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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