Eigenvalues of a Sturm-Liouville problem and inequalities of Lyapunov type

Abstract:

We consider the eigenvalue problem $u''+\lambda u +p(x)u=0$ in $(0,\pi )$, $u(0)=u(\pi )=0$, where $p\in L^{1}(0,\pi )$ keeps a fixed sign and $\|p\|_{L^{1}}> 0$, and we obtain some lower and upper bounds for $\|p\|_{L^{1}}$ in terms of its nonnegative eigenvalues $\lambda$. Two typical results are: (1) $\|p\|_{L^{1}}>\sqrt {\lambda } |\sin {\sqrt {\lambda } \pi }|$ if $\lambda > 1$ and is not the square of a positive integer; (2) $\|p\|_{L^{1}}\le 16/\pi$ if $\lambda =0$ is the smallest eigenvalue.