Some comparison theorems for Sturm-Liouville eigenvalue problems (Q5952273)

Let \(p,p_1,q,q_1\in C([a, b],(0,\infty))\), \(p,p_1\in C^1[a, b]\), and \(\rho,\rho_1\in C([a, b],[0,\infty))\). Let \(u(z)> 0\) be an eigenfunction to the eigenvalue problem NEWLINE\[NEWLINE(pu')'(z)-\rho u(z)+\lambda qu(z)= 0,\quad u(a)= u(b)= 0,NEWLINE\]NEWLINE corresponding to the first eigenvalue \(\lambda\). Let \(v(z)> 0\) be an eigenfunction to the eigenvalue problem NEWLINE\[NEWLINE(pv')'(z)- \rho v(z)+\lambda qv(z)= 0,\quad v(a)= v(b)= 0,NEWLINE\]NEWLINE corresponding to the first eigenvalue \(\lambda_1\). Let \(z_0\) be the first maximum point of \(u(z)\) in \((a,b)\) and \(z_1\) the first maximum point of \(v(z)\) in \((a,b)\). The author gets some comparison theorems on the relative positivity of \(z_0\) and \(z_1\) by means of the variational method.