An estimate of the first eigenvalue in a multi-point boundary value problem (Q1780222)

The author has obtained lower and upper bounds for the first eigenvalue \(\lambda_1\) of the multipoint boundary value problem \[ \begin{aligned} &y^{(n)}+\lambda p(x)y=0,\quad n>1,\;a\leq x\leq b,\\ &y(x_i)=y'(x_i)=\dots=y^{(r_i-1)}(x_i)=0,\\ &i=1,\dots,s,\;x_i<x_j\text{ for }i<j,\;x_1=s, x_s=b,\\ &r_1+\dots+r_s=n,\;r_i\geq1,\end{aligned} \] under the assumption that \(\int_a^b| p(x)|^\alpha dx=1,\;\alpha\neq0\), where \(p\in C([a,b],\mathbb{R}),\) \(-\infty<a<b<\infty\) and \(\alpha\) is a given real number. Following results are obtained: Theorem 1: If \(0<\alpha<1/n\) then \(\lambda_1\leq C\), where \(C\) is a constant independent of \(p(x)\) but depends on \(\alpha, n, b-a\) and the numbers \(x_{i+1}-x_i\), \(i=1,\dots,s-1\). Theorem 2: If \(\int_a^b| p(x)| dx\leq1\), then \(\lambda_1\geq C\), where \(C\) is a constant which depends on \(n\) and \(b-a\).