Eigenvalue Comparisons for Differential Equations on a Measure Chain

The theory of u0-positive operators with respect to a cone in a Banach space is applied to eigenvalue problems associated with the second order ∆-differential equation (often referred to as a differential equation on a measure chain) given by y(t) + λp(t)y(σ(t)) = 0, t ∈ [0, 1], satisfying the boundary conditions y(0) = 0 = y(σ2(1)). The existence of a smallest positive eigenvalue is proven and then a theorem is established comparing the smallest positive eigenvalues for two problems of this type. 1. Background In this paper, we are concerned with comparing the smallest positive eigenvalues for second order ∆-differential equations satisfying conjugate boundary conditions. Much recent attention has been given to differential equations on measure chains, and we refer the reader to [4, 8, 15] for some historical works as well as to the more recent papers [1, 9, 10] and the book [17] for excellent references on these types of equations. Before introducing the problems of interest for this paper, we present some definitions and notation which are common to the recent literature. Our sources for this background material are the two papers by Erbe and Peterson [9, 10]. Definition 1.1. Let T be a closed subset of R, and let T have the subspace topology inherited from the Euclidean topology on R. The set T is referred to as a measure chain or, in some places in the literature, a time scale. For t < supT and r > inf T , define the forward jump operator, σ, and the backward jump operator, ρ, respectively, by σ(t) = inf{τ ∈ T | τ > t} ∈ T, ρ(r) = sup{τ ∈ T | τ < r} ∈ T, for all t, r ∈ T . If σ(t) > t, t is said to be right scattered, and if ρ(r) < r, r is said to be left scattered. If σ(t) = t, t is said to be right dense, and if ρ(r) = r, r is said to be left dense. 1991 Mathematics Subject Classification. 34B99, 39A99 .