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

Under consideration is a class of even-ordered linear differential equations with Sturm-Liouville boundary conditions αiχδ2i (0) − βi+χδ2i (ω) = 0, yi+1χδ2i (ω(1)) + δi+1χδ2i (ω(1)) = 0, for 0 ≤ i ≤ 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.