Fuzzy Sets and Systems 127 (2002) 345–352www.elsevier.com/locate/fss

Common "xed point theorems in fuzzy metric spacesSushil Sharma

Department of Mathematics, Madhav Science College, Ujjain (MP) 456010, India

Received 31 March 1999; received in revised form 16 March 2001; accepted 3 April 2001

Abstract

In this paper we prove common "xed point theorems for six mappings in fuzzy metric space. Our main results extend,generalize and fuzzify some known results in fuzzy metric spaces, probabilistic metric spaces and uniform spaces. c© 2002Published by Elsevier Science B.V.

Keywords: Fuzzy metric spaces; Common "xed point; Compatible mappings of type (�)

1. Introduction

The concept of fuzzy sets was introduced initiallyby Zadeh [34] in 1965. Since then, to use this conceptin topology and analysis many authors have expan-sively developed the theory of fuzzy sets and appli-cations. Especially, Deng [9], Erceg [11], Kaleva andSeikkala [23], Kramosil and Michalek [24] have in-troduced the concept of fuzzy metric space in di=erentways.

Grabiec [14] followed Kramosil and Michalek [24]and obtained the fuzzy version of Banach contrac-tion principle. Moreover, it appears that the study ofKramosil and Michalek [24] of fuzzy metric spacespaves the way for developing a smoothing machineryin the "eld of "xed point theorems, in particular forthe study of contractive type maps.

E-mail address: sksharma2005@yahoo.com (S. Sharma).

Fang [12] proved some "xed point theorems in fuzzymetric spaces, which improve, generalize, unify andextend some main results of Banach [2], Edelstein[10], Istratescu [17], Sehgal and Bharucha-Reid[29].

Sessa [30] de"ned a generalization of commuta-tivity, which is called weak commutativity. FurtherJungck [21] introduced more generalized commutativ-ity, so called compatibility. Mishra et al. [26] obtainedcommon "xed point theorems for compatible maps onfuzzy metric spaces.

Recently, Jungck et al. [22] introduced the con-cept of compatible mappings of type (�) in metricspaces, which is equivalent to the concept of com-patible mappings under some conditions and provedcommon "xed point theorems in metric spaces. Cho[8] introduced the concept of compatible mappings oftype (�) in fuzzy metric spaces.

Many authors have studied the "xed point theory infuzzy metric spaces. The most interesting references in

0165-0114/02/$ - see front matter c© 2002 Published by Elsevier Science B.V.PII: S 0165 -0114(01)00112 -9

346 S. Sharma / Fuzzy Sets and Systems 127 (2002) 345–352

this direction are [12,19,18,14,15,1,6,26,32] and fuzzymappings [3,25,4,5,7,26].

Recently, George and Veeramani [13] modi"ed theconcept of fuzzy metric space introduced by Kramosiland Michalek [24] and de"ned the Hausdor= topologyon the fuzzy metric spaces. They showed also thatevery metric induces a fuzzy metric.

In this paper, we prove common "xed point the-orems for six mappings satisfying some conditionsin fuzzy metric spaces in the sense of Kramosil andMichalek [24]. Our main theorems extend, general-ize and fuzzify some known results in fuzzy metricspaces, probabilistic metric spaces and uniform spaces[27,20,22,14,16,26,31,33,32,8]. We also give an ex-ample to illustrate our main theorem.

2. Preliminaries

Now, we begin with some de"nitions:

De�nition 2.1 (Schweizer and Sklar [28]). A binaryoperation ∗ : [0; 1]× [0; 1]→ [0; 1] is called a contin-uous t-norm if ([0; 1]; ∗) is an abelian topologicalmonoid with unit 1 such that a∗b6c∗d whenevera6c and b6d for all a; b; c; d∈ [0; 1].

Examples of t-norm are a∗b= ab and a∗b=min{a; b}.

De�nition 2.2 (Kramosil and Michalek [24]). The3-tuple (X;M; ∗) is called a fuzzy metric space(shortly, FM-space) if X is an arbitrary set, ∗ is acontinuous t-norm and M is a fuzzy set in X 2x[0;∞)satisfying the following conditions: for all x; y; z ∈Xand s; t¿0,

(FM-1) M (x; y; 0) = 0,(FM-2) M (x; y; t) = 1, for all t¿0 if and only if x=y,(FM-3) M (x; y; t) =M (y; x; ; t),(FM-4) M (x; y; t)∗M (y; z; s)6M (x; z; t + s),(FM-5) M (x; y; :) : [0; 1) → [0; 1] is left continuous.

Note that M (x; y; t) can be thought of as the de-gree of nearness between x and y with respect to t.We identify x=y with M (x; y; t) = 1 for all t¿0 andM (x; y; t)= 0 with ∞ and we can "nd some topologi-cal properties and examples of fuzzy metric spaces in[13]. In the following example, we know that every

metric induces a fuzzy metric.

Example 2.1 (George and Veeramani [13]): Let(X; d) be a metric space. De"ne a∗b= ab (or a∗b=min{a; b}) and for all x; y∈X and t¿0,

M (x; y; t) =t

t + d(x; y): (1a)

Then (X;M; ∗) is a fuzzy metric space. We call thisfuzzy metric M induced by the metric d the standardfuzzy metric. On the other hand, note that there existsno metric on X satisfying (1a).

Lemma 2.1 (Grabiec [14]). For all x; y∈X; M (x; y; :)be nondecreasing.

De�nition 2.3 (Grabiec [14]). Let (X;M; ∗) be afuzzy metric space:(1) A sequence {xn} in X is said to be convergent to

a point x∈X (denoted by limn→∞ xn = x), if

limn→∞ M (xn;;x; t) = 1;

for all t¿0.(2) A sequence {xn} in X is called a Cauchy sequence

if

limn→∞M (xn+p; xn; t) = 1;

for all t¿0 and p¿0.(3) A fuzzy metric space in which every Cauchy se-

quence is convergent is said to be complete.

Remark 2.1. Since ∗ is continuous, it follows from(FM-4) that the limit of the sequence in FM-space isuniquely determined.

Let (X;M; ∗) be a fuzzy metric space with the fol-lowing condition:

(FM-6) limt→∞M (x; y; t) = 1 for all x; y∈X:

Lemma 2.2 (Cho [8] and Mishra et al. [26]). Let {yn}be a sequence in a fuzzy metric space (X;M; ∗)with the condition (FM-6). If there exists a number

S. Sharma / Fuzzy Sets and Systems 127 (2002) 345–352 347

k ∈ (0; 1) such that

M (yn+2; yn+1; kt) ¿ M (yn+1; yn; t) (1b)

for all t¿0 and n= 1; 2; : : : then {yn} is a Cauchysequence in X.

Lemma 2.3 (Mishra et al. [26]). If for all x; y∈X;t¿0 and for a number k ∈ (0; 1);

M (x; y; kt) ¿ M (x; y; t)

then x=y.

3. Compatible mappings of type (�)

In this section, we give the concept of compatiblemappings of type (�) in fuzzy metric spaces and someproperties of these mappings for our main results.

De�nition 3.1 (Mishra et al. [26]). Let A and B bemappings from a fuzzy metric space (X;M; ∗) into it-self. The mappings A and B are said to be compatibleif

limn→∞ M (ABxn; BAxn; t) = 1

for all t¿0, whenever {xn} is a sequence in X suchthat

limn→∞Axn = lim

n→∞Bxn = z

for some z ∈X .

De�nition 3.2 (Cho [8]). Let A and B be mappingsfrom a fuzzy metric space (X;M; ∗) into itself. Themappings A and B are said to be compatible of type(�) if

limn→∞ M (ABxn; BBxn; t) = 1

and

limn→∞ M (BAxn; AAxn; t) = 1

for all t¿0, whenever {xn} is a sequence in X suchthat

limn→∞ Axn = lim

n→∞ Bxn = z

for some z ∈X .

Remark 3.1. In [22,21], we can "nd the equivalentformulations of De"nitions 3.1 and 3.2 and their ex-amples in metric spaces. Such mappings are indepen-dent of each other and more general than commutingand weakly commuting mappings [20,30].

Proposition 3.1 (Cho [8]). Let (X;M; ∗) be a fuzzymetric space with t∗t¿t for all t ∈ [0; 1] and A andB be continuous mappings from X into itself. ThenA and B are compatible if and only if they are com-patible of type (�).

Proposition 3.2 (Cho [8]). Let (X;M; ∗) be a fuzzymetric space with t∗t¿t for all t ∈ [0; 1] and A andB be mappings from X into itself. If A and B arecompatible of type (�) and Az=Bz for some z ∈X;then ABz=BBz=BAz=AAz.

Proposition 3.3 (Cho [8]). Let (X;M; ∗) be a fuzzymetric space with t∗t¿t for all t ∈ [0; 1] and A andB be mappings from X into itself. If A and B arecompatible of type (�) and {xn} is a sequence in Xsuch that(1) limn→∞ Axn = limn→∞ Bxn = z for some z ∈X;then(2) limn→∞ BAxn =Az; if A is continuous at z;(3) ABz=BAz and Az=Bz; if A and B are continu-

ous at z.

4. Common �xed point theorems

In this section, we prove some common "xed pointtheorems for six mappings satisfying some conditions.

Theorem 4.1. Let (X;M; ∗) be a complete fuzzy met-ric space with t∗t¿t for all t ∈ [0; 1] and the condition(FM-6). Let A; B; S; T; P and Q be mappings from Xinto itself such that(4:1) P(X )⊂AB(X ); Q(X )⊂ ST (X );(4:2) AB=BA; ST=TS; PB=BP; QS=SQ; QT

=TQ;

348 S. Sharma / Fuzzy Sets and Systems 127 (2002) 345–352

(4:3) A; B; S and T are continuous;(4:4) the pairs (P; AB) and (Q; ST ) are compatible

of type (�);(4:5) there exists a number k ∈ (0; 1) such that

M (Px; Qy; kt)¿M (ABx; Px; t)∗M (STy; Qy; t)∗M (STy; Px; �t)∗M (ABx; Qy; (2 − �)t)∗M (ABx; STy; t)

for all x; y ∈ X; �∈ (0; 2) and t¿0.Then A; B; S; T; P and Q have a unique common

9xed point in X.

Proof. By (4:1), since P(X )⊂AB(X ), for any x0 ∈X ,there exists a point x1 ∈X such that Px0 =ABx1. SinceQ(X )⊂ ST (X ), for this point x1, we can choose apoint x2 ∈X such that Qx1 = ST x2.

Inductively, we can "nd a sequence {yn} in X asfollows:

y2n = Px2n = ABx2n+1 and

y2n+1 = Qx2n+1 = STx2n+2;

for n= 0; 1; 2; : : : By (4:5), for all t¿0 and �= 1−qwith q ∈ (0; 1), we have

M (y2n+1; y2n+2; kt)

= M (Px2n+1; Qx2n+2; kt)

¿ M (ABx2n+1; Px2n+1; t)∗M (STx2n+2; Qx2n+2; t)

∗M (STx2n+2; Px2n+1; �t)∗M (ABx2n+1;

Qx2n+2; (2 − �)t)∗M (ABx2n+1; STx2n+2; t)

= M (y2n; y2n+1; t)∗M (y2n+1; y2n+2; t)

∗M (y2n+1; y2n+1; (1 − q)t)∗M (y2n; y2n+2; (1 + q)t)∗M (y2n; y2n+1; t)

¿ M (y2n; y2n+1; t)∗M (y2n+1; y2n+2; t)

∗1∗M (y2n; y2n+1; t)

∗M (y2n+1; y2n+2; qt)∗M (y2n; y2n+1; t)

¿ M (y2n; y2n+1; t)∗M (y2n+1; y2n+2; t)

∗M (y2n; y2n+1; qt) (2a)

Since the t-norm ∗ is continuous and M (x; y; :) is leftcontinuous, letting q→ 1 in (2a), we have

M (y2n+1; y2n+2; kt)

¿ M (y2n; y2n+1; t)∗M (y2n+1; y2n+2; t) (2b)

Similarly we have also

M (y2n+2; y2n+3; kt)

¿ M (y2n+1; y2n+2; t)∗M (y2n+2; y2n+3; t) (2c)

Thus from (2b) and (2c), it follows that

M (yn+1; yn+2; kt)

¿ M (yn; yn+1; t)∗M (yn+1; yn+2; t)

for n= 1; 2; : : : and so, for positive integers n; p

M (yn+1; yn+2; kt)

¿ M (yn; yn+1; t)∗M (yn+1; yn+2; t=kp):

Thus, since M (yn+1; yn+2; t=kp)→ 1 as p→∞, wehave

M (yn+1; yn+2; kt) ¿ M (yn; yn+1; t):

By Lemma 2.2, {yn} is a Cauchy sequence in X . SinceX is complete, {yn} converges to a point z ∈X . Since{Px2n}; {Qx2n+1}; {ABx2n+1} and {STx2n+2} are sub-sequences of {yn}, they also converge to the point z,that is, as n→∞; Px2n; Qx2n+1; STx2n+1 → z. Since Aand B are continuous and the pair {P; AB} is compat-ible mappings of type (�), by Proposition 3.3(1), wehave as n→∞P(AB)x2n+1 → ABz; (AB)2x2n+1 → ABz:

Similarly, since S and T are continuous and the pair{Q; ST} is compatible mappings of type (�), by Propo-sition 3.3(1), we have as n→∞Q(ST )x2n+2 → STz and (ST )2x2n+2 → STz:

By putting x= (AB)x2n+1 and y= x2n+2 with �= 1 in(4:5), we have

M (P(AB)x2n+1; Qx2n+2; kt)

¿ M ((AB)2x2n+1; P(AB)x2n+1; t)

S. Sharma / Fuzzy Sets and Systems 127 (2002) 345–352 349

∗M (STx2n+2; Qx2n+2; t)

∗M (STx2n+2; P(AB)x2n+1; t)

∗M ((AB)2x2n+1; Qx2n+2; t)

∗M ((AB)2x2n+1; STx2n+2; t)

which implies that as n→∞M (ABz; z; kt)

¿ 1∗1∗M (z; ABz; t)M (ABz; z; t)∗M (ABz; z; t)

¿ M (ABz; z; t):

Therefore, by Lemma (2.3), we have ABz= z.By putting x=Px2n and y= x2n+1 with �= 1 in (4:5),we have

M (P(Px2n); Qx2n+1; kt)

¿ M (AB(Px2n); P(Px2n); t)

∗M (STx2n+1; Qx2n+1; t)

∗M (STx2n+1; P(Px2n); t)

∗M (AB(Px2n); Qx2n+1; t)

∗M (AB(Px2n); STx2n+1; t)

taking the limit n→∞, we have

M (Pz; z; kt)

¿ M (z; Pz; t)∗M (z; z; t)∗M (z; Pz; t)

∗M (z; z; t)∗M (z; z; t)

= M (z; Pz; t)∗1∗M (z; Pz; t)∗1∗1

¿ M (Pz; z; t)

By Lemma (2.3), we have Pz= z. ThereforeABz= z=Pz.

Now, we show that Bz= z. By putting x=Bz andy= x2n+1 with �= 1 in (4:5), and using (4:2), wehave

M (P(Bz); Qx2n+1; kt)

¿ M (AB(Bz); P(Bz); t)∗M (STx2n+1; Qx2n+1; t)

∗M (STx2n+1; P(Bz); t)∗M (AB(Bz); Qx2n+1; t)

∗M (AB(Bz); STx2n+1; t)

which implies that, as n→∞

M (Bz; z; kt)¿M (Bz; Bz; t)∗M (z; z; t)∗M (z; Bz; t)

∗M (Bz; z; t)∗M (Bz; z; t)

= 1∗1∗M (z; Bz; t)∗M (Bz; z; t)

∗M (Bz; z; t)

¿M (Bz; z; t):

Therefore, by Lemma 2.3, we have Bz= z. SinceABz= z, therefore Az= z.

By putting x= z and y= STx2n+2 with �= 1 in(4:5), we have

M (Pz; Q(ST )x2n+2; kt)

¿ M (ABz; Pz; t)∗M ((ST )2x2n+2; Q(ST )x2n+2; t)

∗M ((ST )2x2n+2; Pz; t)∗M (ABz; Q(ST )x2n+2; t)

∗M (ABz; (ST )2x2n+2; t)

taking the limit as n→∞, we have

M (z; STz; kt)

¿ M (z; z; t)∗M (STz; STz; t)∗M (STz; z; t)

∗M (z; STz; t)∗M (z; STz; t)

= 1∗1∗M (STz; z; t)∗M (z; STz; t)∗M (z; STz; t)

¿ M (z; STz; t):

By Lemma 2.3, this implies that STz= z.Now by putting x= z and y=Qx2n+1 with �= 1 in

(4:5) and using (4:2), we have

M (Pz; Q(Qx2n+1); kt)

¿ M (ABz; Pz; t)∗M (ST (Qx2n+1); Q(Qx2n+1); t)

∗M (ST (Qx2n+1); Pz; t)∗M (ABz; Q(Qx2n+1); t)

∗M (ABz; ST (Qx2n+1); t)

350 S. Sharma / Fuzzy Sets and Systems 127 (2002) 345–352

taking the limit n→∞, we have

M (z; Qz; kt)¿M (z; z; t)∗M (z; Qz; t)∗M (z; z; t)

∗M (z; Qz; t)∗M (z; z; t)

= 1∗M (z; Qz; t)∗1∗M (z; Qz; t)∗1

¿M (z; Qz; t):

Therefore, by Lemma 2.3, this implies that Qz= z andhence STz= z=Qz. Finally, we show that Tz= z. Byputting x= z and y=Tz with �= 1 in (4:5) and using(4:2), we have

M (Pz; Q(Tz); kt)

¿ (ABz; Pz; t)∗M (ST (Tz); Q(Tz); t)

∗M (ST (Tz); Pz; t)∗M (ABz; Q(Tz); t)

∗M (ABz; ST (Tz); t):

= M (z; z; t)∗M (Tz; Tz; t)∗M (Tz; z; t)∗M (z; Tz; t)

∗M (z; Tz; t)

= 1∗1∗M (Tz; z; t)∗M (z; Tz; t)∗M (z; Tz; t)

¿ M (Tz; z; t):

Therefore, by Lemma 2.3, we have Tz= z. SinceSTz= z, threrefore Sz= z.

By combining the above results, we have

Az = Bz = Sz = Tz = Pz = Qz = z;

that is z is a common "xed point of A; B; S; T; Pand Q.

For uniqueness, let w (w = z) be another common"xed point of A; B; S; T; P and Q and �= 1, then by(4:5), we write

M (Pz; Qw; kt)

¿ M (ABz; Pz; t)∗M (STw;Qw; t)∗M (STw; Pz; t)

∗M (ABz; Qw; t)∗M (ABz; STw; t)

it follows that

M (z; w; kt)¿M (z; z; t)∗M (w; w; t)∗M (w; z; t)

∗M (z; w; t)∗M (z; w; t)

= 1∗1∗M (w; z; t)∗M (z; w; t)

∗M (z; w; t)

¿M (z; w; t)

Therefore, by using Lemma 2.3, we have z=w.This completes the proof of the theorem.

Remark 4.1. In Theorem 4.1, if we put P=Q, ourtheorem reduces to the result due to Cho [8].

If we put B=T = Ix (the identity map on X ) inTheorem 4.1, we have the following:

Corollary 4.2. Let (X;M; ∗) be a complete fuzzy met-ric space with t∗t¿t for all t ∈ [0; 1] and the condi-tion (FM-6): Let A; S; P and Q be mappings fromX into itself such that

(4:6) P(X ) ⊂ A(X ); Q(X ) ⊂ S(X )(4:7) the pairs {P; A} and {Q; S} are compatible

of type (�);(4:8) A and S are continuous;(4:9)There exists a number k ∈ (0; 1) such that

M (Px; Qy; kt)

¿ M (Ax; Px; t)∗M (Sy; Qy; t)∗M (Sy; Px; �t)

∗M (Ax; Qy; (2 − �)t)∗M (Ax; Sy; t)

for all x; y∈X; �∈ (0; 2) and t¿0.

Then A; S; P and Q have a unique common "xedpoint in X .

If we put A=B= S =T = Ix in Theorem 4.1, wehave the following:

Corollary 4.3. Let (X;M; ∗) be a complete fuzzymetric space with t∗t¿t for all t ∈ [0; 1] and thecondition (FM-6): Let P and Q be mappings from Xinto itself. If there exists a constant k ∈ (0; 1) suchthat

S. Sharma / Fuzzy Sets and Systems 127 (2002) 345–352 351

M (Px; Qy; kt)¿M (x; Px; t)∗M (y;Qy; t)

∗M (y; Px; �t)∗M (x; Qy; (2 − �)t)∗M (x; y; t)

for all x; y∈X; �∈ (0; 2) and t¿0:

Then P and Q have a unique common "xed pointin X:

If we put P=Q; A= S and B=T = Ix in Theo-rem 4:1; we have the following:

Corollary 4.4. Let (X;M; ∗) be a complete fuzzy met-ric space with t∗t¿t for all t ∈ [0; 1] and the condi-tion (FM-6): Let P; S be compatible maps of type(�) on X such that P(X )⊂ S(X ): If S is continuousand there exists a constant k ∈ (0; 1) such that

M (Px; Py; kt)

¿ M (Sx; Px; t)∗M (Sy; Py; t)∗M (Sy; Px; �t)

∗M (Sx; Py; (2 − �)t)∗M (Sx; Sy; t)

for all x; y∈X; �∈ (0; 2) and t¿0:

Then P and S have a unique common "xed pointin X .

Example 4.1. Let the set X = [0; 1] with the metric dde"ned by

d(x; y) = |x − y| and for each t ∈ [0; 1] de"ne

M (x; y; t) =t

t + |x − y|M (x; y; 0) = 0

for all x; y∈X:

Clearly (X;M; ∗) is a complete fuzzy metric spacewhere ∗ is de"ned by a∗b= ab:

Let A; B; S; T; P and Q be de"ned as

Ax = x; Bx = x=2; Sx = x=5; Tx = x=3;

Px = x=6 and Qx = 0

for all x∈X:

Then P(X ) = [0; 1=6]⊂ [0; 1=2] =AB(X ) andQ(X ) = {0}⊂ [0; 1=15] = ST (X ):

If we take k = 1=2; t= 1 and �= 1, we see thatcondition (4:5) of Theorem 4.1 is satis"ed. Clearly,conditions (4:2) and (4:3) of Theorem 4.1 are alsosatis"ed.

Further the pair {P; AB} is compatible mappings oftype (�):

if limn→∞ xn = 0 where {xn} is a sequence in X;

such that

limn→∞ Pxn = lim

n→∞AB(xn) = 0 for some 0 ∈ X:

Similarly, the pair {Q; ST} is also compatible map-pings of type (�):

Thus all the conditions of Theorem 4.1, are sat-is"ed and 0 is the unique common "xed point ofA; B; S; T; P and Q:

We give following metric version of the condi-tion (4:5).

For self mappings A; B; S; T; P and Q of a metricspace (X; d) there exists a number k ∈ (0; 1) such that

d(Px; Qy)6 k max{d(ABx; Px); d(STy; Qy);1=2(d(ABx; Qy) + d(STy; Px));

d(ABx; STy)} (2d)

for all x; y∈X:If we put P=Q in (2d), we have

d(Px; Py)

6 k max{d(ABx; Px); d(STy; Py);1=2(d(ABx; Py) + d(STy; Px)); d(ABx; STy)}

(2e)

for all x; y∈X and k ∈ (0; 1):If we put B=T = Ix and A=B= S =T = Ix in (2e),

respectively, we have

d(Px; Py)6 k max{d(Ax; Px); d(Sy; Py);1=2(d(Ax; Py) + d(Sy; Px));

d(Ax; Sy)} (2f)

352 S. Sharma / Fuzzy Sets and Systems 127 (2002) 345–352

and

d(Px; Py)

6 k max{d(x; Px); d(y; Py); 1=2(d(x; Py)

+d(y; Px)); d(x; y)} (2g)

for all x; y∈X , where k ∈ (0; 1), we can "nd many"xed point theorems for three or four or "ve mappingson metric spaces satisfying the conditions (2e), (2f )and (2g) in many papers.

Acknowledgements

The author thanks Professor Y. J. Cho for supplyingmaterials during the preparation of this paper.

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