Common New Fixed Point Theorem in Modified Intuitionistic Fuzzy Metric Spaces Using Implicit Relation ()
1. 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 common 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].
2. Preliminaries
Lemma 2.1. [11] Consider the set L* and the operation ≤L* defined by

for every (x1, x2), (y1,y2)
L*. Then (L*, ≤L*) is a complete lattice.
We denote its units by 0L* = (0,1) and 1L* = (1,0).
Definition 2.1. [12] A triangular norm (t-norm) on L* is a mapping
satisfying the following conditions:
(1)
for all x
L*(2)
for all x, y
L*(3)
for all x, y, z
L*(4) If for all x,
, y,
L*,
and
implies
.
Definition 2.2. [11,12] A continuous t-norm F on L* is called continuous t-representable iff there exist a continuous t-norm * and a continuous t-conorm
on [0, 1] such that for all 

Definition 2.3. [4] Let M, N are fuzzy sets from
such that
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,
is a continuous t-representable and
is a mapping
satisfying the following conditions for every x, y
X and t, s > 0:
(a)
;
(b)
;
(c)
;
(d)
;
(e)
is continuous.
In this case,
is called an Intuitionistic fuzzy metric. Here,
.
In the sequel, we will call
to be just a Modified Intuitionistic fuzzy metric space.
Remark 2.1. [13] In Modified Intuitionistic fuzzy metric space
,
is non decreasing and
is non-increasing for all x, y
X. Hence
is non-decreasing with respect to t for all x, y
X.
Definition 2.4. [4] A sequence {xn} in a Modified Intuitionistic fuzzy metric space
is called a Cauchy sequence if for each
and t > 0, there exists
such that
for each
and for all t.
Definition 2.5. [4] A sequence {xn} in a Modified Intuitionistic fuzzy metric spac
is said to be convergent to x
X, denoted by
if
for all t.
A Modified Intuitionistic fuzzy metric space
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 commuting if
for all
and 
Definition 2.7. [4] A pair of self mappings (f, g) of Modified Intuitionistic fuzzy metric space
is said to be compatible if
whenever
is a sequence in X such that
for some z
X.
Definition 2.8. [13] Two self-mappings f and g are called non-compatible if there exists at least one sequence
such that
for some z
X but either
or the limit does not exist for all z
X.
Definition 2.9. [15] A pair of self mappings (f, g) of Modified Intuitionistic fuzzy metric space
is said to be weakly compatible if they commute at coincidence points i.e. if fu = gu for some u
X, then fgu = gfu.
Definition 2.10. [16] A pair of self mappings (f, g) of Modified Intuitionistic fuzzy metric space
is said to be occasionally weakly compatible (owc) 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
in X with
for
and for all t > 0,
.
Definition 2.12. [10] Let f and g be maps from a Modified Intuitionistic fuzzy metric space
into itself. The maps f and g are said to be reciprocally continuous if for a sequence
in X
, whenever 
for some
and for all
.
Definition 2.13. [17] Let f and g be two maps from modified intuitionistic fuzzy metric space
into itself. The maps f and g are said to be subsequentially continuous if there exist a sequence
in X such that
for some 
and
for all
.
3. Main Results
Implicit relations play important role in establishing of common fixed point results.
Let M6 be the set of all continuous functions
satisfying the following conditions (for all
,
and
):
(A) 
and
(B) 
for all
.
Example 3.1. Define
as
where
is increasing and continuous function such that
for all
Clearly,
in M6.
We begin with following observation:
Theorem 3.1. Let A, B, S and T be four self mappings of a Modified Intuitionistic fuzzy metric space
. If the pairs (A, S) and (B, T) are compatible and subsequentially continuous mappings, then
(3.1) the pair (A, S) has a coincidence point
(3.2) 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 following:
(3.3) for any
,
in M6 and for all t > 0,

Proof. Since the pairs (A, S) and (B, T) are compatible and subsequentially continuous mappings, therefore there exist sequences {xn} and {yn} in X such that
,
for some
and

and 
so that Az = Sz and Bw = Tw i.e. z is a coincidence point of A and S where as w is a coincidence point of B and T, which proves (3.1) and (3.2).
Now, we prove that z = w, if not, then by using (3.3), we have

which on making
reduces to

a contradiction to (A) so that z = w.
Now, we assert that Az = z, if not, then by (3.3), we get

taking the limit as
, 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 common 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
. If the pairs (A,S) and (B,T) are subcompatible 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 condition (3.3).
Proof. Proof easily follows on same lines of Theorem 3.1 and using definition of reciprocally continuous and subcompatible mappings. □
Corollary 3.1. The conclusions of Theorem 3.1 and Theorem 3.2 remain true if we replace the inequality (3.3) by any one of the following:
(3.6) 
where
is increasing and continuous function such that
for all
.
(3.7) 
where

where
is increasing and continuous function such that
for all
and
is a Lebesgue integrable function which is summable and satisfies
, for all
.
By setting A = B in Theorems 3.1, 3.2, we derive the following corollaries for three mappings.
Corollary 3.2. Let A, S and T be three self mappings of a Modified Intuitionistic fuzzy metric space
. If the pairs (A,S) and (A,T) are compatible and subsequentially continuous mappings, then
(3.8) the pair (A, S) has a coincidence point
(3.9) the pair (A, 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 following:
(3.10) for any
,
in M6 and for all t > 0,

Corollary 3.3. Let A, S and T be three self mappings of a Modified Intuitionistic fuzzy metric space
. If the pairs (A,S) and (A,T) are subcompatible and reciprocally 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 3.1, 3.2, we derive the following corollaries for three mappings.
Corollary 3.4. Let A, B and S be three self mappings of a Modified Intuitionistic fuzzy metric space
. If the pairs (A,S) and (B,S) are compatible 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
,
in M6 and for all t > 0,

Corollary 3.5. Let A, B and S be three self mappings of a Modified Intuitionistic fuzzy metric space
. If the pairs (A,S) and (B,S) are subcompatible and reciprocally continuous mappings, then (3.11) and (3.12) satisfied. Further, A, B and S have a unique common fixed point provided A, B and S satisfy the condition (3.13).
Finally, by setting A = B and S = T in Theorems 3.1 and 3.2, we derive the following corollaries:
Corollary 3.6. Let A and S be four self mappings of a modified intuitionistic fuzzy metric space
. If the pair (A,S) is compatible and subsequentially continuous mappings, then
(3.14) the pair (A, S) has a coincidence point.
Further, A and S have a unique common fixed point provided A and S satisfy the following:
(3.15) for any
,
and
in M6 and for all t > 0,

Corollary 3.7. Let A and S be pair of self mappings of a modified intuitionistic fuzzy metric space
. If the pair (A,S) is subcompatible and reciprocally continuous mappings, then (3.14) is satisfied. Further, A and S have a unique common fixed point provided A and S satisfy the condition (3.15).
Example 3.1. [10]
Let
be a Modified Intuitionistic fuzzy metric space where 
and define
as
where
is increasing and continuous function such that
for all
Clearly,
satisfies all conditions (A) and (B). Define A, B, S and T by

and 
where
for all
, t > 0. Clearly, for the sequence
, (A,S) and (B,T) are compatible as well as subsequentially continuous. Therefore, all the conditions of Theorem 3.1 are satisfied. Evidently, z = 0 is a coincidence as well as unique common fixed point of A, B, S and T.
Example 3.2. [10]
Let
be a Modified Intuitionistic fuzzy metric space where
and define 
as 
where
is increasing and continuous function
such that
for all 
Clearly,
satisfies all conditions (A) and (B). Define A, B, S and T by


and
for all
, t > 0. Clearly, for the sequence
, (A,S) and(B,T) are subcompatible as well as reciprocally continuous. Therefore, all the conditions of Theorem 3.2 are satisfied. Evidently, z = 0 is a coincidence as well as unique common fixed point of A, B, S and T.
4. Acknowledgements
The authors thank the referee for his/her careful reading and useful suggestions of the manuscript.