Common fixed point theorems using faintly compatible

Computer Engineering and Intelligent Systems www.iiste.org
ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online)
Vol.5, No.7, 2014
15
Common Fixed Point Theorems Using Faintly Compatible
Mappings In Fuzzy Metric Spaces
Kamal Wadhwa and Ved Prakash Bhardwaj
Govt. Narmada P.G. College Hoshangabad, (M.P.) India
E-mail: ved_bhar2@rediffmail.com
Abstract: In this paper we prove common fixed point theorems using faintly compatible mappings in fuzzy
metric space. Our results extend and generalized the results of A. Jain et.al. [5].
Keywords: Fuzzy Metric Spaces, non compatible mappings, faintly compatible mappings and sub sequentially
continuous mappings.
1. Introduction: Weak compatibility is one of the weaker forms of the commuting mappings. Many researchers
use this concept to prove the existence of unique common fixed point in fuzzy metric space. Al-Thagafi and
Shahzad [2] introduced the concept of occasionally weakly compatible (owc) and weaken the concept of
nontrivial weakly compatible maps.
Recently, R.K. Bist and R. P. Pnat [3] criticize the concept of owc as follows “Under contractive conditions the
existence of a common fixed point and occasional weak compatibility are equivalent conditions, and
consequently, proving existence of fixed points by assuming owc is equivalent to proving the existence of fixed
points by assuming the existence of fixed points”. Therefore use of owc is a redundancy for fixed pint theorems
under contractive conditions.
This redundancy can be also seen in recent result of A. Jain et.al. [5]. To remove this we used faintly compatible
mapping in our paper which is weaker than weak compatibility or semi compatibility. Faintly compatible maps
introduced by Bisht and Shahzad [4] as an improvement of conditionally compatible maps, Pant and Bisht [8],
introduced the concept of conditional compatible maps. This gives the existence of a common fixed point or
multiple fixed point or coincidence points under contractive and non-contractive conditions.
The aim of this paper is remove redundancy of results of A. Jain et.al. [5], and prove the existence of common
fixed point using faintly compatible maps in fuzzy metric space.
2. Preliminaries:
In this section, we recall some definitions and useful results which are already in the literature.
Definition 2.1[10]: A binary operation *: [0, 1] ´ [0, 1] → [0, 1] is continuous t- norm if * satisfies the
following conditions:
(i) * is commutative and associative; (ii) * is continuous; (iii) a *1 = a "aÎ[0; 1];
(iv) a * b ≤ c*d whenever a ≤c and b ≤d "a, b, c, dÎ[0,1].
Example of continuous t-norm 2.2[10]: a * b = min {a, b}, minimum t-norm.
George and Veeramani modified the nothing of fuzzy metric space of Kramosil and Michalek as follows:

Vol.5, No.7, 2014
16
Definition 2.3: The 3-tuple (X, M, *) is called a fuzzy metric space if X is an arbitrary set, * is a continuous t-
norm and M is a fuzzy set on X2
´(0,¥) satisfying the following conditions: "x, y, z ÎX, t, s > 0;
(GV - 1) M(x, y, t) > 0;
(GV - 2) M(x, y, t) = 1 iff x = y;
(GV - 3) M(x, y, t) = M(y, x, t);
(GV - 4) M(x, y, t)*M(y, z, s) ≤ M(x, z, t + s);
(GV - 5) M(x, y, ·): [0,¥) → [0, 1] is continuous.
Definition 2.4: A pair of self-maps (A, S) on a fuzzy metric space (X, M, *) is said to be
(a) Non-compatible: if (A, S) is not compatible, i.e., if there exists a sequence {xn} in X such that limn→¥Axn =
limn→¥Sxn = x, for some x Î X, and limn→¥M(ASxn, SAxn, t) ¹ 1 or non-existent "t > 0.
(b) Conditionally compatible [8]: if whenever the set of sequences {xn} satisfying limn→¥Axn = limn→¥Sxn, is
non-empty, there exists a sequence {zn} in X such that limn→¥Azn = limn→¥Szn = t, for some tÎX and
limn→¥M(ASxn, SAxn, t) = 1 for all t > 0.
(c) Faintly compatible [4]: if (A, S) is conditionally compatible and A and S commute on a non-empty subset of
the set of coincidence points, whenever the set of coincidence points is nonempty.
(d) Satisfy the property (E.A.) [1]: if there exists a sequence {xn} in X such that lim n→¥Axn = limn→¥Sxn = x,
for some x ÎX.
(e) Sub Sequentially continuous [11]: iff there exists a sequence {xn} in X such that lim n→¥Axn = limn→¥Sxn =
x, xÎX and satisfy limn→¥ASxn= Ax, limn→¥SAxn = Sx.
Note that, compatibility, non- compatibility and faint compatibility are independent concepts. Faintly
compatibility is applicable for mappings that satisfy contractive and non contractive conditions.
(f) Semi-compatible [5]: if limn→¥ASxn= Sx, whenever is a sequence such that limn→¥Axn = limn→¥Sxn = xÎX.
Lemma 2.5[6]: Let (X, M, *) be a fuzzy metric space and for all x, yÎX, t > 0 and if there exists a constant k
Î(0, 1) such that M(x, y, kt) ³ M(x, y, t) then x = y.
A. Jain et.al. [5], proved the following:
Theorem 2.1[5]: Let A, B, S and T be self mappings of a complete fuzzy metric space (X, M, *). Suppose that
they satisfy the following conditions:
(2.1.1) A(X)ÌT(X), B(X)ÌS(X);
(2.1.2) the pair (A, S) is semi-compatible and (B, T) is occasionally weakly compatible;
(2.1.3) there exists kÎ(0, 1) such that "x, yÎX and t>0,
M(Ax, By, kt) ³ min{M(By, Ty, t), M(Sx, Ty, t), M(Ax, Sx, t)}.
Then A, B, S and T have a unique fixed point in X.
Now we prove some common fixed point theorems for pair of faintly compatible mappings.
3. Main Results:
Theorem 3.1: Let (X, M,*) be a fuzzy metric space and let A, B, S, T, P and Q be self mappings of X such that
(3.1.1) the pairs (A, SP) and (B, TQ) are non compatible, sub sequentially continuous faintly compatible;

Vol.5, No.7, 2014
17
(3.1.2) Pair (A, P), (S, P), (B, Q), (T, Q) are commuting;
(3.1.3) there exists kÎ(0,1) such that " x, yÎX and t > 0,
³ ;
where a,b,c,d,e,f≥0 with a&b, c&d and e&f cannot be simultaneously 0.
Then A, B, S, T, P and Q have a unique common fixed point in X.
Proof: Non compatibility of (A, SP) and (B, TQ) implies that there exist sequences {xn} and {yn} in X such that
limn→∞Axn=limn→∞(SP)xn=t1 for some t1 X, and M(A(SP)xn,(SP)Axn,t)¹1 or nonexistent "t > 0; Also
limn→∞Bxn=limn→∞(TQ)xn=t2 for some t2 X, and M(B(TQ)xn,(TQ)Bxn,t)¹1 or nonexistent "t>0.
Since pairs (A, SP) and (B, TQ) are faintly compatible therefore conditionally compatibility of (A, SP) and (B,
TQ) implies that there exist sequences {zn} and {zn'} in X satisfying
limn→∞Azn = limn→∞(SP)zn = u for some u X, such that M(A(SP)zn, (SP)Azn, t)=1;
Also limn→∞Bzn' = limn→∞(TQ)zn' = v for some v X, such that M(B(TQ)zn',(TQ)Bzn', t)=1.
As the pairs (A, SP) and (B, TQ) are sub sequentially continuous, we get
limn→∞A(SP)zn = Au, limn→∞(SP)Azn = (SP)u
and so Au = (SP)u i.e. (u is coincidence point of A and (SP));
Also limn→∞B(TQ)zn' = Bv, limn→∞(TQ)Bzn' = (TQ)v
and so Bv = (TQ)v i.e. (v is coincidence point of B and (TQ)).
Since pairs (A, SP) and (B, TQ) are faintly compatible, we get
A(SP)u=(SP)Au & so AAu=A(SP)u=(SP)Au=(SP)(SP)u;
and Also B(TQ)v=(TQ)Bv & so BBv=B(TQ)v=(TQ)Bv=(TQ)(TQ)v.
Now we show that Au=Bv, AAu= Au, BBv=Bv, PAu=Au and QAu=Au.
By taking x=u and y=v in (3.1.3),
³ ;
³ ;
³ ;
, lemma (2.5) Þ Au=Bv.
By taking x=Au and y=v in (3.1.3),
³ ;
³ ;
³ ;

Vol.5, No.7, 2014
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, lemma (2.5) Þ AAu=Bv=Au.
By taking x=u and y=Bv in (3.1.3),
³ ;
³ ;
³ ;
, lemma (2.5) Þ Au=BBv Þ BBv=Au=Bv.
Now we have AAu=(SP)Au=Au, Au= BBv=BAu and Au= BBv=(TQ)Bv=(TQ)Au since Bv=Au.
Hence AAu=(SP)Au=BAu=(TQ)Au=Au
i.e. Au is a common coincidence point of A, B, SP and TQ.
By taking x=PAu and y=Au in (3.1.3),
³ ;
³ ;
Since (A, P) and (S, P) are commuting, therefore
³ ;
³ ;
, lemma (2.5) Þ PAu= .
By taking x=Au and y=QAu in (3.2.2),
³ ;
Since (B, Q) and (T, Q) are commuting, therefore
³ ;
, lemma (2.5) Þ Au= .
Therefore AAu=(SP)Au=BAu=(TQ)Au=Au Þ AAu=SPAu=SAu and BAu=TQAu=TAu.

Vol.5, No.7, 2014
19
Hence AAu=BAu=SAu=TAu=PAu=QAu=Au,
i.e. Au is a common fixed point of A, B, S, T, P and Q in X.
The uniqueness follows from (3.3.2).This completes the proof of the theorem.
If we take P=Q=I (the identity map on X) in theorem 3.1 then condition (3.1.2) trivially satisfied and we get the
following corollary:
Corollary 3.2: Let (X, M,*) be a fuzzy metric space and let A, B, S, T, P and Q be self mappings of X such that
(3.2.1) the pairs (A, S) and (B, T) are non compatible, sub sequentially continuous faintly compatible;
³ ;
where a,b,c,d,e,f≥0 with a&b, c&d and e&f cannot be simultaneously 0;
Then A, B, S and T have a unique common fixed point in X.
Proof: The proof is similar to the proof of theorem 3.1 without required condition (3.1.2).
Remark 3.2.1: If we take a=c=e=0 and P= Q=I in theorem 3.1 then we get the result of A. Jain et.al. [5], for
faintly compatibility and sequentially continuous map.
³f
where a,b,c,d,e,f≥0 with a&b, c&d and e&f cannot be simultaneously 0 and f :[0,1]→[0,1] such that f(t) > t " 0
< t < 1;
Proof: The prove follows from theorem 3.1.
Now we are giving more improved form of theorem 3.1 as follows:
³f ;
where a, b, c, d, e, f ≥ 0 with a & b, c & d and e & f cannot be simultaneously 0 and
f : [0, 1]3
→ [0, 1] such that f(1, t, 1) > t " 0 < t < 1;
Proof: Non compatibility of (A, SP) and (B, TQ) implies that there exist sequences {xn} and {yn} in X such that
limn→∞Axn=limn→∞(SP)xn=t1 for some t1 X, and M(A(SP)xn,(SP)Axn,t)¹1 or nonexistent "t > 0; Also

Vol.5, No.7, 2014
20
limn→∞Bxn=limn→∞(TQ)xn=t2 for some t2 X, and M(B(TQ)xn,(TQ)Bxn,t)¹1 or nonexistent "t>0.
Since pairs (A, SP) and (B, TQ) are faintly compatible therefore conditionally compatibility of (A, SP) and (B,
TQ) implies that there exist sequences {zn} and {zn'} in X satisfying
limn→∞Azn = limn→∞(SP)zn = u for some u X, such that M(A(SP)zn, (SP)Azn, t)=1;
Also limn→∞Bzn' = limn→∞(TQ)zn' = v for some v X, such that M(B(TQ)zn',(TQ)Bzn', t)=1.
As the pairs (A, SP) and (B, TQ) are sub sequentially continuous, we get
limn→∞A(SP)zn = Au, limn→∞(SP)Azn = (SP)u
and so Au = (SP)u i.e. (u is coincidence point of A and (SP));
Also limn→∞B(TQ)zn' = Bv, limn→∞(TQ)Bzn' = (TQ)v
and so Bv = (TQ)v i.e. (v is coincidence point of B and (TQ)).
Since pairs (A, SP) and (B, TQ) are faintly compatible, we get
A(SP)u=(SP)Au & so AAu=A(SP)u=(SP)Au=(SP)(SP)u;
and Also B(TQ)v=(TQ)Bv & so BBv=B(TQ)v=(TQ)Bv=(TQ)(TQ)v.
Now we show that Au=Bv, AAu= Au., PAu=Au and QAu=Au.
By taking x=u and y=v in (3.4.3),
³f ;
³f ;
³f ;
, lemma (2.5) Þ Au=Bv.
By taking x=Au and y=v in (3.4.3),
³f ;
³f ;
³f ;
, lemma (2.5) Þ AAu=Bv=Au.
Similarly we can show BBv=Bv By taking x=u and y=Bv in (3.4.3).
Now we have AAu=(SP)Au=Au, Au= BBv=BAu and Au= BBv=(TQ)Bv=(TQ)Au since Bv=Au.
Hence AAu=(SP)Au=BAu=(TQ)Au=Au
i.e. Au is a common coincidence point of A, B, SP and TQ.
By taking x=PAu and y=Au in (3.4.3),
³f ;

Vol.5, No.7, 2014
21
³f ;
Since (A, P) and (S, P) are commuting, therefore
³f ;
³f ;
f
, lemma (2.5) Þ PAu= .
Similarly we can show Au= , by taking x=Au and y=QAu in (3.4.3).
Therefore AAu=(SP)Au=BAu=(TQ)Au=Au Þ AAu=SPAu=SAu and BAu=TQAu=TAu.
Hence AAu=BAu=SAu=TAu=PAu=QAu=Au,
i.e. Au is a common fixed point of A, B, S, T, P and Q in X.
The uniqueness follows from (3.4.3).This completes the proof of the theorem.
Conclusion: Our theorem 3.1 is an improvement and generalization of theorem 3.1 of A. Jain et.al. [5], in the
following way:
(i) Requirement of the semi-compatibility replaced by weaker form faintly compatibility.
(ii) Completeness of the space has been removed completely.
(iii) Our results never require the containment of the ranges.
(iv) In the light of [3], owc mappings have been replaced by faintly compatible mappings.
Open Problem: In this paper, we used weaker form of reciprocal continuity, namely sub-sequentially continuity.
Are the results true without any continuity condition?
References:
[1] M. Aamri and D. El. Moutawakil, Some new common fixed point theorems under strict contractive
conditions, J. Math. Anal. Appl., 270 (2002), 181-188.
[2] M.A. Al-Thagafi and N. Shahzad, Generalized I-nonexpansive selfmaps and invariant approximations,
Acta Mathematica Sinica, English Series, 24 (5) (2008), 867-876.
[3] R.K. Bisht and R.P. Pant, A critical remark on “Fixed point theorems for occasionally weakly
compatible mappings”, J. of the Egyptian Mathematical Soci.,(2013) 21,273-275.
[4] R.K. Bisht and N. Shahzad, Faintly compatible mappings and common fixed points, Fixed point theory
and applications, 2013, 2013:156.
[5] A. Jain., V.K. Gupta, V.H. Badshah and R.S. Chandelkar, Fixed point theorem in fuzzy metric space
using semi-compatible mappings, Adv. Inequal. Appl., 2014:19, 1-10.
[6] S.N. Mishra, N. Sharma and S.L. Singh, Common fixed points of maps on fuzzy metric spaces,
Internat. J. Math. Math. Sci., 17 (2) (1994), 253-258.
[7] R.P. Pant, Common fixed points of four mappings, Bull. Calcutta Math. Soc., 90 (1998), 281-286.

Vol.5, No.7, 2014
22
[8] R.P. Pant and R.K. Bisht, Occasionally weakly compatible mappings and fixed points. Bull. Belg.
Math. Soc. Simon Stevin, 19 (2012), 655-661.
[9] V. Pant and R.P. Pant, Common fixed points of conditionally commuting maps, Fixed Point Theory, 11
(1) (2010), 113-118.
[10]B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math., 10 (1960), 313-334.
[11] K. Wadhwa, F. Beg and H. Dubey, Common fixed point theorem for compatible and sub sequentially
continuous maps in fuzzy metric space using implicit relation, IJRRAS, 2011, 87-92.

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