tailieunhanh - Báo cáo hoa học: " Research Article Existence of Weak Solutions for Second-Order Boundary Value Problem of Impulsive Dynamic Equations on Time Scales"

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: Research Article Existence of Weak Solutions for Second-Order Boundary Value Problem of Impulsive Dynamic Equations on Time Scales | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 907368 16 pages doi 2009 907368 Research Article Existence of Weak Solutions for Second-Order Boundary Value Problem of Impulsive Dynamic Equations on Time Scales Hongbo Duan and Hui Fang Department of Applied Mathematics Kunming University of Science and Technology Kunming Yunnan 650093 China Correspondence should be addressed to Hui Fang kmustfanghui@ Received 9 April 2009 Accepted 28 June 2009 Recommended by Victoria Otero-Espinar We study the existence of weak solutions for second-order boundary value problem of impulsive dynamic equations on time scales by employing critical point theory. Copyright 2009 H. Duan and H. Fang. 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 the following boundary value problem -uAA t f ơ t uơ ff t e 0 T t t tj j 1 2 . p - - u tj u tj Aju tj j 1 2 . . . p J J uAj - uA tp BjuA tp Ij u t-P j 1 2 . p u 0 0 u T where T is a time scale 0 T T 0 T n T ơ 0 0 and ơ T T f 0 T T X R R is a given function Ij e C R R Aj Bj are real sequences with Bj 1 Aj -1 - 1 and 2fc 1 Ak 1 the impulsive points tj e 0 T T are right-dense and 0 to t1 tp tp 1 T limh 0 uA tj h and limh 0 uA tj - h represent the right and left limits of uA t at t tj in the sense of the time scale that is in terms of h 0 for which tj h tj - h e 0 T t whereas if tj is left-scattered we interpret uA t- uA tj and u t- u tj . 2 Advances in Difference Equations The theory of time scales which unifies continuous and discrete analysis was first introduced by Hilger 1 . The study of boundary value problems for dynamic equations on time scales has recently received a lot of attention see 2-16 . At the same time there have been significant developments in impulsive differential .