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
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 
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
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
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
satisfying the following:
(
1) 
is continuous and non-decreasing; (
2) 
for all
.
Obviously, 
is stronger than
.
Example 2.1. Define 
as follow:
Obviously, 
, but since
, so
.
The following is the main conclusion in this paper.
Theorem 2.2. Let 
be a 2-metric space,
four mappings satisfying that
and
.
Suppose that for each
,
(1)
where 
and
. If one of
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:
For any
,
(2)
If
for some
, then
, hence we have that
Hence we can assume now that
for all
.
If
for some
, then (2) becomes that
which is a contradiction since
. Hence we have that
for all
.
If 
for some
, then from (2),
(3)
If 
for some
, then from (2),
(4)
If
, then
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 
. (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
(If 
is complete, then there exists 
,hence the conclusions remains the same).
Since
and 
is Cauchy sequence and
, we know that
.
For any
,
Let
, then by Lemma 1.8, the above becomes
If 
for some
, then we obtain that
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
,
Let
, then we obtain that
If 
for some
, then the above becomes that
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
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 
anwd
. Suppose that for each
,
(10)
where 0 < q < 1 and
. If one of 
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
,
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
,
where 
and
. Then 
and 
have an unique common fixed point in
.
NOTES
#Corresponding author.