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

摘要

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 $ rac {y_{1}^{'} (0) }{y_{1} (0) }= cot lpha_{1} mbox{ and } rac{y_{1}^{'} (1) }{y_{1} (1) } =rac{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 $ rac{y_{2}^{'} (0) }{y_{2} (0)}= cot lpha_{2} mbox{ and } rac{y_{2}^{'} (1) }{y_{2} (1)} =rac{ 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 $lpha_i$ 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 eq 0 $ for $i,j=1,2$. Results are given on asymptotics, oscillation of eigenfunctions and location of eigenvalues.