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A New Extension Theorem for Concave Operators
Fixed Point Theory and Applications volume 2009, Article number: 571546 (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
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ2_HTML.gif)
the graph of is a set in
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ3_HTML.gif)
and the epigraph of is a set in
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ5_HTML.gif)
The epigraph of is a set in
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ7_HTML.gif)
An operator is called a sublinear operator, if for all
and all real number
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ9_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ10_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ11_HTML.gif)
and . It follows from the hypothesis that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ12_HTML.gif)
It follows from the concavity of on
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ13_HTML.gif)
That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ14_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ15_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ16_HTML.gif)
By (3.7),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ17_HTML.gif)
By (3.8),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ18_HTML.gif)
We may relabel by
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ19_HTML.gif)
Define a map from
to
as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ20_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ21_HTML.gif)
That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ22_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F571546/MediaObjects/13663_2008_Article_1156_Equ23_HTML.gif)
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
.
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Peng, Jw., Rong, Wd. & Yao, JC. A New Extension Theorem for Concave Operators. Fixed Point Theory Appl 2009, 571546 (2009). https://doi.org/10.1155/2009/571546
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DOI: https://doi.org/10.1155/2009/571546