Open Access

# A New Extension Theorem for Concave Operators

Fixed Point Theory and Applications20092009:571546

DOI: 10.1155/2009/571546

Accepted: 25 February 2009

Published: 8 March 2009

## 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 .

## 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 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 [912], Elster and Nehse [13], Wang [14], Shi [15], and Brumelle [16] generalized the Hahn-Banach theorem to the partially ordered linear space. Yang [17] proved a Hahn-Banach theorem in which a linear map is weakly dominated by a set-valued map which is convex. Meng [18] obtained Hahn-Banach theorems by using concept of efficient for -convex set-valued maps. Chen and Wang [19] proved a Hahn-Banach theorems in which a linear map is dominated by a -set-valued map. Peng et al. [20] proved some Hahn-Banach theorems in which a linear map or an affine map is dominated by a -set-valued map. Peng et al. [21] also proved a Hahn-Banach theorem in which an affine-like set-valued map is dominated by a -set-valued map. The various geometric forms of Hahn-Banach theorems (i.e., Hahn-Banach separation theorems) were presented by Eidelheit [22], Rockafellar [23], Deumlich et al. [24], Taylor and Lay [25], Wang [14], Shi [15], and Elster and Nehse [26] in different spaces.

Hahn-Banach theorems play a central role in functional analysis, convex analysis, and optimization theory. For more details on Hahn-Banach theorems as well as their applications, please also refer to Jahn [2729], Kantorovitch and Akilov [30], Lassonde [31], Rudin [32], Schechter [33], Aubin and Ekeland [34], Yosida [35], Takahashi [36], and the references therein.

The purpose of this paper is to present some new and interesting extension results for concave operators.

## 2. Preliminaries

Throughout this paper, unless other specified, we always suppose that and are real linear spaces, is the zero element in both and with no confusion, is a pointed convex cone, and the partial order on a partially ordered linear space (in short, PL space) is defined by if and only if . If each subset of which is bounded above has a least upper bound in , then is order complete. If and are subsets of a PL space , then means that for each and . Let be a subset of , then the algebraic interior of is defined by
(2.1)

If , then is called to be absorbed (see [14]).

The relative algebraic interior of is denoted by , that is, is the algebraic interior of with respect to the affine hull aff of .

Let be a set-valued map, then the domain of is
(2.2)
the graph of is a set in :
(2.3)
and the epigraph of is a set in :
(2.4)

A set-valued map is -convex if its epigraph is a convex set.

An operator is called a convex operator, if the domain of is a nonempty convex subset of and if for all and all real number
(2.5)
The epigraph of is a set in :
(2.6)

It is easy to see that an operator is convex if and only if is a convex set.

An operator is called a concave operator if is a nonempty convex subset of and if for all and all real number
(2.7)
An operator is called a sublinear operator, if for all and all real number ,
(2.8)

It is clear that if is a sublinear operator, then must be a convex operator, but the converse is not true in general.

For more detail about above definitions, please see [68, 16, 18, 20, 21, 2730, 34] and the references therein.

## 3. An Extension Theorem with Applications

The following lemma is similar to the generalized Hahn-Banach theorem [7, page 105] and [4, Lemma 1].

Lemma 3.1.

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 .

Proof.

The theorem holds trivially if . Assume that . Since is a proper subspace of , there exists . Let
(3.1)
It is clear that is a subspace of , and the above representation of in the form is unique. Since , there exists such that . And so there exist such that and such that . We define the sets and as follows:
(3.2)

It is clear that both and are nonempty.

Moreover, for all and for all , we have . In fact, let and , then there exist such that and . Let , then . Since is a convex set, we have
(3.3)
and . It follows from the hypothesis that
(3.4)
It follows from the concavity of on that
(3.5)
That is,
(3.6)

That is, .

Since is an order-complete PL space, there exist the supremum of denoted by and the infimum of denoted by . Since , taking , then we have
(3.7)
(3.8)
By (3.7),
(3.9)
By (3.8),
(3.10)
We may relabel by , then
(3.11)
Define a map from to as
(3.12)

Then , that is, is an extension of to . Since is a concave operator, it is easy to verify that is also a concave operator.

From (3.9) and (3.11), we know that satisfies
(3.13)
That is,
(3.14)

Let be the collection of all ordered pairs , where is a subspace of that contains and is a concave operator from to that extends and satisfies whenever and .

Introduce a partial ordering in as follows: if and only if for all . If we can show that every totally ordered subset of has an upper bound, it will follow from Zorn's lemma that has a maximal element . We can claim that is the desired map. In fact, we must have . For otherwise, we have shown in the previous proof of this lemma that there would be an such that and . This would violate the maximality of the .

Therefore, it remains to show that every totally ordered subset of has an upper bound. Let be a totally ordered subset of . Define an ordered pair by
(3.15)

This definition is not ambiguous, for if and are any of the elements of , then either or . At any rate, if , then . Clearly, . Hence, it is an upper bound for , and the proof is complete.

As a generalization of Lemma 3.1, we now present the main result as follows.

Theorem 3.2.

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 .

Proof.

Consider aff . Because is a linear space.

If , then . By Lemma 3.1, the result holds.

If . Of course, . Taking , we have that . By Lemma 3.1, we can find a concave operator such that , and for all . Taking a linear subspace of such that (i.e., and ) and defined by for all verifies the conclusion.

By Theorem 3.2, we can obtain the following new and interesting Hahn-Banach extension theorem in which a concave operator is dominated by a -convex set-valued map.

Corollary 3.3.

Let be a real linear space, and let be a real order complete PL space. Let be a -convex set-valued map. Let be a real linear proper subspace of , with . 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 .

Proof.

Let . Then is a convex set, , and . Since is a concave operator satisfying whenever and , we have that whenever and . Then by Theorem 3.2, there exists a concave operator such that (i) is an extension of , that is, for all , and (ii) for all . Since , we have for all .

Let be replaced by a single-valued map in Corollary 3.3, then we have the following Hahn-Banach extension theorem in which a concave operator is dominated by a convex operator.

Corollary 3.4.

Let be a real linear space, and let be a real order complete PL space. Let be a convex operator. Let be a real linear proper subspace of , with . Let be a concave operator such that whenever . Then there exists a concave operator such that (i) is an extension of , that is, for all , and (ii) for all .

Since a sublinear operator is also a convex operator, so from corollary 3.4, we have the following result.

Corollary 3.5.

Let be a real linear space, and let be a real order complete PL space. Let be a sublinear operator, and let be a real linear proper subspace of . Let be a concave operator such that whenever . Then there exists a concave operator such that (i) is an extension of , that is, for all , and (ii) for all .

## Authors’ Affiliations

(1)
College of Mathematics and Computer Science, Chongqing Normal University
(2)
Department of Mathematics, Inner Mongolia University
(3)
Department of Applied Mathematics, National Sun Yat-Sen University

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