Author Topic: Common Fixed Point Theorems For Multivalued Compatible Maps in IFMS  (Read 1920 times)

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Author : Anil Rajput , Namrata Tripathi, Sheel Kant Gour, Seema Chouhan, Rajamani S
International Journal of Scientific & Engineering Research Volume 2, Issue 10, October-2011
ISSN 2229-5518
Download Full Paper : PDF

Abstract-The aim of this paper is to obtain the notion of multivalued weakly compatible (mwc) maps and prove common fixed point theorems for single and multi valued maps by using a contractive condition of integral type in intuitionistic fuzzy metric spaces.

Index Terms- Fixed Points , intuitionistic fuzzy metric space, multivalued weakly compatible maps, compatible maps. 
 
1.   Introduction and Preliminaries
A fundamental result in fixed point theory is  intuitionistic fuzzy metric spaces which is stated in theorem Through out the paper X will represent the intuitionistic  fuzzy metric space (X, M, N, *,  ) and CB( X) , the set of all non-empty closed and bounded sub-sets of X . For A, B  CB( X )and for every t>0,
denote H(A, B, t)=sup{M(a, b, t);a A, b B} and H(A, B, t)=inf{N(a, b, t);a A, b B}
and δM(A, B, t)=Inf{ M(a, b, t);a A, b B},
δN(A, B, t)=sup{N(a, b, t);a A, b B}
If A consists of a single point a, we write
δM(A, B, t)= δM(a, B, t) and δN(A, B, t)= δN(a, B, t).  If B also consists of a single point b, we write
δM(A, B, t)= M(A, B, t) and δN(A, B, t)= N(A, B, t)
It follows immediately from definition that
δM(A, B, t)= δM(B, A, t)≥0 and
δN(A, B, t)= δN(B, A, t)≥0
δM(A, B, t)=1  A=B={a}
δN(A, B, t)=0  A=B={a} for all A,B  CB(X)
 Definition: Maps A :X →X  and B: X→ CB (X)  are said to be multivalued weakly compatible (mwc) if there exists some point x  X such that
Ax  Bx  and ABx  BAx.
Clearly weakly compatible maps are multivalued weakly compatible (mwc).

2. Main Result
Now, we prove our main result.
Theorem 1. Let (X,M, N, *,  ) be a complete intuitionistic fuzzy metric space with  continuous
t-norm * and continuous t-corm   defined by t*t=t and (1 - t) (1 - t) ≤ (1 - t) for all t   [0, 1] such that, A :X →X  and B: X→ CB (X) be single and multi valued mappings respectively such that the maps (A,S) and (B,T) are (mwc) and satisfy the inequality for all x, y   X    where φ :[0,1]→[0,1]

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