This paper presents common fixed point theorems for weakly compatible maps within intuitionistic fuzzy metric spaces, focusing on both finite self-mappings and their properties. The authors define necessary terms and establish theorems that include conditions under which unique common fixed points exist. The study builds on earlier research in fuzzy set theory and includes a thorough examination of related concepts and definitions.

1. International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 4 Issue 10 || December. 2016 || PP-14-21
www.ijmsi.org 14 | Page
Common Fixed Point Theorem for Weakly Compatible Maps in
Intuitionistic Fuzzy Metric Spaces
Madhu Shrivastava1
, Dr.K.Qureshi2
, Dr.A.D.Singh3
1
TIT Group of Instititution,Bhopal
2
Ret.Additional Director,Bhopal
3
Govt.M.V.M.College, Bhopal
ABSTRACT: In this paper, we prove some common fixed point theorem for weakly compatible maps in
intuitionistic fuzzy metric space for two, four and six self mapping.
KEYWORDS: Intuitionistic fuzzy metric space, weakly compatible mappings.
I. INTRODUCTION
The introduction of the concept of fuzzy sets by Zadeh [1] in1965.Many authors have introduced the concept of
fuzzy metric in different ways. Atanassov [2] introduced and studied the concept of intuitionistic fuzzy sets. In
1997 Coker [3 ] introduced the concept of intuitionistic fuzzy topological spaces .Park [20] defined the notion of
intuitionistic fuzzy metric space with the help of continuous t-norm. Alaca et al. [4] using the idea of
intuitionistic fuzzy sets, they defined the notion of intuitionistic fuzzy metric space. Turkoglu et al. [34]
introduced the cocept of compatible maps and compatible maps of types and in intuitionistic fuzzy
metric space and gave some relations between the concepts of compatible maps and compatible maps of types
. Gregory et al. [12 ], Saadati et al [28],Singalotti et al [27], Sharma and Deshpande [30], Ciric etal
[9], Jesic [13], Kutukcu [16] and many others studied the concept of intuitionistic fuzzy metric space and its
applications. Sharma and Deshpandey [30] proved common fixed point theorems for finite number of mappings
without continuity and compatibility on intuitionistic fuzzy metric spaces. R.P. Pant [23] has initiated work
using the concept of R-weakly commuting mappings in 1994.
II. PRELIMINARIES
We begin by briefly recalling some definitions and notions from fixed point theory literature that we will use in
the sequel.
Definition 2.1 [Schweizer st. al.1960] - A binary operation is a continuous t-norms if
satisfying conditions:
(i) is commutative and associative;
(ii) is continuous;
(iii)
(iv) wherever and for all
Example of t-norm are and
Definition 2.2[Schweizer st. al.1960] A binary operation is a continuous t-norms if
satisfying conditions:
(i) is commutative and associative;
(ii) is continuous;
(iii) for all
(iv) wherever and for all
Example of t-norm are and
Definition 2.3 [Alaca, C. et. Al. 2006] A 5-tuple is said to be an intuitionistic fuzzy metric space
if X is an arbitrary set, is a continuous t -norm, ◊ is a continuous
t -norm and are fuzzy sets on satisfying the following conditions:
(i) for all and

2. Common Fixed Point Theorem for Weakly Compatible Maps in Intuitionistic Fuzzy Metric Spaces
www.ijmsi.org 15 | Page
(ii) for all ;
(iii) for all and if and only if ;
(iv) for all and
(v) for all and ;
(vi) For all is left continuous ;
(vii)
(viii) for all ;
(ix) for all and if and only if
(x) for all and
(xi) for all and
(xii) for all is right continuous;
(xiii)
is called an intuitionistic fuzzy metric on . The functions denote the degree of
nearness and the degree of non-nearness between and with respect to respectively.
Remark 2.4 [Alaca, C. et. Al. 2006 ]. An intuitionistic fuzzy metric spaces with continuous t-norm and
Continuous t -conorm ◊ defined by and for all ,
Then for all is non-decreasing and, is non-increasing.
Remark 2.5[Park, 2004]. Let be a metric space .Define t-norm and t-
conorm and for all , and
Then is an intuitionistic fuzzy metric space induced by the metric. It is obvious that
Alaca, Turkoglu and Yildiz [Alaca, C. et. Al. 2006] introduced the following notions:
Definition 2.6. Let be an intuitionistic fuzzy metric space. Then
(i) a sequence is said to be Cauchy sequence if, for all
(ii) a sequence in X is said to be convergent to a point if ,for all
Since and ◊ are continuous, the limit is uniquely determined from (v) and (xi) of respectively.
Definition 2.7. An intuitionistic fuzzy metric space is said to be complete if and only if every
Cauchy sequence in is convergent.
Definition 2.8. A pair of self-mappings , of an intuitionistic fuzzy metric space is said to
be compatible if and for every
,whenever is a sequence in X such that
Definition 2.9. A pair of self-mappings , of an intuitionistic fuzzy metric space is said to be
non compatible if or nonexistence and or
nonexistence for every ,
whenever is a sequence in such that for some
In 1998, Jungck and Rhoades [Jungck et. al. 1998] introduced the concept of weakly compatible maps as
follows:
Definition 2.10. Two self maps and are said to be weakly compatible if they commute at coincidence
points.

3. Common Fixed Point Theorem for Weakly Compatible Maps in Intuitionistic Fuzzy Metric Spaces
www.ijmsi.org 16 | Page
Definition 2.11 [Alaca, C. et. Al. 2006 ]. Let be an intuitionistic fuzzy metric space then
are said to be weakly compatible if they commute at coincidence points.
Lemma 2.12[ Alaca, C. et. Al. 2006]. Let be an intuitionistic fuzzy metric space and be a
sequence in X if there exist a number such that,
i)
ii)
for all and then is a cauchy sequence in
Lemma 2.13 Let be an IFM-space and for all and if for a number
then
III. MAIN RESULTS
Theorem 3.1 – Let be an intuitionstic fuzzy metric space with continuous t-norm and continuous
t-norm ◊ defined by and Let be weakly compatible
self mapping in s.t.
Where , and is continuous function s.t. and ,for each
and with
c) If one of is complete
Then have a unique common fixed point.
Proof – Let be any arbitrary point. Since ,choose
Such that .
Then by (b),
..............(1)
As , for each
and
As
For all
and
and ,
Hence by lemma (2.12) , is a Cauchy sequence in X. by completeness of X, = is convergent,
call ,
Then
Now suppose is complete, so there exist a point
Now from (b),

4. Common Fixed Point Theorem for Weakly Compatible Maps in Intuitionistic Fuzzy Metric Spaces
www.ijmsi.org 17 | Page
As ,
............................(3)
And
As ,
.............................(4)
From (3) and (4),we have
As are weakly compatible ,therefore
Now , we show that is a fixed point of from (b),
As ,
.....................(5)
and
As ,
....................(6)
From (5) and (6),
Hence is common fixed point of .
Uniqueness – Let be another fixed point of ,then by (b) ,
.........................(7)
and
.....................(8)
From (7) and (8),
Therefore is unique common fixed point of .
Theorem – 3.2 Let be an intuitionstic fuzzy metric space with continuous t-norm and continuous
t-norm ◊ defined by Let A, B, S and T be self mappings in
X s.t.
a)
b) There exist a constant

5. Common Fixed Point Theorem for Weakly Compatible Maps in Intuitionistic Fuzzy Metric Spaces
www.ijmsi.org 18 | Page
is continuous function s.t.
, for each and
with
c) If one of the is complete subspace of X,
Then and have a coincidence point
More over ,if the pair and are weakly compatible ,then A, B, S and T have a unique common fixed
point.
Proof – Let be any arbitrary point there is a point
Again since for this and so on. Then we get a sequence
s.t. and
Putting in (b) we have,
As for each
and
As
For all
and
and
Hence by lemma (2.12) , is a Cauchy sequence in
Now suppose is a complete subspace of X.note that the sequence is contained in and has a
limit in say so we get we shall use the fact that subsequence also convergence to .
now putting , in (b)
and taking
.......................(3)
Also,
i.e, ................................(4)
from (3) and (4),
Since is the coincidence point of
As let then
By putting in (b),we get,
As
1 ...................(5)
and
As

6. Common Fixed Point Theorem for Weakly Compatible Maps in Intuitionistic Fuzzy Metric Spaces
www.ijmsi.org 19 | Page
= ..........................(6)
From (5) and (6),we get,
Since we have thus is the coincidence point of and If one assume to be
complete ,then an analogous argument establish this claim.
The remaining two cases pertain essentially to the previous cases. Indeed if is complete then
and if is complete then Thus (c) is completely established.
Since the pair are weakly compatible ,i.e.
and
Putting in (b),we get
As
................................(7)
and
As
..............................(8)
From (7) and (8), implies that
Similarly by putting in (b) and as
We have
i.e is a common fixed point of and
Uniqueness- let be another common fixed point of and
Then by putting
....................(9)
and
. ......................(10)
From (9) and (10)
for all and
Therefore is the unique common fixed point of and
Theorem – 3.3 Let be an intuitionstic fuzzy metric space with continuous t-norm and continuous
t-norm ◊ defined by
Let and be self mappings in s.t.
a)
b) There exist a constant
is continuous function s.t.
, for each and
with
(c) If one of the is a complete subspace of then

7. Common Fixed Point Theorem for Weakly Compatible Maps in Intuitionistic Fuzzy Metric Spaces
www.ijmsi.org 20 | Page
have a coincidence point.
(d)
Moreover if the pair are weakly compatible ,then have a unique
common fixed point.
Proof – Let be an arbitrary point. since there exist s.t. again
since for this there is s.t.
and so on. Inductively we get a sequence and in s.t.
Putting in (b) we have,
........................(1)
..........................(2)
Hence we have from (1) and (2)
and .
Similarly we also have
and .
and ,
Hence by lemma (2.12) , is a Cauchy sequence in
Now suppose is a complete subspace of .note that the sequence is contained in and has
a limit in say so we get we shall use the fact that subsequence also convergence to
. now putting , in (b)
and taking , we have
As
............................(3)
As
.........................(4)
From (3) and (4) , . Since thus we have that is is coincidence point of
and . Since implies that
let then
Putting and in (b), we have
As
........................(5)
..............................(6)
From (5) and (6),we have .

8. Common Fixed Point Theorem for Weakly Compatible Maps in Intuitionistic Fuzzy Metric Spaces
www.ijmsi.org 21 | Page
Again Putting and in (b) and as
..................(7)
............................(8)
From (7) and (8), So
By putting in (b) and taking , hence .
Now putting in (b) and using (d) ,we have
and thus
Since therefore
To prove , , we put in (b) and using (d),
We have and Thus
Since therefore
By combining the above result we have
that is is a common fixed point of
Uniqueness – Let be another common fixed point of and then
By putting we have and
Hence for all and
Therefore is the unique common fixed point of and
REFERENCES
[1] Zadeh, L. A (1965), Fuzzy sets, Inform. and Control, 8, 338–353.
[2] Atanassov K.(1986), Intuitionistic fuzzy sets, Fuzzy Sets and System, 20, 87- 96.
[3] Coker, D.(1997), An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and System, 88, 81–89.
[4] C. Alaca, D. Turkoglu and C. Yildiz, Fixed points in intuitionistic fuzzy metric spaces, Choas, Solitons & Fractals, 29 (2006),
1073–1078.
[5] C. Alaca, I. Altun and D. Turkoglu, On compatible mappings of type (I) and (II) in intuitionistic fuzzy metric spaces, Commun.
Korean Math. Soc., 23(2008), 427–446.
[6] C. Alaca, D. Turkoglu and C. Yildiz, Common fixed points of compatible maps in intuitionistic fuzzy metric spaces, Southeast
Asian Bull. Math., 32(2008), 21–33.
[7] M. Aamri and D. EI Moutawakil, Some new common fixed point theorems under strict contractive conditions, J.Math. Anal. Appl.,
270(2002), 181–188.
[8] [S. Banach, Theoriles operations Linearies Manograie Mathematyezne, Warsaw, Poland, 1932.
[9] L. Ciric, S. Jesic and J. Ume, The existence theorems for fixed and periodic points of nonexpansive mappings in intuitionistic fuzzy
metric spaces, Choas, Solitons & Fractals, 37(2008), 781–791.
[10] M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc., 37(1962), 74– 79.
[11] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, 27(1988), 385–389.
[12] V. Gregory, S. Romaguera and P. Veeramani, A note on intuitionistic fuzzy metric spaces, Chaos, Solitons & Fractals, 28 (2006),
902–905.
[13] S. Jesic, Convex structure, normal structure and a fixed point theorem in intuitionistic fuzzy metric spaces, Chaos, Solitons &
Fractals, 41(2009), 292–301.
[14] O. Karmosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11(1975), 326–334.
[15] E. P. Klement, R. Mesiar and E. Pap, Triangular Norms, Kluwer Academic Pub. Trends in Logic 8, Dordrecht 2000.
[16] S. Kutukcu, Weak Compatibility and common coincidence points in intuitionistic fuzzy metric spaces, Southeast Asian Bulletin of
Mathematics, 32(2008), 1081–1089.
[17] S. Kutukcu, C. Yildiz and D. Turkoglu, Fixed points of contractive mappings in intuitionistic fuzzy metric spaces, J. Comput. Anal.
Appl., 9(2007), 181–193.
[18] K. Menger, Statistical metrices, Proc. Nat. Acad. Sci., 28(1942), 535–537.
[19] S. N. Mishra, N. Sharma and S. L. Singh, Common fixed points of maps on fuzzy metric spaces, Internat. J. Math. Math. Sci.,
17(1994), 253–258.
[20] J. H. Park, Intuitionistic fuzzy metric spaces, Choas, Solitons & Fractals, 22(2004), 1039–1046.
[21] R. P. Pant, Common fixed points of contractive maps, J. Math. Anal. Appl., 226(1998), 251–258.
[22] R. P. Pant, Noncompatible mappings and common fixed points, Soochow J. Math., 26(2000), 29–35.
[23] R. P. Pant, Common fixed points of noncommuting mappings, J. Math. Anal. Appl., 188(1994), 436–440.
[24] R. P. Pant, Common fixed points of Lipschitz type mapping Pairs, J. Math. Anal. Appl., 240(1999), 280–283.
[25] V. Pant, Some fixed point theorems in fuzzy metric space, Tamkang J. Math., 40(2009), 59–66.
[26] H. K. Pathak, Y. J. Cho and S. M. Kang, Remarks on R-weakly commuting mappings and common fixed point theorems, Bull.
Korean Math. Soc., 34(1997), 247–257.

9. Common Fixed Point Theorem for Weakly Compatible Maps in Intuitionistic Fuzzy Metric Spaces
www.ijmsi.org 22 | Page
[27] LDJ Sigalottiand A. Mejias, OnEl Naschie’sconjugate complex time, fractalE(∞) space-timeand faster-thanlight particles, Int. J.
Nonlinear Sci. Number Simul., 7, (2006), 467–472.
[28] R. Saadati, S. Sedghi and N. Shobe, Modified intuitionistic fuzzy metric spaces and some fixed point theorems, Chaos, Solitons &
Fractals, 38(2008), 36–47.
[29] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific Journal Math., 10(1960), 314–334.
[30] Sushil Sharma and B. Deshpande, Commonfixed point theorems for finite number of mappings without continuity and compatibility
on intuitionistic fuzzy metric spaces, Choas, Solitons & Fractals, 40(2009), 2242– 2256.
[31] Sushil Sharma and B. Deshpande, Compatible mappings of type (I) and (II) on intuitionistic in consideration of common fixed
point, Comm. Korean Math. Soc., 24(2009), 197–214.
[32] Sushil Sharmaand B. Deshpande,Discontinuity and weak compatibility in fixed point consideration on non-complete fuzzy metric
spaces, J. Fuzzy Math., 11(2003), 671–686.
[33] Sushil Sharma, S. Kutukcu and A. Pathak, Common fixed point theorems for weakly compatible mappings in intuitionistic fuzzy
metric spaces, J. Fuzzy Math., 17(2009), 225–240.
[34] D. Turkoglu, C. Alaca and C. Yildiz, Compatible maps and compatible maps of types (α) and (β) in intuitionistic fuzzy metric
spaces, Demonstratio Math., 39 (2006), 671–684.
[35] D. Turkoglu, C. Alaca, Y. J. Cho and C. Yildiz, Common fixed point theorems in intuitionistic fuzzy metric spaces, J. Appl. Math.
and Computing, 22 (2006), 411–424.
[36] R. Vasuki, Common fixed point for R-weakly commuting maps in fuzzy metric spaces, Indian J. Pure Appl. Math., 30(1999), 419–
423.
[37] R. R. Yager, On a class of weak triangular norm operators, InformationSciences, 96(1997), 47–78.