Fixed Point Theory and Applications
Volume 2009 (2009), Article ID 571546, 8 pages
doi:10.1155/2009/571546
Research Article

A New Extension Theorem for Concave Operators

Jian-wen Peng,1 Wei-dong Rong,2 and Jen-Chih Yao3

1College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China
2Department of Mathematics, Inner Mongolia University, Hohhot, Inner Mongolia 010021, China
3Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan

Received 5 November 2008; Accepted 25 February 2009

Academic Editor: Anthony Lau

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.

Abstract

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⊂X×Y be convex. Let X0 be a real linear proper subspace of X, with θ∈(AX−X0)ri, where AX={x∣(x,y)∈A for some y∈Y}. Let g0:X0→Y be a concave operator such that g0(x)≤z whenever (x,z)∈A and x∈X0. Then there exists a concave operator g:X→Y such that (i) g is an extension of g0, that is, g(x)=g0(x) for all x∈X0, and (ii) g(x)≤z whenever (x,z)∈A.