COMMON FIXED POINT FOR SIX SELF MAPS IN METRIC SPACE

COMMON FIXED POINT FOR SIX SELF MAPS IN METRIC SPACE BATHINI RAJU AND V. NAGARAJU Department of Mathematics, University Post Graduate College, Secunderabad Osmania University, Hyderabad-500003 (Telangana), India Department of Mathematics, University College of Engineering (Autonomous) Osmania University, Hyderabad-500007 (Telangana), India Copyright © 2018 Raju and Nagaraju. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract: In this paper we obtain a common fixed point theorem for six self maps in a metric space without completeness. We also give an example in support of our result.


Introduction and Preliminaries
In 1986, G. Jungck [1] introduced the concept of compatible maps as follows.[1]: Two self maps E and F of a metric space (X, d) are said to be compatible mappings if lim

=
= for some t X  .Further Jungck and Rhoades [4] defined weaker class of maps called weakly compatible maps and is defined as follows.[4]: Two self maps E and F of a metric space (X, d) are said to be weakly compatible if they commute at their coincidence point.i.e, if Eu = Fu for some uX  then EFu = FEu.

Weakly Compatible mappings
It is clear that every pair of compatible maps is weakly compatible but not conversely.

Example:
Let X= (-1, 1] with the usual metric ( , )   d x y x y = − for all x, yX.Define self mappings E and F of X by Hence E and F are weakly compatible.
therefore E and F are but not compatible.
1.4 Associated sequence [6]: Suppose E,F,G,H,I and J are six self maps of a metric space ( , ) Xdsuch that ( ) .Then for an arbitrary 0 , there exists 1 x , there is a point 2 x X  such that for all x,y in X where , 0, 1.
Furtherif X is complete, then for any 0 x X  and for any of its associated sequence converges to some point p in X .
Proof: From the conditions (2.1) and ( 2. ) That is Now for any positive integer k, we have y is a Cauchy sequence in X.Since X is a complete, it converges to some pointp in X.

Remark:
The converse of the above Lemma is not true.That is, if E, F, G, H, I and J are self maps of metric space(X, d) satisfying (2.1), (2.2) and even if for any 0 x in X and for any of its associated sequence of converges.The metric space need not be complete.This can be seen from the following example.

Example:
Let X=(-1, 1] with the usual metric ( , )   d x y x y = − for all x, yX.Define self mappings E, F, G, H, I and J of X by  Then Also the inequality (2.2) can easily be verified for appropriate values of , 0, 1.

   
 +  Moreover if we take .Note that (X, d) is not complete.
The following theorem was proved in [5].
2.8 Theorem: Let P, Q, S and T be self mappings from a complete metric space ( , )  Xd into itself satisfying the following conditions Now we extend and generalize the above Theorem to six self maps as follows.

Main result
3.1 Theorem: If E, F, G, H, I and J are self maps of a metric space ( , ) Xd satisfying the conditions Further if there is a point 0 x X  and its associated sequence ..........} Jp  , since α, β  0, α + β < 1 and this implies Jp = p.Thus IJp = p  Ip =p.