tailieunhanh - Báo cáo hóa học: "Research Article A New Extension Theorem for Concave Operators"

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 A New Extension Theorem for Concave Operators | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 571546 8 pages doi 2009 571546 Research Article A New Extension Theorem for Concave Operators Jian-wen Peng 1 Wei-dong Rong 2 and Jen-Chih Yao3 1 College of Mathematics and Computer Science Chongqing Normal University Chongqing 400047 China 2 Department of Mathematics Inner Mongolia University Hohhot Inner Mongolia 010021 China 3 Department of Applied Mathematics National Sun Yat-Sen University Kaohsiung 804 Taiwan Correspondence should be addressed to Jian-wen Peng jwpeng6@ Received 5 November 2008 Accepted 25 February 2009 Recommended by Anthony Lau We present a new and interesting extension theorem for concave operators as follows. Let X be a real linear space and let Y K be a real order complete PL space. Let the set A c X X Y be convex. Let X0 be a real linear proper subspace of X with 0 e Ax - X0 ri where AX x x y e A for some y e Y . Let g0 X0 Y be a concave operator such that g0 x z whenever x z e A and x e X0. Then there exists a concave operator g X Y such that i g is an extension of g0 that is g x go x for all x e X0 and ii g x z whenever x z e A. Copyright 2009 Jian-wen Peng 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 A very important result in functional analysis about the extension of a linear functional dominated by a sublinear function defined on a real vector space was first presented by Hahn 1 and Banach 2 which is known as the Hahn-Banach extension theorem. The complex version of Hahn-Banach extension theorem was proved by Bohnenblust and Sobczyk in 3 . Generalizations and variants of the Hahn-Banach extension theorem were developed in different directions in the past. Weston 4 proved a Hahn-Banach extension theorem in which a real-valued linear .