Some Integral Type Fixed Point Theorems in Dislocated Metric Space

In this article, we establish a common fixed point theorem satisfying integral type contractive condition for two pairs of weakly compatible mappings with E. A. property and also generalize Theorem (2) of B.E. Rhoades [1] in dislocated metric space.


Introduction
In 1986, S. G. Matthews [2] introduced some concepts of metric domains in the context of domain theory.In 2000, P. Hitzler and A.K. Seda [3] introduced the concept of dislocated topology where the initiation of dislocated metric space was appeared.Since then, many authors have established fixed point theorems in dislocated metric space.In the literature, one can find many interesting recent articles in the field of dislocated metric space (see for examples [4]- [10]).
The study of fixed point theorems of mappings satisfying contractive conditions of integral type has been a very interesting field of research activity after the establishment of a theorem by A. Branciari [11].The purpose of this article is to establish a common fixed point theorem for two pairs weakly compatible mappings with E. A. property and to generalize a result of B.E. Rhoades [1] in dislocated metric space.

Preliminaries
We start with the following definitions, lemmas and theorems.
Definition 1 [3] Let X be a non empty set and let Then d is called dislocated metric (or d-metric) on X and the pair ( ) , X d is called the dislocated metric space (or d-metric space).[3] A sequence in d-metric space converges with respect to d (or in d) if there exists x X ∈ such that ( ) Lemma 1 [3] Limits in a d-metric space are unique.Definition 5 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 6 [12] 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 = for some x X ∈ implies .ASx SAx = Definition 7 [13] Let A and S be two self mappings defined on a metric space ( )

Main Results
Now we establish a common fixed point theorem for two pairs of weakly compatible mappings using E. A. property.
Theorem 1 Let (X, d) be a dislocated metric space.Let , , , : A B S T X X → satisfying the following conditions where : is a Lebesgue integrable mapping which is summable, non-negative and such that ( ) for some u X ∈ .Since ( ) ( ) From condition (2) we have where T X is closed, then there exits v X ∈ such that Tv u = .We claim that Av u = .Now from condition (2) which is a contradiction.Hence ( ) This proves that v is the coincidence point of ( ) Again, since ( ) ( ) Now we claim that Bw u = .From condition (2) where This establishes the uniqueness of the common fixed point of four mappings.Now we have the following corollaries: If we take T = S in Theorem (1) the we obtain the following corollary Corollary 1 Let (X,d) be a dislocated metric space.Let , , : A B S X X → satisfying the following conditions is a Lebesgue integrable mapping which is summable, non-negative and such that ( ) where : is a Lebesgue integrable mapping which is summable, non-negative and such that ( ) Hence, from (20), ( 23), ( 24), ( 25), ( 26), ( 27) and (28) , , is a Cauchy sequence.Hence there exists a point z X ∈ such that the sequence { } n y and its subsequences converge to z.From the condition (18) This completes the proof of the theorem.
, M v w d Sw Av d Tv Sw d Tv Av d Bw Sw d Tv Bw d u u d u u d u u d Bw u d u Bw d u u d Bw u Bw u Sw = = .This represents that w is the coincidence point of the maps B and S. Hence, u Bw Sw Tv Av = = = = Since the pairs ( ) We claim Bu u = .From condition (2) z d Sz Au d Tu Sz d Tu Au d Bz Sz d Tu Bz d z u d u z d u u d z z d u z d u z d u u d z z we get the contradiction, since , M x y d Sy Ax d Sx Sy d Sx Ax d By Sy d Sx By = 1.The pairs ( ) , A S or ( ) , B S satisfy E. A. property.2. The pairs ( ) , A S and ( ) , B S are weakly compatible.if S(X) is closed then 1) the maps A and S have a coincidence point 2) the maps B and S have a coincidence point 3) the maps A, B and S have an unique common fixed point.If we take B = A in Theorem (1) we obtain the following corollary.Corollary 2 Let (X, d) be a dislocated metric space.Let , , : A S T X X → satisfying the following conditions Let z and w two fixed point fixed points of the function f.If maximum of the given expression in the set is Similarly for other cases also we get the contradiction.Hence z = w.