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, x(DeltaDelta)(t) + lambdap(t)f(x(sigma)(t)) = 0, t is an element of [t(1), t(2)], satisfying the boundary conditions alphax(t(1)) - betax(Delta) = 0 and gammax(sigma(t(2))) + delta(x(Delta)(sigma)(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), sigma(t(2))] of the measure chain. The methods involve applications of a fixed point theorem for operators on a cone in a Banach space. (C) 2002 Elsevier Science B.V. All rights reserved.