Uniqueness of Common Fixed Points for a Family of Mappings with Φ-Contractive Condition in 2-Metric Spaces ()
1. Introduction and Preliminaries
There have appeared many unique common fixed point theorems for self-maps 
with some contractive condition on 2-metric spaces. But most of them held under subsidiary conditions [1-4], for examples: commutativity of 
or uniform boudedness of 
at some point, and so on. In [5], the author obtained similar results under removing the above subsidiary conditions. The result generalized and improved many same type unique common fixed point theorems. Recently, the author discussed unique common fixed point theorems for a family of contractive or quasi-contractive type mappings on 2-metric spaces, see [6-8], these results improve the above known common fixed point theorems.
In this paper, in order to generalize and unify further these results, we will prove that a family of self-maps 
satisfying 
-contractive condition on 2-metric spaces have an unique common fixed point if 
satisfy the condition 2.
The following definitions are well known results.
Definition 1.1. [4] 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. [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. [4,5] A sequence 
is said to be convergent to
, if for each
, 
. And write 
and call 
the limit of
.
Definition 1.4. [4,5] 2-metric space 
is said to be complete, if every cauchy sequence in 
is convergent.
Let 
denotes a family of mappings such that each
, 
is continuous and increasing in each coordinate variable, and 
for all
.
There are many functions 
which belongs to
:
Example 1.5. Let 
be defined by
Then obviously, 
Example 1.6. Let 
be defined by
Then obviously, 
is continuous and increasing in each coordinate variable, and
Hence 
The following two lemmas are known.
Lemma 1.7. [1-4] 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. [1-4] If 
is a 2-metric space and sequence
, then 
for each
.
2. Main Result
The following theorem is the main result in this present paper.
Theorem 2.1. Let 
be a complete 2-metric space, 
a family of maps from 
into itself, 
a family of positive integers, and 
and 
for each
. If the following 
- contractive conditions hold
(1)
and 
for all 
with
. Then 
have an unique common fixed point in X.
Proof Fix 
and let 
for each
, then (1) becomes the following
(2)
Take an 
and define a sequence as follows
Then
(3)
If
, then
(4)
which is a contradiction since
, hence
. And therefore, (3) becomes
(5)
If there exists an 
such that
, then (5) becomes
which is a contradiction since 
and 
, hence he have that 
for all
. In this case, (5) becomes
(6)
(6) implies that 
is a cauchy sequence by Lemma 1, hence by the completeness of
, 
converges to some element
. (7)
Now, we prove that 
is the unique common fixed point of
. In fact, for any fixed 
and any 
with 
and any
,
Let
, then by Lemma 2, the continuity of 
and (7), the above becomes
.
But
, hence 
for all
, and therefore, 
for all
.
This completes that 
is a common fixed point of
.
Let 
be a common fixed point of
, If there exists an 
such that
, then
which is a contradiction since
, hence 
for all
, and therefore
. This completes that 
has an unique common fixed point 
for all
.
Next, we will prove that 
is the unique common fixed point of 
for each fixed
. Indeed, for fixed
, Since 
for each
, hence
for each
, which means that 
is a fixed point of 
for each
. Now, fix 
and let 
with
, if there exists an 
such that 
, then
which is a contradiction since
, hence
for all
, and therefore
. This means that
is a common fixed point of
. But 
is the unique common fixed point of
, hence
for all
, which means that 
is a common fixed point of 
for all
.
If 
is a common fixed point of
, then 
for all
, which means that 
is a common fixed point of
. But 
is the unique common fixed point of
, hence
. This completes that 
has the unique common fixed point 
for each
.
Finally, we will prove that 
for all
. In fact, for any fixed 
with
, since 
and
, hence
by condition 2). Which means that 
is a common fixed point of 
for all
. But the unique common fixed point of 
is
, hence
for all
, this means that 
is a common fixed point of
, and therefore
since 
is the unique common fixed point of
. Let
, then 
is the unique common fixed point of
.
The following is a particular form of Theorem 2.1:
Theorem 2.2. Let 
be a complete 2-metric space, 
a family of maps from 
into itself and 
and
. If the following 
-contractive condition holds
then 
has an unique common fixed point in
.
Next theorem is the main result in [5].
Theorem 2.3. Let 
be a complete 2-metric space, 
a family of maps from 
into itself. If there exist a family non-negative integers 
and nonnegative real numbers 
with 
such that for all 
and all natural numbers 
with
, the following holds
Then 
have an unique common fixed point in
.
Remark. Obviously, Theorem 2.3 is a very particular form of Theorem 2.1. In fact, Let
, and take
satisfying
, then 
and 
satisfy all conditions of Theorem 2.1. Hence we sure that our main result generalized and improve many corresponding common fixed point theorems in 2-metric spaces.
NOTES