Positive eigenvalues of second order boundary value problems and a theorem of M. G. Krein

Abstract

Conditions are given which guarantee that the least real eigenvalue is positive for certain boundary value problems for the vector-matrix equation − y ′ ′ + p ( x ) y = γ w ( x ) y -y^{\prime \prime }+p(x)y=\gamma w(x)y . This leads to conditions which guarantee the stable boundedness, according to Krein, for solutions of y ′ ′ + λ p ( x ) y = 0 y^{\prime \prime }+\lambda p(x)y=0 with certain real values of λ \lambda . As a consequence, a result first stated by Krein is proven.