Advances in Pure Mathematics
Vol.3 No.2(2013), Article ID:28692,5 pages DOI:10.4236/apm.2013.32039
Four Mappings Satisfying Ψ-Contractive Type Condition and Having Unique Common Fixed Point on 2-Metric Spaces*
Department of Mathematics, College of Science, Yanbian University, Yanji, China
Email: hljin98@ybu.edu.cn, pyj6216@hotmail.com
Received November 7, 2012; revised January 21, 2013; accepted January 30, 2013
Keywords: 2-metric space; class Ψ; Cauchy sequence; coincidence point; common fixed point
ABSTRACT
In this paper, we introduce a class Ψ of real functions defined on the set of non-negative real numbers, and obtain a new unique common fixed point theorem for four mappings satisfying Ψ-contractive condition on a non-complete 2-metric space and give the versions of the corresponding result for two and three mappings.
1. Introduction and Preliminaries
Using subsidiary conditions [1,2] such as commutability of mappings or uniform boundless of mappings at some point and so on, many authors have discussed and obtained many unique common fixed point theorems of mappings with some contractive or quasi-contractive condition on 2-metric spaces. The author [3-7] obtained similar results for infinite mappings with contractive conditions or quasicontractive conditions under removing the above subsidiary conditions. These results generalized and improved many same type unique common fixed point theorems. Recently, the author [8] discussed the existence of coincidence points and common fixed points for four mappings with -contractive conditions on 2-metric spaces and give some corresponding results.
Here, by introducing a new class of real functions defined on
, we will discuss the existence problem of unique common fixed points for four mappings with
-contractive type condition on non-complete 2-metric spaces and give some corresponding forms.
The following definitions and lemmas are well known.
Definition1.1. ([3]) 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]) 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. ([3,4]) A sequence is said to be convergent to
, if for each
,
. And we write that
and call
the limit of
.
Definition 1.4. ([3,4]) A 2-metric space is said to be complete, if every Cauchy sequence in
is convergent.
Definition 1.5. ([9,10]) Let and
be self-maps 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. ([11]) Two mappings are weakly compatible, if for every
holds
whenever
.
Lemma 1.7. ([5-7]) 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. ([5-7]) If is a 2-metric space and sequence
, then
for each
.
Lemma 1.9. ([9,10]) Let be weakly compatible. If
have a unique point of coincidence
, then
is the unique common fixed point of
.
2. Main Results
Denoted by the set of functions
satisfying the following:
is continuous and non-decreasing,
for all
.
Remark if and only if
is continuous and increasing in each coordinate variable and satisfy that
and
for all
, see [8]. Obviously, the set
is vary different from the set
.
Example Let be defined by
Then, obviously,.
The following is the main result in this paper.
Theorem 2.1. Let be a 2-metric space,
four single valued 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 fixed, by (1) and
and (iv)
in definition 1.1, we obtain that
Suppose that.
Take, then by (1) and definition 1.1 and
, we obtain that
which is a contradiction since.
Hence, so we have that
(2)
If for some
, then (2) becomes that
This is a contradiction. Hence for all, so we have that
Similarly, we can obtain that
.
Hence we have that
.
So is a Cauchy sequence by Lemma 1.7.
Suppose that is complete, then there exists
and
such that
. (If
is complete, there exists
, then the conclusions remains the same). Since
and is Cauchy sequence and
,we know that
.
For any,
Let,then by
and Lemma 1.8, the above becomes
(3)
If for some
, then we obtain from (3) 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
.
Since, there exists
such that
. For any
,
Let, then by
and Lemma 1.8, we obtain that
If for some
, then the above becomes that
, which is a contradiction since
, so
for all
. Hence
, i.e.,
is a point of coincidence of
and
, and
is coincidence point of
and
. Suppose that
is another point of coincidence of
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,
is also the unique point of coincidence of
and
.
By Lemma 1.9, is the unique common fixed point of
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 will omit this part.
Using Theorem 2.1 and in Example, we will obtain the next particular result.
Theorem 2.2. Let be a 2-metric space
four single valued mappings satisfying that
and
. Suppose that for each
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
.
The following two theorems are the contractive and quasi-contractive versions of theorem 2.1 for two mappings.
Theorem 2.3. 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
.
Proof Let, then by Theorem 2.1, there exist
such that
is the unique point of coincidence of
and
. But obviously
and
are weakly compatible, so
is the unique fixed point of
by Lemma 1.9. Similarly,
is also unique fixed point of
, hence
is the unique common fixed point of
and
.
Theorem 2.4. Let be a complete 2-metric space,
two subjective mappings satisfying that for each
,
where and
. Then
and
have an unique fixed point in
.
Proof Let, then by Theorem 2.1, there exist
such that
is the unique point of coincidence of
and
. But obviously
and
are weakly compatible, so
is the unique fixed point of
by Lemma 1.9. Similarly,
is also unique fixed point of
, hence
is the unique common fixed point of
and
.
Finally we give two coincidence point theorems for three mappings.
Theorem 2.5. Let be a 2-metric space,
three mappings satisfying that
. Suppose that for each
,
where and
. If one of
and
is complete, then
and
and
have an unique point of coincidence in
. Further,
is one to one mapping, then
have an unique point of coincidence.
Proof Let, then by Theorem 2.1, there exist a unique element
and
such that
and
, hence
, which implies that
, so we obtain that
. This means that
is point of coincidence of
. If
is also point of coincidence of
, then
is also point of coincidence of
, hence by uniqueness of points of coincidence of
and
, we have that
. Hence
is the unique point of coincidence of
.
Theorem 2.6. Let be a 2-metric space,
three mappings satisfying that
. Suppose that for each
,
where and
. If one of
and
is complete, then
and
and
have an unique point of coincidence. Further,
is one to one mapping, then
have an unique point of coincidence.
Proof The proof is similar to that of Theorem 2.5. So we will omit it.
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NOTES
*Project supported by NNSFC (No. 11261062).