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

We study the existence of eigenvalue intervals for the second-order differential equation on a measure chain, xΔΔ(t)+λp(t)f(xσ(t))=0, t∈[t1,t2], satisfying the boundary conditions αx(t1)−βxΔ(t1)=0 and γx(σ(t2))+δxΔ(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.