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Abstract: The aim of this paper is to present common fixed point theorem in fuzzy metric spaces for occasionally weakly compatible mappings with integral type inequality by reducing its minimum value.

Keywords: Fuzzy metric space, occasionally weakly compatible (owc) mappings, common fixed point.

1. Introduction:

Fuzzy set was defined by Zadeh [26]. Kramosil and Michalek [14] introduced fuzzy metric space, many authors extend their views, Grorge and Veermani [6] modified the notion of fuzzy metric spaces with the help of continuous t-norms Grabiec[7], Subramanyam[28],Vasuki[25], Pant and Jha,[20] obtained some analogous results proved by Balasubramaniam et al. Subsequently, it was developed extensively by many authors and used in various fields, Jungck [10] introduced the notion of compatible maps for a pair of self maps. Several papers have come up involving compatible mapping proving the existence of common fixed points both in the classical and fuzzy metric spaces.

The theory of fixed point equations is one of the basic tools to handle various physical formulations. Fixed point theorems in fuzzy mathematics has got a direction of vigorous hope and vital trust with the study of Kramosil and Michalek [14], who introduced the concept of fuzzy metric space. Later on this concept of fuzzy metric space was modified by George and Veermani [6] Sessa [27] initiated the tradition of improving commutative condition in fixed point theorems by introducing the notion of weak commuting property. Further Juncgk [10] gave a more generalized condition defined as compatibility in metric spaces.

Jungck and Rhoades [11] introduced the concept of weakly compatible maps which were found to be more generalized than compatible maps.

Grabiec [7] obtained fuzzy version of Banach contraction principle. Singh and M.S. Chauhan [29] brought forward the concept of compatibility in fuzzy metric space. Pant [18, 19, 20] introduced the new concept reciprocally continuous mappings and established some common fixed point theorems. Balasubramaniam et al. [4], have shown that Rhoades [22] open problem on the existence of contractive definition which generates a fixed point but does not force the mappings to be continuous at the fixed point, posses an affirmative answer. Recent literature in fixed point in fuzzy metric space can be viewed in [1, 2, 9, 16, 24].

This paper offers the fixed point theorems on fuzzy metric spaces which generalize extend and fuzzify several known fixed point theorems for occasionally compatible maps on metric space by making use of integral type inequality.

2. Preliminary Notes:

Definition 2.1 [26] A fuzzy set A in X is a function with domain X and values in [0,1].

Definition 2.2 [23] A binary operation : [0, 1] ï¿½ [0, 1] → [0, 1] is a continuous t-norms

if it satisfies the following conditions:

(i) *is associative and commutative

(ii) *is continuous;

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

a*b ≤ c*d whenever a ≤ c and b ≤ d, and a, b, c, d [0,1].

Definition 2.3 [6] A 3-tuples (X,M,) is said to be a fuzzy metric space (shortly FM Space) if X is an arbitrary set, * is a continuous t-norm and M is a fuzzy set on

ï¿½ [0, ∞) satisfying the following conditions, for all x, y, z X and s, t > 0 ;

(FM 1): M( x,y,t ) > 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) ≤ M(x, z, t s)

(FM 5): M(x, y, .) : [0,) → (0,1] is left continuous.

(X, M,*) denotes a fuzzy metric space, (x,y,t) can be thought of as

degree of nearness between x and y with respect to t. We identify x = y with M(x, y, t) = 1

for all t > 0 . In the following example every metric induces a fuzzy metric.

Example 2.4 Let X = [0,1], t-norm defined by a*b = min{a,b} where a,b [0,1] and M is the fuzzy set on X²ï¿½ (0, ∞) defined by M(x,y,t) = for all x,y X, t > 0. Then (X,M,) is a fuzzy metric space.

Example 2.5 (Induced fuzzy metric [6]) Let (X, d) be a metric space, denote a * b = a.b & for all a,b [0,1] and let M_dbe fuzzy sets on X²ï¿½ (0, ∞) defined as follows

(x,y,t) =

Then (X,M,) is a fuzzy metric space. We call this fuzzy metric induced by a metric d as the standard intuitionistic fuzzy metric.

Definition 2.6 [11] Two self mappings f and g of a fuzzy metric space (X,M,*) are called compatible if M (fg,gf, t) = 1 wherever {} is sequence in X such that

f = g = x for some x in X

Definition 2.7 [5] Two self maps f and g of a fuzzy metric space (X, M,*) are called reciprocally continuous on X if f = fx and gf = gx wherever {} is sequence in X such that

f = g = x for some x in X.

Definition 2.8 [6] : Let (X,M,) be a fuzzy metric space. Then

(a) A sequence {x_n}in X is said to converges to x in X if for each > 0 and each t > 0,

there exist n_o N such that M ( x_n, x,t ) > 1 ï¿½ for all n ≥ n_o..

(b) A sequence {x_n} in X is said to be Cauchy if for each > 0 and each t > 0, there exist n_o N such that M ( x_n, x_m,t ) > 1 ï¿½ for all n, m ≥ n_o..

Definition 2.9 Two self maps f and g of a set X are occasionally weakly compatible (owc) iff there is a point x in X which is a coincidence point of f and g at which f and g commute.

A. Al-Thagafi and Naseer Shahzad [3] shown that occasionally weakly is weakly

compatible but converse is not true.

Definition 2.10 Let (X,d) be a compatible metric space, α [0,1], f:X X a mapping such that for each x,y X

≤ α where : is lebesgue integral mapping which is summable,

> 0, > 0

nonnegative and such that, for each. Then f has a unique common fixed z X such that for each x X, = z
Rhodes[30], extended this resul by replacing the above condition by the following

≤

Ojha et al.(2010). Let (X,d) be a metric space and let be single and a multi valued map respectively, suppose that and are occasionally weakly commutative (owc) and satisfy the inequality

≤

For all x,y in X, where p ≥ 2 is an integer a ≥ o and then f and F have unique common fixed point in X.

Example 2.11 [3] Let R be the usual metric space. Define S, T: R → R by Sx = 2x and Tx = x²for all x R. Then Sx = Tx for x = 0, 2 but ST0 = TS0, and ST2 ≠ TS2. Hence S and T are occasionally weakly compatible self maps but not weakly compatible

Lemma 2.12 [12] Let X be a set, f, g owc self maps of X. If f and g have a unique point of coincidence, w = fx = gx, then w is the unique common fixed point of f and g.

Lemma 2.13 Let (X,M,) be a fuzzy metric space. If there exist q (0, 1) such that

M (x, y, qt) ≥ M (x, y, t) for all x, y X & t > 0 then x = y

3. Main Results:

Theorem 3.1 Let (X,M,) be a complete fuzzy metric space and let F, G, S and T are selfï¿½mapping of X. Let the pairs {F,S} and {G ,T} be owc . If there exists q ( 0,1) such that

≥

(3.1)

for all x, y X and for all t > 0, then there exists a unique point w X such that Fw = Sw = w and a unique point z X such that Gz = Tz = z Moreover , z = w , so that there is a unique common fixed point of F,G,S and T.

Proof: Let the pairs { F,S} and { G ,T} be owc so there are points x,y X such that Fx = Sx and

Gy = Ty. We claim that Fx = Gy. If not by inequality (3.1)

≥

Therefore Fx = Gy , i.e. Fx = Sx = Gy =Ty. Suppose that z such that Fz = Sz then by (1) we have Fz = Sz = Gy = Ty so Fx = Fz and w = Fx = Sx is the unique point of coincidence of F and S.

Similarly there is a unique point z X such that z = Gz = Tz.

Assume that w ≠ z .We have

= ≥

Therefore we have z = w by Lemma 2.14 and z is a common fixed point of F, G, S and T. The uniqueness of fixed point holds from (3.1)

Theorem3.2.1: Let (X,M,) be complete fuzzy metric space and let F,G,S and T be self mappings of X .let the pairs {F,S} and{G,T} be owc . If there exists q (0,1) such that

≥

(3.2)

for all x , y X and ï¿½:[0,1] [0,1] such that ï¿½(t) > t for all 0 < t < 1 ,then there exist a unique common fixed point of F, G,S and T

Proof: From equation (3.2)

≥

≥ from theorem 3.1

Now proof fallows by (3.1)

Theorem3.2.2: Let (X,M,) be a complete fuzzy metric space and let F,G,S and T are self mappings of X.

Let the pairs {F,S}and {G,T} be owc . If there exists

q ( 0,1) such that

≥

(3.3)

for all x, y X and ï¿½: such that ï¿½ (t,1,t,t) > t for all ,0 < t < 1 then there exists a unique common fixed point of F,G,S and T.

Proof: Let the pairs {F,S}and {G,T} are owc ,there are points x, y X such that Fx = Sx and Gy = Ty are

Claim that Fx = Gy. By inequality (3.3) we have

[∵ M(Fx,Fx,t)=1,M(Gy,Gy,t) =1]

a contradiction , therefore Fx = Gy i.e. Fx = Sx = Gy = Ty

suppose that there is another point z such that Fz = Sz then by (3.3) we have

Fz = Sz = Gy = Ty so Fx = Fz and w = Fx = Tx is unique point of coincidence of F and T.

By Lemma 2.14 w is a unique common fixed point of F and S, similarly there is a unique point z X such that z = Gz = Tz..Thus z is common fixed point of F,G,S and T. The uniqueness of fixed point holds from (3.3)

Theorem3.2.3: Let (X,M,) be complete fuzzy metric space and let F,G,S and T be self mappings of X, let the pairs {F,S} and {G,T} are owc. If there exists a points q (0,1) for all x, y X and t > 0

≥

(3.4)

then there exists a unique common fixed points of F,G ,S and T.

Proof: Let the points {F, S} and {G,T} are owc and there are points x , y X such that Fx = Sx and Gy = Ty and claim that Fx = Gy

By inequality (3.4)

We have

≥

Thus we have Fx = Gy i.e. Fx = Sx = Gy = Ty suppose that there is another point z such that Fz=Sz then by (3) we have

Fz = Sz = Gy = Ty so Fx = Fz and w = Fx = Sx is unique point of coincidence of F and S.

Similarly there is a unique point z X such that z = Gz = Tz. Thus w is a common fixed point of F,G ,S, and T.

Corollary3.2.4:Let (X,M,*) be a complete fuzzy metric space and let F,G,S and T be self mapping of X

Let the pairs {F,S} and {G,T} are owc.

If there exists a point q (0,1) for all x, y X and t > 0

≥

(3.5)

then there exists a unique common fixed point of F,G,S and T.

Proof: We have

≥

[ Fx = Sx and Gy = Ty]

and therefore from Theorem 3.2.3 , F, G, S and T have common fixed point.

Corollary3.2.5: Let (X,M,*) be complete fuzzy metric space and let F,G,S and T be self-mapping of X. Let the pairs {F,S} and {G,T} are owc. If there exist point q (0,1) for all x, y X and t > 0

≥ (3.6)

then there exists a unique common fixed point of F,G,S and T.

Proof: The proof follows from Corollary 3.2.4

Theorem 3.2.6: Let (X,M,*) be complete fuzzy metric space . Then continues self mappings S and T of X have a common fixed point in X if and only if there exists a self mapping F of X such that the following conditions are satisfied

(i) FX TX ∩ SX

(ii) pairs { F,S} and {F,T} are weakly compatible,

(iii) there exists a point q (0,1) such that for every

x, y X and t > 0

(3.7)

then F, S and T have a unique common fixed point.

Proof: Since compatible implies owc, the result follows from (Theorem3.2.3)

Theorem 3.2.7: Let (X,M,*) be a complete fuzzy metric space and let F and G be self mapping of X .Let F and G are owc. If there exists a point q (0,1) for all x, y X and t

(3.8)

for all x,y X , where α, β > 0 , αβ >1. Then F and S have a unique common fixed point.

Proof: Let the pairs {F,S}be owc, so there is a point x X such that Fx = Sx. Suppose that there exist

another point y X for which Fy = Sy We claim that Sx = Sy by equation (8) we have

A contradiction, since (α+β) > 1therefore Sx = Sy. Therefore Fx = Fy and Fx is unique.

From lemma 2.14, F and S have a unique fixed point.

References:

[1] C.T. Aage , J.N. Salunke, ï¿½Common Fixed Point Theorems in Fuzzy Metric Spacesï¿½ , International Journal of pure and Applied Mathematics 56(2), 2009, pp 155-164.

[2] C.T. Aage, J.N. Salunke, ï¿½Some Fixed Point Theorems in Fuzzy Metric Spacesï¿½ , International Journal of pure and Applied Mathematics 56(3), 2009, pp 311-320.

[3] A.Al-Thagafi and Naseer Shahzad, ï¿½Generalized I-Nonexpansive Selfmaps and Invarient Approximationsï¿½, Acta Mathematica Sinica, English Series May, 2008, Vol. 24, No.5pp 867-876.

[4] P. Balasubrmaniam , S. Muralisnkar ,R.P. pant, ï¿½Common Fixed Points of four mappings in a fuzzy metric spaces ï¿½, J Fuzzy math. 10(2) (2002)

[5] Y.J. Cho , H.K. Pathak , S.M. Kang , J.S. Jung ï¿½Common Fixed Points of compatible maps of type (A) on fuzzy metric spacesï¿½ , Fuzzy Sets and Systems 93 (1998), 99-111

[6] A George, P. Veeramani , ï¿½On some results in fuzzy metric spacesï¿½ , Fuzzy Sets and Systems, 64 (1994), 395-399.

[7] M. Grabiec, ï¿½Fixed points in fuzzy metric spacesï¿½, Fuzzy Sets and Systems 27 ( 1988) , 385-389.

[8] O. Hadzic , ï¿½Common Fixed point theorems for families of mapping in complete metric spaceï¿½, Math.Japon.29 (1984), 127- 134.

[9] Mohd. Imdad and Javid Ali, ï¿½Some Fixed Point Theorems in Fuzzy Metric Spacesï¿½ , Mathematical Communications 11(2006) , 153-163 153.

[10] G. Jungck, ï¿½Compatible mappings and common fixed points (2)ï¿½ Inter-nat. J.math. Math. Sci.(1988), 285-288.

[11] G. Jungck and B.E. Rhoades, ï¿½Fixed point for Set valued functions without Continuityï¿½ , Indian J. Pure Appl. Math, 29 (3), (1998), pp.771-779.

[12] G. Jungck and B.E. Rhoades , ï¿½Fixed Point Theorems for Occasionally Weakly compatible Mappingsï¿½ , Fixed point Theory, Volume 7, No. 2, 2006, 287-296.

[13] G. Jungck and B.E. Rhoades, ï¿½Fixed Point Theorems for Occasionally Weakly compatible Mappingsï¿½, Erratum, Fixed point Theory, Volume9,No.1, 2008,383-384.

[14] O. Kramosil and J. Michalek, ï¿½Fuzzy metric and statistical metric spacesï¿½, Kybernetka, 11(1975), 326-334.

[15] S. Kutukcu ï¿½A fixed point theorem for contraction type mappings in Menger spacesï¿½, Am. J.Appl. Sci. 4(6) (2007), 371-373.

[16] Servet Kutukcu , Sushil Shrma and Hanifi Tokogoz , ï¿½A Fixed Point Theorem in Fuzzy Metric Spacesï¿½ Int. Journal of Math. Analysis, Vol.1,2007, no. 18,861-872.

[17] S.N. Mishra, ï¿½Common Fixed points of compatible mapping in PM- spacesï¿½ , Math . Japon.36 (1991), 283-289.

[18] R.P. Pant, ï¿½Common Fixed points of four mappingsï¿½, Bull.Math. Soc.90 (1998) , 281-286.

[19] R.P. Pant, ï¿½Common Fixed points of for contractive mappsï¿½, J. Math. Anal. Appl. 226(1998), 251-258

[20] R.P. Pant, K. Jha, ï¿½A remark on common fixed points of four mappings in a fuzzy metric spaceï¿½, J Fuzzy Math. 12(2) (2004) ,433-437.

[21] H. K.Pathak and Prachi Sing ,ï¿½Common fixed Point Theorem for Weakly Compatible Mappingï¿½, International Mathematical Forum, 2, 2007, no.57,2831-2839.

[22] B. E. Rhoades, ï¿½Contractive definitionsï¿½ , Contemporary Math.72 (1988) , 233-245.

[23] B . Schweizer and A. Sklar , ï¿½Statistical metric spacesï¿½ , Pacific J.Math. (1960), 313-334.

[24] Seong Hoon Cho, ï¿½On common Fixed point in fuzzy metric spaceï¿½, Int . Math. Forum, 1,2006, 10 471-479.

[25] R. Vasuki, ï¿½Common fixed points for R-weakly commuting Maps in fuzzy metric spacesï¿½. Indian J .Pure Appl. Math 30 (1999) ,419-423.

[26] L.A. Zadeh , ï¿½Fuzzy sets, Inform and Controlï¿½ 8(1965), 338-353.

[27] S.Sessa, ï¿½On weak commutativity condition of mapping in fixed point considerationï¿½. Publ. Inst. Math(Beograd) N.S.,32(46),(1982),149-153.

[28] P.V. Subramanayam, ï¿½Common fixed points theorem in fuzzy metric spacesï¿½, Information Science 83(1995) 105-112.

[29] Brijendra Singh & M.S. Chauhan, ï¿½Common fixed points of compatible maps, in fuzzy metric spaceï¿½. Fuzzy sets and systems 115(2000), 471-475.

[30] B.E. Rhodes ï¿½Two fixed point theorem for mapping satisfying a general contractive condition of integral typeï¿½. Int. J. Math. Sci., 3: pp4007-4013.