Common Fixed Point Theorems in b-Metric Spaces

In 1922, Banach [1] proved the Banach contraction principle. Since then, several works have been done about fixed point theory regarding different classes of contractive conditions in some spaces such as: quasi-metric spaces [2,3], cone metric spaces [4,5], partially order metric spaces [6,7,8], G-metric spaces [9]. The concept of b − metric space was introduced by Czerwik in [10]. After that, several papers have been published on the fixed point theory of various classes of single-valued and multi-valued operators in b − metric spaces (see [2,11,12]). Aydi et al. in [13] proved common fixed point results for single-valued and multi-valued mappings satisfying a weak φ − contraction in b − metric spaces. Starting from the results of Berinde [14], Pacurar [15] proved the existence and uniqueness of fixed point of φ − contractions on bmetric spaces. Using a contraction condition defined by means of a comparison function, [16] established results regarding the common fixed points of two mappings. Hussain and Shah in [17] introduced the notion of a cone bmetric spaces, generalizing both the notions of bmetric spaces and cone metric spaces, they considered topological properties of cone b − metric spaces and results on KKM mappings in the setting of cone bmetric spaces. The aim of this paper is to consider and establish some common fixed point results for two mappings satisfying contraction conditions in complete b-metric spaces. Meanwhile, two examples are presented to support our results.

The concept of b − metric space was introduced by Czerwik in [10]. After that, several papers have been published on the fixed point theory of various classes of single-valued and multi-valued operators in b − metric spaces (see [2,11,12]). Aydi et al. in [13] proved common fixed point results for single-valued and multi-valued mappings satisfying a weak φ − contraction in b − metric spaces. Starting from the results of Berinde [14], Pacurar [15] proved the existence and uniqueness of fixed point of φ − contractions on b-metric spaces. Using a contraction condition defined by means of a comparison function, [16] established results regarding the common fixed points of two mappings. Hussain and Shah in [17] introduced the notion of a cone b-metric spaces, generalizing both the notions of b-metric spaces and cone metric spaces, they considered topological properties of cone b − metric spaces and results on KKM mappings in the setting of cone b-metric spaces.
The aim of this paper is to consider and establish some common fixed point results for two mappings satisfying contraction conditions in complete b-metric spaces. Meanwhile, two examples are presented to support our results.

Preliminaries
Let  and +  denote the sets of all real numbers and nonnegative numbers respectively.
 is upper semicontinuous and nondecreasing in each coordinate variable satisfying condition ( , , , , , , , , ) ( ) t t t t t t t t t t t ψ ψ = < }. In order to obtain our main results, we need to introduce some definitions and lemmas. Definition. Let X be a nonempty set and : There are examples of b − metric spaces which are not metric spaces. (see [18])

Main Results
Now we are ready to prove our main results.
for all , x y X ∈ , then A and B have a unique common which is a contradiction. Therefore, x is a common fixed point of A and B.
which is a contradiction. It follows that * x is a unique common fixed point in X. This completes the proof.
If A B = in Theorem 1, then we get that: .  is a Cauchy sequence in X . As in the proof of Theorem 3.1, we obtain that inequalities (9),(10) hold, and The triangle inequality in b-metric space and contraction condition (11)