
1. Introduction
As is known to all, the notion of metric spaces has been generalized by many scholars, of which the most essential work is the 2-metric spaces. In 1963, the notion of 2-metric spaces was first introduced by Gähler in [1] , from then on, many scholars had proved the common fixed points theorems in this spaces [1] [2] [3] . In 1993, Czerwik introduced the notion of b-metric spaces [4] , and proved theorems of common fixed points in this space. The two metric spaces above have obviously generalized the traditional metric spaces. Therefore, the fixed point theory has been developed a lot.
On the other hand, many scholars generalized the fixed point theorems by improving the contraction or expansive conditions,such as the φ-contraction, quasi-contraction [5] [6] [7] . Recently, Zead Mustafa has introduced the notion of b2 metric spaces [8] which is a generalization of both 2 and b metric spaces. Some fixed point theorems were then obtained under various contractive conditions in this spaces [9] [10] .
The purpose of this paper is to consider the common fixed points of a self-mappings family in the b2-metric spaces satisfying the φ contractions to generalize the fixed points theorems in 2-metric spaces and improve the theorems in b2 metric spaces.
2. Preliminary Notes
Before stating our main results, we introduce some necessary definitions as follows.
Definition 2.1 [1] Let X be a non-empty set and let
be a map satisfying the following conditions:
1) For every pair of distinct points
, there exists a point
such that
.
2) If at least two of three points
are the same, then
.
3) The symmetry:
for all
.
4) The rectangle inequality:
for all
.
Then d is called a 2-metric on X and
is called a 2-metric space.
Definition 2.2 [4] Let X be a non-empty set and
be a given real number. A function
is a b-metric on X if for all
, the following conditions hold:
1)
if and only if
.
2)
.
3)
.
In this case, the pair
is called a b-metric space.
Definition 2.3 [6] we call the function as the comparison function if it satisfies the following conditions.
1)
.
2) φ is nondecreasing, sequentially continuous from the right.
3) for each
,
.
Without loss of generality, we mark
.
Definition 2.4 [8] Let X be a non-empty set,
be a real number and let d:
be a map satisfying the following conditions:
1) For every pair of distinct points
, there exists a point
such that
.
2) If at least two of three points
are the same, then
.
3) The symmetry:
for all
.
4) The rectangle inequality:
for all
.
Then d is called a b2-metric on X and
is called a b2-metric space with parameter s. Obviously, for s = 1, b2-metric reduces to 2-metric.
Definition 2.5 [8] Let
be a sequence in a b2-metric space
.
1) A sequence
is said to be b2-convergent to
, written as
, if for all
,
.
2)
is Cauchy sequence if and only if
, when
.
3)
is said to be b2-complete if every b2-Cauchy sequence is a b2-convergent sequence.
3. Main Results
These are the main results of the paper.
Lemma 3.1 [6] For each nonnegative sequence
satisfying the condition:
,
, then
Proof. For each t > 0,
Since φ is sequentially continuous from the right, we can get
which shows that
.
Theorem 3.2 Let (X,d) be a b2 metric spaces,
are two self mappings on X, and satisfy the condition:
(1)
with s > 1, if fX or gX is complete, then f and g have an unique common fixed point.
Proof.
, from the condition (1), we can construct a sequence as follow:
(2)
we can easily get:
(3)
In fact, from the condition (1,2), we know that:
holds for each
Suppose that
Then, we can get:
which is a contradiction.
Therefore,we can get
Hence, we can get:
(4)
From now on, we will proof the sequence we construct is a cauchy sequence through two cases as follows.
Case 1: if
Meanwhile, we notice that:
(5)
In this case, with condition (5), we can get:
(6)
with condition (4, 6), we can easily get the following holds:
(7)
From condition (4, 7), suppose that
we can get:
which shows
,
since
we can get
Hence, we can get
holds for each
and
is a constant sequence, obviously, it is a cauchy sequence.
Case 2: if
Meanwhile, we noticed that:
In this case, we can get:
(8)
From the condition (4) and (8), we can get:
(9)
From condition (4, 9), we can get:
(10)
Let
, we can mark the condition (10) as:
, and from Lemma 3.1, we can get:
, which shows that
In this case, as
We can also get:
Therefore, for each
and
,
(11)
From now on, we will prove the sequence in case 2 is a cauchy sequence through mathematical induction.
From the condition(11), we can get: For arbitrary
, when
, the following holds:
(12)
Then, we use the mathematical induction for m to prove
,
holds.
1) When
,
holds
2) Suppose that when
,
holds, from this, we will prove the condition holds for
.
From the condition (12) and the inductive hypothesis, we can easily get:
holds for all
From the rectangle inequality, we can get:
(13)
From the condition (13) and inductive principle, we claim that:
holds for all
and
Therefore, from the definition 2.5, we can get the conclusion that the sequence
we construct in case 2 is a cauchy sequence.
Therefore, from the proof above, we can get the conclusion that the sequence we construct in condition (2) is a cauchy sequence.
If fX is complete, then from the condition (2), we can get:
From the rectangle inequality, we can get:
Since the
is a cauchy sequence, let
, we can get:
Therefore, we can get
Then, we will prove the point u is the unique common fixed point for the mappings f and g.
Let
, suppose that
, we can get:
which is a contradiction.
Therefore, we claim that
holds for
, which means
.
Suppose that
, we can get:
which is a contradiction. Therefore, we claim that
holds for
,which means
.
If there exist another
,
, suppose that
we can get:
which is a contradiction. Therefore, we can get:
holds for
, which means
.
Therefore, we claim that u is the unique common fixed point for the mappings f and g.
The proof is in the similar way for the case if gX is complete.
Now, we will generalize the theorem 3.2 into family of mappings.
Let
,
, Let
. If for each
, then we call
is pairwise commuting.
Theorem 3.3 Let (X,d) be a b2 metric space,
is the mapping family, let
, and satisfy the following condition:
if fX or gX is complete,
is pairwise commuting, then
has an unique common fixed point.
Proof. From theorem 3.2, we know u is the unique common fixed point for
. Since
is pairwise commuting, we can get: For each
Therefore,
is the common fixed point for
, since u is the unique common fixed point for
, we can obtain
, we can prove
in the similar way. Therefore, u is the common fixed point for
, if v is the common fixed point for
, obviously, v is the common fixed point for
, therefore,
is the unique common fixed point for
.
Example 3.4 [8] Let
, and let
denote the square of the area of triangle with vertices
, e.g.
It is easy to check that d is a b2-metric with parameter s = 2.
Consider the mappings
given by:
and let the comparison function
, in order to prove the mappings f and g satisfy the condition (1),we will divide it into 3 cases.
case 1
, we can easily check
case 2
, we can easily check
case 3
, we can easily check
Now, we have proved the mappings f and g satisfy condition (1) and we can obtain the point
is the unique common fixed point for f and g by using theorem 3.2.
Since b2-metric space is a generalization of 2-metric space, we can obtain the following theorem.
Theorem 3.5 Let (X,d) be a 2 metric spaces,
are two self mappings on X, and satisfy the condition:
if fX or gX is complete, then f and g have an unique common fixed point.
Proof. In the proof of theorem 3.2, let s = 1, then the b2 metric space turn to 2 metric space. Meanwhile, we can easily check the proof for theorem 3.2 holds for 2-metric space, therefore, we get the conclusion.
Theorem 3.6 Let (X,d) be a 2 metric spaces,
are two self mappings on X, and satisfy the condition:
, if fX or gX is complete, then f and g have an unique common fixed point.
Proof. Let
, we can easily check
satisfy the comparison function’s condition, from the theorem 3.5, we can get the conclusion.
4. Conclusion
In this paper, we introduce the concept of φ-contraction, and proved that mappings or family of mappings satisfying the contraction have the unique common fixed point. Meanwhile, we give an example for the theorem we proved. The results we obtained generalized many results in the b2 metric spaces.