A New Extension Theorem for Concave Operators

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 [9–12], 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 [27–29], 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 [6–8, 16, 18, 20, 21, 27–30, 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 .