Common Fixed Points for a Countable Family of Set-Valued Mappings with Quasi-Contractive Conditions on Metrically Convex Spaces ()
1. Introduction
There have appeared many fixed point theorems for a single-valued self map of a closed subset of a Banach space. However, in many applications, the mapping under considerations is not a self-mapping on a closed subset. In 1976, Assad [1] gave sufficient condition for such single valued mapping to obtain a fixed point by proving a fixed point theorem for Kannan mappings on a Banach space and putting certain boundary conditions on the mapping. Similar results for multi-valued mappings were respectively given by Assad [2] and Assad and Kirk [3]. On the other hand, many authors discussed common fixed point problems [4-7] for finite single or multi-valued mappings on a complete 2-metric convex space or a complete cone metric space respectively. And some authors also discussed common fixed point problems [8-13] for a countable family of self-single-valued mappings with contractive or quasi-contractive conditions on a metric space or a metrically convex space respectively. These results improved and generalized many previous works.
In this paper, we will discuss the existent problems of common fixed points for a countable family of surjective set-valued mappings, which satisfy certain quasi-contractive condition, defined on a complete metrically convex space and obtain some important theorems. The main results in this paper further generalize and improve many common fixed point theorems for single valued or multi-valued mappings with quasi-contractive type conditions.
Through this paper,
(or
) is a metric space. Let
denote the families of all bounded closed subset of
.
Let
, the distance between
and
.
Definition 1.1. ([8-10]) A metric space
is said to be metrically convex, if any
with
, there exists
such that
,
and
.
Lemma 1.1. ([3,8]) If
is a nonempty closed subset of a complete metrically convex space
, then for any
and
, there exists
which satisfies
.
Lemma 1.2. ([13]) If
is a complete metric space and
, then
is continuous on
. Moreover, we have :
1)
;
2)
if and only if
,
;
3) for any
,
.
2. Main Results
Theorem 2.1. Let
be a nonempty closed subset of a complete metrically convex space
with
,
a countable family of surjective set-valued mappings with nonempty values such that for any
with
, any
,
(1)
where
and
is a constant number.
Furthermore, if
for all
, and for each
and
and any
, there exists
such that
, then
has a unique common fixed point in
.
Proof Take
. We will construct two sequences
and
in the following manner. Since
is on-to, there exists
such that
. If
, then put
; if
, then by Lemma 1.1 there exits
such that
. For
, since
is on-to, there exists
such that
. If
, then put
; if
, then by Lemma 1.1 there exists
such that
. Continuing this way, we obtain
and
:
1)
;
2) if
, then put
;
3) if
, then by Lemma 1.1 there exists
such that ![](https://www.scirp.org/html/3-5300618x\463ce339-b085-4d1f-8aa0-718bffba72ec.jpg)
4)
for all ![](https://www.scirp.org/html/3-5300618x\f8e75010-c023-44ba-950e-df7f16e00082.jpg)
Let
and
. If there exists
such that
, then
In fact, By 3) and the definition of
, we have that
,
,
. If
, then
. On the other hand, since
and
, hence
which is a contradiction. If
, then
and
, hence
, so
, which is another contradiction.
By the definitions and properties of
and
, we can estimate
into three cases:
Case I.
. In this case,
,
,
and
. And we have
![](https://www.scirp.org/html/3-5300618x\79d8cf07-d754-4ae5-85be-095484d54448.jpg)
where
![](https://www.scirp.org/html/3-5300618x\eb5dfd10-30d7-4295-ace4-acf53ac8d303.jpg)
If
then
![](https://www.scirp.org/html/3-5300618x\116fc80a-3c08-4b04-afc0-937ddb305818.jpg)
hence
![](https://www.scirp.org/html/3-5300618x\8f998fa1-2e78-4411-bc15-6e8284cdeebc.jpg)
If
, then
![](https://www.scirp.org/html/3-5300618x\6b7885b2-bbd6-40d3-ab78-c79529dd248f.jpg)
hence
![](https://www.scirp.org/html/3-5300618x\67965991-2809-4f99-85c6-9499ac88e255.jpg)
Therefore, in any situation, we have
![](https://www.scirp.org/html/3-5300618x\0408fec7-7062-4b2c-b439-527eb080e81b.jpg)
Case II.
and
. In this case,
,
and
and
. And we have
![](https://www.scirp.org/html/3-5300618x\1740cf3f-2dd7-4db1-b29f-f01606109412.jpg)
where
![](https://www.scirp.org/html/3-5300618x\6899fec1-ff5d-4615-9f95-cb4d0b681bf7.jpg)
If
then
![](https://www.scirp.org/html/3-5300618x\6a8b16dd-3fb9-44b3-a3fe-6434c530e875.jpg)
hence
![](https://www.scirp.org/html/3-5300618x\6735759b-c660-40a6-90f1-39f5b7f0ace5.jpg)
If
, then
![](https://www.scirp.org/html/3-5300618x\0c7e6a91-be38-4230-9447-b498833923e1.jpg)
hence
![](https://www.scirp.org/html/3-5300618x\aeac6cde-3e10-4dd8-b1ba-50697dabef8c.jpg)
Therefore, in any situation, we have
![](https://www.scirp.org/html/3-5300618x\91e1682d-a7c0-4312-847d-3fe94c074e69.jpg)
But
, hence we obtain
![](https://www.scirp.org/html/3-5300618x\b1da09a1-d281-423b-aae4-a136f7baacbd.jpg)
Case III.
and
. In this case,
by the property of
and
, and
,
,
and
. And we have
![](https://www.scirp.org/html/3-5300618x\a2a3f3de-e8d8-42b1-b86f-321dddd3df1a.jpg)
where
![](https://www.scirp.org/html/3-5300618x\00719c5f-553b-480b-b346-98eb70bd2dbd.jpg)
Here, we give two basic properties:
1) since
so
and hence ![](https://www.scirp.org/html/3-5300618x\baaac9c4-731b-40f7-8d21-378d52dd4291.jpg)
2) since
hence ![](https://www.scirp.org/html/3-5300618x\5d257f5d-f78e-4e09-86e2-c304ea61e750.jpg)
If
then
![](https://www.scirp.org/html/3-5300618x\716ae024-acd3-425e-be7b-db4f3b4353a5.jpg)
hence by 2),
![](https://www.scirp.org/html/3-5300618x\a8da6cef-97f3-42b2-92c6-6f1ea0a69afa.jpg)
So by Case II, we obtain
![](https://www.scirp.org/html/3-5300618x\496ccd69-47c6-4c1c-8a72-e26fe4ef7ff5.jpg)
If
, then
![](https://www.scirp.org/html/3-5300618x\2e33c49b-6c00-4bb8-b098-0687c546876d.jpg)
hence by 2),
![](https://www.scirp.org/html/3-5300618x\8a63a6a8-7061-4137-a3ed-96523d70cd46.jpg)
So by Case II again, we obtain
![](https://www.scirp.org/html/3-5300618x\c779ff35-71fb-4e80-8f1d-131887426816.jpg)
Hence in any situation, we have
![](https://www.scirp.org/html/3-5300618x\c1b6ee9a-74c2-411c-a292-22514957f928.jpg)
Therefore, from Case I, Case II and Case III, we obtain
![](https://www.scirp.org/html/3-5300618x\297e82c3-d445-49e1-96b4-31527606043c.jpg)
Let
, then
since
, hence we have
![](https://www.scirp.org/html/3-5300618x\3710688b-a810-4df2-bf78-304b48c30d55.jpg)
so
![](https://www.scirp.org/html/3-5300618x\7c563a50-b4bd-4214-98f7-1dc416dd01d3.jpg)
Let
, then for
,
as
. Hence
is a Cauchy sequence. Since
is complete,
has a limit
. But
is closed and
for all
, hence
.
By the property of
and
, we can see that there exists an infinite subsequence
of
such that
, hence
and ![](https://www.scirp.org/html/3-5300618x\7b0de37c-46f8-4a32-87b5-cf7e31000467.jpg)
Next, we will prove that
is a common fixed point of
. Fix any
, for each fixed
, there exists
such that
. Take an enough large
such that
and
. By Lemma 1.2 3) and (1), we have
![](https://www.scirp.org/html/3-5300618x\a18edeaf-3db3-41d1-bb2b-4bfe55885638.jpg)
and
![](https://www.scirp.org/html/3-5300618x\b1422400-6fa3-40b3-867b-49d3259eaa0d.jpg)
where
![](https://www.scirp.org/html/3-5300618x\d2182c29-59e4-45a5-80f1-224107461769.jpg)
If
then
![](https://www.scirp.org/html/3-5300618x\47be54d2-da3d-4fb2-9eea-51f20f2764ca.jpg)
Let
, then
since
, hence
. So
since
, therefore
by Lemma 1.2 1).
If
, then
![](https://www.scirp.org/html/3-5300618x\f513df5c-ed9f-4c07-8cfd-a157e80fdae4.jpg)
Let
, then
since
, hence similarly, ![](https://www.scirp.org/html/3-5300618x\f48e243c-6d06-48a1-87f1-72cadbd115e8.jpg)
So in any situation,
for all
, so
is a common fixed point of
.
If
and
are all common fixed points of
, then we will have
![](https://www.scirp.org/html/3-5300618x\35129875-38f3-4b7b-86e2-3863b2b24a9b.jpg)
where
![](https://www.scirp.org/html/3-5300618x\a5cbf139-269d-41b4-95e1-3c9a85b60b79.jpg)
If
, then
, hence
;
If
, then
hence
since
, so
.
Hence in any situation,
. So
is the unique common fixed points of ![](https://www.scirp.org/html/3-5300618x\fa3734ba-698a-4de1-b45d-efc7debf8643.jpg)
If the mappings in Theorem 2.1 are all single-valued, then Theorem 2.1 becomes the next form.
Theorem 2.2. Let
be a nonempty closed subset of a complete metrically convex space
with
,
a countable family of surjective single-valued mappings such that for any
with
, any
,
(2)
where
and
is a constant number.
Furthermore, if
for all
, and for each
and
, there exists
such that
, then
has a unique common fixed point in
.
From Theorem 2.2, we can obtain the following more generalized common fixed point theorem.
Theorem 2.3. Let
be a nonempty closed subset of a complete metrically convex space
with
,
a family of subjective single-valued mappings,
a family of positive integral numbers such that for any
,
,
(3)
where
and
is a constant number. Furthermore, if 1)
for all
, 2) for each
and
there exists
such that
, 3) for each
with
,
. Then
has a unique common fixed point in
.
Proof Fix
, and let
, then
satisfies all of the conditions of Theorem 2.2, hence
has an unique common fixed point
in
. Now, we will prove that
is also unique common fixed point of
. In fact, for any fixed
,
. This means that
is a fixed point of
. For any
with
, there exists
such that
by 2), and by (3) we have that
![](https://www.scirp.org/html/3-5300618x\f25d2c75-de95-4234-82b8-da630c8dacf3.jpg)
where
![](https://www.scirp.org/html/3-5300618x\95d31417-23b7-41ea-b81b-b71972924429.jpg)
If
, then
, hence
;
If
, then
, hence ![](https://www.scirp.org/html/3-5300618x\26c74e12-1b55-4d1d-9365-fbece471cc0d.jpg)
Hence in any situation, we have that
is a fixed point of
for each
with
. So
is a common fixed point of
. By uniqueness of common fixed points of
, we have
for each
. Hence
is a common fixed point of
.
If
and
are all common fixed points of
, then they are also common fixed points of
, hence by the uniqueness of common fixed points of
, we obtain
. This means that for each
has a unique common fixed point
.
Now, we prove
for each
. In fact, for any
with
, since
and
, so
, hence
by 3). Therefore,
is a fixed point of
for each
i.e.,
is a common fixed point of
. But
has a unique common fixe point
, hence
for each
, and therefore
is a common fixed point of
. But ![](https://www.scirp.org/html/3-5300618x\0600e9f7-e760-4cbc-92a4-8f7019f1841d.jpg)
has a unique common fixed point
, hence
. Let
, then
is the common fixed point of
. The uniqueness of common fixed points of
is obvious.
Funding
This work was supported by the National Natural Science Foundation of China (No. 11361064).
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NOTES
*Corresponding author.