Fixed point theorems in complex valued b -metric spaces

. In this paper, we have proved common fixed point theorems using Hardy and Rogers type contraction condition in complex-valued b -metric spaces The results of the paper extend the results proved in S. Ali [1].

It is further observed that Hardy and Rogers [10] have extended Banach fixed point theorem in complete metric spaces. Hardy and Rogers' notions have also been generalized by various researchers. Recently, Mukheimer [13] has proved a uniqueness common fixed point in complete complex valued b-metric spaces.
In this paper we have proved some common fixed theorems using Hardy and Rogers type contraction mappings. Our theorems have generalized the available results in [1]. Definition 1. A complex valued metric on a non-empty set X is a mapping d : X × X → C such that for all x, y, z ∈ X, the following conditions holds: (i) 0 ≾ d(x, y) and d(x, y) = 0 if and only if x = y, . Then the pair (X, d) is called a complex valued metric space. Definition 2 ( [14]). A complex valued metric on a non-empty set X is a mapping d : X × X → C, such that for all x, y, z ∈ X, the following conditions holds: Example 1 ( [14]). Let X = [0, 1]. Define the mapping d : X × X → C by d(x, y) = |x − y| 2 + i|x − y| 2 , for all x, y ∈ X. Then (X, d) is a complex valued b-metric space with s = 2.

Definition 3 ([14]
). Let (X, d) be a complex valued b-metric space and A ⊂ X. We recall the following definitions: (i) a ∈ A is called an interior point of the set A whenever there is 0 ≺ r ∈ C, such that x ∈ X.0 ≺ r} is a sub-basis for a topology on X. The topology is denoted by τ . It is to be noted that this topology τ is Hausdorff topology.

Definition 4 ([14]
). Let (X, d) be a complex valued b-metric space and {x n } be a sequence in X and x ∈ X. We call (i) the sequence {x n } converges to x if for every c ∈ C with 0 ≺ c there is N ∈ N such that for all n > N, d(x n , x) ≺ c. We write this as lim n→∞ x n = x or, x n → x as n → ∞; (ii) The sequence {x n } is a Cauchy sequence if for every c ∈ C with 0 ≺ c there is N ∈ N such that for all n > N and m ∈ N, d(x n , x m ) ≺ c; (iii) The metric space (X, d) is a complete complex valued b-metric space if every Cauchy sequence is convergent in X.

Main Results
Our main results are as follows.
Theorem 1. Let (X, d) be a complete complex valued b-metric space with coefficient s ≥ 1 and f, g : X → X be self-maps satisfying the following condition: where α + β + sγ < 1, α, β, γ ≥ 0. Then f and g have unique common fixed point in X.
Proof. Let x 0 ∈ X be an arbitrary. We construct a sequence {x n } in X such that x Therefore, Again let, n, m ∈ N, n ≥ m. Then, Thus {x n } is a Cauchy sequence. Since X is a complete complex valued b-metric space, there exists an u ∈ X such that Therefore, lim  Thus u is a common fixed point of f and g. Let, v be another common fixed point f and g. Then, i.e., u = v.
Thus f and g have unique common fixed point in X. □ Corollary 1. Let (X, d) be a complete complex valued b-metric space with coefficient s ≤ 1 and f, g : X → X be self-maps satisfying the following condition: where 0 ≤ β < 1. Then f and g have unique common fixed point in X. This result is Theorem 1 of S. Ali [1].
Corollary 2. Let (X, d) be a complete complex valued b-metric space with coefficient s ≥ 1 and f : X → X be self-map satisfying the following condition: where α + β + sγ < 1, α, β, γ ≥ 0. Then f have unique fixed point in X.  This result is Banach Theorem in complete complex valued b-metric space.
Example 2. Let X = C and d : X ×X → C be defined by d(x, y) = i|x−y| 2 . Also let f, g : X → X be given by f Now consider the sequence {x n }, where x n = 1 n+1 for i = 0, 1, 2, . . . , with initial approximation x 0 = 1 given by x n = f x n−1 and x n+1 = gx n . Again, Also, min{d(x, gy), d(y, f x)} ≾ d(x, y). Therefore the condition of (1) is satisfied. So by Theorem 1, f and g have unique common fixed point ′ 0 + i0 ′ .
Theorem 2. Let (X, d) be a complete complex valued b-metric space with coefficient s ≥ 1 and f : X → X be self-map satisfying the following condition: where each of α i ≥ 0 and α 1 + sα 2 + α 3 + 2sα 4 + sα 5 < 1. Then f have a unique fixed point in X. Proof. Let x 0 ∈ X be an initial point. We construct a sequence {x n } ∈ X such that x n = f x n−1 for all n ∈ N. At first we show that lim n→∞ |d(x n , x n+1 )| = 0. Since, Therefore, lim n→∞ |d(x n , x n+1 )| = 0. Now let, n, m ∈ N and n ≥ m. Then .
Taking modulus and limit as n → ∞, we get Thus {x n } is a Cauchy sequence in X. Since the space is complete, there exists an x ∈ X such that lim n→∞ |d(x n , x)| = 0. Now we show that x is a fixed point of f. Again, Therefore, From (2), we get Again if d(x, f x) ⪯ d(x n−1 , f x), then from (3), we get Therefore, . . .
Thus we get from (2), To show that x is unique let, y be another fixed point of f. This result is Banach contraction condition in complex valued b-metric space. This result is Chatterjea contraction condition in complex valued b-metric space.
Corollary 8. Let (X, d) be a complete complex valued b-metric space with coefficient s ≥ 1 and f : X → X be self-map satisfying the following condition: d(f x, f y) ⪯ α 1 d(x, y) + α 2 d(x, f x) + α 3 d(y, f y), where each of α i ≥ 0 and α 1 + sα 2 + α 3 < 1. Then f have a unique fixed point in X.
This result is Reich contraction condition in complex valued b-metric space.

Conclusion
In this article we have extended Hardy and Roger's [10] result in complexvalued b-metric spaces. This result has also extended the results of Kannan, Chatterjea, Reich, etc. We have provided an example in support of condition used in our theorems.
Further, the obtained results scope for extension of many results available in the literature in future.