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Hindawi Publishing Corporation Boundary Value Problems Volume 2011, Article ID 567054, 14 pages

Hindawi Publishing Corporation Boundary Value Problems Volume 2011, Article ID 567054, 14 pages doi:10.1155/2011/567054 Research Article New Fixed Point Theorems of Mixed Monotone Operators and Applications to Singular Boundary Value Problems on Time Scales Huiye Xu College of Economics and Management, North University of China, Taiyuan, Shanxi 030051, China Correspondence should be addressed to Huiye Xu, silviahsu2005@yahoo.com.cn Received 3 July 2010; Accepted 13 December 2010 Academic Editor: Daniel Franco Copyright q 2011 Huiye Xu. 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. | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 567054 14 pages doi 10.1155 2011 567054 Research Article New Fixed Point Theorems of Mixed Monotone Operators and Applications to Singular Boundary Value Problems on Time Scales Huiye Xu College of Economics and Management North University of China Taiyuan Shanxi 030051 China Correspondence should be addressed to Huiye Xu silviahsu2005@yahoo.com.cn Received 3 July 2010 Accepted 13 December 2010 Academic Editor Daniel Franco Copyright 2011 Huiye Xu. 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. Some new existence and uniqueness theorems of fixed points of mixed monotone operators are obtained and then they are applied to a nonlinear singular second-order three-point boundary value problem on time scales. We prove the existence and uniqueness of a positive solution for the above problem which cannot be solved by using previously available methods. 1. Introduction The study of mixed monotone operators has been a matter of discussion since they were introduced by Guo and Lakshmikantham 1 in 1987 because it has not only important theoretical meaning but also wide applications in microeconomics the nuclear industry and so on see 1-4 . Recently some new and interesting results about these kinds of operators have emerged and they are used extensively in nonlinear differential and integral equations see 5-9 . In this paper we extend the main results of 9 to mixed monotone operators. Without demanding compactness and continuity conditions and the existence of upper and lower solutions we study the existence uniqueness and iterative convergence of fixed points of a class of mixed monotone operators. Then we apply these results to the following singular second-order three-point boundary value problem on time scales -x t w t f1 x t f2 x t