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 [
-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. ([
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. ([
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. ([
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
.
the set of functions 
satisfying the following: 
is continuous and non-decreasing, 
for all
.
if and only if 
is continuous and increasing in each coordinate variable and satisfy that 
and 
for all
, see [
is vary different from the set
.
be defined by 
.
be a 2-metric space, 
four single valued mappings satisfying that 
and
. Suppose that for each 
and
.
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
.
, then in view of the conditions 
and
, we can construct two sequences 
and 
as follows:
.
, by (1) and 
and (iv)
.
, then by (1) and definition 1.1 and
, we obtain that
.
, so we have that
for some
, then (2) becomes that
, so we have that
.
.
is a Cauchy sequence by Lemma 1.7.
is complete, then there exists 
and 
such that 
. (If 
is complete, there exists
, then the conclusions remains the same). Since
is Cauchy sequence and
,we know that
.
,
，then by 
and Lemma 1.8, the above becomes
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
.
, there exists 
such that
. For any
,
, then by 
and Lemma 1.8, we obtain that
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
for all
, hence
, i.e., 
is the unique point of coincidence of 
and
. Similarly, 
is also the unique point of coincidence of 
and
.
is the unique common fixed point of 
and 
respectively, hence 
is the unique common fixed point of
.
or 
is complete, then we can also use similar method to prove the same conclusion. We will omit this part.
in Example, we will obtain the next particular result.
be a 2-metric space 
four single valued mappings satisfying that 
and
. Suppose that for each 
and
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
.
be a 2-metric space, 
two mappings satisfying that for each
,
and
. If one of 
and 
is complete, then 
and 
have an unique common fixed point in
.
, 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
.
be a complete 2-metric space, 
two subjective mappings satisfying that for each
,
and
. Then 
and 
have an unique fixed point in
.
, 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
.
be a 2-metric space, 
three mappings satisfying that 
. Suppose that for each 
,
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.
, 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
.
be a 2-metric space, 
three mappings satisfying that 
. Suppose that for each 
,
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.