# A New Extension Theorem for Concave Operators

- Jian-wen Peng
^{1}Email author, - Wei-dong Rong
^{2}and - Jen-Chih Yao
^{3}

**2009**:571546

**DOI: **10.1155/2009/571546

© Jian-wen Peng et al. 2009

**Received: **5 November 2008

**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 [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

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 .

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

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

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.

It is clear that both and are nonempty.

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

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 .

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

## References

- Hahn H:
**Über lineare Gleichungssysteme in linearen Räumen.***Journal für die Reine und Angewandte Mathematik*1927,**157:**214–229.MATHGoogle Scholar - Banach S:
*Théorie des Opérations Linéaires*. Subwncji Funduszu Narodowej, Warszawa, Poland; 1932.MATHGoogle Scholar - Bohnenblust HF, Sobczyk A:
**Extensions of functionals on complex linear spaces.***Bulletin of the American Mathematical Society*1938,**44**(2):91–93. 10.1090/S0002-9904-1938-06691-8MathSciNetView ArticleMATHGoogle Scholar - Weston JD:
**A note on the extension of linear functionals.***The American Mathematical Monthly*1960,**67**(5):444–445. 10.2307/2309293MathSciNetView ArticleMATHGoogle Scholar - Hirano N, Komiya H, Takahashi W:
**A generalization of the Hahn-Banach theorem.***Journal of Mathematical Analysis and Applications*1982,**88**(2):333–340. 10.1016/0022-247X(82)90196-2MathSciNetView ArticleMATHGoogle Scholar - Chen G-Y, Craven BD:
**A vector variational inequality and optimization over an efficient set.***Mathematical Methods of Operations Research*1990,**34**(1):1–12. 10.1007/BF01415945MathSciNetView ArticleMATHGoogle Scholar - Day MM:
*Normed Linear Space*. Springer, Berlin, Germany; 1962.View ArticleGoogle Scholar - Peressini AL:
*Ordered Topological Vector Spaces*. Harper & Row, New York, NY, USA; 1967:x+228.MATHGoogle Scholar - Zowe J:
*Konvexe Funktionen und Konvexe Dualitätstheorie in geordneten Vektorräumen, Habilitation thesis*. University of Würzburg, Würzburg, Germany; 1976.Google Scholar - Zowe J:
**Linear maps majorized by a sublinear map.***Archiv der Mathematik*1975,**26**(6):637–645.MathSciNetView ArticleMATHGoogle Scholar - Zowe J:
**Sandwich theorems for convex operators with values in an ordered vector space.***Journal of Mathematical Analysis and Applications*1978,**66**(2):282–296. 10.1016/0022-247X(78)90232-9MathSciNetView ArticleMATHGoogle Scholar - Zowe J:
**A duality theorem for a convex programming problem in order complete vector lattices.***Journal of Mathematical Analysis and Applications*1975,**50**(2):273–287. 10.1016/0022-247X(75)90022-0MathSciNetView ArticleMATHGoogle Scholar - Elster K-H, Nehse R:
**Necessary and sufficient conditions for order-completeness of partially ordered vector spaces.***Mathematische Nachrichten*1978,**81**(1):301–311. 10.1002/mana.19780810116MathSciNetView ArticleMATHGoogle Scholar - Wang SS:
**A separation theorem for a convex cone on an ordered vector space and its applications.***Acta Mathematicae Applicatae Sinica*1986,**9**(3):309–318.MathSciNetMATHGoogle Scholar - Shi SZ:
**A separation theorem for convex sets in a complete vector lattice, and its application.***Chinese Annals of Mathematics. Series A*1985,**6**(4):431–438.MathSciNetMATHGoogle Scholar - Brumelle SL:
**Convex operators and supports.***Mathematics of Operations Research*1978,**3**(2):171–175. 10.1287/moor.3.2.171MathSciNetView ArticleMATHGoogle Scholar - Yang XQ:
**A Hahn-Banach theorem in ordered linear spaces and its applications.***Optimization*1992,**25**(1):1–9. 10.1080/02331939208843803MathSciNetView ArticleMATHGoogle Scholar - Meng ZQ:
**Hahn-Banach theorem of set-valued map.***Applied Mathematics and Mechanics*1998,**19**(1):55–61.MathSciNetMATHGoogle Scholar - Chen GY, Wang YY:
**Generalized Hahn-Banach theorems and subdifferential of set-valued mapping.***Journal of Systems Science and Mathematical Sciences*1985,**5**(3):223–230.MathSciNetMATHGoogle Scholar - Peng JW, Lee HWJ, Rong WD, Yang XM:
**Hahn-Banach theorems and subgradients of set-valued maps.***Mathematical Methods of Operations Research*2005,**61**(2):281–297. 10.1007/s001860400397MathSciNetView ArticleMATHGoogle Scholar - Peng J, Lee HWJ, Rong W, Yang XM:
**A generalization of Hahn-Banach extension theorem.***Journal of Mathematical Analysis and Applications*2005,**302**(2):441–449. 10.1016/j.jmaa.2004.03.038MathSciNetView ArticleMATHGoogle Scholar - Eidelheit M:
**Zur Theorie der konvexen Mengen in linearen normierten Räumen.***Studia Mathematica*1936,**6:**104–111.MATHGoogle Scholar - Rockafellar RT:
*Convex Analysis, Princeton Mathematical Series, no. 28*. Princeton University Press, Princeton, NJ, USA; 1970:xviii+451.Google Scholar - Deumlich R, Elster K-H, Nehse R:
**Recent results on separation of convex sets.***Mathematische Operationsforschung und Statistik. Series Optimization*1978,**9**(2):273–296. 10.1080/02331937808842491MathSciNetView ArticleMATHGoogle Scholar - Taylor AE, Lay DC:
*Introduction to Functional Analysis*. 2nd edition. John Wiley & Sons, New York, NY, USA; 1980:xi+467.Google Scholar - Elster K-H, Nehse R:
**Separation of two convex sets by operators.***Commentationes Mathematicae Universitatis Carolinae*1978,**19**(1):191–206.MathSciNetMATHGoogle Scholar - Jahn J:
*Mathematical Vector Optimization in Partially Ordered Linear Spaces, Methoden und Verfahren der Mathematischen Physik*.*Volume 31*. Peter D Lang, Frankfurt am Main, Germany; 1986:viii+310.Google Scholar - Jahn J:
*Introduction to the Theory of Nonlinear Optimization*. 2nd edition. Springer, Berlin, Germany; 1996:viii+257.View ArticleMATHGoogle Scholar - Jahn J:
*Vector Optimization: Theory, Applications, and Extensions*. Springer, Berlin, Germany; 2004:xiv+465.View ArticleMATHGoogle Scholar - Kantorvitch L, Akilov G:
*Functional Analysis in Normed Spaces*. Fizmatgiz, Moscow, Russia; 1959.Google Scholar - Lassonde M:
**Hahn-Banach theorems for convex functions.**In*Minimax Theory and Applications, Nonconvex Optimization and Its Applications 26*. Edited by: Ricceri B, Simons S. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1998:135–145.Google Scholar - Rudin W:
*Functional Analysis, McGraw-Hill Series in Higher Mathematic*. McGraw-Hill, New York, NY, USA; 1973:xiii+397.Google Scholar - Schechter M:
*Principles of Functional Analysis*. Academic Press, New York, NY, USA; 1971:xix+383.Google Scholar - Aubin J-P, Ekeland I:
*Applied Nonlinear Analysis, Pure and Applied Mathematics*. John Wiley & Sons, New York, NY, USA; 1984:xi+518.MATHGoogle Scholar - Yosida K:
*Functional Analysis*. Springer, New York, NY, USA; 1965.View ArticleMATHGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis. Fixed Point Theory and Its Applications*. Yokohama, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.