Comparison of eigenvalues for Sturm-Liouville boundary value problems on a measure chain

Under consideration is a class of even-ordered linear differential equations (-1)(m) x(Delta2m) (t) = lambda Sigma(i=0)(m-1) pi (t)x(Delta2i) (sigma(t)), with Sturm-Liouville boundary conditions alpha(i+1)x(Delta2i) (0) - beta(i+1)x(Delta2i+1) (0) = 0, gamma(i+1)x(Delta2i) (sigma(1)) + delta(i+1)x(Delta2i+1) (sigma(1)) = 0, for 0 less than or equal to i less than or equal to M - 1. The derivative in this dynamic equation is the generalized delta-derivative defined on a measure chain. For a pair of eigenvalue problems for this dynamic equation, we first verify the existence of smallest positive eigenvalues and then establish a comparison between the smallest eigenvalues of each eigenvalue problem. (C) 2003 Elsevier Science Ltd. All rights reserved.