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 Exact Multiplicity of Positive Solutions for a Class of Second-Order Two-Point Boundary Problems with Weight Function | Hindawi Publishing Corporation Boundary Value Problems Volume 2010 Article ID 207649 16 pages doi 2010 207649 Research Article Exact Multiplicity of Positive Solutions for a Class of Second-Order Two-Point Boundary Problems with Weight Function Yulian An1 and Hua Luo2 1 Department of Mathematics Shanghai Institute of Technology Shanghai 200235 China 2 School of Mathematics and Quantitative Economics Dongbei University of Finance and Economics Dalian 116025 China Correspondence should be addressed to Yulian An anyulian@ Received 6 March 2010 Revised 18 July 2010 Accepted 11 August 2010 Academic Editor Raul F. Manasevich Copyright 2010 Y. An and H. Luo. 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. An exact multiplicity result of positive solutions for the boundary value problems u Xa t f u 0 t e 0 1 u 0 0 u 1 0 is achieved where A is a positive parameter. Here the function f 0 to 0 to is C2 and satisfies f 0 f s 0 f u 0 for u e 0 s u s to for some s e 0 to . Moreover f is asymptotically linear and f can change sign only once. The weight function a 0 1 0 to is C2 and satisfies a t 0 3 a t 2 2a t a t for t e 0 1 . Using bifurcation techniques we obtain the exact number of positive solutions of the problem under consideration for A lying in various intervals in R. Moreover we indicate how to extend the result to the general case. 1. Introduction Consider the problem u Xa t f u 0 t e 0 1 u 0 0 u 1 0 where A 0 is a parameter and a e C2 0 1 is a weight function. The existence and multiplicity of positive solutions for ordinary differential equations have been studied extensively in many literatures see for example 1-3 and references therein. Several different approaches such as the Leray-Schauder theory the fixed-point theory the lower and upper solutions theory and the shooting method etc .
