Some Fixed Point Theorems For Mappings Satisfying Contractive Conditions Of Integral Type

: In this paper, we establish some common fixed-point theorems for two pairs of weakly compatible mappings satisfying integral type contractive conditions integral type in dislocated metric spaces by using E.A. which improves and extends similar known results in the literature

In 1922, S. Banach proved a fixed point theorem for contraction mapping in metric space.Since then a number of fixed point theorems have been proved by different authors, and many generalizations of this theorem have been established.Jungck [7] generalized the Banach contraction principle by introducing a contractive condition for a pair of commuting self-mappings on metric space and pointed out the potential of commuting mappings for generalizing fixed point theorems in metric spaces.Jungck's [7] results have been further generalized.
Sessa [24], initiated the tradition of improving commutativity conditions in metrical common fixed point theorems.While doing so Sessa [24] introduced the notion of weak commutativity.Motivated by Sessa [24], Jungck [8] defined the concept of compatibility of two mappings, which includes weakly commuting mappings as a proper subclass.Jungck and Rhoades [10] introduced the notion of weakly compatible (coincidentally commuting) mappings and showed that compatible mappings are weakly compatible but not conversely.Many interesting fixed point theorems for weakly compatible maps satisfying contractive type conditions have been obtained by various authors.The concept of compatible mappings was frequently used to show the existence of common fixed points.However, the study of the existence of common fixed points for noncompatible mappings is also very interesting.Aamri and Moutawakil [4] gave a notion (E.A) which generalizes the concept of noncompatible mappings in metric spaces.
Branciari [2] introduced the notion of contraction of integral type and proved first fixed point theorem for this class of mapping.Further results on this class of mappings were obtained by Rhoades [22], Aliouche [3], Djoudi and Merghadi [6] and many others.
. Matthews [11] introduced some concepts of metric domains in the context of domain theory The notion of a dislocated metric (d-metric) space was introduced by Pascal Hitzler in [12] as a part of the study of logic programming semantics.The study of common fixed point mappings in dislocated metric space satisfying certain contractive conditions has been at the center of vigorous research activity, see for example in [13][14][15][16][17][18][19][20][21].
In this article, we have established some common fixed point results of integral type contractive conditions using the concept of weakly compatible mappings with (E.A.) property in dislocated metric (d-metric) space.Our obtained results generalizes some well known results of the literature.

Preliminary Notes
We begin by recalling some basic concepts of the theory of dislocated metric (d-metric) spaces.
Definition 2.1 Let X be a non empty set and let be a function satisfying the following conditions: Then is called dislocated metric (or simply d-metric) on .Obviously a commuting pair is weakly commuting but its converse need not be true as is evident from the following example., whenever is a sequence in such that for some .
Very recently concept of weakly compatible obtained by Jungck-Rhoades [10] stated as the pair of mappings is said to be weakly compatible if they commute at their coincidence point.Definition 2.9: Let S and T be two self mappings of a d-metric space .We say that S and T satisfy the property (E.A) if there exist a sequence such that ,for some u .
Proposition 2.1 Let S and T be compatible mappings from a d-metric space into itself.Suppose that for some .
if S is continuous then .
Theorem 2.1 Let be a complete d-metric space and let be a contraction mapping, then T has a unique fixed point.

Main Results
Now, we establish a common fixed point theorem for two pairs of weakly compatible mappings using E. A. property.Theorem 3.1 Let be a complete dislocated metric space.Let satisfying the following conditions (i) and …(1) for all where is a Lebesgue integrable mapping which is summable, non-negative and such that for all .… (3) …( 4) (iii) The pairs or satisfy E.A. property.
(iv) The pairs and are weakly compatible.
If is closed then the mappings AB,ST,I and J have a unique common fixed point in X.Furthermore, if the pairs (A,B), (A,I), (B,I), (S,T), (S,J) and (T,J) are commuting mappings then A, B, S, T, I and J have a unique common fixed point in X.

Proof:
Assume that the pair satisfy E. A. property, so there exists a sequence Such that …(5) For some .Since , so there exists a sequence such that .Hence, … (6) From condition (2), we have , … where Taking limit as , we get Since Therefore, we have , which is a contradiction, since .Hence .Since , so there exists a sequence such that .Hence, we have Assume is closed, then there exists such that .We claim that .Now from condition (2), , where …(9) Since So, taking limit as in ( 9), we conclude that … (10) which is a contradiction, since .Hence, .
Now, we have … (11) This proves that is the coincidence point of .
Again so there exists such that Now, we claim that .From condition (2), we have , where Hence, , which is a contradiction, since .
Hence, .Therefore, .This represents that is the coincidence point of the maps and .Hence, Since the pair and are weakly compatible so, , Since and , we claim that .From condition (2), we have , where Hence, , which is a contradiction, since .
Hence, .Therefore, .Similarly, .Hence, .This represents that is a common fixed point of the mappings and .

Uniqueness:
If possible, let be other common fixed point of the mappings, then by the condition (2) , where

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Hence, , which is a contradiction, since .
Hence, .This establishes the uniqueness of the common fixed point of mappings and .Finally, we prove that is also a common fixed point of A, B, S, T, I and J. Let both the pairs and have a unique common fixed point .Then, which implies that has common fixed points which are and .We get thereby .Similarly, using the commutativity of and , can be shown.Now, we need to show that .By using condition (2), we have where , Therefore, , which is a contradiction, since .
Hence, .Similarly, can be shown.Consequently, is a unique common fixed point of A, B, S, T, I and J.
If we put AB = A, ST = B in Theorem (3.1), we get the following, which generalize the result of Panthi and Subedi [20] in dislocated metric spaces.We have, .Therefore, .Similarly, .Hence, .This represents that is a commpn fixed point of the mappings and .

Uniqueness:
If possible, let be other common fixed point of the mappings, then by the condition (13) , where M(u, z) Hence Since and So if or or we get the contradiction, since or and .
We have, .This establishes the uniqueness of the common fixed point of mappings and .Finally, we prove that is also a common fixed point of A, B, S, T, I and J.
Let both the pairs and have a unique common fixed point .Then, which implies that has common fixed points which are and .We get thereby .Similarly, using the commutativity of and , can be shown.Now, we need to show that .By using condition ( 13

Example 2 . 1 .Definition 2 . 8 .
Consider the set X = [0, 1] with the usual metric.Let and for every .Then for all , .Hence .Thus S and T do not commute.Again , and so, S and T commute weakly.Obviously, the class of weakly commuting is wider and includes commuting mappings as subclass.Two self mappings S and T from a d-metric space into itself are called compatible if and only if
. Consequently, is a unique common fixed point of A, B, S, T, I and J.If we put AB = A, ST = B in Theorem (3.2), we get the following, which generalize the result of Panthi and Kumari[20] in dislocated metric spaces.integrable mapping which is summable, non-negative and such that for all .(iii)The pairs or satisfy E.A. property.(iv) The pairs and are weakly compatible.