- Research Article
- Open access
- Published:
Approximate Fixed Point Theorems for the Class of Almost
-
Mappings in Abstract Convex Uniform Spaces
Fixed Point Theory and Applications volume 2009, Article number: 791514 (2009)
Abstract
We use a concept of abstract convexity to define the almost -
property, al-
-
family, and almost
-spaces. We get some new approximate fixed point theorems and fixed point theorems in almost
-spaces. Our results extend some results of other authors.
1. Introduction and Preliminaries
In 1929, Knaster et al. [1] proved the well-known theorem for an
-simplex. Ky Fan's generalization of the
theorem to infinite dimensional topological vector spaces in 1961 [2] proved to be a very versatile tool in modern nonlinear analysis with many far-reaching applications.
Chang and Yen [3] undertook a systematic study of the property, and Chang et al. [4] generalized this property as well as the notion of a
family of [4] to the wider concepts of the
-
property and its related
-
family.
Among the many contributions in the study of the property and related topics, we mention the work by Amini et al. [5] where the classes of
and
-
mappings have been introduced in the framework of abstract convex spaces. The authors of [5] also define a concept of convexity that contains a number of other concepts of abstract convexities and obtain fixed point theorems for multifunctions verifying the
-
property on
-spaces that extend results of Ben-El-Mechaiekh et al. [6] and Horvath [7], motivated by the works of Ky Fan [2] and Browder [8]. We refer for the study of these notions to Ben-El-Mechaiekh et al. [9], and more recently, to Park [10], and Kim and Park [11].
In this paper, we use a concept of abstract convexity to define the almost -
property, the corresponding notion of almost
-
family as well as the concept of almost
-spaces.
Let and
be two sets, and let
be a set-valued mapping. We will use the following notations in the sequel;
(i)
(ii)
(iii)
(iv) and
(v)if is a nonempty subset of
, then
denotes the class of all nonempty finite subsets of
.
For the case where and
are two topological spaces, a set-valued map
is said to be closed if its graph
is closed.
is said to be compact if the image
of
under
is contained in a compact subset of
.
Definition 1.1.
An abstract convex space consists of a nonempty topological space
, and a family
of subsets of
such that
and
belong to
and
is closed under arbitrary intersection. This kind of abstract convexity was widely studied; see [5, 9, 12, 13].
Suppose that is a nonempty subset of an abstract convex space
. Then
(i)a natural definition of -convex hull of
is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F791514/MediaObjects/13663_2009_Article_1177_Equ1_HTML.gif)
(ii)we say that is
-convex if for each
,
.
Remark 1.2.
It is clear that if , then
is
-convex. That is, each member of
is
-convex.
Definition 1.3.
We list some properties of a uniform space. A uniformity [14] for a set is a nonempty family
of subsets of
such that
(i)each member of contains the diagonal
where the diagonal
denotes the set of all pairs
for
in
;
(ii)if , then
;
(iii)if , then
for some
;
(iv)if , then
;
(v)if and
, then
.
The pair is called a uniform space. Every member
in
is called an entourage. An entourage is said to be symmetric if
whenever
.
Definition 1.4.
If is an abstract convex space with a uniformity
, then we say that
is an abstract convex uniform space.
Definition 1.5.
Let be a nonempty subset of an abstract convex uniform space
which has a uniformity
, and
has a symmetric basis
. Then
is called almost
-convex if, for any
and for any
, there exists a mapping
such that
for all
and
. Moreover, we call the mapping
a
-convex-inducing mapping.
Remark 1.6.
It is clear that every -convex set must be almost
-convex, but the converse is not true. And in general, the
-convex-inducing mapping is not unique. If
and
, then
can be regarded as
. If
, then
can be regarded as
.
Recently, Amini et al. [5] introduced the class of multifunctions with the property in abstract convex spaces.
Definition 1.7 (see [5]).
Let be a nonempty set,
an abstract convex space, and
a topological space. If
,
and
are three multifunctions satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F791514/MediaObjects/13663_2009_Article_1177_Equ2_HTML.gif)
then is called a
-
mapping with respect to
. If the multifunction
satisfies the requirement that for any
-
mapping
with respect to
, the family
has the finite intersection property where
denotes the closure of
, then
is said to have the
-
property with respect to
. We define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F791514/MediaObjects/13663_2009_Article_1177_Equ3_HTML.gif)
We extended the property to the almost
property, as follows.
Definition 1.8.
Let be a nonempty set, let
be an almost
-convex subset of an abstract convex uniform space
which has a uniformity
and
has a symmetric basis
, and let
be a topological space. If
,
and
are three multifunctions satisfying for each
, each
, and each
, there exists a
-convex-inducing mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F791514/MediaObjects/13663_2009_Article_1177_Equ4_HTML.gif)
then is called an almost
-
mapping with respect to
. If the multifunction
satisfies the requirement that for any almost
-
mapping
with respect to
, the family
has the finite intersection property, then
is said to have the almost
-
property with respect to
. We define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F791514/MediaObjects/13663_2009_Article_1177_Equ5_HTML.gif)
From the above definitions, we have the following proposition of the family.
Proposition 1.9.
Let be a nonempty set, let
be an almost
-convex subset of an abstract convex uniform space
, let
and
be two topological spaces, and let
be a multifunction. If
and if
is continuous, then
The -mappings and the
-spaces, in an abstract convex space setting, were also introduced by Amini et al. [5].
Definition 1.10 (see [5]).
Let be an abstract convex space, and
a topological space.
map
is called a
-mapping if there exists a multifunction
such that
(i)for each ,
implies
, and
(ii).
The mapping is called a companion mapping of
.
Furthermore, if the abstract convex space which has a uniformity
and
has a symmetric basis
, then
is called a
-space if for each entourage
, there exists a
-mapping
such that
.
Remark 1.11.
(i)If is a
-mapping, then for each nonempty subset
of
,
is also a
-mapping.
(ii)It is easy to see that if and
, then
is also a
-space.
In order to establish the main result of this paper for the multifunctions with the almost property, we need the following definitions concerning the almost
-mappings and the almost
-spaces.
Definition 1.12.
Let be an almost
-convex subset of an abstract convex uniform space
which has a uniformity
and
has a symmetric base family
, and
a topological space. A map
is called an almost
-mapping if there exists a multifunction
such that
(i)for each ,
and
, there exists a
-convex-inducing
such that
, and
(ii)
The mapping is called an almost companion mapping of
.
Furthermore, is called an almost
-space, if, for each entourage
, there exists an almost
-mapping
such that
.
Definition 1.13.
Let be an almost
-space, and let
. We say that
has the approximate fixed point property if, for each
, there exists
such that
.
2. Main Results
Using the above introduced concepts and definitions, we now state our main theorem.
Theorem 2.1.
Let be an almost
-space, and let
be a surjective single-valued function. If
is compact, then
has the approximate fixed point property.
Proof.
Let be a symmetric basis of the uniform structure, and let
. Take
such that
. Then, by the definition of the almost
-space, there exists an almsot
-mapping
such that
. Since
is an almsot
-mapping, there exists an almost companion mapping
such that
.
Let . Then
is compact, since
is compact. Hence there exists
such that
. Since
is surjective, there exists a finite subset
of
such that
.
Now, we define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F791514/MediaObjects/13663_2009_Article_1177_Equ6_HTML.gif)
By the definition of , we obtain that
is not an almost
mapping with respect to
. Hence, there exist
and
such that for any
-convex-inducing
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F791514/MediaObjects/13663_2009_Article_1177_Equ7_HTML.gif)
So, for any -convex-inducing
, there exist
and
such that
. Consequently,
, and so
for all
. Since
is an almost
-mapping, there exists a
-convex-inducing
such that
. So
and
. Thus, there exists
such that
. Since
is an almost
-space, we have
, and so
, that is,
.
Therefore, . The proof is finished.
Remark 2.2.
In the case, if is a
-space and
, then the above theorem reduces to Amini et al. [5, Theorem 2.5]
From Theorem 2.1 above, we obtain immediately the following fixed point theorem.
Theorem 2.3.
Suppose that all of the assumptions of Theorem 2.1 hold. If is closed, then
has a fixed point in
.
Proof.
By Theorem 2.1, for each , there exist
such that
. Since
is compact, without loss of generality, we may assume that
converges to some
in
; then
also converges to
since
is a Hausdorff uniform space and
for each
. By the closedness of
, we have that
.
Corollary 2.4.
Let be an almost
-space, and let
be a surjective single-valued function. Suppose
such that
is totally bounded. Then
has the approximate fixed point property.
Corollary 2.5.
Suppose that all of the assumptions of the above Corollary 2.5 hold. If is closed, then
has a fixed point in
.
In case is an almost convex subset of Hausdorff topological vector spaces and for each
, we have
(i), and
(ii).
This allows us to state the following results.
Theorem 2.6.
Let be a Hausdorff locally convex space, let
be an almost convex subset of
, and let
be a surjective function. Assume that
is compact and closed, then
has a fixed point in
.
Proof.
Let be the family of all convex subsets of
, and let
be a local basis of
such that each
is symmetric and convex for each
. For each
, we set
. Noting that
. Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F791514/MediaObjects/13663_2009_Article_1177_Equ8_HTML.gif)
Then is a basis of a uniformity of
. For each
,
, we define the two set-valued mappings
by
for each
. Then we have
-
(i)
for each
,
, and
(ii).
So, is an almost companion mapping of
. This implies that
is an almost
-mapping such that
. Therefore,
is an almost
-space.
All conditions of Theorems 2.1 and 2.3 are therefore fulfilled; the result follows from an argument similar to those in the proofs of Theorems 2.1 and 2.3.
Theorem 2.7.
Let be a topological vector space, let
be an almost convex subset of
, and let
be a surjective function. Suppose that
is compact, then for any symmetric convex neighborhood
of
in
, there is
such that
.
Proof.
Let be the family of all convex subsets of
, and let
be a new local basis of
. We will use
to construct a weaker topology on
such that
becomes a new topological vector space. For each
, we set
. Noting that
. Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F791514/MediaObjects/13663_2009_Article_1177_Equ9_HTML.gif)
Then is a basis of a uniformity of
. In vein of the reasonings similar to those of Theorems 2.1 and 2.6, we complete the proof.
References
Knaster B, Kuratowski C, Mazurkiewicz S: Ein Beweis des Fixpunksatzes fur -dimensionale Simplexe. Fundamenta Mathematicae 1929, 14: 132–137.
Fan K: A generalization of Tychonoff's fixed point theorem. Mathematische Annalen 1961, 142: 305–310. 10.1007/BF01353421
Chang T-H, Yen C-L: KKM property and fixed point theorems. Journal of Mathematical Analysis and Applications 1996,203(1):224–235. 10.1006/jmaa.1996.0376
Chang T-H, Huang Y-Y, Jeng J-C: Fixed-point theorems for multifunctions in S-KKM class. Nonlinear Analysis: Theory, Methods & Applications 2001,44(8):1007–1017. 10.1016/S0362-546X(99)00318-1
Amini A, Fakhar M, Zafarani J: Fixed point theorems for the class S-KKM mappings in abstract convex spaces. Nonlinear Analysis: Theory, Methods & Applications 2007,66(1):14–21. 10.1016/j.na.2005.11.005
Ben-El-Mechaiekh H, Deguire P, Granas A: Points fixes et cocidences pour les fonctions multivoques (applications de type et
). II. Comptes Rendus de l'Académie des Sciences 1982,295(5):381–384.
Horvath CD: Contractibility and generalized convexity. Journal of Mathematical Analysis and Applications 1991,156(2):341–357. 10.1016/0022-247X(91)90402-L
Browder FE: The fixed point theory of multi-valued mappings in topological vector spaces. Mathematische Annalen 1968, 177: 283–301. 10.1007/BF01350721
Ben-El-Mechaiekh H, Chebbi S, Florenzano M, Llinares J-V: Abstract convexity and fixed points. Journal of Mathematical Analysis and Applications 1998,222(1):138–150. 10.1006/jmaa.1998.5918
Park S: Fixed points of better admissible maps on generalized convex spaces. Journal of the Korean Mathematical Society 2000,37(6):885–899.
Kim J-H, Park S: Comments on some fixed point theorems in hyperconvex metric spaces. Journal of Mathematical Analysis and Applications 2004,291(1):154–164. 10.1016/j.jmaa.2003.10.022
Kay DC, Womble EW: Axiomatic convexity theory and relationships between the Caratheodory, Helly, and Radon numbers. Pacific Journal of Mathematics 1971, 38: 471–485.
Llinares J-V: Abstract convexity, some relations and applications. Optimization 2002,51(6):797–818. 10.1080/0233193021000015587
Kelly JL: General Topology. Van Nostrand, Princeton, NJ, USA; 1955.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Chang, TH., Chen, CM. & Huang, YH. Approximate Fixed Point Theorems for the Class of Almost -
Mappings in Abstract Convex Uniform Spaces.
Fixed Point Theory Appl 2009, 791514 (2009). https://doi.org/10.1155/2009/791514
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/791514