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: AN EXISTENCE RESULT FOR A MULTIPOINT BOUNDARY VALUE PROBLEM ON A TIME SCALE | AN EXISTENCE RESULT FOR A MULTIPOINT BOUNDARY VALUE PROBLEM ON A TIME SCALE BASANT KARNA AND BONITA A. LAWRENCE Received 31 January 2006 Revised 15 April 2006 Accepted 19 April 2006 We will expand the scope of application of a fixed point theorem due to Krasnosel skii and Zabreiko to the family of second-order dynamic equations described by uAA t f u t t G 0 1 n T with multipoint boundary conditions u 0 0 u ơ2 1 ỵ n 1aiu ni andy n 1ai 1 for the purpose of establishing existence results. We will determine sufficient conditions on our function f such that the assumptions of the fixed point theorem are satisfied which in return gives us the existence of solutions. Copyright 2006 B. Karna and B. A. Lawrence. 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 Consider a very general family of dynamic equations of the form uAA t f u t t G 0 1 n T a second-order dynamic equation defined on a time scale T that is a subset closed by definition of the interval 0 1 . For more information on time scales calculus see the books of Bohner and Peterson 4 5 andHilger 12 . Assume that you know the behavior of the solution at an initial value as well as the relationship between the values of the solution at several other points. In particular let us consider boundary conditions n n u 0 0 u ơ2 1 aiu pi ai 1 where n G 0 1 n T. A standard method for establishing the existence of solutions for such dynamic boundary value problems is the application of a fixed point theorem. Many Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID 63208 Pages 1-8 DOI ADE 2006 63208 2 Existence of positive solution such theorems such as Guo-Krasnosel skii found in 8 Leggett-Williams 15 and more recently Avery-Henderson 3 have been used to develop the various existence results .
