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 Solutions for Fourth-Order Four-Point Boundary Value Problem on Time Scales | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 491952 20 pages doi 2009 491952 Research Article Existence of Solutions for Fourth-Order Four-Point Boundary Value Problem on Time Scales Dandan Yang 1 Gang Li 1 and Chuanzhi Bai2 1 School of Mathematical Science Yangzhou University Yangzhou 225002 China 2 Department of Mathematics Huaiyin Normal University Huaian Jiangsu 223300 China Correspondence should be addressed to Chuanzhi Bai czbai8@ Received 11 April 2009 Revised 8 July 2009 Accepted 28 July 2009 Recommended by Irena Rachunkova We present an existence result for fourth-order four-point boundary value problem on time scales. Our analysis is based on a fixed point theorem due to Krasnoselskii and Zabreiko. Copyright 2009 Dandan Yang et al. 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 Very recently Karaca 1 investigated the following fourth-order four-point boundary value problem on time scales yA4 Í - q f yA2 ficyW - 2 f y ơ4 b 0 ay a - fyA a 0 YyA2 Ỗ1 - ổyA3 Ỗ1 0 zyA2 nyA3 0 for t e a b c T a ị1 2 Ơ b and f e C Ịa b X R X R X R . And the author made the following assumptions A1 a fi Y ô z q 0 and a Z1 Z2 Ơ b A2 q f 0. If qf 0 then Y z 0. The following key lemma is provided in 1 . 2 Boundary Value Problems Lemma see 1 Lemma . Assume that conditions A1 and A2 are satisfied. If h e C a b then the boundary value problem yAi t - q t yA2 ơ t h fi t e a b y ơ4 b 0 ay a - fyA a 0 ryA2 Ỗ1 - 5yA3 Ỗ1 0 yl nyA3 0 has a unique solution ơ4 b Ỉ2 y t G1 l ị Gfifisyh As Aị a Ỉ1 where 1 f ơ4 b - ơ s a t - a 6 t s G1 t s 4r . d ơ4 b - f a ơ s - a 6 t Ơ s 1 y ơ sf ự fi t s G2 f s Di L5 D y t ự ơ s t ơ s . Here D z ẻ1 - n A ẻ1 ổ ẻ2 Ỵy fi2 d 6 a ơ4 b - a and y fi y t are given as follows p t n Z t - Ỗ1 q s ơ s As At Ỉ1 Ỉ1 . nỈ2 q
