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

Abstract

Under consideration is a class of even-ordered linear differential equations (-1) mχ δ 2m (t)=α ∑ i=0 m-1 p i(t)χ δ 2i (ω(t)) with Sturm-Liouville boundary conditions α iχ δ 2i (0) − β i+χ δ 2i (ω) = 0, y i+1χ δ 2i (ω(1)) + δ i+1χ δ 2i (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.