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 Fixed Points for Discontinuous Monotone Operators | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 926209 11 pages doi 2010 926209 Research Article Fixed Points for Discontinuous Monotone Operators Yujun Cui1 and Xingqiu Zhang2 1 Department of Applied Mathematics Shandong University of Science and Technology Qingdao 266510 China 2 School of Mathematics Liaocheng University Liaocheng 252059 China Correspondence should be addressed to Yujun Cui cyj720201@ Received 24 September 2009 Accepted 21 November 2009 Academic Editor Tomas Dominguez Benavides Copyright 2010 Y. Cui and X. Zhang. 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. We obtain some new existence theorems of the maximal and minimal fixed points for discontinuous monotone operator on an order interval in an ordered normed space. Moreover the maximal and minimal fixed points can be achieved by monotone iterative method under some conditions. As an example of the application of our results we show the existence of extremal solutions to a class of discontinuous initial value problems. 1. Introduction Let X be a Banach space. A nonempty convex closed set P c X is said to be a cone if it satisfies the following two conditions i x e P A 0 implies Ax e P ii x e P -x e P implies x 0 where 0 denotes the zero element. The cone P defines an ordering in E given by x y if and only if y - x e P. Let D uo Vo be an ordering interval in X and A D X an increasing operator such that u0 Au0 Av0 v0. It is a common knowledge that fixed point theorems on increasing operators are used widely in nonlinear differential equations and other fields in mathematics 1-7 . But in most well-known documents it is assumed generally that increasing operators possess stronger continuity and compactness. Recently there have been some papers that considered the existence of .