L2 spaces and boundary value problems on time-scales

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Abstract

In this paper we consider a second-order Sturm-Liouville-type boundary value operator of the formL u : = - [p u?]? + q us, 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 ? Crd1 (T?), q ? Crd0 (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.

Original languageEnglish
Pages (from-to)1217-1236
Number of pages20
JournalJournal of Mathematical Analysis and Applications
Volume328
Issue number2
DOIs
Publication statusPublished - 15 Apr 2007

Keywords

  • Boundary value problem
  • Lebesgue integration
  • Sobolev spaces
  • Time-scales

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