common fixed point theorems for finite number of mappings without continuity and compatibility on...

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Chaos, Solitons and Fractals 40 (2009) 2242–2256

www.elsevier.com/locate/chaos

Common fixed point theorems for finite numberof mappings without continuity and compatibility

on intuitionistic fuzzy metric spaces

Sushil Sharma a, Bhavana Deshpande b,*

a Department of Mathematics, Madhav Science College, Ujjain (M.P.), Indiab Department of Mathematics, Government Arts and Science P.G. College, Sukhakarta,

90 Rajiv Nagar (Near Kasturba Nagar), Ratlam 457001 (M.P.), India

Accepted 8 October 2007

Abstract

The purpose of this paper is to prove some common fixed point theorems for finite number of discontinuous, non-compatible mappings on noncomplete intuitionistic fuzzy metric spaces. Our results extend, generalize and intuitionisticfuzzify several known results in fuzzy metric spaces. We give an example and also give formulas for total number ofcommutativity conditions for finite number of mappings.� 2007 Elsevier Ltd. All rights reserved.

1. Introduction

The theory of fuzzy sets has evolved in many directions after investigation of notion of fuzzy sets by Zadeh [68] andis finding applications in wide variety of fields in which the phenomenon under study are too complex or too ill definedto be analyzed by the conventional techniques. In applications of fuzzy set theory the field of engineering has undoubt-edly been a leader. All engineering disciplines such as civil engineering, electrical engineering, mechanical engineering,robotics, industrial engineering, computer engineering, nuclear engineering, etc. have already been affected to variousdegrees by the new methodological possibilities opened by fuzzy sets.

There are large number of authors who studied applications of fuzzy set theory in different engineering branches. Weare mentioning some of them. Fetz [16,17] Fetz et al. [18], Halder and Reddy [23], Lessmann et al. [38], Moller [43] andmany others applied fuzzy set theory in civil engineering. A method utilizing the mathematics of fuzzy sets has beenshown to be effective in solving engineering problems such as aircraft gas turbine [37], car body structure NVH design[40], multiobjective system optimization [46], preliminary passenger vehicle structure [51], computational tools for pre-liminary engineering design [66], knowledge base system design [69], intelligent system design support [70], machine flex-ibility [63] and many others.

0960-0779/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2007.10.011

* Corresponding author.E-mail addresses: [email protected] (S. Sharma), [email protected] (B. Deshpande).

mailto:[email protected]

mailto:[email protected]

S. Sharma, B. Deshpande / Chaos, Solitons and Fractals 40 (2009) 2242–2256 2243

The inventory of successful applications of fuzzy set theory has been growing steadily, particularly in 1990s. Med-icine, Economics, interpersonal communication, psychology, physics, chemistry, biology, ecology, political science,geology, etc. are fields in which the applicability of fuzzy set theory was recognized. For applications of fuzzy set theoryin physics one can refer [10–13].

The theory of fixed points is one of the basic tools to handle various physical formulations. Fixed point theorems infuzzy mathematics are emerging with vigorous hope and vital trust. There have been several attempts to formulate fixedpoint theorems in fuzzy mathematics. From amongst several formulations of fuzzy metric spaces [7,14,33,36,19], Gra-biek [21] followed Kramosil and Michalek [36] and obtained fuzzy version of Banach contraction principle.

Jungck [28] established common fixed point theorem for commuting maps generalizing the Banach’s fixed pointtheorem.

Sessa [52] defined a generalization of commutativity. Further Jungck [29] introduced more generalized commutativ-ity so called compatibility. Jungck and Rhoades [32] introduced the notion of weak compatible maps and proved thatcompatible maps are weakly compatible but converse is not true.

The notion of compatible maps in fuzzy metric spaces has been introduced by Mishra et al. [42], compatible maps oftype (a) by Cho [5] and compatible maps of type (b) by Cho et al. [6].

Chang et al. [4], Cho [5], Cho et al. [6], Fang [15], George and Veeramani [20], Jung et al. [26], Jung et al. [27], Mishraet al. [42], Sharma [53,54], Sharma and Deshpande [55–58], Subrahmanyam [61] and many others studied fixed pointtheorems in fuzzy metric spaces.

Motivated by the potential applicability of fuzzy topology to quantum particle physics particularly in connectionwith both string and e(1) theory developed by El Naschie [10,11]. Park introduced and discussed in [44] a notion ofintuitionistic fuzzy metric spaces which is based both on the idea of intuitionistic fuzzy set due to Atanassov [1] andthe concept of a fuzzy metric space given by George and Veeramani in [19]. Actually, park’s notion is useful in mod-elling some phenomena where it is necessary to study relationship between two probability function. It has a directphysics motivation in the context of the two slit experiment as foundation of E-infinity of high energy physics, recentlystudied by El Naschie in [12,13].

Alaca et al. [2] using the idea of intuitionistic fuzzy sets, they defined the notion of intuitionistic fuzzy metric space asPark [44] with the help of continuous t-norms and continuous t-conorms as a generalization of fuzzy metric space due toKramosil and Michalek [36]. Further, they introduced the notion of Cauchy sequences in intuitionistic fuzzy metricspaces and proved the well known fixed point theorems of Banach [3] and Edelstein [9] extended to intuitionistic fuzzymetric spaces with the help of Grabiec [21]. Turkoglu et. al [64] introduced the concept of compatible maps and com-patible maps of type (a) and (b) in intuitionistic fuzzy metric spaces and gave some relations between the concepts ofcompatible maps of type (a) and (b).

Turkoglu et al. [64] gave generalization of Jungck’s common fixed point theorem [29] to intuitionistic fuzzy metricspaces.

Gregori et al. [22], Saadati and Park [49] studied the concept of intuitionistic fuzzy metric spaces and its applications.Most of the fixed point theorems in intuitionistic fuzzy metric spaces deal with conditions of continuity and com-

patibility or compatibility of type ( a) or compatible of type (b).There are maps which are not continuous but have fixed points. Also weakly compatible maps defined by Jungck

and Rhoades [32] are weaker than that of compatibility.These observations motivated us to prove common fixed point theorem for ten noncompatible, discontinuous map-

pings in noncomplete intuitionistic fuzzy metric spaces. We also extend our results for finite number of mappings. Ourmain theorems extend, improve, generalize and intuitionistic fuzzify several known results in fuzzy metric spaces. Wegive an example to validate our result. To prove existence of common fixed point for finite number of mappings somecommutativity conditions are required. How many commutativity conditions are necessary? We give answer of thisquestion by giving formulas.

2. Preliminaries

Definition 1 (cf. [50]). A binary operation *: [0,1]! [0,1] is continuous t-norm if * is satisfying the followingconditions:

(i) * is commutative and associative,(ii) * is continuous,

(iii) a * 1 = a for all a 2 [0,1],

2244 S. Sharma, B. Deshpande / Chaos, Solitons and Fractals 40 (2009) 2242–2256

(iv) a * b 6 c * d whenever a 6 c and b 6 d, for all a, b, c, d 2 [0,1].

Definition 2 (cf. [50]). A binary operation e: [0,1] · [0,1]! [0,1] is continuous t-conorm if e is satisfying the follow-ing conditions:

(i) e is commutative and associative,(ii) e is continuous,

(iii) a e 0 = a for all a 2 [0,1],(iv) a e b 6 c e d whenever a 6 c and b 6 d for all a, b, c, d 2 [0,1].

Remark 1. The concept of triangular norms (t-norms) and Triangular conorms (t-conorms) are known as the axiomaticskeletons that we use for characterizing fuzzy intersections and unions respectively. These concepts were originallyintroduced by Menger [41] in his study of statistical metric spaces. Several examples for these concepts were proposedby many authors [8,34,35,67].

Definition 3 (cf. [2]). A 5-tuple (X,M,N, *,e) is said to be an intuitionistic fuzzy metric space if X is an arbitrary set * isa continuous t-norm, e is continuous t-conorm and M,N are fuzzy sets on X2 · [0,1) satisfying the followingconditions:

(i) M(x,y, t) + N(x,y, t) 6 1 for all x,y 2 X and t > 0,(ii) M(x,y, 0) = 0 for all x,y 2 X,

(iii) M(x,y, t) = 1 for all x,y 2 X and t > 0 if and only if x = y,(iv) M(x,y, t) = M(y,x, t) for all x,y 2 X, t > 0,(v) M(x,y, t) * M(y,z, s) 6M(x,z, t + s) for all x, y, z 2 X and s, t > 0,

(vi) for all x,y 2 X,M(x,y, Æ):[0,1)! [0,1] is left continuous,(vii) limt!1M(x,y, t) = 1 for all x, y 2 X and t > 0,

(viii) N(x,y, 0) = 1 for all x,y 2 X,(ix) N(x,y, t) = 0 for all x,y 2 X and t > 0 iff x = y,(x) N(x,y, t) = N(y,x, t) for all x,y 2 X and t > 0,

(xi) N(x,y, t) e N(y,z, s) P N(x,z, t + s) for all x, y, z 2 X and s, t > 0,(xii) for all x,y 2 X,N(x,y, Æ):[0,1)! [0,1] is right continuous,

(xiii) limt!1N(x,y, t) = 0 for all x, y 2 X.

Then (M,N) is called an intuitionistic fuzzy metric on X. The function M(x,y, t) and N(x,y, t) denote the degree ofnearness and the degree of non nearness between x and y with respect to t, respectively.

Remark 2. Every fuzzy metric space (X,M, *) is an intuitionistic fuzzy metric space of the form (X,M,1 �M, *,e) suchthat t-norm * and t-conorm e are associated [39], i. e. x e y = 1 � ((1 � x)*(1 � y)) for all x, y 2 X.

Example 1. Let (X,d) be a metric space. Define t-norm a * b = min{a,b} and t-conorm a e b = max{a,b} and for all x,y 2 X and t > 0,

Mdðx; y; tÞ ¼t

t þ dðx; yÞ ; Ndðx; y; tÞ ¼dðx; yÞ

t þ dðx; yÞ

Then (X,M,N, *,e) is an intuitionistic fuzzy metric space. We call this intuitionistic fuzzy metric (M,N) induced by themetric d the standard intuitionistic fuzzy metric.

Remark 3. In intuitionistic fuzzy metric space (X,M,N, *,e), M(x,y, Æ) is non-decreasing and N(x,y, Æ) is non increasingfor all x,y 2 X.

Definition 4 [2]. Let (X,M,N, *,e) be an intuitionistic fuzzy metric space. Then

(i) a sequence {xn} in X is said to be Cauchy sequence if for all t > 0 and p > 0,


limn!1

Mðxnþp; xn; tÞ ¼ 1; limn!1

Nðxnþp; xn; tÞ ¼ 0

(ii) a sequence {xn} in X is said to be convergent to a point x 2 X if for all t > 0,

limn!1

Mðxn; x; tÞ ¼ 1; limn!1

Nðxn; x; tÞ ¼ 0:

Since * and e are continuous, the limit is uniquely determined from (v) and (xi) respectively.

Definition 5 (cf. [2]). An intuitionistic fuzzy metric space (X,M,N, *,e) is said to be complete if and only if every Cau-chy sequence in X is convergent.

The proof of the following lemma is on the lines of Cho [5].

Lemma A. Let (X,M, N,*,e) be an intuitionistic fuzzy metric space and {yn} be a sequence in X. If there exists a number

k 2 (0,1) such that

(I) M(yn+2,yn+1,kt) P M(yn+1,yn, t)(II) N(yn+2,yn+1,kt) 6 N(yn+1,yn, t)

for all t > 0 and n = 1, 2, . . . then {yn} is a Cauchy sequence in X.

Proof. By simple induction with the condition (I) and with the help of Alaca et. al [2], we have for all t > 0 and n = 1,2, . . . ,

Mðynþ1; ynþ2; tÞP M y1; y2;t

kn

� �; Nðynþ1; ynþ2; tÞ 6 N y1; y2;

tkn

� �ðIIIÞ

Thus by (III) and Definition 3(v) and (xi), for any positive integer p and real number t > 0, we have

Mðyn; ynþp; tÞP M yn; ynþ1;tp

� �� . . . p-times . . . �M ynþp�1; ynþp;

tp

� �

P M y1; y2;t

pkn�1

� �� p-times . . . �M y1; y2;

t

pknþP�2

� �

and

Nðyn; ynþp; tÞ 6 N yn; ynþ1;tp

� �� . . . p-times � � ��N ynþp�1; ynþp;

tp

� �

6 N y1; y2;t

pkn�1

� �� p-times � � ��N y1; y2;

t

pknþP�2

� �:

Therefore by Definition 3(vii) and (xiii), we have

limn!1

Mðyn; ynþp; tÞP 1 � � � � p-times � � � � 1 P 1

and

limn!1

Nðyn; ynþp; tÞP 0 � � � � p-times � � �� 0 6 0;

which implies that {yn} is a Cauchy sequence in X. This completes the proof. h

The proof of the following lemma is on the lines of Mishra et al. [42].

Lemma B. Let (X, M,N,*,e) be an intuitionistic fuzzy metric space and for all x,y 2 X, t > 0 and if for a number k 2 (0,1)

Mðx; y; ktÞP Mðx; y; tÞ and Nðx; y; ktÞ 6 Nðx; y; tÞ ðIVÞ

then x = y.

Proof. Since t > 0 and k 2 (0,1), we get t > kt. Using Remark 3, we have

Mðx; y; tÞP Mðx; y; ktÞ and Nðx; y; tÞ 6 Nðx; y; ktÞ:

Using (IV) and the definition of intuitionistic fuzzy metric, we have x = y. h


Definition 6 (cf. [65]). Let A and B be maps from an intuitionistic fuzzy metric space (X,M,N, *,e) into itself. Themaps A and B are said to be compatible if for all t > 0, limn!1M(ABxn, BAxn, t) = 1 and limn!1N(ABxn,BAxn, t) = 0whenever {xn} is a sequence in X such that limn!1Axn = limn!1 Bxn = z for some z 2 X.

Definition 7 (cf. [65]). Let A and B be maps from an intuitionistic fuzzy metric space (X,M,N, *,e) into itself. Themaps A and B are said to be compatible of type (a) if for all t > 0,

limn!1

MðABxn;BBxn; tÞ ¼ 1 and limn!1

NðABxn;BBxn; tÞ ¼ 0;

limn!1

MðBAxn;AAxn; tÞ ¼ 1 and limn!1

NðBAxn;AAxn; tÞ ¼ 0

whenever {xn} is a sequence in X such that limn!1Axn = limn!1Bxn = z for some z 2 X.

Definition 8 (cf. [65]). Let A and B be maps from an intuitionistic fuzzy metric space (X,M,N, *,e) into itself. Themaps A and B are said to be compatible of type (b) if for all t > 0,

limn!1

MðAAxn;BBxn; tÞ ¼ 1 and limn!1

NðAAxn;BBxn; tÞ ¼ 0;

whenever {xn} is a sequence in X such that limn!1Axn = limn!1 Bxn = z for some z 2 X.

Definition 9 [32]. Two self maps A and B on a set X are said to be weakly compatible if they commute at coincidencepoint.

Remark 4

(i) In [28,31,45] we can find the equivalent formulations of definitions of compatible maps, compatible maps of type(a) and compatible maps of type (b). Such maps are independent of each other and more general than commutingand weakly commuting maps [28,52].

(ii) Compatible or compatible of type (a) or compatible of type (b) maps are weakly compatible but converse neednot true [57].

In our theorems and corollaries (X,M,N, *,e) will denote an intuitionistic fuzzy metric space (IFM-space)with continuous t-norm * and continuous t-conorm e defined by t * t P t and (1 � t) e (1 � t) 6 (1 � t) for allt 2 [0,1].

3. Main results

Theorem 1. Let (X,M,N,*,e) be an IFM-space. Let A, B, S, T, I, J, L, U, P and Q be mappings from X into itself such

that

(1.1) P(X) � ABIL(X), Q(X) � STJU(X),(1.2) there exists a constant k 2 (0,1) such that

½1þ aMðSTJUx;ABILy; ktÞ� �MðPx;Qy; ktÞP a½MðPx; STJUx; ktÞ �MðQy;ABILy; ktÞ þMðQy; STJUx; ktÞ�MðPx;ABILy; ktÞ� þMðABILy; STJUx; tÞ �MðPx; STJUx; tÞ�MðQy;ABILy; tÞ �MðQy; STJUx; atÞ �MðPx;ABILy; ð2� aÞtÞ

for all x, y 2 X, a P 0, a 2 (0,2) and t > 0 and

½1þ aNðSTJUx;ABILy; ktÞ��NðPx;Qy; ktÞ 6 a½NðPx; STJUx; ktÞ � NðQy;ABILy; ktÞþ NðQy; STJUx; ktÞ�NðPx;ABILy; ktÞ�þ NðABILy; STJUx; tÞ�NðPx; STJUx; tÞ�NðQy;ABILy; tÞ� NðQy; STJUx; atÞ�NðPx;ABILy; ð2� aÞtÞ


for all x, y 2 X, a P 0, a 2 (0,2) and t > 0,

(1.3) if one of P(X), ABIL(X), STJU(X), Q(X) is a complete subspace of X then

(i) P and STJU have a coincidence point and(ii) Q and ABIL have a coincidence point.

Further if

(1.4) AB = BA, AI = IA, AL = LA, BI = IB, BL = LB, IL = LI,QL = LQ, QI = IQ, QB = BQ, ST = TS, SJ = JS,SU = US,TJ = JT, TU = UT, JU = UJ, PU = UP, PJ = JP, PT = TP,

(1.5) the pairs {P,STJU} and {Q,ABIL} are weakly compatible, then

(iii) A, B, S, T, I, J, L, U, P and Q have a unique common fixed point in X.

Proof. By (1.1) since P(X) � ABIL(X) for any point x0 2 X there exists a point x1 in X such that Px0 = ABILx1. SinceQ(X) � STJU(X), for this point x1 we can choose a point x2 in X such that Qx1 = STJUx2 and so on. Inductively, wecan define a sequence {yn} in X such that for n = 0, 1, 2, . . . .

y2n ¼ Px2n ¼ ABILx2nþ1 and y2nþ1 ¼ Qx2nþ1 ¼ STJUx2nþ2:

By (1.2), for all t > 0 and a = 1 � q with q 2 (0,1), we have

½1þ aMðy2n; y2nþ1; ktÞ� �Mðy2nþ1; y2nþ2; ktÞP a½Mðy2nþ2; y2nþ1; ktÞ �Mðy2nþ1; y2n; ktÞ þMðy2nþ1; y2nþ1; ktÞ�Mðy2nþ2; y2n; ktÞ� þMðy2n; y2nþ1; tÞ �Mðy2nþ2; y2nþ1; tÞ�Mðy2nþ1; y2n; tÞ �Mðy2nþ1; y2nþ1; ð1� qÞtÞ �Mðy2nþ2; y2n; ð1þ qÞtÞ;

P a½Mðy2n; y2nþ1; ktÞ �Mðy2nþ1; y2nþ2; ktÞ� þMðy2n; y2nþ1; tÞ�Mðy2nþ1; y2nþ2; tÞ �Mðy2n; y2nþ1; qtÞ

and

½1þ aNðy2n; y2nþ1; ktÞ��Nðy2nþ1; y2nþ2; ktÞ 6 a½Nðy2nþ2; y2nþ1; ktÞ � Nðy2nþ1; y2n; ktÞþ Nðy2nþ1; y2nþ1; ktÞ�Nðy2nþ2; y2n; ktÞ�þ Nðy2n; y2nþ1; tÞ�Nðy2nþ2; y2nþ1; tÞ�Nðy2nþ1; y2n; tÞ � Nðy2nþ1; y2nþ1; ð1� qÞtÞ � Nðy2nþ2; y2n; ð1þ qÞtÞ;

6 a½Nðy2n; y2nþ1; ktÞ � Nðy2nþ1; y2nþ2; ktÞ�þ Nðy2n; y2nþ1; tÞ�Nðy2nþ1; y2nþ2; tÞ � Nðy2n; y2nþ1; qtÞ:

Thus it follows that

Mðy2nþ1; y2nþ2; ktÞP Mðy2n; y2nþ1; tÞ �Mðy2nþ1; y2nþ2; tÞ �Mðy2n; y2nþ1; qtÞ ð1aÞ

and

Nðy2nþ1; y2nþ2; ktÞ 6 Nðy2n; y2nþ1; tÞ � Nðy2nþ1; y2nþ2; tÞ � Nðy2n; y2nþ1; qtÞ ð1bÞ

Since the t-norm * and the t-conorm e are continuous and M(x,y, Æ) is left continuous and N(x,y, Æ) is right continuousletting q! 1 in (1a) and (1b), we have

Mðy2nþ1; y2nþ2; ktÞP Mðy2n; y2nþ1; tÞ �Mðy2nþ1; y2nþ2; tÞ ð1cÞ

and

Nðy2nþ1; y2nþ2; ktÞ 6 Nðy2n; y2nþ1; tÞ � Nðy2nþ1; y2nþ2; tÞ: ð1dÞ

Similarly, we also have

Mðy2nþ2; y2nþ3; ktÞP Mðy2nþ1; y2nþ2; tÞ �Mðy2nþ2; y2nþ3; tÞ ð1eÞ

and

Nðy2nþ2; y2nþ3; ktÞ 6 Nðy2nþ1; y2nþ2; tÞ � Nðy2nþ2; y2nþ3; tÞ: ð1fÞ


Thus from (1c)–(1f), it follows that for m = 1, 2, . . ..

Mðymþ1; ymþ2; ktÞP Mðym; ymþ1; tÞ � Mðymþ1; ymþ2; tÞ and

Nðymþ1; ymþ2; ktÞ 6 Nðym; ymþ1; tÞ � Nðymþ1; ymþ2; tÞ:

Consequently, it follows that for m = 1, 2, . . . , p = 1, 2, . . .

Mðymþ1; ymþ2; ktÞP Mðym; ymþ1; tÞ �Mðymþ1; ymþ2; t=kpÞ

and

Nðymþ1; ymþ2; ktÞ 6 Nðym; ymþ1; tÞ�Nðymþ1; ymþ2; t=kpÞ:

By noting that M(ym+1,ym+2, t/kp)! 1 as p!1 and N(ym+1,ym+2, t/kp)! 0 we have for m = 1, 2, . . . .

Mðymþ1; ymþ2; ktÞP Mðym; ymþ1; tÞ and

Nðymþ1; ymþ2; ktÞ 6 Nðym; ymþ1; tÞ:

Hence by Lemma A, {yn} is a Cauchy sequence in X. Now suppose STJU(X) is complete. Note that the subsequence{y2n+1} is contained in STJU(X) and has a limit in STJU(X) call it z. Let w 2 STJU�1(z). Then STJUw = z. We shalluse the fact that subsequence {y2n} also converges to z. By putting x = w,y = x2n+1 in (1.2), with a = 1 and taking limitas n!1 we have

MðPw; z; ktÞP MðPw; z; tÞ and NðPw; z; ktÞ 6 NðPw; z; tÞ:

Therefore by Lemma B, we have Pw = z. Since STJUw = z thus we have Pw = z = STJUw that is w is coincidence pointof P and STJU. This proves (i).

Since P(X) � ABIL(X), Pw = z implies that z 2 ABIL(X). Let v 2 ABIL�1z. Then ABILv = z.By putting x = x2n+2, y = v in (1.2), with a = 1 and taking limit as n!1 we have

MðQv; z; ktÞP MðQv; z; tÞ and NðQv; z; ktÞ 6 NðQv; z; tÞ:

Therefore by Lemma B, we have Qv = z. Since ABILv = z, we have Qv = z = ABILv that is v is coincidence point of Q

and ABIL. This proves (ii).The remaining two cases pertain essentially to the previous cases. Indeed if P(X) or Q(X) is complete then by (1.1),

z 2 P(X) � ABIL(X) or z 2 Q(X) � STJU(X). Thus (i) and (ii) are completely established.Since the pair {P,STJU} is weakly compatible therefore P and STJU commute at their coincidence point that is

P(STJUw) = (STJU)Pw or Pz = STJUz.Since the pair {Q,ABIL} is weakly compatible therefore Q and ABIL commute at their coincidence point that is

Q(ABILv) = (ABIL)Qv or Qz = ABILz. By putting x = z, y = x2n+1 in (1.2), with a = 1 and taking limit as n!1 wehave

MðPz; z; ktÞP MðPz; z; tÞ and NðPz; z; ktÞ 6 NðPz; z; tÞ:

Therefore by Lemma B, we have Pz = z. So Pz = STJUz = z. By putting x = x2n+2, y = z in (1.2), with a = 1 and takinglimit as n!1 we have

Mðz;Qz; ktÞP MðQz; z; tÞ and Nðz;Qz; ktÞ 6 NðQz; z; tÞ:

Therefore by Lemma B, we have Qz = z, so Qz = ABILz = z. By putting x = z, y = Lz in (1.2) with a = 1 and using(1.4), we have

Mðz; Lz; ktÞP MðLz; z; tÞ � 1 � 1 �MðLz; z; tÞ �MðLz; z; tÞP MðLz; z; tÞ and

Nðz; Lz; ktÞ 6 NðLz; z; tÞ � 0 � 0 � NðLz; z; tÞ � NðLz; z; tÞ 6 NðLz; z; tÞ:

Therefore by Lemma B, we have Qz = z, so Qz = ABILz = z. By putting x = z, y = Lz in (1.2) with a = 1 and using(1.4), we have

Mðz; Lz; ktÞP MðLz; z; tÞ � 1 � 1 �MðLz; z; tÞ �MðLz; z; tÞP MðLz; z; tÞ and

Nðz; Lz; ktÞ 6 NðLz; z; tÞ � 0 � 0 � NðLz; z; tÞ � NðLz; z; tÞ 6 NðLz; z; tÞ:

Therefore by Lemma B, we have Iz = z. Since ABIz = z therefore ABz = z. Now to prove Bz = z we put x = z, y = Bz in(1.2), with a = 1 and using (1.4), we have


MðIz; z; ktÞP MðIz; z; tÞ � 1 � 1 �MðIz; z; tÞ �Mðz; Iz; tÞP MðIz; z; tÞ and

NðIz; z; ktÞ 6 NðIz; z; tÞ � 0 � 0�NðIz; z; tÞ � Nðz; Iz; tÞ 6 NðIz; z; tÞ:

Therefore by Lemma B, we have Bz = z. Since ABz = z therefore Az = z. To prove Uz = z, we put x = Uz, y = z in(1.2), with a = 1 and using (1.4), we have

MðUz; z; ktÞP MðUz; z; tÞ � 1 � 1 �MðUz; z; tÞ �MðUz; z; tÞP MðUz; z; tÞ

and

NðUz; z; ktÞ 6 NðUz; z; tÞ � 0 � 0 � NðUz; z; tÞ � NðUz; z; tÞ 6 NðUz; z; tÞ:

Therefore by Lemma B, we have Uz = z. Since STJUz = z therefore STJz = z. To prove Jz = z put x = Jz, y = z in (1.2)with a = 1 and using (1.4), we have

MðJz; z; ktÞP MðJz; z; tÞ � 1 � 1 �MðJz; z; tÞ �MðJz; z; tÞP MðJz; z; tÞ

and

NðJz; z; ktÞ 6 NðJz; z; tÞ � 0 � 0 � NðJz; z; tÞ � NðJz; z; tÞ 6 NðJz; z; tÞ:

Therefore by Lemma B, we have Jz = z. Since STJz = z therefore STz = z. To prove Tz = z put x = Tz, y = z in (1.2),with a = 1 and using (1.4), we have

MðTz; z; ktÞP MðTz; z; tÞ � 1 � 1 �MðTz; z; tÞ �MðTz; z; tÞP MðTz; z; tÞ

and

NðTz; z; ktÞ 6 NðTz; z; tÞ � 0 � 0�NðTz; z; tÞ�NðTz; z; tÞ 6 NðTz; z; tÞ:

Therefore by Lemma B, we have Tz = z. Since STz = z therefore Sz = z. By combining the above results we haveAz = Bz = Sz -= Tz = Iz = Jz = Lz = Uz = Pz = Qz = z. That is z is a common fixed point of A, B, S, T, I, J, L, U,P and Q. The uniqueness of the common fixed point of A, B, S, T, I, J, L, U, P and Q follows easily from (1.2). Thiscompletes the proof.

From Theorem 1, with a = 0, we have the following result: h

Corollary 2. Let (X,M,N,*,e) be an IFM-space. Let A, B, S, T, I, J, L, U, P and Q be mappings from X into itself satisfy

condition (1.2) with a = 0.

If conditions (1.1) and (1.3) are satisfied then conclusions (i) and (ii) of Theorem 1 hold. Further if conditions (1.4) and (1.5)

are satisfied then conclusion (iii) of Theorem 1 holds.

If we put P = Q in Theorem 1, we have the following result:

Corollary 3. Let (X,M, N,*,e) be an IFM-space. Let A, B, S, T, I, J, L, U and P be mappings from X into itself such that

(3.1) P(X) � ABIL(X), P(X) � STJU(X),(3.2) there exists a constant k 2 (0,1) such that

½1þ aMðSTJUx;ABILy; ktÞ� �MðPx; Py; ktÞP a½MðPx; STJUx; ktÞ �MðPy;ABILy; ktÞ þMðPy; STJUx; ktÞ�MðPx;ABILy; ktÞ� þMðABILy; STJUx; tÞ �MðPx; STJUx; tÞ�MðPy;ABILy; tÞ �MðPy; STJUx; atÞ �MðPx;ABILy; ð2� aÞtÞ

for all x, y 2 X, a P 0, a 2 (0,2) and t > 0

and

½1þ aNðSTJUx;ABILy; ktÞ��NðPx; Py; ktÞ 6 a½NðPx; STJUx; ktÞ � MðPy;ABILy; ktÞ þ NðPy; STJUx; ktÞ�NðPx;ABILy; ktÞ� þ NðABILy; STJUx; tÞ � NðPx; STJUx; tÞ�NðPy;ABILy; tÞ � NðPy; STJUx; atÞ�NðPx;ABILy; ð2� aÞtÞ


(3.3) if one of P(X), ABIL(X), STJU(X) is a complete subspace of X then

(i) P and STJU have a coincidence point and

(ii) P and ABIL have a coincidence point.


Further if

(3.4) AB = BA, AI = IA, AL = LA, BI = IB, BL = LB, IL = LI, PL = LP, PI = IP, PB = BP, ST = TS, SJ = JS,

SU = US, TJ = JT, TU = UT, JU = UJ, PU = UP, PJ = JP, PT = TP,

(3.5) the pairs {P,STJU} and {P, ABIL} are weakly compatible, then

(iii) A, B, S, T, I, J, L, U and P have a unique common fixed point in X.

From Corollary 3 with a = 0, we have the following:

Corollary 4. Let (X,M,N,*,e) be an IFM-space. Let A, B, S, T, I, J, L, U and P be mappings from X into itself satisfy


If conditions (3.1) and (3.3) are satisfied then conclusions (i) and (ii) of Corollary 3 hold. Further if conditions (3.4) and

(3.5) are satisfied then conclusion (iii) of Corollary 3 holds.

If we put L = U = IX (The identity map on X) in Theorem 1, we have the following:

Corollary 5. Let (X,M, N,*,e) be an IFM-space. Let A, B, S, T, I, J, P and Q be mappings from X into itself such that

(5.1) P(X) � ABI(X), Q(X) � STJ(X),(5.2) there exists a constant k 2 (0,1) such that

½1þ aMðSTJx;ABIy; ktÞ� �MðPx;Qy; ktÞP a½MðPx; STJx; ktÞ �MðQy;ABIy; ktÞ þMðQy; STJx; ktÞ �MðPx;ABIy; ktÞ�þMðABIy; STJx; tÞ �MðPx; STJx; tÞ �MðQy;ABIy; tÞ �MðQy; STJx; atÞ�MðPx;ABIy; ð2� aÞtÞ


and

½1þ aNðSTJx;ABIy; ktÞ��NðPx;Qy; ktÞ 6 a½NðPx; STJx; ktÞ � NðQy;ABIy; ktÞ þ NðQy; STJx; ktÞ�NðPx;ABIy; ktÞ�

þ NðABIy; STJx; tÞ � NðPx; STJx; tÞ�NðQy;ABIy; tÞ � NðQy; STJx; atÞ

�NðPx;ABIy; ð2� aÞtÞ


(5.3) if one of P(X), ABI(X), STJ(X), Q(X) is a complete subspace of X then

(i) P and STJ have a coincidence point and

(ii) Q and ABI have a coincidence point.

Further if

(5.4) AB = BA, AI = IA, BI = IB, QI = IQ, QB = BQ,ST = TS, SJ = JS, TJ = JT, PJ = JP, PT = TP,

(5.5) the pairs {P,STJ} and {Q,ABI} are weakly compatible, then
(iii) A, B, S, T, I, J, P and Q have a unique common fixed point in X. If we put a = 0 in Corollary 5, we get the following:
Corollary 6. Let (X,M,N,*,e) be an IFM-space. Let A, B, S, T, I, J, P and Q be mappings from X into itself satisfy




If we put P = Q in Corollary 5 we get the following:

Corollary 7. Let (X,M, N,*,e) be an IFM-space. Let A, B, S, T, I, J and P be mappings from X into itself such

that


(7.1) P(X) � ABI(X), P(X) � STJ(X),(7.2) there exists a constant k 2 (0,1) such that

½1þ aMðSTJx;ABIy; ktÞ� �MðPx; Py; ktÞP a½MðPx; STJx; ktÞ �MðPy;ABIy; ktÞ þMðPy; STJx; ktÞ �MðPx;ABIy; ktÞ�þMðABIy; STJx; tÞ �MðPx; STJx; tÞ �MðPy;ABIy; tÞ �MðPy; STJx; atÞ�MðPx;ABIy; ð2� aÞtÞ


and

½1þ aNðSTJx;ABIy; ktÞ��NðPx; Py; ktÞ 6 a½NðPx; STJx; ktÞ � NðPy;ABIy; ktÞ þ NðPy; STJx; ktÞ�NðPx;ABIy; ktÞ�þ NðABIy; STJx; tÞ�NðPx; STJx; tÞ�NðPy;ABIy; tÞ� NðPy; STJx; atÞ � NðPx;ABIy; ð2� aÞtÞ


(7.3) if one of P(X), ABI(X), STJ(X) is a complete subspace of X then

(i) P and STJ have a coincidence point and

(ii) P and ABI have a coincidence point.

Further if

(7.4) AB = BA, AI = IA, BI = IB, PI = IP, PB = BP,ST = TS, SJ = JS, TJ = JT, PJ = JP, PT = TP,

(7.5) the pairs {P, STJ} and {P, ABI} are weakly compatible, then

(iii) A, B, S, T, I, J and P have a unique common fixed point in X.

If we put a = 0 in Corollary 7, we get the following:

Corollary 8. Let (X,M,N,*,e) be an IFM-space. Let A, B, S, T, I, J and P be mappings from X into itself satisfy con-

dition (7.2) with a = 0.



Remark 5. Theorem 1 and Corollary 2–8 improve, extend, generalize and intuitionistic fuzzify the results of Cho [5],Cho et al. [6], Grabiec [21], Iseki [24], Istratescu [25], Jungck [28–30], Jungck et al. [31], Mishra et al. [42], Rhoades[47,48], Sharma [53,54], Sharma and Deshpande [55–58], Singh [59], Singh and Kasahara [60], Sing and Ram [47],Tiwari and Singh [62].

Remark 6.

(i) From Corollary 5, with I = J = IX (the identity map on X), we obtain the intuitionistic version of the result ofSharma and Deshpande [58].

(ii) From Corollary 6, with I = J = IX (the identity map on X),we obtain the intuitionistic version of the result ofSharma and Deshpande [57].

(iii) From Corollary 7, with I = J = IX (the identity map on X),we obtain the intuitionistic version of the result ofSharma and Deshpande [58].

(iv) From Corollary 8, with I = J = IX (the identity map on X),we obtain the intuitionistic version of the result ofSharma and Deshpande [57].

(v) From Corollary 5, with I = J = B = T = IX (the identity map on X), we obtain the intuitionistic version of theresult of Sharma and Deshpande [58].

(vi) From Corollary 5, with I = J = B = T = IX (the identity map on X) and a = 0, we obtain the intuitionistic versionof the result of Sharma and Deshpande [57].

(vii) If we put I = J = B = T = IX (the identity map on X) and A = S in Corollary 5, we obtain result for three map-pings A, P and Q. In addition if we put a = 0 then condition (5.2) changes accordingly.

(viii) From Corollary 5, with I = J = B = T = A = S = IX (the identity map on X),we obtain intuitionistic version ofthe result of Cho et al. [6].


Example 2. Let X = [0,15) with the metric d defined by d(x,y) = |x � y| and for each t 2 [0,1] define

Mðx; y; tÞ ¼ tt þ jx� yj and Nðx; y; tÞ ¼ jx� yj

t þ jx� yj

M(x,y,0) = 0 and N(x,y,0) = 1 for all x, y 2 X. Clearly (X,M,N, *,e) is a noncomplete intuitionistic fuzzy metric spacewhere * is defined by a * b = ab and e is defined by a e b = min {1,a + b}.

Define A, S, P and Q: X! X by

AðX Þ ¼0 if x ¼ 0

3 if 0 < x 6 10

x� 3 if 10 < x < 15

8><>: SðX Þ ¼

0 if x ¼ 0

6 if 0 < x 6 10

x� 7 if 10 < x < 15

8><>:

P ðX Þ ¼0 if x ¼ 0

3 if 0 < x < 15

�QðX Þ ¼

0 if x ¼ 0

7 if 0 < x < 15

�

If we take t = 1, k = 0.5 and a = 1 we see that A, S, P and Q satisfy all the conditions of Remark 6(v) and (vi) and havea unique common fixed point 0 2 X. It may be noted in this example that the mappings P and S commute at coincidencepoint 0 2 X. So P and S are weakly compatible maps. Similarly Q and A are weakly compatible maps. To see the pairs{P,S} and {Q,A} are noncompatible. Let us consider a sequence {xn} such that xn! 10. Then limn!1 Pxn = 3,limn!1 Sxn = 3 but

limn!1

MðPSxn; SPxn; tÞ ¼t

t þ j3� 6j–1 and limn!1

NðPSxn; SPxn; tÞ ¼j3� 6j

t þ j3� 6j–0:

Thus the pair{P,S} is noncompatible. Also limn!1 Qxn = 7, limn!1 Axn = 7 but

limn!1

MðQAxn;AQxn; tÞ ¼t

t þ j7� 3j–1 and limn!1

NðQAxn;AQxn; tÞ ¼j7� 3j

t þ 7� 3–0:

So the pair {Q,A} is noncompatible. It can be easily verified in this example that the pair {P,S} and {Q,A} are neithercompatible of type (a) nor compatible of type (b). All the mappings involved in this example are discontinuous even atthe common fixed point x = 0.

If we put I = J = B = T = A = S = IX (the identity map on X) in Corollary 7 we have the following:

Corollary 9. Let (X,M,N,*,e) be an IFM-space. Let P be mapping from X into itself such that

(9.1) there exists a constant k 2 (0,1) such that

½1þ aMðx; y; ktÞ� �MðPx; Py; ktÞP a½MðPx; x; ktÞ �MðPy; y; ktÞ þMðPy; x; ktÞ �MðPx; y; ktÞ� þMðy; x; tÞ�MðPx; x; tÞ �MðPy; y; tÞ �MðPy; x; atÞ �MðPx; y; ð2� aÞtÞ


and

½1þ aNðx; y; ktÞ��NðPx; Py; ktÞ 6 a½NðPx; x; ktÞ � NðPy; y; ktÞ þ NðPy; x; ktÞ�NðPx; y; ktÞ�þ Nðy; x; tÞ � NðPx; x; tÞ�NðPy; y; tÞ � NðPy; x; atÞ � NðPx; y; ð2� aÞtÞ


(9.2) if P(X) is a complete subspace of X then

P has a unique common fixed point in X.

Corollary 10 (Intuitionistic fuzzy Banach contraction theorem). Let (X, M,N,*,e) be an IFM-space. Let A, P and Q be

mappings from X into itself such that


M(Px,Py,kt)PM(x,y, t)

for all x, y 2 X and t > 0,


N(Px,Py, kt) 6 N(x,y, t)

for all x, y 2 X and t > 0.

(10.2) if P(X) is a complete subspace of X then

P has a unique common fixed point in X.

Remark 7. In Corollary 10, we use condition (10.2) that is P(X) is a complete subspace of X, while Alaca et al. [2] usedcompleteness of the whole space (X,M,N, *,e). However Alaca et al. [2] did not require t * t P t and(1 � t) e (1 � t) 6 (1 � t) for all t 2 [0,1].

Now we extend Theorem 1 for finite number of mappings in the following way:

Theorem 11. Let (X,M,N,*,e) be an IFM-space. Let A1, A2, . . . ,An, S1, S2, . . . ,Sn, P and Q be mappings from X into

itself such that

(11.1) P(X) � A1A2. . .An(X), Q(X) � S1S2. . .Sn(X),


½1þ aMðS1S2 . . . Snx;A1A2 . . . Any; ktÞ� �MðPx;Qy; ktÞP a½MðPx; S1S2 . . . Snx; ktÞ �MðQy;A1A2 . . . Any; ktÞþMðQy; S1S2 . . . Snx; ktÞ �MðPx;A1A2 . . . Any; ktÞ�þMðA1A2 . . . Any; S1S2 . . . Snx; tÞ �MðPx; S1S2 . . . Snx; tÞ�MðQy;A1A2 . . . Any; tÞ �MðQy; S1S2 . . . Snx; atÞ�MðPx;A1A2 . . . Any; ð2� aÞtÞ


and

there exists a constant k 2 (0,1) such that

½1þ aNðS1S2 . . . Snx;A1A2 . . . Any; ktÞ��NðPx;Qy; ktÞ 6 a½NðPx; S1S2 . . . Snx; ktÞ�NðQy;A1A2 . . . Any; ktÞ

þ NðQy; S1S2 . . . Snx; ktÞ � NðPx;A1A2 . . . Any; ktÞ�

þ NðA1A2 . . . Any; S1S2 . . . Snx; tÞ�NðPx; S1S2 . . . Snx; tÞ

�NðQy;A1A2 . . . Any; tÞ � NðQy; S1S2 . . . Snx; atÞ

�NðPx;A1A2 . . . Any; ð2� aÞtÞ


(11.3) if one of P(X), A1A2. . .An(X), S1S2. . .Sn(X), Q(X) is a complete subspace of X then(i) P and S1S2. . .Sn have a coincidence point and

(ii) Q and A1A2. . .An have a coincidence point.

Further if

(11.4) A1 commutes with A2, A3, . . . ,An,
A2 commutes with A3,A4, . . . ,An,A3 commutes with A4,A5, . . . ,An,...
An�1 commutes with An,similarly S1 commutes with S2, S3, . . . ,Sn,

S2 commutes with S3,S4, . . . ,Sn,S3 commutes with S4,S5. . ., Sn,...

Sn�1 commutes with Sn,P commutes with S2, S3, . . . ,Sn, Q commutes with A2, A3, . . . ,An.

(11.5) the pairs {P, S1S2. . .Sn} and {Q,A1A2. . .An} are weakly compatible, then

(iii) A1, A2, . . . ,An, S1, S2, . . . ,Sn, P and Q have a unique common fixed point in X.


Proof. Since P(X) � A1A2 . . . An(X), for any point x0 2 X there exists a point x1 2 X such that Px0 = A1A2 . . . Anx1.Since Q(X) � S1S2 . . . Sn(X), for this point x1 we can choose a point x2 2 X such that Qx1 = S1S2 . . . Snx2 and soon. Inductively, we can define a sequence {yn} in X such that for n = 0, 1, 2, . . .

y2n ¼ Px2n ¼ A1A2 . . . Anx2nþ1;

y2nþ1 ¼ Qx2nþ1 ¼ S1S2 . . . Snx2nþ2:

By using the method of proof of Theorem 1, we can see that conclusions (i), (ii) and (iii) hold.From Theorem 11, with a = 0, we have the following: h

Corollary 12. Let (X,M, N,*,e) be an IFM-space. Let A1, A2, . . ., An, S1, S2, . . . Sn, P and Q be mappings from X into

itself satisfy (11.2) with a = 0.

If conditions (11.1) and (11.3) are satisfied then conclusions (i) and (ii) of Theorem 11 hold. Further if conditions (11.4)

and(11.5) are satisfied then conclusion (iii) of Theorem 11 holds.

4. Discussions and auxiliary results

In view of above results it is very much clear that we extend, improve and generalize many results in metric spaces,fuzzy metric spaces and intuitionistic fuzzy metric spaces. We prove common fixed point theorems for finite number ofmappings in intuitionistic fuzzy metric spaces. This is the first effort in existing literature. To prove common fixed pointtheorems for contractive type condition with more than four mappings some commutative conditions for mappings arealways essential. How many commutative conditions are necessary? As an answer of this question we are giving thefollowing formulas:

(i) If the number of mappings are even and finite in above theorems and corollaries then there will be n2�2n�84

commutativity conditions, where n = 4, 6, 8, 10, 12, . . .. . . up to finite values. For example if n = 10 then 18commutativity conditions are required (see (1.4)).

(ii) If the number of mappings are odd and finite in above theorems and corollaries then there will be n2�94

com-mutativity conditions, where n = 5, 7, 9, 11 . . .. . . up to finite values. For example if n=7 then 10 commu-tativity conditions are required (see (7.4)).

(iii) If n = 1, 2, 3, 4 then any commutativity condition is not required. See Remark 6(v) to (viii) and Corollary 9and Corollary 10.

Our results apply to a wider class of mappings than the results on compatible or compatible of type (a) or compatibleof type (b ) maps since compatible or compatible of type (a) or compatible of type (b) maps constitute a proper subclassof weakly compatible maps.

We point out that common fixed point theorems for finite number of maps can be proved without continuity of anymappings.

In our all results we replace the completeness of the whole space with a set of alternative conditions.In this way we prove common fixed point theorems for finite number of maps in intuitionistic fuzzy metric spaces by

relaxing, replacing and omitting some conditions in the analogous results.Our results contain so many results in the existing literature and will be helpful for the workers in the field.

Acknowledgements

Authors extend thanks to Professor B. Fisher, Department of Mathematics and Computer Science of Leicester,Leicester, England for the kind help during preparation of this paper. Authors are also grateful to the learned refereefor his useful suggestions.

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