SOME CONDITIONS IMPLYING THE EXISTENCE OF COINCIDENCE POINTS OF A PAIR OF INTUITIONISTIC FUZZY MAPPINGS

In this article, two novel conditions implying the existence of a pair of intuitionistic fuzzy mappings with an integral type contractive conditions are discussed. An example validating the main result is given. As an application of our result, an existence theorem of solution for a class of nonlinear Volterra equation of the first type is established. This way, the work extends a few existing results in the literature .


Introduction
Let (X, d) = X be a metric space and CB(X) be the set of all nonempty closed and bounded subsets of X.Let A, B ∈ CB(X) A fuzzy set A in X is an object of the form where µ A : X −→ [0, 1] is called the membership function of A and µ A (x) is the degree of membership of x in A. The α-level set of a fuzzy set A is denoted by [A] α and is defined as: where B denotes the closure of a crisp set B.
A fuzzy mapping T is a fuzzy set-valued mapping of X into I X .A fuzzy mapping T is a fuzzy subset of X × Y with membership function T (x)(y).The function value T (x)(y) is the grade of membership of y in T (x).An x * ∈ X is called a fuzzy fixed point of T if there exists an α ∈ (0, 1] such that x * ∈ [T x * ] α .An element x * of X is called a fixed point of T if T (x * )(x * ) ≥ T (x * )(x), ∀x ∈ X.Similarly, x * is known as Heilpern fixed point of T if {x * } ⊂ T (x * ).For some of these modification of classical fixed point, see [6,8] and the reference therein.
The earliest most known result on fixed points for contractive-type mappings is the Banach's theorem introduced in 1922 (see, [8,13] ) .The Banach's contraction theorem has been extended in different directions (see, [1,2,14,15,18], and the reference therein).The idea of intuitionistic fuzzy set (IFS) was first studied in 1986 by Atanassov [7], using the concepts of t-norm and t-conorm as an extension of fuzzy set initiated by Zadeh [22] in 1965.Intuitionistic fuzzy sets do not only characterize the degree of membership of an element, but also tell the degree of nonmembership.As a result, (IFS) gained applications in the areas of modeling real life problems such as psychological investigation, career determination, to mention but a few.For some development in the field of (IFS), see [10,11,12].Thereafter, the idea of fuzzy mappings was introduced by Heilpern [16] which is also fuzzy extension of Banach contraction mapping and a direct expansion of the concept of fuzzy sets.
Along this development , Akbar [2] used generalized contractive conditions involving a rational inequality to study common fixed point theorems for fuzzy set-valued mappings.The result was further developed in [5] with particular applications in function space.As a result, some new common intuitionistic fixed point theorems for a pair of intuitionistic fuzzy mappings in a complete metric space with (α, β )-cut set of intuitionistic fuzzy sets were derived.
In this paper, we study two conditions guaranteing the existence of coincidence point of a given pair of intuitionistic fuzzy mappings.The main contractive condition used here is an extension of the work of [3,5] into integral form.In this direction, it is hoped this work will attract in a way, interests in the field of fuzzy fixed point theory.

Preliminaries
For the use of our main results, the following salient preliminary definitions and results are recalled.
Let Ψ represents the class of functions ϕ : [0, ∞) −→ [0, ∞) satisfying the following conditions: Definition 2.1.[7] Let X be a universal set.An intuitionistic fuzzy set (IFS) A in X is an object of the form : where the function µ A : X −→ [0, 1] and v A : X −→ [0, 1] define the degree of membership and nonmembership of the element x ∈ X to the set A, respectively, such that for every x ∈ X, Definition 2.2.[7] Let A be an intuitionistic fuzzy set and x ∈ X.The α-level set of A is denoted by [A] α and is defined as: of A is defined as: Definition 2.4.[21] Let X be an arbitrary set and Y a metric space.A mapping T : Lemma 2.5.[19] Let A and B be nonempty closed and bounded subsets of a metric space X.If Lemma 2.6.[19] Let A and B be nonempty closed and bounded subsets of a metric space X.
Then, for any ε > 0, and x ∈ A, there exists y ∈ B such that

Main Results
Now, we discuss the two conditions for the existence of point of coincidence of a pair of intuitionistic fuzzy mappings with an integral type contractive conditions.
Suppose that for each x ∈ X, there exists (α, β ) Sx , (α, is complete.If there exists γ ∈ (0, 1) and any ρ > 0 such that for all x, y ∈ X, Proof.Let x 0 be an arbitrary element of X. Suppose that y . This implies Hence, . This means y is the required coincidence point of S and For this . Again, by Lemma 2.6 , we can choose Continuing this process repeatedly produces x n ∈ X such that and Now, from ineqs.( 1) and (2), we get ) = 0, then by similar arguments, the conclusion is obtained.

APPLICATION
In the following result, we apply Theorem 3.1 with the completeness property of the function space (C[a, b], R) to establish an existence theorem of solution for a class of nonlinear Volterra equation of the first type.This technique is borrowed from [2].
Then, by assumption, there exists y ∈ X such that r = h • y.Consequently, Now, from eqs. ( 5) and ( 6), we obtain Notice that for γ = λ (b − a) and recalling that ρ was arbitrary, all the hypotheses of Theorem 3.1 are satisfied to find a continuous function This shows that z is a solution of the integral equation (5).
Below, as an additional application, we obtain a solution of an initial value problem(IVP) using Theorem 3.1.