# A New Extension Theorem for Concave Operators

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

**2009**:571546

https://doi.org/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 .

## Keywords

## 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

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