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Some estimates for the first eigenvalue of the Sturm-Liouville problem with a weight integral condition. (English) Zbl 1265.34313
Summary: Let \(\lambda _1(Q)\) be the first eigenvalue of the Sturm-Liouville problem \[ y''-Q(x)y+\lambda y=0,\;y(0)=y(1)=0,\quad 0<x<1. \] 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 \(\int _0^1x^\alpha (1-x)^\beta Q^{\gamma }(x)\, \text{d} x=1\) \((\alpha ,\beta ,\gamma \in \mathbb {R},\gamma \neq 0)\).

MSC:
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34B24 Sturm-Liouville theory
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