TWO-PARAMETER UNIFORMLY ELLIPTIC STURM–LIOUVILLE PROBLEMS WITH EIGENPARAMETER-DEPENDENT BOUNDARY CONDITIONS

[摘要] We consider the two-parameter Sturm–Liouville system$$ -y_1''+q_1y_1=(lambda r_{11}+mu r_{12})y_1quadext{on }[0,1], $$with the boundary conditions$$ frac{y_1'(0)}{y_1(0)}=cotalpha_1quadext{and}quadfrac{y_1'(1)}{y_1(1)}=frac{a_1lambda+b_1}{c_1lambda+d_1}, $$and$$ -y_2''+q_2y_2=(lambda r_{21}+mu r_{22})y_2quadext{on }[0,1], $$with the boundary conditions$$ frac{y_2'(0)}{y_2(0)} =cotalpha_2quadext{and}quadfrac{y_2'(1)}{y_2(1)}=frac{a_2mu+b_2}{c_2mu+d_2}, $$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 $c_{i}eq0$ for $i,j=1,2$. Results are given on asymptotics, oscillation of eigenfunctions and location of eigenvalues.

[学科分类] 数学（综合）

[关键词] Primary 34B08;Secondary 34B24;Sturm–Liouville equations;definiteness conditions;eigencurves;oscillation theorems