Eigenvalue intervals for a two-point boundary value problem on a measure chain

摘要

We study the existence of eigenvalue intervals for the second-order differential equation on a measure chain, x~(ΔΔ)(t) + λp(t)f(x~σ(t)) = 0, t ∈ [t_1,t_2], satisfying the boundary conditions αx(t_1) - βx~Δ(t_1) = 0 and γx(σ(t_2)) + δx~Δ(σ(t_2)) = 0, where f is a positive function and p a nonnegative function that is allowed to vanish on some subintervals of [t_1,σ(t_2)] of the measure chain. The methods involve applications of a fixed point theorem for operators on a cone in a Banach space.