Some estimates for the first eigenvalue of the Sturm-Liouville problem with a weight integral condition

Telnova, Maria

Mathematica Bohemica, Tome 137 (2012) no. 2, pp. 229-238.

Résumé

Let $\lambda _1(Q)$ be the first eigenvalue of the Sturm-Liouville problem $$ y''-Q(x)y+\lambda y=0,\quad y(0)=y(1)=0,\quad 01. $$ We give some estimates for $m_{\alpha ,\beta ,\gamma }=\inf _{Q\in T_{\alpha ,\beta ,\gamma }}\lambda _1(Q)$ and $M_{\alpha ,\beta ,\gamma }=\sup _{Q\in T_{\alpha ,\beta ,\gamma }}\lambda _1(Q)$, where $T_{\alpha ,\beta ,\gamma }$ is the set of real-valued measurable on $\left [0,1\right ]$ $x^\alpha (1-x)^\beta $-weighted $L_\gamma $-functions $Q$ with non-negative values such that $\int _0^1x^\alpha (1-x)^\beta Q^{\gamma }(x) {\rm d} x=1$ $(\alpha ,\beta ,\gamma \in \mathbb {R},\gamma \neq 0)$.

Détail

DOI : 10.21136/MB.2012.142868

Classification : 34B24, 34L15
Keywords: first eigenvalue; Sturm-Liouville problem; weight integral condition