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: ON LOWER AND UPPER SOLUTIONS WITHOUT ORDERING ON TIME SCALES | ON LOWER AND UPPER SOLUTIONS WITHOUT ORDERING ON TIME SCALES PETR STEHLÍK Received 31 January 2006 Revised 16 May 2006 Accepted 16 May 2006 In order to enlarge the set of boundary value problems on time scales for which we can use the lower and upper solutions technique to get existence of solutions we extend this method to the case when the pair lacks ordering. We use the degree theory and a priori estimates to obtain the existence of solutions for the second-order Dirichlet boundary value problems. To illustrate a wider application of this result we conclude with an example which shows that a combination of well- and nonwell- ordered pairs can yield the existence of multiple solutions. Copyright 2006 Petr Stehlik. 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 The method of lower and upper solutions is a widely used concept in the study of nonlinear boundary value problems further abbreviated by BVP . Three quarters of the century after the pioneering work of Dragoni 8 this method still belongs among the basic tools and is frequently employed in applied analysis or mechanics. Dragoni s basic idea was to transform the BVP with an unbounded right-hand side into a problem with a bounded right-hand side this transformation is possible thanks to the existence of lower and upper solutions and in the second step to show that a solution of the modified problem is also a solution of the original problem. Together with the later introduced Nagumo conditions for the derivative dependent right-hand sides this basic scheme forms the foundations of this method. On the other hand the time scales calculus with its concept to unify and extend discrete and continuous worlds is a recent idea the seminal work is due to Hilger see . 9 . In spite of this this calculus is already broadly used. It is not .
