- Research Article
- Open access
- Published:
Some Maximal Elements' Theorems in
-Spaces
Journal of Inequalities and Applications volume 2009, Article number: 905605 (2009)
Abstract
Let be a finite or infinite index set, let
be a topological space, and let
be a family of
-spaces. For each
, let
be a set-valued mapping. Some new existence theorems of maximal elements for a set-valued mapping and a family of set-valued mappings involving a better admissible set-valued mapping are established under noncompact setting of
-spaces. Our results improve and generalize some recent results.
1. Introduction
It is well known that many existence theorems of maximal elements for various classes of set-valued mappings have been established in different spaces. For their applications to mathematical economies, generalized games, and other branches of mathematics, the reader may consult [1–12] and the references therein.
In most of the known existence results of maximal elements, the convexity assumptions play a crucial role which strictly restrict the applicable area of these results. In this paper, we will continue to study existence theorems of maximal elements in general topological spaces without convexity structure. We introduce a new class of generalized -majorized mappings
for each
which involve a set-valued mapping
. The notion of generalized
-majorized mappings unifies and generalizes the corresponding notions of
-majorized mappings in [4];
-majorized mappings in [2, 13];
-majorized mappings in [14]. Some new existence theorems of maximal elements for generalized
-majorized mappings are proved under noncompact setting of
-spaces. Our results improve and generalize the corresponding results in [2, 4, 14–16].
2. Preliminaries
Let and
be two nonempty sets. We denote by
and
the family of all subsets of
and the family of all nonempty finite subsets of
, respectively. For each
, we denote by
the cardinality of
. Let
denote the standard
-dimensional simplex with the vertices
. If
is a nonempty subset of
we will denote by
the convex hull of the vertices
.
Let and
be two sets, and let
be a set-valued mapping. We will use the following notations in the sequel:
(i),
(ii),
(iii).
For topological spaces and
, a subset
of
is said to be compactly open (resp., compactly closed) if for each nonempty compact subset
of
,
is open (resp., closed) in
. The compact closure of
and the compact interior of
(see [17]) are defined, respectively, by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ1_HTML.gif)
It is easy to see that ,
is compactly open (resp., compactly closed) in
if and only if
(resp.,
). For each nonempty compact subset
of
,
and
, where
(resp.,
)) denotes the closure (resp., interior) of
in
. A set-valued mapping
is transfer compactly open valued on
(see [17]) if for each
and
, there exists
such that
. Let
be transfer compactly open valued, then
. It is clear that each transfer open valued correspondence is transfer compactly open valued. The inverse is not true in general.
The definition of -space and the class
of better admissible mapping were introduced by Ding in [8]. Note that the class
of better admissible mapping includes many important classes of mappings, for example, the class
in [18],
in [19] and so on as proper subclasses. Now we introduce the following definition.
Definition 2.1.
An -space
is said to be an
-space if for each
, there exists a compact
-subspace
of
containing
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_IEq85_HTML.gif)
be a -convex space, let the notion of
-convex space was introduced by Ding in [4].
Lemma 2.2 ([8]).
Let be any index set. For each
, let
be an
-space,
and
. Then
is also an
-space.
Let be a topological space, and let
be any index set. For each
, let
be an
-space, and let
such that
is an
-space defined as in Lemma 2.2. Let
and for each
, let
be a set-valued mapping. For each
,
(1) is said to be a generalized
-mapping if
(a)for each and
,
, where
is the projection of
onto
and
;
(b) is transfer compactly open in
for each
;
(2) is said to be a generalized
-majorant of
at
if
is a generalized
-mapping and there exists an open neighborhood
of
in
such that
for all
;
(3) is said to be a generalized
-majorized if for each
with
, there exists a generalized
-majorant
of
at
, and for any
, the mapping
is transfer compactly open in
;
(4) is said to be a generalized
-majorized if for each
, there exists a generalized
-majorant
of
at
.
Then is said to be a family of generalized
-mappings (resp.,
-majorant mappings) if for each
is a generalized
-mapping (resp.,
-majorant mapping).
If for each , let
be a
-convex space, a family of
-mappings (resp.,
-majorant mappings) were introduced by Ding in [4]. Clearly, each family of generalized
-mappings must be a family of generalized
-majorant mappings. If
is a single-valued mapping and
is an
-subspace of
for each
, then condition
for each
implies that condition (a) in (1) holds. Indeed, if (a) is false, then there exist
,
, and
such that
and hence
for each
. It follows from
that
where
. It follows from
being an
-subspace of
that
which contradicts condition
for each
. Hence each
-mapping (resp.,
-majorant mapping) introduced by Deguire et al. (see [2, page 934]) must be a generalized
-mapping (resp.,
-majorant mapping). The inverse is not true in general.
3. Maximal Elements
In order to obtain our main results, we need the following lemmas.
Lemma 3.1 ([3]).
Let and
be topological spaces, let
be a nonempty compact subset of
and let
be a set-valued mapping such that
for each
. Then the following conditions are equivalent:
(1) have the compactly local intersection property;
(2)for each , there exists an open subset
of
(which may be empty) such that
and
;
(3)there exists a set-valued mapping such that for each
is open or empty in
,
and
;
(4)for each , there exists
such that
and
;
(5) is transfer compactly open valued on
.
Lemma 3.2 ([8]).
Let be a topological space, and let
be an
-space,
and
such that
(i)for each and for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ2_HTML.gif)
(ii) is transfer compactly open valued;
(iii)there exists a nonempty set and for each
, there exists a compact
-subspace
of  
containing
such that
is empty or compact in
, where
denotes the complement of
.
Then there exists a point such that
.
Theorem 3.3.
Let be a topological space, let
be a nonempty compact subset of
, and let
be an
-space,
and
be a generalized
-mapping such that
(i)for each , there exists a compact
-subspace
of
containing
such that for each
.
Then there exists a point such that
.
Proof.
Suppose that for each
. Since
is a generalized
-mapping,
is transfer compactly open valued. By Lemma 3.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ3_HTML.gif)
Since is compact, there exists a finite set
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ4_HTML.gif)
By condition (i) and , there exists a compact
-subspace
of
containing
and
is compact in
, and hence we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ5_HTML.gif)
By using similar argument as in the proof of Lemma 3.2, we can show that there exists such that
. Condition (i) implies that
must be in
. This completes the proof.
Remark 3.4.
Theorem 3.3 generalizes in [4, Theorem  2.2] in the following several aspects: (a) from -convex space to
-space without linear structure; (b) from
-mappings to generalized
-mappings.
Theorem 3.5.
Let be a topological space, and let
be an
-space. Let
and
be a generalized
-majorized mapping such that
(i)there exists a paracompact subset of
such that
;
(ii)there exists a nonempty set and for each
, there exists a compact
-subspace
of
containing
such that the set
is empty or compact.
Then there exists a point such that
.
Proof.
Suppose that for each
. Since
is a generalized
-majorized, for each
, there exists an open neighborhood
of
in
and a generalized
-mapping
such that
(a) for each
,
(b)for each and
,
(c) is transfer compactly open in
,
(d)for any , the mapping
is transfer compactly open in
.
Since for each
, it follows from condition (i) that
is paracompact. By Dugundji in [20, Theorem  VIII.1.4], the open covering
has an open precise locally finite refinement
, and for each
since
is normal. For each
, define a mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ6_HTML.gif)
Then for each , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ7_HTML.gif)
Hence is transfer compactly open in
by (c).
Now define a mapping by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ8_HTML.gif)
We claim that is a generalized
-mapping and
for each
. Indeed, for any nonempty compact subset
of
and each
with
, we may take any fixed
. Since
is locally finite, there exists an open neighborhood
of
in
such that
is a finite set. If
, then
, and hence
for all
which implies that
for all
. It follows that for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ9_HTML.gif)
For any nonempty compact subset of
and each
, if
. Since
is open in
, it follows from (d) that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ10_HTML.gif)
This proves that is transfer compactly open valued in
.
On the other hand, for each and
, if
, then
. Since there exists
such that
and
, we have
, and hence
by (b). Hence we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ11_HTML.gif)
for each and
. This shows that
is a generalized
-mapping.
For each , if
, then there exists an
such that
and
. It follows from (a) that
. Hence we have
for each
. By condition (ii), there exists a nonempty set
and for each
, there exists a compact
-subspace
of
containing
such that the set
is empty or compact. Note that
for each
implies
for each
. Hence
and
is empty or compact. By Lemma 3.2, there exists a point
such that
, and hence
which contradicts the assumption that
for each
. Therefore, there exists
such that
.
Theorem 3.6.
Let be a topological space, let
be a nonempty compact subset of
and
be an
-space. Let
and
be a generalized
-majorized mapping such that
(i)there exists a paracompact subset of
such that
;
(ii)for each , there exists a compact
-subspace
of
containing
such that for each
.
Then there exists such that
.
Proof.
Suppose that for each
. By using similar argument as in the proof of Theorem 3.5, we can show that there exists a generalized
-mapping
such that
for each
. It follows from condition (ii) that for each
. By Theorem 3.3, there exists
such that
, and hence
which contradicts the assumption that
for each
. Therefore, there exists
such that
. Condition (ii) implies
. This completes the proof.
Remark 3.7.
Theorem 3.5 generalizes [4, Theorem  2.3] in several aspects: Section 1(1) from -convex space to
-space without linear structure; Section 1(2) from a
-majorized mapping to a generalized
-majorized mapping; Section 1(3) condition (ii) of Theorem 3.5 is weaker than condition (ii) of [4, Theorem  2.3]. If
is compact, condition (i) is satisfied trivially. If
is a compact
-space, then by letting
for all
, conditions (i) and (ii) are satisfied automatically. Theorem 3.6 unifies and generalizes Shen's [14, Theorem  2.1, Corollary  2.2 and Theorem  2.3] in the following ways: Section 2(1) from
-convex space to
-space without linear structure; Section 2(2) from
-majorized correspondences to generalized
-majorized mapping; Section 2(3) condition (ii) of Theorem 3.6 is weaker than that in the corresponding results of Shen in [14]. Theorem 3.6 also generalizes in [4, Theorem  2.4], Ding in [15, Theorem  5.3], and Ding and Yuan in [16, Theorem  2.3] in several aspects.
Corollary 3.8.
Let be a compact topological space, and let
be an
-space. Let
and
be a generalized
-majorized mapping. Then there exists a point
such that
.
Proof.
The conclusion of Corollary 3.8 follows from Theorem 3.6 with .
Corollary 3.9.
Let be a topological space, and let
be an
-space. Let
be a compact mapping and
be a generalized
-majorized mapping. Then there exists a point
such that
.
Proof.
Since is a compact mapping, there exists a compact subset
of
such that
. The mapping
be the restriction of
to
. It is easy to see that
is also generalized
-majorized. By Corollary 3.8, there exists
such that
.
Remark 3.10.
Corollary 3.8 generalizes Deguire et al. [2, Theorem  1] in the following ways: (1.1) from a convex subset of Hausdorff topological vector space to an -space without linear structure; (1.2) from a
-majorized mapping to a generalized
-majorized mapping. Corollary 3.8 also generalizes [4, Corollary  2.3] from
-convex space to
-space and from a
-majorized mapping to a generalized
-majorized mapping. Corollary 3.9 generalizes [2, Theorem  2] and [4, Corollary  2.4] in several aspects.
Theorem 3.11.
Let be a topological space, and let
be any index set. For each
, let
be an
-space, and let
such that
is an
-space defined as in Lemma 2.2. Let
such that for each
,
(i)let be a generalized
-majorized mapping;
(ii);
(iii)there exists a paracompact subset of
such that
;
(iv)there exists a nonempty set and for each
, there exists a compact
-subspace
of
containing
such that the set
is empty or compact, where
.
Then there exists such that
for each
.
Proof.
For each ,
. Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ12_HTML.gif)
Then for each ,
if and only if
. Let
with
, then there exists
such that
. By condition (ii), there exists
such that
. Since
is generalized
-majorized, there exist an open neighborhood
of
in
and a generalized
-majorant
of
at
such that
(a) for all
,
(b)for each and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ13_HTML.gif)
(c) is transfer compactly open in
,
(d)for each , the mapping
is transfer compactly open in
.
Without loss of generality, we can assume that . Hence,
for each
. Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ14_HTML.gif)
We claim that is a generalized
-majorant of
at
. Indeed, we have
()for each ,
,
()for each and
, if
, then
. It is easy to see that
, so that
, i.e.,
and hence
by (b). It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ15_HTML.gif)
()for each , we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ16_HTML.gif)
is transfer compactly open in by (c).
Hence is a generalized
-majorant of
at
.
For each and
, by (3.15), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ17_HTML.gif)
It follows from (d) that is transfer compactly open in
.
Hence is generalized
-majorized. By condition (iii), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ18_HTML.gif)
By condition (iv), there exists a nonempty set and for each
, there exists a compact
-subspace
of
containing
. By the definition of
, for each
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ19_HTML.gif)
It follows from condition (iv) that is empty or compact and hence all conditions of Theorem 3.5 are satisfied. By Theorem 3.5, there exists
such that
which implies
, that is,
for each
.
Theorem 3.12.
Let be a topological space, and let
be any index set. For each
, let
be an
-space, and let
. Let
be a compact mapping such that for each
,
(i)let be a generalized
-majorized mapping;
(ii).
Then there exists such that
for each
.
Proof.
Since for each , let
be an
-space, then for each
, there exists a compact
-subspace
of
containing
. Let
and
, then
is a compact
-subspace of
for each
,
is a compact
-subspace of
containing
. Hence
is also an
-space.
For each ,
. Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ20_HTML.gif)
Then for each ,
if and only if
. By using similar argument as in the proof of Theorem 3.11, we can show that
is a generalized
-majorized mapping. By Corollary 3.9, there exists
such that
, and so
. Hence, we have
for each
.
Theorem 3.13.
Let be a topological space, let
be a nonempty compact subset of
and let
be any index set. For each
, let
be an
-space, and let
such that
is an
-space defined as in Lemma 2.2. Let
such that for each
,
be a generalized
-mapping such that
(i)for each and
, there exists a compact
-subspace
of
containing
and for each
, there exists
satisfying
.
Then there exists such that
for each
.
Proof.
Suppose that the conclusion is not true, then for each , there exists
such that
. Since
is a generalized
-mapping,
is transfer compactly open valued. By Lemma 3.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ21_HTML.gif)
Since is compact, there exists a finite set
such that for each
, there exists
with
. It follows that for each
, there exists a
such that
. We may take any fixed
. For each
, let
. By condition (i), for each
, there exists a compact
-subspace
of
containing
and for each
, there exists
satisfying
. Hence for each
, there exists
such that
. Let
, then
is a compact
-subspace of
and hence it is also a compact
-space. Let
, then
is compact in
. Define
by
. For each
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ22_HTML.gif)
Since is transfer compactly open valued in
for each
and
, so that we claim that
is transfer open valued in
. Noting that each
is a generalized
-mapping, for each
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F905605/MediaObjects/13660_2009_Article_2030_Equ23_HTML.gif)
where .
Hence for each ,
is a generalized
-mapping and hence it is also a generalized
-majorized mapping. All conditions of Corollary 3.8 are satisfied. By Corollary 3.8, there exists
such that
for each
, so we have
which contradicts the fact that for each
there exists
such that
. Therefore, there exists
such that
for each
.
Remark 3.14.
Theorem 3.11 generalizes [4, Theorem  2.5] in several aspects. Theorem 3.12 improves [2, Theorem  3] from convex subsets of topological vector spaces to -spaces without linear structure and from a family of
-majorized mappings to the family of generalized
-majorized mappings. Theorem 3.13 generalizes [4, Theorem  2.6] in several aspects: (1.1) from
-convex spaces to
-spaces without linear structure; (1.2) from a
-mapping to a generalized
-mapping; (1.3) condition (i) of Theorem 3.13 is weaker than condition (i) of [4, Theorem  2.6]. Theorem 3.13 improves and generalizes [2, Theorem  7] in the following ways: (2.1) from nonempty convex subsets of Hausdorff topological vector spaces to
-space without linear structure; (2.2) from the family of
-majorized mappings to the family of generalized
-majorized mappings.
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Acknowledgment
This work is supported by a Grant of the Natural Science Development Foundation of CUIT of China (no. CSRF200709).
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He, RH., Zhang, Y. Some Maximal Elements' Theorems in -Spaces.
J Inequal Appl 2009, 905605 (2009). https://doi.org/10.1155/2009/905605
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DOI: https://doi.org/10.1155/2009/905605