Common Fixed Point Theorems of Compatible Maps in Complex Valued b -Metric Spaces

In a paper Mukheimer established some results regarding common fixed point of two self mappings in a complex valued b -metric space satisfying a rational type contractive condition. In this paper we have established a common fixed point theorem by making use of compatibility and weak compatibility of self mappings satisfying generalized rational inequality of four mappings as opposed to two self mappings in a complex valued b – metric space. Some concrete examples have also been presented to verify the effectiveness and applicability of established results. Towards the end, an application to solution of Urysohn’s integral equations has also been presented to substantiate the usability of the obtained results. Results of this paper generalize the results of Mukheimer and some other earlier results. Theorems approved in this article will be supportable for researchers to work on rational contractions with more than two pair of self mappings.


Introduction
A basic and fundamental result namely Banach contraction principle (BCP) was established by Banach [1]. He proved that a contraction map on a complete metric space always possess a unique fixed point. After this interesting result and its various applications, a huge number of generalization of this result are available in the literature by using different types of contractive conditions in various abstract spaces. By generalizing the Banach contraction principle, Jungck [2] set out tradition of common fixed point of mappings for two commuting mappings on complete metric space. After the result of Jungck [2] many authors introduced many concepts namely weak commutativity, compatibility, weak compatibility of maps (Sessa [3], Jungck [4,5], Jungck and Rhoades [6] etc.) and established results regarding common fixed point theory. In fact commutativity of maps weak commutativity of maps Ɓcompatibility of maps weak compatibility of maps, but the converse of these implications is not true. In 2011, Azam et al. [7] by introducing the notion of complex valued metric space, gave sufficient condition for the existence of some common fixed point for a pair of maps satisfying rational inequality.
As a generalization of metric space the structure of -metric space was developed by Bakhtin [8]. An analogous to the structure of -metric space, Rao et al. [9] developed the structure of complex valued -metric space and initiated the study of common fixed point of maps. After that number of researchers has proved several results regarding fixed point in context of complex valued -metric space [10][11][12][13].
In a paper, Bairagi et al. [12] by generalizing the results of Azam et al. [7], Bhatt et al. [14], Rouzkard et al. [15] and others, proved some common fixed point theorem in complex valued -metric space for a pair of mappings satisfying contractive conditions involving rational inequalities.
The aim of this paper is to prove some results regarding common fixed point of maps, by using the notion of compatibility and weak compatibility of maps in complex valued -metric space satisfying contractive conditions involving rational expression. Further we applied our results to find the solution of Uryshon's integral equations.

Preliminaries
We recall some basic definition and results which will be utilized in our subsequent discussion.
) is a complex valued -metric space with Definition 9. [9] Let ( ) be a complex valued b-metric space. Consider the following: (i) A point is called interior point of a set whenever there exists such that ( ) * ( ) + (ii) A point is called a limit point of a set whenever, for every is called open whenever each element of is an interior point of .
(iv) A subset is called closed whenever each limit point of belongs to . (v) The family * ( ) and + is a sub basis for a topology on . This topology is denoted by . Indeed, the topology is Hausdorff.

Definition 10. [9]
Let ( ) be a complex valued b-metric space and * +, a sequence in and Consider the following: (i) If for every with there is such that, for all ( ) then * + is said to be convergent, * + converges to , and is the limit point of * + We denote this and * + as (ii) If for every with there is such that, for all ( ) where then * + is said to be a Cauchy sequence. (iii) If every Cauchy sequence is convergent in ( ), then ( ) is said to be a complete complex valued b-metric space.
Proposition 18. Let and be two self mappings defined on a complex valuedmetric space ( ). Then the weak commutativity of and implies compatibility but the converse is not always true.
Proof. If and are two self maps of a complex valued -metric space. If S and T are weakly commuting maps then | ( )| | ( )| Now we take a sequence * + such that as for some Then | ( )| | ( )| as i.e. S and T are compatible maps. For the converse part, we consider the following example: Define self maps an on by and Then we see that

Main Results
Here by using the notion of compatibility and weak compatibility maps, we generalize the above results by taking four maps as opposed to two maps.
Theorem 23. Let ( ) be a complete complex valued -metric space and mappings and satisfying.
Therefore for all , we have

) Now for any and by (CVbM 3), we have
+ is a Cauchy sequence in Since is complete, therefore * + converges to point in and its subsequences * + * + * + * + are also converge to Case I Suppose that is continuous. Then = . Also by the compatibility of and from Lemma 20 Using (ii), we have

is the common fixed point of and
Case II For the 'or' part let is continuous. Then . Also by the compatiblility of and , from Lemma (2.20) .
there exist a point in such that Then from (ii), we have ( ) , yields Since and are weakly compatible on and and Using (ii), we have . Hence Therefore, is a common fixed point of and Now for the uniqueness of , suppose that be another common fixed point of and Then, from (ii), we have By the definition of self mappings we get condition (i) of the theorem 23. Now consider Since the commutativity of pairs ( ) and ( ) yields the compatibility of ( ) and weak compatibility of ( )

Urysohn Integral Equations
In this section, we studied [7,18] and some other papers and we apply our result (Theorem 23) to the existence and uniqueness of a common solution of the system of the Urysohn's integral equations. Hence the pair ( ) is weakly compatible. Similarly we can show that ( ) is weakly compatible. Thus all the conditions of Theorem (23) are satisfied. Therefore there exists a unique common fixed point of and in and consequently there exist a unique common solution of the system of integral equations (4).

Conclusion
In this article, we extended the study of fixed point theory by using the notions of compatibility and weakly compatibility of self mappings satisfying the new generalized rational type contractive conditions for four self mappings in the complete complex valued b-metric spaces. Our results generalized some earlier results exists in the literature. An illustrative example is also given to substantiate our newly proved results. Moreover we demonstrated an application in support of our main result. This idea is expected to bring wider applications of fixed point theorems which will be helpful for researchers to work in the development of fixed point theory.