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: EXISTENCE AND NONEXISTENCE OF POSITIVE SOLUTIONS TO A RIGHT-FOCAL BOUNDARY VALUE PROBLEM ON TIME SCALES | EXISTENCE AND NONEXISTENCE OF POSITIVE SOLUTIONS TO A RIGHT-FOCAL BOUNDARY VALUE PROBLEM ON TIME SCALES ILKAY YASLAN KARACA Received 10 October 2005 Revised 19 January 2006 Accepted 30 January 2006 We are concerned with proving the existence of one or more than one positive solution of an n-point right-focal boundary value problem for the nonlinear dynamic equation _ 1 n 1 xA n t Ar t f t xa t . We will also obtain criteria which lead to nonexistence of positive solutions. Here the independent variable t is in a time scale. We will use fixed point theorems for operators on a Banach space. Copyright 2006 Ilkay Yaslan Karaca. 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 Motivated by the work of Anderson 3 on discrete third-order three-point right-focal boundary value problems in this paper we will study an nth-order n-point right-focal boundary problem on time scales. This paper also gives nonexistence and multiplicity results for positive solutions to the time scale boundary value problem _1 n_1xAn t Ar t f t xa t Vt e t1 p tn x t1 xA t2 xAn_1 tn 0 where n 2 t1 t2 tn_1 tn A is a real parameter and x x t is a desired solution. The arguments are similar to those used in 9 13 . In the third section we obtain multiplicity results for this problem with A 1. In the fourth section existence nonexistence and multiplicity results are given for the eigenvalue problem. To understand this so-called dynamic equation on a time scale T we need some preliminary definitions. Definition . Let T be a nonempty closed subset of R and define the forward jump operator Ơ t at t for t sup T by Ơ t inf s t s e T Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID 43039 Pages 1-13 DOI ADE 2006 43039 2 A right-focal boundary value problem on time scales
