Common Fixed Point Theorem in Intuitionistic Fuzzy Metric Spaces Satisfying Integral Type Inequality

We prove common fixed point theorem for weakly compatible maps in Intuitionistic fuzzy metric space satisfying integral type inequality but without using the completeness of space or continuity of the mappings involved. We prove by using the concept of E A property.


Introduction
In the study of common fixed points of compatible mappings we often require assumption on completeness of the space or continuity of mappings involved besides some contractive condition but the study of fixed points of non comapatible mappings can be extend to the class of non expansive or Lipschitz type mapping pairs even without assuming the continuity of the mappings involved or completeness of the space. Aamri and El Moutawakil [6] generalized the concepts of non comapatibility by defining the notion of (E.A) property and proved common fixed point theorems under strict contractive condition.
We prove common fixed point theorems for weakly compatible maps in Intuitionistic fuzzy metric space by using the concept of (E.A) property, however, without assuming either the completeness of the space or continuity of the mappings involved.

Preliminaries
Definition:-A binary operation is continuous t-norm if is satisfying the following condition: (i) * is commutative and associative ; (ii) * is continuous ; (iii) ; (iv)  Then is an intuitionistic fuzzy metric space.
Definition: -Let and be two self maps of an intuitionistic fuzzy metric space are said to be compatible if and whenever } is a sequence in X such that as ,for some Definition :-Two self maps U and V of intuitionistic fuzzy metric space are said to be weakly compatible if they commute at their coincidence point, i.e. whenever

Definition:-Let A and B be two self-maps of a intuitionistic fuzzy metric space We say that A and B satisfy the property (E.A) if there exists a sequence { } such that
Note that weakly compatible and property (E.A) are independent to each other.

Bulletin of Mathematical Sciences and Applications Vol. 12
Definition: -Let f and g be two self maps of a metric space and and to be weakly commuting if for all.
It can be seen that commuting maps are weakly compatible, but converse is false.
In 2002, A.Branciari [1] analyzed the existence of fixed point for mapping T defined on a complete metric space satisfying a general contractive condition of integral type in the following theorem.

Theorem[1]
:-Let (X; d) be a complete metric space, and let be a mapping such that for each where is a Lesbesgue-integrable mapping which is summable (i.e. with finite integral) on each compact subset of [0;+∞), non-negative, and such that for each ε > 0, then T has a unique fixed point such that for each , After the paper of Branciari [1], a lot of research works have been carried out on generalizing contractive conditions of integral type for different contractive mappings satisfying various known properties. A fine work has been done by Rhoades [2] extending the result of Branciari [1] by replacing the condition (i) by the following

Main Results run as
Theorem:-Let and be two weak -compatible self maps of intuitionistic fuzzy metric space satisfying the property E.A. and Whenever If the range of or is complete subspace of X, then f and g have a common fixed point This is contradiction and so Hence is a common fixed point of .
The case when is a complete subspace X is similar to the above since .

Conclusion
We prove fixed point theorem for weakly compatible maps in intuitionistic fuzzy metric space satisfying integral type inequality by using E.A. property.