Common Fixed Points of Single and Multivalued Maps in Fuzzy Metric Spaces

In this paper we introduce the notion of common property (EA) in fuzzy metric spaces. Further we prove some common fixed points theorems for hybrid pair of single and multivalued maps under hybrid contractive conditions. Our results extend previous ones in fuzzy metric spaces.


Introduction
In 1965 Zadeh [1] introduced the theory of fuzzy sets.Many authors introduced the notion of fuzzy metric space in different ways.George and Veeramani [2] modified the concept of fuzzy metric space introduced by Kramosil and Michalek [3] and defined Haussdorf topology in fuzzy metric space.Several authors [4][5][6][7][8][9][10][11] studied and developed the concept in different directions and proved fixed point theorems in fuzzy metric spaces.
In 1986 Jungck [12] introduced the concept of compatible mappings and utilized it to improve and generalize the commutativity conditions employed in common fixed point theorems.This induced interest in non-compatible mappings initiated by Pant [13].Recently Aamri and Moutawakil [14] and Liu et al. [15] respectively defined the property (E.A) and the common property (E.A) as a generalization of non-compatibility and proved some common fixed point theorems in metric spaces.The aim of this paper is to define the common property (E.A) in the settings of fuzzy metric space and utilize the same to obtain some common fixed point theorems in fuzzy metric spaces.
We begin with some definitions and preliminary concepts.

 
, ,0 for all if and only if We denote the set of all coincidence points of f and T by .

 
,T : C f Definition 2.11.[16] Maps f X X  and are weakly compatible if they commute at their coincidence points,that is, if Definition 2.12.[18] Maps

Main Results
We begin with the following definition.Definition 3.1.[11] Let be a fuzzy metric space and  , ,* X M  , ., : and G have a common fixed point provided that both and  are true.implies that Copyright © 2011 SciRes.

AM
Taking the limit as , we get

M fv A t M gw Gw t M fv Gw t M gw A t M A Gw t M fv gw t
But from (2.5), we have , ,  Combining the inequalities ( 3) and ( 4) we get Hence  On the other hand by condition (2), we have

M fv Fv t M gy Gy t M fv Gy t M gy Fv t M Fv Gy t M fv gy t
Taking limit as , we get and let F be a map from X into   , ,  , ,  , ,  , , > min  , , ,  ,  2 2

M fx Fx t M fy Fy t M fx Fy t M fy Fx t M Fx Fy t M fx fy t
and let F and be two maps from

M fx Fx t M fy Gy t M fx Gy t M fy Fx t M Fx Gy t M fx fy t
Pro et = f g , then the result follows.
F and are single valued maps in theorem 3.3, then we have the following corollary.
Corollary 3.6.Let and be four selfmaps of the fuzzy    . Let , f be two self maps of Theorem 3. 7 the and  ,  g G satisfy the common property   EA , there exist two sequences   fuzzy metric on X .The functions The maps pair   , f F and   , g G satisfy the common property