L2 spaces and boundary value problems on time-scales

In this paper we consider a second-order Sturm–Liouville-type boundary value operator of the form Lu := − pu + quσ , on an arbitrary, bounded time-scale T, for suitable functions p, q, together with suitable boundary conditions. Operators of this type on time-scales have normally been considered in a setting involving Banach spaces of continuous functions on T. In this paper we introduce a space L2(T) of square-integrable functions on T, and Sobolev-type spaces Hn(T), n 1, consisting of L2(T) functions with nth-order generalised L2(T)-type derivatives. We prove some basic functional analytic results for these spaces, and then formulate the operator L in this setting. In particular, we allow p ∈ H1(T), while q ∈ L2(T) — this generalises the usual conditions that p ∈ C1 rd(Tκ ), q ∈ C0 rd(Tκ2 ). We give some immediate applications of the functional analytic results to L, such as ‘positivity’, injectivity, invertibility and compactness of the inverse.We also construct a Green’s function for L. The analogues of these results on real intervals are well known, and are fundamental to the usual Sturm–Liouville theory on such intervals. © 2006 Elsevier Inc. All rights reserved.