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 
4) 
for all 
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
where
If 
then
hence
If
, then
hence
Therefore, in any situation, we have
Case II. 
and
. In this case, 
, 
and 
and
. And we have
where
If 
then
hence
If
, then
hence
Therefore, in any situation, we have
But
, hence we obtain
Case III. 
and
. In this case, 
by the property of 
and
, and
, 
, 
and
. And we have
where
Here, we give two basic properties:
1) since 
so 
and hence 
2) since
hence 
If 
then
hence by 2),
So by Case II, we obtain
If
, then
hence by 2),
So by Case II again, we obtain
Hence in any situation, we have
Therefore, from Case I, Case II and Case III, we obtain
Let
, then 
since
, hence we have
so
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 
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
and
where
If 
then
Let
, then 
since
, hence
. So 
since
, therefore 
by Lemma 1.2 1).
If
, then
Let
, then 
since
, hence similarly, 
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
where
If
, then
, hence
;
If
, then
hence 
since
, so
.
Hence in any situation,
. So 
is the unique common fixed points of 
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
where
If
, then
, hence
;
If
, then
, hence 
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 
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.