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: THE FORMULATION OF SECOND-ORDER BOUNDARY VALUE PROBLEMS ON TIME SCALES | THE FORMULATION OF SECOND-ORDER BOUNDARY VALUE PROBLEMS ON TIME SCALES FORDYCE A. DAVIDSON AND BRYAN P. RYNNE Received 6 February 2006 Revised 21 April 2006 Accepted 15 May 2006 We reconsider the basic formulation of second-order two-point Sturm-Liouville-type boundary value problems on time scales. Although this topic has received extensive attention in recent years we present some simple examples which show that there are certain difficulties with the formulation of the problem as usually used in the literature. These difficulties can be avoided by some additional conditions on the structure of the time scale but we show that these conditions are unnecessary since in fact a simple amended formulation of the problem avoids the difficulties. Copyright 2006 F. A. Davidson and B. P. Rynne. 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 In the time scale literature there has been considerable interest in Sturm-Liouville boundary value problems of the form - puA A t q t ua t f A t u t a b n TK2 for suitable functions p q and f and real parameter A together with boundary conditions which are generally taken to have the form au a ịỉuA a 0 yu ơ2 b SuA ơ b 0 for some points a b e T with a b and a I Ddyl d 0 see . 1 2 6 7 9 11 15 and in the case of systems 12 although there are other formulations of the boundary conditions see Remark . The formulation - covers both linear eigenvalue problems and nonlinear problems. Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID 31430 Pages 1-10 DOI ADE 2006 31430 2 Boundary value problems At first sight - seems to be a reasonable formulation of a boundary value problem and indeed it has received considerable attention. However in this paper we show that for general time scales there are
