A New Extension Theorem for Concave Operators

Jian-Wen Peng,Jen-Chih Yao,Wei-Dong Rong

doi:10.1155/2009/571546

Abstract

We present a new and interesting extension theorem for concave operators as follows. Let be a real linear space, and let be a real order complete PL space. Let the set be convex. Let be a real linear proper subspace of , with , where for some . Let be a concave operator such that whenever and . Then there exists a concave operator such that (i) is an extension of , that is, for all , and (ii) whenever .

Highlights

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
Weston 4 proved a Hahn-Banach extension theorem in which a real-valued linear functional is dominated by a real-valued convex function
Hirano et al 5 proved a Hahn-Banach theorem in which a concave functional is dominated by a sublinear functional in a nonempty convex set

Summary

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. Weston 4 proved a Hahn-Banach extension theorem in which a real-valued linear functional is dominated by a real-valued convex function. Hirano et al 5 proved a Hahn-Banach theorem in which a concave functional is dominated by a sublinear functional in a nonempty convex set. Chen and Craven 6 , Day 7 , Peressini 8 , Zowe 9–12 , Elster and Nehse , Wang , Shi , and Brumelle generalized the Hahn-Banach theorem to the partially ordered linear space. Yang proved a Hahn-Banach theorem in which a linear map is weakly dominated by a set-valued map which is convex. Chen and Wang proved a Hahn-Banach theorems in which a linear map is dominated by a K-set-valued map. Peng et al proved a Hahn-Banach theorem in which an affine-like set-valued map is dominated by a Kset-valued map. The purpose of this paper is to present some new and interesting extension results for concave operators

Preliminaries

An Extension Theorem with Applications