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Báo cáo hóa học: "EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR SOME THREE-POINT NONLINEAR BOUNDARY VALUE PROBLEMS"

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 MULTIPLICITY OF SOLUTIONS FOR SOME THREE-POINT NONLINEAR BOUNDARY VALUE PROBLEMS | EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR SOME THREE-POINT NONLINEAR BOUNDARY VALUE PROBLEMS XU XIAN AND DONAL O REGAN Received 30 September 2004 Accepted 20 October 2004 We study the existence and multiplicity of solutions for the three-point nonlinear boundary value problem u t Aa t f u 0 0 t 1 u 0 0 u 1 - Yu n where n e 0 1 Y e 0 1 a t and f u are assumed to be positive and have some singularities and A is a positive parameter. Under certain conditions we prove that there exists A 0 such that the three-point nonlinear boundary value problem has at least two positive solutions for 0 A A at least one solution for A A and no solution for A A . Copyright 2006 X. Xian and D. O Regan. 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 In this paper we consider the following second-order three-point boundary value prob lem BVP u t Aa t f u 0 0 t 1 u 0 0 u 1 - yu n 1.1a where n e 0 1 Y e 0 1 a e C 0 1 0 to and f e C R 0 R here A is a positive parameter and R 0 to . Now a t may have a singularity at t 0 and t 1 f u may have a singularity at u 0 so the BVP 1.1a is a singular problem. The BVP 1.1a in the case when Y 0 can be reduced to the Dirichlet BVP u t Aa t f u 0 0 t 1 u 0 0 u 1 . 1.2a The BVP 1.2a has been studied extensively in the literature see 1 2 5 9 12 and the references therein. Choi 1 studied the particular case where f u eu a e C1 0 1 a 0 in 0 1 and a can be singular at t 0 but is at most O 1 t2-8 as t 0 for some 8. Using the shooting method he established the following result. Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 79653 Pages 1-17 DOI 10.1155 JIA 2006 79653 2 Solutions of three-point nonlinear BVPs Theorem 1.1 see 1 . There exists Ao 0 such that the BVP 1.2a has a solution in c2 0 1 n C 0 1 for 0 A A0 while there is no solution