Common New Fixed Point Theorem in Modified Intuitionistic Fuzzy Metric Spaces Using Implicit Relation

In this paper, we prove some common fixed point theorems for two pair of compatible and subsequentially continuous mappings satisfying an implicit relation in Modified Intuitionistic fuzzy metric spaces. Consequently, our results improve and sharpen many known common fixed point theorems available in the existing literature of metric fixed point theory.


Introduction
The concept of fuzzy set was introduced in 1965 by Zadeh [1].In 1986, with similar endeavour, Atanasov [2] introduced and studied the concept of Intuitionistic fuzzy sets (IFS).Using the idea of IFS, a generalization of fuzzy metric space was introduced by Park [3] which is known as Intuitionistic fuzzy metric space.Since the Intuitionistic fuzzy metric space has extra conditions (see [2]), Saadati et al. [4] reframed the idea of Intuitionistic fuzzy metric space and proposed a new notion under the name of Modified Intuitionistic fuzzy metric space by introducing idea of continuous t-representable.
In 1986, Jungck [5] introduced the notion of compatible maps for a pair of self mappings.Jungck et al. [6] initiated the study of weakly compatible maps in metric space.With a view to improve commutativity conditions in common fixed point theorems, Sessa [7] introduced the notion of weakly commuting pair.Most recently, Bouhadjera et al. [8] (see also [9]) introduced two new notions namely: subsequential continuity and subcompatibility.
In this paper, we prove some common fixed point theorems for two pair of compatible and subsequentially continuous mappings satisfying an implicit relation in modified Intuitionistic fuzzy metric spaces.Consequently, our results improve and sharpen many known com-mon fixed point theorems available in the existing literature of metric fixed point theory and generalize the results of D. Gopal et al. [10, Theorem 3.1 and Theorem 3.2].

Definition 2.1. [12]
A triangular norm (t-norm) on L* is a mapping satisfying the following conditions: for all x, y X and t > 0. The 3-tuple is said to be a Modified Intuitionistic fuzzy metric space if X is an arbitrary non empty set, F is a continuous t-representable and satisfying the following conditions for every x, y X and t, s > 0: , , 1 iff , , , ,


x y t M x y t N x y t   .
In the sequel, we will call   , , , for all t.
A Modified Intuitionistic fuzzy metric space F  is said to be complete iff every Cauchy sequence is converges to a point of it.Definition 2.6.[14] Let f and g be maps from a Modified Intuitionistic fuzzy metric space into itself.The maps f and g are said to be weakly com- Definition 2.8.[13] Two self-mappings f and g are called non-compatible if there exists at least one se- such that lim lim

Main Results
Implicit relations play important role in establishing of common fixed point results.
Let M 6 be the set of all continuous functions   , and a contradiction to (A) so that z = w.Now, we assert that Az = z, if not, then by (3.3), we get , , which is a contradiction to (B).Therefore, Az = z = Sz.
Similarly, we prove that Bz = z = Tz by using (3.3).Therefore, in all, z = Az = Bz = Sz = Tz.i.e. z is common fixed point of A, B, S and T. The uniqueness of mmon fixed point is an easy consequence of the inequality (3.3).This completes the proof of the theorem.□ Theorem 3.2.Let A, B, S and T be four self mappings of a Modified Intuitionistic fuzzy metric space

 
, ( , , ) * 0 0 ( ) ( ) ; for all and summable and satisfies 1 By setting A = B in Theorems 3.1, 3.2, we deri following corollaries for three mappings.(3.10) for any , x y X  ,  in M 6 and for all t > 0, the pair (A procally continuous mappings, then (3.8) and (3.9) satisfied.Further, A, S and T have a unique common fixed point provided A, S and T satisfy the condition (3.10).
Alternatively, by setting S = T in Theorems 3.  F .If the pairs (A,S) and (B,S) are compati- ble and subsequentially continuous mappings, then (3.11) the pair (A, S) has a coincidence point, (3.12) the pair (B, S) has a coincidence point.Further, A, B and S have a unique common fixed point provided A, B and S satisfy the following: (3.13) for any , x y X  ,  in M 6 and for all t > 0, , , , , , ,

Lemma 2 . 1 .
[11] Consider the set L* and the operation ≤ L* defined by


is said to be compatible if * L satisfying the following conditions (for all Copyright © 2013 SciRes.

F.
If the pairs (A,S) and (B,T) are subcom , M N tible and reciprocally continuous mappings, then (3.4) the pair (A, S) has a coincidence point, (3.5) the pair (B, T) has a coincidence point.Further, A, B, S and T have a unique common fixed point provided A, B, S and T satisfy the cond -Proof easily follows on same lines of Theorem 3.1 and using definition of reciprocally continu mappings.□ Corollary 3.1.The conclusions of Theorem 3.1 eorem 3.2 remain true if we replace the inequ any one of the following:

.
If the pairs (A,S) and (A,T) are compatiquentially continuous mappings, then (3.8) the pair (A, S) ha oint, he pai Further, A, B, S and T have a unique common fixed point provided A, B, S and T satisfy the following:

3 . 4 .
) and (A,T) are subcom e derive the following corollaries for three mappings.Corollary Let A, B and S be three self mappings of ntuitionistic fuzzy metric space

F
S) and (B,T) are compatible as well as subus.Therefore, all the conditions of 3.1 are satisfied.Evidently, z = 0 is a coincis well as unique common fixed point of A, B, S and T. be a Modified Intuitionistic fuzzy

Definition 2.4. [4] A sequence {x n } in a Modified In- tuitionistic fuzzy metric space  
Fis said to be weakly compatible if they commute at coincidence points i.e. if fu = gu for some u  X, then fgu = gfu.) if the pair (f, g) commutes at least one coincidence point i.e. there exists at least one point x  X such that fx = gx and fgx = gfx.
Definition 2.11.[9] Let f and g be maps from a Modified Intuitionistic fuzzy metric space into itself.The maps f and g are said to be subcompatible if there exist a sequence   , , , X  F into itself.The maps f and g are said to be reciprocally continuous if for a sequence   n x in X then (3.14) is satisfied.Further, A and S have a unique common fixed point provided A and S satisfy the condition (3.15).