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

Abstract

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