A Common Fixed Point Theorem for Two Pairs of Mappings in Dislocated Metric Space

Dislocated metric space differs from metric space for a property that self distance of a point needs not to be equal to zero. This property plays an important role to deal with the problems of various disciplines to obtain fixed point results. In this article, we establish a common fixed point theorem for two pairs of weakly compatible mappings which generalize and extend the result of Brain Fisher [1] in the setting of dislocated metric space with replacement of contractive constant by contractive modulus for which continuity of mappings is not necessary and compatible mappings by weakly compatible mappings.


Introduction
In 1922, S. Banach [2] established a fixed point theorem for contraction mapping in metric space.Since then a number of fixed point theorems have been proved by many authors and various generalizations of this theorem have been established.In 1982, S. Sessa [3] introduced the concept of weakly commuting maps and G. Jungck [4] in 1986, initiated the concept of compatibility.In 1998, Jungck and Rhoades [5] initiated the notion of weakly compatible maps and pointed that compatible maps were 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.In 1986, S. G. Matthews [6] introduced the concept of dislocated metric space under the name of metric domains in domain theory.In 2000, P. Hitzler and A. K. Seda [7] generalized the famous Banach Contraction Principle in dislocated metric space.The study of dislocated metric plays very important role in topology, logic programming and in electronics engineering.
The purpose of this article is to establish a common fixed point theorem for two pairs of weakly compatible mappings in dislocated metric spaces which generalize and improve similar results of fixed point in the literature.

Preliminaries
We start with the following definitions, lemmas and theorems.Definition 1. [7] Let X be a non empty set and let × → ∞ be a function satisfying the following conditions: 2) ( ) ( ) Then d is called dislocated metric (or d-metric) on X and the pair ( )  [7] Limits in a d-metric space are unique.Theorem 1. [7] Let ( ) , X d be a complete d-metric space and let : T X X → be a contraction mapping, then T has a unique fixed point.Definition 6.Let A and S be two self mappings on a set X. Mappings A and S are said to be commuting if ASx SAx = x X ∀ ∈ .Definition 7. Let A and S be two self mappings on a set X.If Ax Sx = for some x X ∈ , then x is called coincidence point of A and S. Definition 8. [5] Let A and S be mappings from a metric space ( ) , X d into itself.Then, A and S are said to be weakly compatible if they commute at their coincident point; that is, Ax Sx It is clear that every continuous function is upper semicontinuous but converse may not be true.In 1983, B. Fisher [1] established the following theorem in metric space.Theorem 2. Suppose that S, P, T and Q are four self maps of a complete metric space (X, d) satisfying the following conditions 2) Pairs (S, P) and (T, Q) are commuting.
3) One of S, P, T and Q is continuous.Then S, P, T and Q have a unique common fixed point z X ∈ .Also, z is the unique common fixed point of pairs (S, P) and (T, Q).

Main Results
Theorem 3. Let (X, d) be a complete d-metric space.Suppose that A, B, S and T are four self mappings of X satisfying the following conditions where φ is an upper semicontinuous contractive modulus and Now by condition ii), we have ( , , , , , , , Since φ is upper semicontinuous, contractive modulus the Equation ( 1) implies that the sequence n n d y y + is monotonic decreasing and continuous.
Hence there exists a real number 0 t ≥ such that ( ) is not a cauchy sequence.Then there exists a real number 0 ε > and subsequences i q and i p such that Taking limit as i → ∞ we have ( ) So by contractive condition ii) and ( 2) p q p q p q q q q p q p p q q p q q p p q p p q q p m x Now taking limit as i → ∞ we get ( ) Therefore from (3) we have   Thus Sv u = .Hence Av Sv u = = which represents that v is the coincidence point of A and S.
φ ≤which is a contradiction, since φ is a contractive modulus.