Abstract:
We establish the attainability of the infimum $m_\gamma$ for the minimal eigenvalues of the boundary-value problems
\begin{gather*}
-y''+qy=\lambda y,
\\
y'(0)=y'(1)=0
\end{gather*}
as the nonnegative potential $q\in L_1[0,1]$ ranges over the unit sphere of the space $L_\gamma[0,1]$, where $\gamma\in (0,1)$. We also establish that, for $\gamma\leqslant 1-2\pi^{-2}$, the equality $m_\gamma=1$ holds and that, otherwise, the inequality $m_\gamma<1$ is valid.
Keywords:Sturm–Liouville problem, infimum, Lagrange finite-increment theorem, minimal eigenvalue, Hölder's inequality, the space $L_\gamma[0,1]$, $\gamma\in (0,1)$.
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