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[t₁,t₂], satisfying the boundary conditions αx(t₁)−βx^Δ(t₁)=0 and γx(σ(t₂))+δx^Δ(σ(t₂))=0, where f is a positive function and p a nonnegative function that is allowed to vanish on some subintervals of [t₁,σ(t₂)] of the measure chain. The methods involve applications of a fixed point theorem for operators on a cone in a Banach space.