An Improvement of a Known Unique Common Fixed Point Result for Four Mappings on 2-Metric Spaces ()
1. Introduction
The second author has obtained an unique common fixed point theorem for four mappings with
-contractive condition [1,2] on 2-metric spaces in [1], where
is a continuous and non-decreasing real function on
satisfying that
for all
. The result generalizes and improves many corresponding results.
Here, we introduce a new class
of real functions defined on
, and reprove the well known unique common fixed point theorem for four mappings with
-contractive condition replaced by
-contractive condition on 2-metric spaces. The method used in this paper is very different from that in [1].
At first, we give well known definitions and results.
Definition 1.1. ([3,4]) A 2-metric space
consists of a nonempty set
and a function
![](https://www.scirp.org/html/1-7401413\f5f5cc78-b282-4473-b788-ae15de65ef5d.jpg)
such that
1) for distant elements
, there exists an
such that
;
2)
if and only if at least two elements in
are equal;
3)
, where
is any permutation of
;
4)
for all
.
Definition 1.2. ([3,4]) A sequence
in 2-metric space
is said to be cauchy sequence, if for each
there exists a positive integer
such that
for all
and
.
Definition 1.3. ([5,6]) A sequence
is said to be convergent to
, if for each
,
.
And write
and call
the limit of
.
Definition 1.4. ([5,6]) A 2-metric space
is said to be complete, if every cauchy sequence in
is convergent.
Definition 1.5. ([7,8]) Let
and
be two selfmappings on a set
. If
for some
, then
is called a coincidence point of
and
, and
is called a point of coincidence of
and
.
Definition 1.6. ([9]) Two mappings
are said to be weakly compatible if, for every
, holds
whenever ![](https://www.scirp.org/html/1-7401413\06d6001e-1717-4353-96ae-7c0b07356024.jpg)
The following three lemmas are known results.
Lemma 1.7. ([3-6]) Let
be a 2-metric space and
a sequence. If there exists
such that
![](https://www.scirp.org/html/1-7401413\4d9ef004-1125-4b86-9bea-061be9a4d246.jpg)
for all
and
, then
for all
, and
is a cauchy sequence.
Lemma 1.8. ([3-6]) If
is a 2-metric space and sequence
, then
![](https://www.scirp.org/html/1-7401413\20ce24fe-94e9-4f16-af6d-8effc87dc5d0.jpg)
for each
.
Lemma 1.9. ([7,8]) Let
be weakly compatible. If
and
have a unique point of coincidence
, then
is the unique common fixed point of
and
.
2. Main Results
Denote by
the set of functions
satisfying the following:
(
1)
is continuous; (
2)
for all
.
Denote by
the set of functions
![](https://www.scirp.org/html/1-7401413\2f0098fe-7367-47f7-8d29-a7031c156380.jpg)
satisfying the following:
(
1)
is continuous and non-decreasing; (
2)
for all
.
Obviously,
is stronger than
.
Example 2.1. Define
as follow:
![](https://www.scirp.org/html/1-7401413\9dcbdb0a-b422-40fc-9ae4-17418ea149cb.jpg)
Obviously,
, but since
, so
.
The following is the main conclusion in this paper.
Theorem 2.2. Let
be a 2-metric space,
![](https://www.scirp.org/html/1-7401413\f4054797-50cd-46f1-9970-27aa1b4848a8.jpg)
four mappings satisfying that
and
.
Suppose that for each
,
(1)
where
and
. If one of
![](https://www.scirp.org/html/1-7401413\1a654353-ce0f-4e81-abbe-cf294d9213f9.jpg)
and
is complete, then
and
,
and
have an unique point of coincidence in
. Further,
and
are weakly compatible respectively, then
have an unique common fixed point in
.
Proof Take any element
, then in view of the conditions
and
, we can construct two sequences
and
as follows:
![](https://www.scirp.org/html/1-7401413\67bac1cb-9e88-444d-a60c-df964ee95788.jpg)
For any
,
(2)
If
![](https://www.scirp.org/html/1-7401413\66fe49e5-d549-4721-941c-197b1d7e2cdb.jpg)
for some
, then
, hence we have that
![](https://www.scirp.org/html/1-7401413\0867f39c-dc8f-425c-aa2c-5dc0740d1d95.jpg)
Hence we can assume now that
![](https://www.scirp.org/html/1-7401413\2dde2f62-0441-40b5-9364-d94b72b88e0b.jpg)
for all
.
If
![](https://www.scirp.org/html/1-7401413\2a61fd27-98eb-4108-9452-5f88588b402f.jpg)
for some
, then (2) becomes that
![](https://www.scirp.org/html/1-7401413\67a8e95a-af39-46d2-8a67-970a3ee2e1b8.jpg)
which is a contradiction since
. Hence we have that
![](https://www.scirp.org/html/1-7401413\ea319172-e6c8-40e0-8bd4-6c50391e51d9.jpg)
for all
.
If
for some
, then from (2),
(3)
If
for some
, then from (2),
(4)
If
, then
![](https://www.scirp.org/html/1-7401413\686f6391-3783-412b-8c58-16e0e93510d0.jpg)
which is a contradiction since
. hence
.
So (4) becomes that
(5)
Hence we obtain that
(6)
By (3) and (6), we obtain that
(7)
Similarly, we can obtain that for each ![](https://www.scirp.org/html/1-7401413\0bbb7d78-b3bb-4f84-b495-62c368c264a2.jpg)
. (8)
Combining (7) and (8), we have that
. (9)
Hence
is Cauchy sequence by Lemma 1.7.
Suppose that
is complete, then there exists
and
such that
![](https://www.scirp.org/html/1-7401413\c0505c2c-c020-4b38-813c-daa973ec9e5c.jpg)
(If
is complete, then there exists ![](https://www.scirp.org/html/1-7401413\f06dbc31-3bab-4a81-8a0a-b4880183eee3.jpg)
,hence the conclusions remains the same).
Since
![](https://www.scirp.org/html/1-7401413\0bc73ae3-3f73-41c4-86d9-86c0b076f078.jpg)
and
is Cauchy sequence and
, we know that
.
For any
,
![](https://www.scirp.org/html/1-7401413\5e35e411-b73f-4e7e-aee3-cd032067d02d.jpg)
Let
, then by Lemma 1.8, the above becomes
![](https://www.scirp.org/html/1-7401413\73da7cc1-acc7-4e46-b7d2-9d59ca35066a.jpg)
If
for some
, then we obtain that
![](https://www.scirp.org/html/1-7401413\48b702d8-af80-4fb4-9caf-f528d7a51f33.jpg)
which is a contradiction since
. Hence
for all
, so
, i.e.,
is a point of coincidence of
and
, and
is a coincidence point of
and
.
On the other hand, since
, there exists
such that
By (1), for any
,
![](https://www.scirp.org/html/1-7401413\95edc7a3-9675-4acb-ba28-844d7d7dda31.jpg)
Let
, then we obtain that
![](https://www.scirp.org/html/1-7401413\5b1699b0-d2ce-4072-b8b5-cb310867e79f.jpg)
If
for some
, then the above becomes that
![](https://www.scirp.org/html/1-7401413\2ddea0df-3ced-48a5-bf24-6cd7bf5c7c16.jpg)
which is a contradiction since 0 < q < 1, so
for all
. Hence
, i.e,
is a point of coincidence of
and
, and
is a coincidence point of
and
.
If
is another point of coincidence of S and
, then there exists
such that
, and we have that
![](https://www.scirp.org/html/1-7401413\a8595aa8-c0d9-4dfc-ae1f-0df8e40b6b33.jpg)
which is a contradiction. So
for all
, hence
, i.e,
is the unique point of coincidence of
and
. Similarly, we can prove that
is also the unique point of coincidence of
and
.
By Lemma 1.9,
is the unique common fixed point
and
respectively, hence
is the unique common fixed point of
.
If
or
is complete, then we can also use similar method to prove the same conclusion. We omit the part.
The following particular form of Theorem 2.2 for
-condition is the main result in [1]. The detailed proof can be found in [1].
Theorem 2.3. Let
be a 2-metric space,
four mappings satisfying that ![](https://www.scirp.org/html/1-7401413\c1ae024c-7fe9-4151-a711-3134f3fbd470.jpg)
anwd
. Suppose that for each
,
(10)
where 0 < q < 1 and
. If one of ![](https://www.scirp.org/html/1-7401413\e4e16a3b-1340-42fa-9f3f-5cc883b8adf7.jpg)
and
is complete, then
and
,
and
have an unique point of coincidence in
.
Further,
and
are weakly compatible respectively, then
have an unique common fixed point in
.
Using Theorem 2.2, we can give many different type fixed point or common fixed point theorems. But we give only the next two contractive or quasi-contractive versions of Theorem 2.2 for two mappings.
Theorem 2.4. Let
be a 2-metric space,
two mappings satisfying that for each
,
![](https://www.scirp.org/html/1-7401413\821c730d-d11b-467d-9f83-1eaddc06db77.jpg)
where
and
. If one of
and
is complete, then
and
have an unique common fixed point in
.
Theorem 2.5. Let
be a complete 2-metric space,
two surjective mappings. If for each
,
![](https://www.scirp.org/html/1-7401413\cef3612e-c386-4d5d-8d0b-f6331a5643a1.jpg)
where
and
. Then
and
have an unique common fixed point in
.
NOTES
#Corresponding author.