Fixed point theory in complex valued controlled metric spaces with an application

: In this article we have introduced a metric named complex valued controlled metric type space, more generalized form of controlled metric type spaces. This concept is a new extension of the concept complex valued b -metric type space and this one is di ﬀ erent from complex valued extended b metric space. Using the idea of this new metric, some ﬁxed point theorems involving Banach, Kannan and Fisher contractions type are proved. Some examples togetheran application are described to sustain our primary results.


Introduction and preliminaries
Fixed point theory, famous due to its vast applications in various areas of engineering, physics and mathematical sciences, has become an interesting area for many researchers. For non-linear analysis, techniques of fixed point act as a pivotal tool. Banach [9] made huge involvement in this area by introducing the concept of contraction mapping for a complete metric type space to find fixed point of the stated function.
The classical Banach contraction theorem [9] has been studied by many mathematicians and researchers in different ways, see [1,2,4,11,12,15,17,26]. During these generalizations, different fixed point theorems were studied for different contractive mappings or sometimes studied after metric space extension. Bakhtin [7] and Czerwick ( [13,14]) introduced a generalized metric with respect to the structure named b-metric space.
Definition 1.1. [2,14] Consider D 0 with a real number t > 1. The functional h b : D × D → [0, ∞) satisfying the following conditions: h b (d, e) = h b (e, d), for all d, e, f ∈ D, called b-metric. Also we call the pair (D, h b ) a b-metric space.
A b-metric is an usual metric in case of t = 1. So, the category of b -metric spaces is appreciably greater than that of classic metric spaces. For instance, see [13,14,25,27,30]. Many other extensions of this space were used in related literature as platforms for fixed point results, see [3,8,16,18,29].
Azam et al. [2] was the first one who introduced and presented the notion complex valued metric space, with more generalized form than a metric space. Ullah et al. gave in [31] the concept of complex valued extended b-metric space, which is extension of notion b-metric space which was introduced by Kamran et al. [19]. For examples and applications see [6,31].
Consider set of complex numbers C and d 1 , d 2 ∈ C. Since we cannot compare in usual way two complex numbers let us add to the complex set C the following partial order , known in related literature as lexicographic order d 1 d 2 ⇐⇒ Re(d 1 ) ≤ Re(d 2 ) or (Re(d 1 ) = Re(d 2 ) and Im(d 1 ) ≤ Im(d 2 ).
Considering the previous definition, we may say that d 1 d 2 if one of the next conditions are satisfied: (1) Re(d 1 ) < Re(d 2 ) and Im(d 1 ) < Im(d 2 ); (2) Re(d 1 ) < Re(d 2 ) and Im(d 1 ) = Im(d 2 ); (3) Re(d 1 ) < Re(d 2 ) and Im(d 1 ) > Im(d 2 ); (4) Re(d 1 ) = Re(d 2 ) and Im(d 1 ) < Im(d 2 ); in continuation of generalizations of metric spaces, another form of b-metric space was introduced and presented by Rao et al. [28] in 2013 and named as complex valued b-metric space. This idea was also studied and generalized by many mathematicians and researchers. In [23] Maliki et al. introduced the idea of controlled metric and gave few important fixed point theorems with respect to this new type of metric. Several researchers proved fixed point theorems using this idea (see [5,20,21,24]).
Also the pair (D, h cm ) is said controlled metric type space. Now moving towards the main definition/concept, in which we have generalized the idea of controlled metric spaces in complex valued spaces as follows.
where k > 0 and h cvc : D × D → C defined as follows We will put in evidence the previous remark by the following example.
For contradiction to complex valued (c.v.) extended b-metric, we have the following inequality: In conclusion, (D, h cvc ) is a complex valued controlled metric space. It can also be seen that it is not complex valued extended b-metric space.
For complex valued controlled (c.v.c.) metric type spaces, we now define Cauchy sequence and also convergent sequence as below. Definition 1.6. Consider (D, h cvc ) is a complex valued controlled (c.v.c)metric space with {d n } n≥0 a sequence in D and d ∈ D. Then (i) A sequence {d n } in D is convergent and converges to d ∈ D if for every 0 ≺ c ∈ C ∃ a natural number N so that h cvc (d n , d) ≺ c for every n ≥ N. Then we say lim n→∞ d n = d or d n → d as n → ∞.
(ii) If, for every 0 ≺ c where c ∈ C ∃ a natural number N so that h cvc (d n , d n+m ) ≺ c for every m ∈ N and n > N. Then {d n } is said to be Cauchy sequence in (D, h cvc ).
(iii) Complex valued controlled (c.v.c) metric type space (D, h cvc ) is said to be Complete, if every Cauchy sequence in D is convergent in D. (ii) The functional/mapping Υ : Obviously, if Υ is taken continuous at d in the complex valued controlled (c.v.c.) metric spaces (D, h cvc ) then d n → d implies Υd n → Υd as n → ∞.
One can prove the following lemmas for the specific case of complex valued controlled (c.v.c) metric space, in a similar way as in [28]. Proof. Let {d n } be a sequence which is Cauchy in D, then for every ε 0, ∃ n 0 ∈ N such that h cvc (d m , d n ) ≺ , for all m, n > n 0 . (1) Since ∈ C, ∃ a, b ∈ R such that = a + ib. We consider δ * = | | = √ a 2 + b 2 , and taking the modulus on both sides of (1), we have |h cvc (d m , d n )| ≺ δ * .
Since h cvc (d m , d n ) ∈ C, ∃ ∈ C such that δ * = | |. Then, we have h cvc (d m , d n ) ≺ , for every > 0. Thus, ∃ n 0 ∈ N such that m, n > n 0 , {d n } is a complex valued Cauchy sequence. Proof. Letting m → ∞ in proof of the above lemma, we can get the conclusion. Proof. Consider the sequence {d n } with two limit points d * and e * ∈ D and lim n→∞ h cvc (d n , d * ) = 0 = lim n→∞ h cvc (d n , e * ). Since {d n } is Cauchy, from (CCMT 3 ), for d m d n , whenever m n, we can write We get |h cvc (d * , e * )| = 0, i.e., d * = e * . Thus, {d n } converges to at most one point.
Remark that, in general an usual b-metric is not considered a continuous functional. Concerning this aspect we will discuss next, the continuity of the complex valued controlled metric with respect to the partial order . Proof. Choosing any two arbitrary complex numbers y and x, such that y x, then we should show that the set h −1 cvc (x, y) given by is open in the product topology on D × D. A basis for this product topology is the collection of all cartesian products of open balls in (D, h cvc ). .
cvc (x, y). Then, the complex valued metric h cvc (d, e) is continuous in the complex valued controlled (c.v.c) metric space (D, h cvc ), with respect to the partial order " ".
According with this result, let us give the following lemma.
The next graph shows the relations between the recently generalizations of complex valued metric space.
In this paper, CVC-metric space will be the notation of complex valued controlled (c.v.c) metric space. Also, N * := N − {0} is notation of the set for all natural nonzero numbers. Then, for operator Υ, FixΥ := {d * ∈ D | d * = Υ(d * )} will be the set of fixed points.
In this paper, we give the notion of complex valued controlled metric space. We will also prove the Banach contraction principle, Kannan and Fisher rational type fixed point theorems in the settings of complex valued controlled metric. Few examples and an application are presented to sustain our main results.

Banach type fixed point result on CVC-metric space
For this section let us give a Banach type contraction principle related to the complex valued controlled metric space.
In addition, for every d ∈ D the limits lim n→∞ ϑ(d n , d) and lim n→∞ ϑ(d, d n ) exists and are finite.
Then Υ has a unique fixed point.
Proof. Suppose the sequence {d n = Υ n d 0 }. From (2) we get ∀ n < m, where n and m are natural numbers, we have By (3) and applying the ratio test, we get that lim m,n→∞ S n exists and so the real sequence {S n } is Cauchy.
At the end, taking the limit in the inequality (5) when m, n → ∞ we found that Results {d n } is Cauchy sequence in the complete CVC-metric space(D, h c vc); then {d n } converges to a d * ∈ D. We now show that d * is fixed point of Υ.
By the continuity of Υ we obtain For uniqueness of fixed point, we suppse that Υ has two fixed points, d * , e * ∈ FixΥ. Thus we have which holds unless h cvc (d * , e * ) = 0; so d * = e * . Hence Υ has a unique fixed point.
If we avoid the continuity condition of the mapping Υ in the previous theorem we get a new general result as follows.
Theorem 2.2. Let (D, h cvc ) be a complete CVC-metric space and Υ : D → D be a mapping such that for all d, e ∈ D, where 0 < ϑ < 1.
In addition, for every d ∈ D the limits lim n→∞ ϑ(d n , d) and lim n→∞ ϑ(d, d n ) exist and are finite.
Then Υ has a unique fixed point.
Proof. Using similar steps as we did in the proof of Theorem 2.1, and taking into account Lemma 1.4, we find a Cauchy sequence {d n } in the complete CVC-metric space (D, h cvc ). Then the sequence {d n } converges to a d * ∈ D. We shall prove that d * is a fixed point of Υ. The triangular inequality follows that Using (8), (9) and (31), we get lim Using again the triangular inequality and (7), Taking the limit n → ∞ and by (9) and (32) we found that h cvc (d * , Υd * ) = 0. Remark that, in view of Lemma 1.3, the sequence {d n } converges uniquely at the point d * ∈ D.
We illustrate this theorem with the help of following example.
Further, let us give another example to put in evidence that, the complex valued controlled space is larger than the one of controlled metric space. Moreover, Then, all hypothesis of Theorem (2.1) hold; then, Υ has a unique fixed point d = 0 + 0i.
In addition, for every d ∈ D the limits lim n→∞ ϑ(d n , d) and lim n→∞ ϑ(d, d n ) exist and are finite.
Then ϑ has a unique fixed point.
Proof. From Theorem 2.1 we have that Υ n has a unique fixed point f . Since Υ n (Υ f ) = Υ(Υ n f ) = Υ f results Υ f is a fixed point of Υ n . Therefore, Υ f = f by the uniqueness of a fixed point of Υ n . Therefore, f is also a fixed point of Υ. Since the fixed point of Υ is also a fixed point of Υ n , then the fixed point of Υ is unique.

Kannan type fixed point results on CVC-metric space
The main result of this section is a Kannan type fixed point theorem for the case of complex valued controlled metric space.
for all d, e ∈ D, where 0 ≤ γ < 1 2 . For d 0 ∈ D we denote d n = Υ n d 0 . Suppose that In addition, assume for every d ∈ D that the limits lim n→∞ ϑ(d n , d) and lim n→∞ ϑ(d, d n ) exist and are finite.
Then Υ has a unique fixed point.
Proof. For d 0 ∈ D, consider a sequence {d n = Υ n d 0 }. If there exists d 0 ∈ N for which d n 0 +1 = d n 0 , then Υd n 0 = d n 0 . So everything will be trivially satisfied. Now we assume that d n+1 d n for all n ∈ N. By using (2) we get . Continuing in the same way, we have Thus, h cvc (d n , d n+1 ) λ n h cvc (d 0 , d 1 ) for all n ≥ 0. For all n < m, where n and m are natural numbers, we have Further, using ϑ(d, e) ≥ 1. Let Condition (3) Then {d n } is a Cauchy sequence in the complete CVC-metric space (D, h cvc ). This means the sequence{d n } converges to some d * ∈ D. We now show that d * is a fix point of Υ.
By the continuity of Υ we obtain For the uniqueness assume that Υ has two fix points d * , e * ∈ FixΥ. Thus, Since h cvc (d * , e * ) = 0 then d * = e * . Hence Υ has a unique fixed point.
If we consider Υ a mapping not necessary continuous, we get the following general fixed point result.
In addition, assume for every d ∈ D that the limits Then Υ has a unique fixed point.
Proof. Using similar steps as followed in the proof of Theorem 3.1 and taking into account Lemma 1.4, we find a Cauchy sequence {d n } in the complete CVC-metric space (D, h cvc ). Then the sequence {d n } converges to a d * ∈ D. We must prove that d * is a fixed point of Υ.
Using again the triangular inequality and (2) we obtain Taking the limit as n → ∞ and by (4) and (32) we deduce that h cvc (d * , Υd * ) = 0. Remark that, in view of Lemma 1.3, the sequence {d n } converges uniquely at the point d * ∈ D.
Further we will present some special cases concerning this new type of results. We will show that there exists a close connections between CVC-metric space and other different types of spaces.
In addition to this, suppose that for every d ∈ D, we have Then Υ has a unique fixed point.
Proof. If we choose ϑ(d, e) = ϑ(e, f ) in Theorem 3.1 we get the conclusion.
In addition to this, suppose that for every p ∈ D, we have lim n→∞ ϑ(d n , d) and lim n→∞ ϑ(d, d n ) exist and are finite.
Then Υ has a unique fixed point.
In addition, assume that for every d ∈ D, we have lim n→∞ ϑ(d n , d) and lim n→∞ ϑ(d, d n ) exist and are finite.
Then Υ has a unique fixed point.
Let us give the illustrative example as follows. Let us consider the self-map Υ on D as Υ(0) = Υ(1) = Υ(2) = 2. Choosing γ = 2 5 , for both cases from the definition of ϑ it is clearly that (18) holds. Also, for any p 0 ∈ D the condition(19) is satisfied.

Fisher type fixed point result on CVC-metric space
In [20] D. Lateef gave some fixed point result for rational functions, Fisher type, in controlled metric space. In this section let us present a generalisation of D. Lateef result in the settings of the CVC-metric space.
In addition, suppose that for every p ∈ D we have lim n→∞ ϑ(d n , d) and lim n→∞ ϑ(d, d n ) exist and are finite.
Then Υ has a unique fixed point.
Proof. For d 0 ∈ D consider the sequence {d n = Υ n d 0 }. If there exists d 0 ∈ N for which d n 0 +1 = d n 0 , then Υd n 0 = d n 0 . So everything will be trivially satisfied. Now we assume that d n+1 d n for all n ∈ N. By using (2), we get In the same way Continuing the same way, we have Thus, h cvc (d n , d n+1 ) η n h cvc (d 0 , d 1 ) for all n ≥ 0. For all n < m, where n and m are natural numbers, we have Further, using ϑ(d, e) ≥ 1. Let Condition (3), by using the ratio test, ensure that lim m,n→∞ S n exists and hence, the real sequence {S n } is a Cauchy sequence. Finally, if we apply the limit in the inequality (34) as m, n → ∞, we deduce that For uniqueness let us assume that d * , e * ∈ FixΥ are two fixed points of Υ. Then we get which holds h cvc (d * , e * ) = 0; then d * = e * . Hence Υ has a unique fixed point.
If we consider a mapping Υ not continuous we get a more general result for rational type mapping as follows.
for all d, e ∈ D, where ξ, ς ∈ [0, 1) such that η = ξ 1−ς < 1. For d 0 ∈ D we denote d n = Υ n d 0 . Suppose that max In addition, assume that for every d ∈ D we have lim n→∞ ϑ(d n , d) and lim n→∞ ϑ(d, d n ) exist and are finite.
Then Υ has a unique fixed point.
Proof. Using similar steps as in the proof of Theorem 4.1 and taking into account Lemma 1.4, we get a Cauchy sequence {d n } which converges to an d * ∈ D. We must show that d * is a fixed point of Υ.
Using again the triangular inequality and (2), for all d, e ∈ D, where ξ, ς ∈ [0, 1) such that η = ξ 1−ς < 1. For d 0 ∈ D we denote d n = D n d 0 . Suppose that In addition to this, suppose that for every d ∈ D, we have lim n→∞ ϑ(d n , d) and lim n→∞ ϑ(d, d n ) exists and finite.
Then Υ has a unique fixed point.
Proof. If we choose ϑ(d, e) = ϑ(e, f ) in Theorem 4.1 we get the conclusion.
In addition to this, suppose that for every p ∈ D, we have In addition to this , suppose that for every d ∈ D, we have lim n→∞ ϑ(d n , d) and lim n→∞ ϑ(d, d n ) exists and finite.
Then Υ has a unique fixed point.

Application to an integral type equation
During this section we suppose the following type of integral equation.
It is easy to conclude that (D, h cvc ) is a complete CVC-metric space. Then the problem (38) can be again resumed to find the element p * ∈ D which one is a fixed point for the operator Υ. Using the hypothesis (ii) we have It is easy to check that, for both cases of the expression of ϑ(p, q), when p, q ∈ [0, 1] and otherwise, the conditions (3) and (4) are true. Then, for 0 < δ = 1 ω < 1, all the hypothesis of Theorem 2.1 holds. In this conditions we get that equation (38) has a unique solution.

Conclusions
In this article the concept of complex valued controlled metric type space is introduced. We study the connection between this new type of space and any other similar spaces with complex values, as complex valued b-metric space, or extended complex valued space, proving by some illustrative examples that our new type of space is larger than the others. The fixed point theory is used in our paper by given some fixed point theorems for Banach, Kannan and Fisher contractions type. Moreover, an application in integral equation type with complex values, is given to sustain our results.

Open questions
(1) A first open problem involve the concept of C * -algebra valued metric spaces. This notion was presented in 2014 by Ma et al. (see [22]), by interchanging the range set R by unital C * -algebra, giving more generalized class than that of metric spaces. Many other generalizations of such a structure were given in related literature, and an interesting point would be to consider the case of controlled C * -algebra valued spaces and to obtain similar fixed point results in this space.
(2) We may also write the interesting results about the concept of cyclic contraction in view to obtain some results for best proximity points in [10]. Similar results can be obtain for the case of CVCmetric space.