambda r_{11} + \mu r_{12}) y_{1} \mbox{ on } [0,1]$ with the boundary conditions $\frac {y_{1}^{'} (0) }{y_{1} (0) } = \cot \alpha_{1} \mbox{ and } \frac{y_{1}^{'} (1) }{y_{1} (1) } = \frac{a_{1} \lambda + b_{1} }{c_{1} \lambda + d_{1} }$ and (2) $-y_{2}^{''} + q_{2} y_{2} = ( \lambda r_{21} + \mu r_{22}) y_{2} \mbox{ on } [0,1]$ with the boundary conditions $\frac{y_{2}^{'} (0) }{y_{2} (0)} = \cot \alpha_{2} \mbox{ and } \frac{y_{2}^{'} (1) }{y_{2} (1)} = \frac{ a_{2} \mu \,\,+\,\, b_{2} }{c_{2} \mu + d_{2} },$ subject to the uniform left definiteness and uniform ellipticity conditions; where $q_i$ and $r_i_j$ 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 $c_i \neq 0$ for $i,j=1,2$. Results are given on asymptotics, oscillation of eigenfunctions and location of eigenvalues.