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Fixed Point Theorems for a Weaker Meir-Keeler Type
-Set Contraction in Metric Spaces
Fixed Point Theory and Applications volume 2009, Article number: 129124 (2009)
Abstract
We define a weaker Meir-Keeler type function and establish the fixed point theorems for a weaker Meir-Keeler type -set contraction in metric spaces.
1. Introduction and Preliminarie
In 1929, Knaster et al. [1] had proved the well-known theorem on
-simplex. Besides, in 1961, Fan [2] had generalized the
theorem to an infinite dimensional topological vector space. Later, Amini et al. [3] had introduced the class of
-type mappings on metric spaces and established some fixed point theorems for this class. In this paper, we define a weaker Meir-Keeler type function and establish the fixed point theorems for a weaker Meir-Keeler type
-set contraction in metric spaces.
Throughout this paper, by we denote the set of all real nonnegative numbers, while
is the set of all natural numbers. We digress briefly to list some notations and review some definitions. Let
and
be two Hausdorff topological spaces, and let
be a set-valued mapping. Then
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
. If
is a nonempty subset of
, then
denotes the class of all nonempty finite subsets of
. And, the following notations are used:
(i),
(ii),
(iii) and
(iv).
Let be a metric space,
and
. Let
, and let
.
Suppose that is a bounded subset of a metric space
. Then we define the following
(i), and
(ii) is said to be subadmissible [3], if for each
,
.
In 1996, Chang and Yen [4] introduced the family on the topological vector spaces and got results about fixed point theorems, coincidence theorems, and its applications on this family. Later, Amini et al. [3] introduced the following concept of the
property on a subadmissible subset of a metric space
.
Let be an nonempty subadmissible subset of a metric space
, and let
a topological space. If
are two set-valued mappings such that for any
,
, then
is called a generalized
mapping with respect to
. If the set-valued mapping
satisfies the requirement that for any generalized
mapping
with respest to
, the family
has finite intersection property, then
is said to have the
property. The class
is denoted to be the set
has the
property
.
Recall the notion of the Meir-Keeler type function. A function is said to be a Meir-Keeler type function (see [5]), if for each
, there exists
such that for
with
, we have
.
We now define a new weaker Meir-Keeler type function as follows.
Definition 1.1.
We call a weaker Meir-Keeler type function, if for each
, there exists
such that for
with
, and there exists
such that
.
A function is said to be upper semicontinuous, if for each
,
. Recall also that
is said to be a comparison function (see [6]) if it is increasing and
. As a consequence, we also have that for each
,
, and
,
is continuous at
. We generalize the comparison function to be the other form, as follows.
Definition 1.2.
We call a generalized comparison function, if
is upper semicontinuous with
and
for all
.
Proposition 1.3.
If is a generalized comparison function, then there exists a strictly increasing, continuous function
such that
, for all
.
Proof.
Let . Since
is an upper semicontinuous function, hence it attains its minimum in any closed bounded interval of
.
For each , we first define four sequences
, and
as follows:
(i),
(ii),
(iii),
(iv) for
, and
(v) for
.
And, we next let a function satisfy the following:
(1)
(2)if  , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F129124/MediaObjects/13663_2009_Article_1117_Equ1_HTML.gif)
(3)if   , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F129124/MediaObjects/13663_2009_Article_1117_Equ2_HTML.gif)
Then by the definition of the function , we are easy to conclude that
is strictly increasing, continuous. We complete the proof by showing that
for all
.
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F129124/MediaObjects/13663_2009_Article_1117_Equ3_HTML.gif)
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F129124/MediaObjects/13663_2009_Article_1117_Equ4_HTML.gif)
So for all
.
Since and
for all
, so
for all
Proposition 1.4.
If is a generalized comparison function, then there exists a strictly increasing, continuous function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F129124/MediaObjects/13663_2009_Article_1117_Equ5_HTML.gif)
Proof.
By Proposition 1.3, there exists a strictly increasing, continuous function such that
, for all
. So, we may assume that
, by letting
for all
.
Remark 1.5.
In the above case, the function is invertible. If for each
, we let
and
for all
, then we have that
; that is,
.
Proof.
We claim that , for
. Suppose that
for some positive real number
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F129124/MediaObjects/13663_2009_Article_1117_Equ6_HTML.gif)
which is a contradiction. So .
We now are going to give the axiomatic definition for the measure of noncompactness in a complete metric space.
Definition 1.6.
Let be a metric space, and let
the family of bounded subsets of
. A map
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F129124/MediaObjects/13663_2009_Article_1117_Equ7_HTML.gif)
is called a measure of noncompactness defined on if it satisfies the following properties:
(i) if and only if
is precompact, for each
,
(ii), for each
,
(iii), for each
,
(iv), for each
.
The above notion is a generalization of the set measure of noncompactness in metric spaces. The following -measure is a well-known measure of noncompactness.
Definition 1.7.
Let be a complete metric space, and let
the family of bounded subsets of
. For each
, we define the set measure of noncompactness
by:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F129124/MediaObjects/13663_2009_Article_1117_Equ8_HTML.gif)
Definition 1.8.
Let be a nonempty subset of a metric space
. If a mapping
with for each
,
and
are bounded, then
is called
(i)a -set contraction, if for each
,
, where
,
(ii)a weaker Meir-Keeler type -set contraction, if for each
,
, where
is a weaker Meir-Keeler type function,
(iii)a generalized comparison (comparison) type -set contraction, if for each
,
, where
is a generalized comparison (comparison) function.
Remark 1.9.
It is clear that if is a
-set contraction, then
is a weaker Meir-Keeler type
-set contraction, but the converse does not hold.
2. Main Results
Using the conception of the weaker Meir-Keeler type function, we establish the following important theorem.
Theorem 2.1.
Let be a nonempty bounded subadmissible subset of a metric space
. If
is a weaker Meir-Keeler type
-set contraction with for each
,
is nonicreasing, then
contains a precompact subadmissible subset
with
.
Proof.
Take , and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F129124/MediaObjects/13663_2009_Article_1117_Equ9_HTML.gif)
Then
(1) is a subadmissible subset of
, for each
;
(2), for each
.
Since is a weaker Meir-Keeler type
-set contraction, then
and
. Hence, we conclude that
.
Since is nonincreasing, it must converge to some
with
; that is,
. We now claim that
. On the contrary, assume that
.Then by the definition of the weaker Meir-Keeler type function, there exists
such that for each
with
, there exists
such that
.Since
, there exists
such that
, for all
. Thus, we conclude that
. So we get a contradiction. So
, and so
.
Let . Then
is a nonempty precompact subadmissible subset of
, and by (2), we have
.
Remark 2.2.
In the process of the proof of Theorem 2.1, we call the set a Meir-Keeler type precompact-inducing subadmissible subset of
.
Applying Proposition 1.3, 1.4, and Remark 1.5, we are easy to conclude the following corollary.
Corollary 2.3.
Let be a nonempty bounded subadmissible subset of a metric space
. If
is a generalized comparison (comparison) type
-set contraction, then
contains a precompact subadmissible subset
with
.
Proof.
The proof is similar to the proof of Theorem 2.1; we omit it.
Remark 2.4.
In the process of the proof of Corollary 2.3, we also call the set a generalized comparison type precompact-inducing subadmissible subset of
.
Corollary 2.5.
Let be a nonempty bounded subadmissible subset of a metric space
. If
is a
-set contraction, then
contains a precompact subadmissible subset
with
.
Following the concepts of the family (see [3]), we immediately have the following Lemma 2.6.
Lemma 2.6.
Let be a nonempty subadmissible subset of a metric space
, and let
a topological spaces. Then
whenever
, and
is a nonempty subadmissible subset of
.
We now concern a fixed point theorem for a weaker Meir-Keeler type -set contraction in a complete metric space, which needs not to be a compact map.
Theorem 2.7.
Let be a nonempty bounded subadmissible subset of a metric space
. If
is a weaker Meir-Keeler type
-set contraction with for each
,
is nonicreasing, and closed with
, then
has a fixed point in
.
Proof.
By the same process of Theorem 2.1, we get a weaker Meir-Keeler type precompact-inducing subadmissible subset of
. Since
and
for each
, we have
for each
. Since
as
, by the above Lemma 2.6, we have that
is a nonempty compact subset of
.
Since and
is a nonempty subadmissible subset of
, by Lemma 2.6,
.
We now claim that for each , there exists an
such that
. If the above statement is not true, then there exists
such that
, for all
. Let
. Then we now define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F129124/MediaObjects/13663_2009_Article_1117_Equ10_HTML.gif)
Then
(1) is compact, for each
, and
(2) is a generalized
mapping with respect to
.
We prove (2) by contradiction. Suppose is not a generalized
mapping with respect to
. Then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F129124/MediaObjects/13663_2009_Article_1117_Equ11_HTML.gif)
Choose and
such that
. From the definition of
, it follows that
, for each
. Since
,
, we have
, which implies that
. Therefore,
. This contradicts to
. Hence,
is a generalized
mapping with respect to
.
Since , the family
has the finite intersection property, and so we conclude that
. Choose
, then
for all
. But, since
and
, so there exists an
such that
. So, we have reached a contradiction.
Therefore, we have proved that for each , there exists an
such that
. Let
. Since
and
is compact, we may assume that
converges to some
, then
also converges to
. Since
is closed, we have
. This completes the proof.
Corollary 2.8.
Let be a nonempty bounded subadmissible subset of a metric space
. If
is a generalized composion type
-set contraction and closed with
, then
has a fixed point in
.
Corollary 2.9.
Let be a nonempty bounded subadmissible subset of a metric space
. If
is a
-set contraction and closed with
, then
has a fixed point in
.
The -spaces, in an abstract convex space setting, were introduced by Amini et al.[7]. 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. Let
be an abstract convex space, and let
a topological space. A map
is called a
-mapping if there exists a multifunction
such that
(i)for each ,
implies
;
(ii).
The mapping is called a companion mapping of
. Furthermore, if the abstract convex space
which has a uniformity
and
has an open symmetric base family
, then
is called a
-space if for each entourage
, there exists a
-mapping
such that
. Following the conceptions of the abstract convex space and the
-space, we are easy to know that a bounded metric space
is an important example of the abstract convex space, and if
and
, then
is also a
-space.
Applying Theorem of Amini et al. [7], we can deduce the following theorem in metric spaces.
Theorem 2.10.
Let be a nonempty subadmissible subset of a metric space
. If
is compact, then for each
, there exists
such that
.
Proof.
Consider the family of all subadmissible subsets of
and for each
,
, we set
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F129124/MediaObjects/13663_2009_Article_1117_Equ12_HTML.gif)
Then is a basis of a uniformity of
. For each
, we define two set-valued mappings
by
for each
. Then we have
(i)for each ,
;
(ii).
So, is a companion mapping of
. This implies that
is a
-mapping such that
. Therefore,
is a
-space.
Now we let be an identity mapping, all of the the conditions of Theorem
of Amini et al. [7] are fulfilled, and we can obtain the results.
Applying Theorems 2.1 and 2.10, we can conclude the following fixed point theorems.
Theorem 2.11.
Let be a nonempty bounded subadmissible subset of a metric space
. If
is a weaker Meir-Keeler type
-set contraction with for each
is noincreasing, and closed with
, then
has a fixed point in
.
Theorem 2.12.
Let be a nonempty bounded subadmissible subset of a metric space
. If
is a generalized comparison (comparison) type
-set contraction and closed with
, then
has a fixed point in
.
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Chen, CM., Chang, TH. Fixed Point Theorems for a Weaker Meir-Keeler Type -Set Contraction in Metric Spaces.
Fixed Point Theory Appl 2009, 129124 (2009). https://doi.org/10.1155/2009/129124
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DOI: https://doi.org/10.1155/2009/129124