Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: BASIC PROPERTIES OF SOBOLEV’S SPACES ON TIME SCALES | BASIC PROPERTIES OF SOBOLEV S SPACES ON TIME SCALES RAVI P AGARWAL VICTORIA OTERO-ESPINAR KANISHKA PERERA AND DOLORES R. VIVERO Received 18 January 2006 Accepted 22 January 2006 We study the theory of Sobolev s spaces of functions defined on a closed subinterval of an arbitrary time scale endowed with the Lebesgue A-measure analogous properties to that valid for Sobolev s spaces of functions defined on an arbitrary open interval of the real numbers are derived. Copyright 2006 Ravi P. Agarwal et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Sobolev s spaces are a fundamental tool in real analysis for instance in the use of variational methods to solve boundary value problems in ordinary and partial differential equations and difference equations. In spite of this theory for functions defined on an arbitrary bounded open interval of the real numbers is well known see 2 and for functions defined on an arbitrary bounded subset of the natural numbers is trivial as far as we know for functions defined on an arbitrary time scale it has not been studied before. The aim of this paper is to give an introduction to Sobolev s spaces of functions defined on a closed interval a b n T of an arbitrary time scale T endowed with the Lebesgue A-measure. In Section 2 we gather together the concepts one needs to read this paper such as the Lp spaces linked to the Lebesgue A-measure and absolutely continuous functions on an arbitrary closed interval of T. The most important part of this paper is Section 3 where we define the first-order Sobolev s spaces as the space of lA a b n T functions whose generalized A-derivative belongs to lA a b n T moreover we study some of their properties by establishing an equivalence between them and the usual Sobolev s spaces defined on an open interval of the real numbers.
