Banach fixed point principle  is simple but forceful, which is a classical tool for many aspects. There are many generalizations of this principle, see     , from which, an interesting generalization is introduced by Suzuki  in 2008.
Many generalized spaces of Metric space have been established. Among them, b-metric  and 2-metric  have been extensively researched. Both of these metrics of those spaces are not continuous functions of its variables. In order to solve this problem, the author of  established the notion of b2-metric space generalizing from both spaces above. And in this paper, we proved a common fixed point result for two maps in b2-metric space  . Our purpose is to present a fixed point result of two maps under a newly Suzuki-type contractive condition in this space, and the fixed point theory in b2-metric space is perfected.
The following definitions will be presented before giving our results.
Definition 2.1.  Let X be a nonempty set, be a real number 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 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 2.2.  Let be a sequence in a b2 metric space .
1) A sequence is said to be b2-convergent to , written as , if all .
2) is Cauchy sequence if and only if , when . for all .
3) is said to be complete if every b2-Cauchy sequence is a b2-convergent sequence.
Definition 2.3.  Let and be two b2-metric spaces and let be a mapping. Then f is said to be b2-continuous, at a point if for a given , there exists such that and for all imply that . The mapping f is b2-continuous on X if it is b2-continuous at all .
Definition 2.4.  Let and be two b2-metric spaces. Then a mapping is b2-continuous at a point if and only if it is b2-sequentially continuous at x; that is, whenever is b2-convergent to x, is b2-convergent to .
Lemma 2.5.  Let be a b2 metric space with and let be a sequence in X such that
for all and all , where . Then is a b2-Cauchy sequence in .
3. Main Results
Theorem 3.1. Let be a complete b2 metric space and in each variable d is continuous. Let be a selfmap and : be defined by:
where is the positive solution of . If there exists such that for each ,
then f has a unique fixed point z in X and the sequence converges to z.
Proof From (3.1) and take , we get the inequality as follows:
from the above relation, we get
, for each (3.3)
Given and construct a sequence letting , for all . Then by taking in (3.3) we get
since , we have , by Lemma 2.6, we get the conclusion that is a Cauchy sequence, so there exists z in X, such that .
Since and , that is and by the continuity of d, we have , for every , so there exists such that , for each , now for such above n and from the assumption (3.2) we get
, for (3.5)
taking we have
In (3.3), take , we have
, for (3.7)
by induction, we have
Now we claim that
, for every (3.9)
this inequality is true for , assume (3.9) holds for some , if , then we have and
if , then we can obtain the following inequality from (3.6), and that is:
By the induction hypothesis (3.9) for some and (3.8), we have
Therefore, (3.9) is true for every .
Now we assume that and consider the two following possible cases to prove that .
Case 1. Take ，therefore . Firstly we claim that
, for all (3.10)
It is obvious for and this follows from (3.8) for .
From (3.9) we have , that is,
Now assume that (3.10) holds for some , then from part 4 of Definition 2.1 and (3.11) we have
and that is , using (3.8), it follows that
By induction with using (3.8) and (3.9), it is easy for us to get the relation (3.10).
Now from and (3.10), we get for each , therefore, (3.6) and (3.8) show that
From part 4 of Definition 2.1 and (3.11), we get
It follows from (3.10) that
There exists , for and such that , for such n, we get
Then taking from (3.12) we have
That is, , and from (3.10), we get
which is impossible except .
Case 2. Take and that is when , we will prove that we can find a subsequence of such that for each ,
we know for each from (3.4), assume that for some
taking , we get a relation which is impossible. Therefore we have
for each . (3.13.4)
In other words, there is a subsequence for such that (3.13) is true for every , but from (3.2) we have
Taking , we have
which is possible only if .
Therefore, z is a fixed point of f. Let w be another fixed point of f, from (3.6), we have
which is a contraction unless , and that is , f has a unique common fixed point .
Corollary Let be a complete b2-metric space and d is continuous in every variable. Let be a selfmap and be defined by (3.1). If there exists such that for each x, y of X,
then f has a unique fixed point z in X and the sequence converges to z, for each .
A known existence theorems of common fixed points for two maps was proved for the generalized Suzuki-type contractions in b2-metric space. The results generalized and improved the field of fixed point theory for metric spaces and perfected the realization of the fixed point theory in this generalized space.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
 Piao, Y.J. (2008) Unique Common Fixed Point for a Family of Self-Maps with Same Type Contractive Condition in 2-Metric Spaces. Analysis in Theory and Application, 24, 316-320.
 Mustafa, Z., Parvaech, V., Roshan, J.R. and Kadelburg, Z. (2014) b2-Metric Spaces and Some Fixed Point Theorems. Fixed Point Theory and Applications, 2014, 144.