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

A new extension theorem for concave operators

Jian-Wen Peng1 , Wei-Dong Rong2 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

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.