A common fixed point theorem for Reich type co-cyclic contraction in dislocated quasi-metric paces

In this paper we proved the existence of coincidence and common fixed points for Reich type co-cyclic contraction in dislocated quasi-metric space and also, show the uniqueness of the common fixed point. Our work extends the main result in (Karapinar and Erhan, 2011). Examples are also provided in support of our results. Keywords/Phrases: Dislocated quasi-metric space, coincidence points, common fixed point, Reich type co-cyclic contraction DOI: http://dx.doi.org/10.4314/ejst.v10i2.1


INTRODUCTION
The Banach Contraction Principle is a very popular tool for solving existence problems in many branches of Mathematical Analysis and its applications.There are many generalizations of this fundamental theorem.Some of the generalizations weaken the contractive nature of the map; for example see (Kannan, 1968;Kannan, 1969;Jungck, 1976;Sessa, 1982;Kirk et al., 2003;Karapinar and Erhan, 2011), and others.In other generalizations the ambient space is weakened; see (Zeyada et al., 2005;Aage and Saluke, 2008;Abbas et al., 2011;Chaipunya et al., 2012;Zoto et al., 2012;Panthi et al., 2015).This celebrated theorem can be stated as follows.
Theorem 1.1 (Banach, 1922): Let be a complete metric space and be a mapping of into itself satisfying: for all (1.1) where Then, T has a unique fixed point Inequality (1.1) implies continuity of T. A natural question is whether we can find contractive conditions which will imply existence of a fixed point in a complete metric space but do not imply continuity.
Furthermore, Zeyada et al. (2005) generalized the results of Hitzler and Seda (2000) and introduced the concept of complete dislocated quasi metric space.Aage andSalunke (2008, 2008a) derived some fixed point theorems in dislocated quasi metric spaces.

PRELIMINARIES
We recall the definition of complete metric space, quasi metric space, dislocated metric space, dislocated quasi metric space, the notion of convergence and other results that will be needed in the sequel.
Definition 2.1 (Zeyada et al., 2005): Let X be a non-empty set.Suppose that the mapping d: satisfies the following conditions: Here we note that every metric space are quasi metric space, dislocated metric space and dislocated quasi metric space but the converse is not necessarily true and every dislocated metric space are dislocated quasi metric space but the converse is not always true (Zeyada et al., 2005).
Definition 2.2 (Zeyada et al., 2005): A sequence { } in a dq-metric space is called Cauchy sequence if for all ε > 0, ∃ such that for , we have .
Definition 2.3 (Zeyada et al., 2005): A sequence in a dq-metric space converges with respect to dq, if there exists in , such that = = 0.In this case, is called a dq limit of { } and we write as .
Definition 2.4 (Zeyada et al., 2005): A dq-metric space is called complete if every Cauchy sequence in it is convergent in with respect to dq.
Definition 2.6 (Zeyada et al., 2005): Let be a dq-metric space.A mapping is called contraction if there exists such that for all Theorem 2.7 (Zeyada et al., 2005): Let be a complete dq-metric space and let be a contraction mapping.Then, has a unique fixed point.
Definition 2.8 (Kirk et al., 2003): Let and be non-empty subset of a dq-metric space and be a self -map. is said to be dq-cyclic map if and only if and and is said to be dq-cyclic contraction if there exists such that for all in and in .
Definition 2.9 (Jungck and Rhoades, 1998): Let X be a non-empty set.Two self-maps : are said to be i.

Commuting if for all in If
for some in then is called coincidence point of and .We denote the set of coincidence points of and by . ii.
Weakly compatible if they commute at their coincidence points .i.e. if in such that then .Example 2.1 Let be equipped with a dq-metric = .
Define by and .
Then for any in , showing that f, T are weakly compatible maps on and x = is a common fixed point of T and f." Proof: Let (fix) there exist such that (say).Since , there exist such that (say).
On continuing this procedure inductively, we get a sequence n in such that for each , where and for each Similarly, let with .By a similar procedure, we obtain (3.12) Taking n in (3.12), we obtain Thus, } is a Cauchy sequence in X.
Since is complete, there exists in such that = z.Example 3.4: .Let and

Jungck ( 1976 )
proved a common fixed point theorem for commuting maps by generalizing the Banach's fixed point theorem.The concept of the commutativity has been generalized in several ways.For this,Sessa (1982) has introduced the concept of weakly commuting mappings andJungck (1986) initiated the concept of compatibility.When two mappings are commuting then they are compatible but not conversely.Jungck and Rhoades (1998) introduced the notion of weakly compatible mappings and showed that compatible maps are weakly compatible but not conversely.The study of common fixed point of mappings satisfying contractive type conditions has been a very active field of research activity.Chaipunya (2012) introduced co-cyclic contractions as follows which is a guarantee for common fixed point theorem of a pair of self-mappings.Definition 1.4 (Chaipunya, 2012): Let be two self-mappings. is said to be cocyclic representation between T and if the following conditions are satisfied : i.Both and are nonempty subsets of ii. .Karapinar and Erham (2011) introduced the following definition and established the theorem following it.Definition 1.6 (Karapinar and Erham, 2011): Let and B be non-empty subsets of a metric space A cyclic map is said to be Reich type cyclic contraction if: b, c are non-negative real numbers satisfying .Theorem 1.7 (Karapinar and Erham, 2011): Let and be non-empty closed subsets of a complete metric space and be a Reich type cyclic contraction.Then has a unique fixed point in .Inspired and motivated by the result of Karapinar and Erhan (2011) in this paper, we prove the existence of coincidence points and common fixed points of a pair of self-mappings satisfying the conditions of Reich type co-cyclic contraction in dislocated quasi-metric space.Also, the uniqueness of the common fixed points has been shown.An example has been provided in support of our main result.
pair is called metric space.If d satisfies d 1 , d 2 and d 4 , then (X, d) is called quasi metric space(Wilson, 1931).If d satisfies d 2 , d 3 and d 4 , then (X, d) is called dislocated metric space(Hitzler and Seda, 2000).If d satisfies d 2 and d 4 , then (X, d) is called dislocated quasi metric space and we denoted it by dq-metric space(Zeyada et al., 2005).

Definition 2 .
10 (Jungck and Rhoades, 1998): Let, : .If , then z is called point of coincidence of and ; and w is called coincidence point of and .If , then is called a common fixed point of and .
are two coincidence points of and .Note that and commute at 0, i.e., but and and so and are not weakly compatible mappings on .Next we state and prove the main result of this paper.Let and be non-empty subsets of a dislocated quasi metric space .The selfmap is said to be a dq-Reich type co-cyclic contraction if there exists a selfmap such that i. is a co-cyclic representation of between and ii. for all x A and y B, where a, b, c are nonnegative numbers such that Remark: In Definition 3.1(ii), if b = c = 0, the pair of maps is said to be -dq co-cyclic contraction and if a = 0, we call them -dq-Kannan type co-cyclic contraction.Theorem 3.2: Let A and B be non-empty subsets of a complete dislocated quasi dq-metric space .Let be a dq-Reich type co-cyclic contraction.If is injective and and are closed subsets of , where and are weakly compatible mappings, then and have a unique common fixed point in .

Now
each n = 0, 1, 2, ⋯, we have is a common fixed point of and in Now, we show the uniqueness of .Let w be another common fixed point of and .(3.19) and (3.20) we obtain .Therefore, is a unique common fixed point of and in Corollary 3.2: Let A and B be non-empty subsets of a complete dislocated quasi (dq)-metric space .Let be a dq co-cyclic contraction.If is injective and and are closed subsets of , where and are weakly compatible mappings, then and have a unique common fixed point in .Proof: Taking and , the proof follows from Theorem 3.2.Corollary 3.3: Let A and B be a non-empty subsets of a complete dislocated quasi dq-metric space .Let be a Kannan type dq co-cyclic contraction.If is injective and and are closed subsets of , where and are weakly compatible mappings, then and have a unique common fixed point in .Proof: Taking and , the proof follows from Theorem 3.2.Remark 1: If is the identity mapping, Theorem 3.2 become Theorem 1.7.This shows that Theorem 3.2 is an extension of Theorem 1.7.The following is an example in support of Theorem 3.2.