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.