COMMON FIXED POINT THEOREMS FOR RATIONAL TYPE CONTRACTION IN PARTIALLY ORDERED METRIC SPACE

In this paper we prove some common fixed point theorems for two and four self-mappings using rational type contraction and some newly notified definitions in partially ordered metric space. In this way we generalized, modify, and extend some recent results due to Chandok and Dinu [14], Shantanwi and Postolache[29] and many others [1, 2, 4, 5, 21, 29, 30], thus generalizing results of Cabrea, Harjani and Sadarangani [12] as well as Dass and Gupta [15] in the context of partial order metric setting.


Introduction
Turinici [31,32] investigated fixed point theorem for monotone mappings on metrizable uniform spaces and quasi ordered metric spaces. Afterwards, Ran and Reuring [26] extended the Banach contraction principle to partially ordered metric space with some application to matrix equations. Further, Nieto and Rodriguez-Lopez [22,23] generalized the theorem for increasing mapping not requiring the necessity of continuity and gave application to existence and uniqueness of a lower solution to first order ordinary differential equation. Thereafter, Nieto and Rodriguez-Lopez's theorems were generalized by many researchers [3,6,13,17,24,33,34].
Sessa [27] introduced the concept of weak commutative condition of mappings which are generalization of commutative maps [20] in metric space. Jungck [18] generalized this idea of weak commutative mappings by introducing compatible mappings. Further, Jungck [19] introduced the concept of weakly compatible mappings and proved some common fixed point results for these mappings.
In recent years many researchers have generalized the existence of fixed point and common fixed point theorems for generalized weak contractions in partial ordered metric space [7-11, 21, 25]. Most recently Shatanawi and Postolache [28] proved common fixed point theorems for dominating and weak annihilator mappings in ordered metric space for four self mappings, meanwhile Cabrera, Harjani and Sadarangani generalized the Dass and Gupta's theorem in the partial ordered metric space. Very recently Alam et al. [5] identified some more natural definitions in view of Turinici [33,34] on ordered metric setting.
Our aim in this paper is to modify some recent common fixed point theorems for self-mappings using some newly identified ordered metric definitions, and rational type contraction.

Preliminaries
Definition 1. (a) A non empty set together with partial order relation (reflexive, anti-symmetric and transitive) is said to be an ordered set or partially ordered set. We say is comparable to if either or and denoted as symbolically.
(b) A set is said to be totally ordered or linearly ordered if every pair of elements of are comparable.
(c) Triplet is said to be an ordered metric space or partially ordered metric space if is a metric endowed with partial ordering in a nonempty set.
By assumption (b), we have , , . ( Thus for all we have . Hence by using (d), and putting , we get , Similarly it can be proved that .
Now for any positive integer and with we have .
Hence we conclude that is a Cauchy sequence. Now suppose is a subsequence of then by assumption (e), there exists such that .
Assume that is -continuous and in view of assumption (f), we have Also, .
Letting in the above inequality, and using (10), we have , A contradiction.
Now and , so by assumption we have and (d) becomes .
Again letting in the above inequality and using (10) Letting , and if then and hence .
If in the above inequality then using (10), we have , (19) or This is a contradiction. Hence . (20) Therefore , that is is a common fixed point of and .
i.e. is a common fixed point of and .
Therefore assuming , in view of (d) we have , , .
Using mathematical induction we have .
Hence i.e. is a coincidence point of and . Again in view of assumption (h), we arrive at equation (24).
For uniqueness, let us assume that , and but . Consider (23)