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 formL u : = - [p u^?]^? + q u^s, 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 ? C_rd¹ (T^?), q ? C_rd⁰ (T^?2). 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.