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 RESULTS FOR φ-LAPLACIAN BOUNDARY VALUE PROBLEMS ON TIME SCALES | EXISTENCE RESULTS FOR -LAPLACIAN BOUNDARY VALUE PROBLEMS ON TIME SCAlEs ALBERTO CABADA Received 24 January 2006 Revised 31 May 2006 Accepted 1 June 2006 This paper is devoted to proving the existence of the extremal solutions of a 0-Laplacian dynamic equation coupled with nonlinear boundary functional conditions that include as a particular case the Dirichlet and multipoint ones. We assume the existence of a pair of well-ordered lower and upper solutions. Copyright 2006 Alberto Cabada. 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 very well-known tool used in the theory of ordinary and partial differential equations. It was introduced by Picard 14 and allows us to ensure the existence of at least one solution of the considered problem lying between a lower solution a and an upper solution p such that a p. Combining these kinds of techniques with the monotone iterative ones see 13 and references therein one can deduce the existence of extremal solutions lying between the lower and the upper ones. In recent years these techniques have been applied to difference equations 7 9 15 . So existence results of suitable boundary value problems are obtained and the differences and the similarities between the discrete and the continuous problems are pointed out. For instance in second-order ordinary differential equations the existence of a p a pair of well-ordered lower and upper solutions of the periodic problem ensures the existence of at least one solution remaining in a p . This result is true for the periodic discrete centered problem A2Uk f t Uk 1 k e 0 1 . N - 1 u 0 u N Au 0 Au N but it is false for the noncentered ones 4 . It is important to consider both situations under the same formulation that is to study equations on time scales. One .
