A Common Fixed Point Theorem For Occasionally Weakly Compatible Mappings In Fuzzy Metric Spaces With The (Clr)-Property

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1 Advances in Fuzzy Mathematics. ISSN X Volume 11, Number 1 (2016), pp Research India Publications A Common Fixed Point Theorem For Occasionally Weakly Compatible Mappings In Fuzzy Metric Spaces With The (Clr)-Property P.Srikanth Rao 1 and *Veena Kulkarni 2 Department of Mathematics, B.V.R.I.T, Vishnupur, Narsapur Dist. Medak(Telangana State) India (psrao9999@gmail.com, veena.pande@gmail.com) Abstract The aim of this paper is that to prove a common fixed point theorem for a two pairs of occasionally weakly compatible mappings in fuzzy metric space by using the property. Our results improvised and extended the results of Chauhan et al.[3] and Sedghi et al.[33]along with several well known results in literature. Mathematics Subject Classification: 47H10, 57H25 Keywords: Fuzzy metric space, weakly compatible maps, common fixed point, occasionally weakly compatible maps and property. 1.Introduction The concept of a fuzzy set is investigated by Zadeh[42] in his seminar paper. In 1975, Kramosil and Michalek[14] introduced the concept of fuzzy metric space, which opened an avenue for further development of analysis in such spaces. Further, George and Veeramani[7] modified the concept of fuzzy metric space introduced by Kramosil and Michalek[14] with a view to obtain a Hausdorff topology which has very important applications in quantum particle physics, particularly in connection with both string and theory (see, [23-25]). Fuzzy set theory also has applications in applied sciences such as neural network theory, stability theory, mathematical programming, modelling theory, engineering sciences, medical sciences (medical genetics, nervous system), image processing, control theory, communication etc. Consequently in due course of time some metric fixed point results were generalized

2 14 P.SrikanthRao and Veena Kulkarni to fuzzy metric spaces by various authors viz Grabiec[8], Cho [5,6], Subrahmanyam[40] and Vasuki[41]. In 2002, Aamri and El-Moutawakil[1] defined the notion of (E.A) property for self mappings which contained the class of non-compatible mappings in metric spaces. It was pointed out that (E.A) property allows replacing the completeness requirement of the space with a more natural condition of closedness of the range as well as relaxes the completeness of the whole space, continuity of one or more mappings and containment of the range of one mapping into the range of other which is utilized to construct the sequence of joint iterates. Many authors have proved common fixed point theorems in fuzzy metric spaces for different contractive conditions. For details, we refer to [4, 9, 10, 15-17, 22, 26-28, 31, 32, 34-36, 38, 39]. Recently, Sintunavarat and Kumam[37] defined the notion of (CLRg) property in fuzzy metric spaces and improved the results of Mihet [21] without any requirement of the closedness of the subspace. In this paper, we prove a common fixed point theorem for two pairs of occasionally weakly compatible mappings by using property in fuzzy metric space.our results improve the results of Sedghi, Shobe and Aliouche[33]. 2. Preliminaries: 2.1 Definition: [30] A binary operation : [0, 1] [0, 1] is a continuous t-norm if it satisfies the following conditions: 1. is associative and commutative, 2. is continuous, 3. for all a [0, 1], 4. whenever and for all a, b, c, d [0, 1] 2.2Definition: [7]A 3-tuple (X, M, ) is said to be a fuzzy metric space if X is an arbitrary set, is a continuous t-norm and M is a fuzzy set on X 2 (0, ) satisfying the following conditions: for all x, y, z X, t, s>0, if and only if x=y 3., : [0, ) is continuous. Then M is called a fuzzy metric on X. Then denotes the degree of nearness between x and y with respect to t. Let (X, M, ) be a fuzzy metric space. For t>0, the open ball with center x X and radius 0<r<1 is defined by

3 A Common Fixed Point Theorem For Occasionally Weakly Compatible Mappings 15 Now let (X, M, ) be a fuzzy metric space and the set of all with x A if and only if there exist t>0 and 0<r<1 such that. Then is a topology on X induced by the fuzzy metric M. In the following example (see [7]), we know that every metric induces a fuzzy metric 2.3 Example: Let (X, d) be a metric space. Denote (or for all and let M d be fuzzy sets on X 2 (0, ) defined as follows: Then (X, M d, ) is a fuzzy metric space and the fuzzy metric M induced by the metric d is often referred to as the standard fuzzy metric. 2.4 Lemma: [8] Let(X, M, ) be a fuzzy metric space. Then M(x, y, t) is nondecreasing for all x, y X. 2.5 Definition: [12] Two self mappings S and T of a non-empty set X are said to be weakly compatible (or coincidentally commuting) if they commute at their coincidence points, i.e. if Sz=Tz some z X, then STz=TSz. 2.6 Definition: [13] Two self mappings S and T of a non-empty set X are said to be occasionally weakly compatible(owc) if and only if there is a point z X which is a coincidence point of S and T at which S and T commute. i.e., there exists a point z X such that Sz=Tz and STz=TSz. 2.7 Definition: [2] A pair of self-mappings S and T of a fuzzy metric space(x, M, ) is said to satisfy the (E. A) property, if there exists a sequence in X such that for some z X. 2.8 Remark: It is noted that weak compatibility and (E.A) property are independent to each other (see [29], Example 2.1, Example 2.2). 2.9 Definition: [2] Two self mappings S and T of a fuzzy metric space (X, M, ) are non-compatible if and only if there exists at least one sequence in X such that for some z X, but for some t>0, is either less than 1 or nonexistent Remark: From Definition 2.9, it is easy to see that any non-compatible self mappings of a fuzzy metric space (X, M, ) satisfy the (E.A) property. But two mappings satisfying the (E.A) property need not be non-compatible (see [29], Remark 4.8).

4 16 P.SrikanthRao and Veena Kulkarni 2.11 Definition: [37] A pair (A, S) of self-mappings of a fuzzy metric space (X, M, ) is said to satisfy the common limit range property with respect to mapping S (briefly, property), if there exists a sequence in X such that z where z S(X) Definition: Two pairs of self-maps (A, S) and (B, T) of fuzzy metric space (X, M, ) is said to i) Satisfy the common property (E.A)[18] if there exist two sequences and in X such that for some z X. ii) Satisfy the common limit range property with respect to mappings S and T (briefly, property), if there exist two sequences and in X such that where z S(X) T(X). Similarly we can define property 2.13 Lemma: [8] If for all x, y X, t>0 and for a number q (0, 1), then x=y 2.14 Lemma: [13] Let X be a set, S and T be occasionally weakly compatible(owc) self maps on X. If S and T have a unique point of coincidence w=sx=tx for x X, then w is the unique common fixed point of S and T. 3.MainResults In 2010, Sedghi, Shobe and Aliouche [33] proved a common fixed point theorem for a pair of weakly compatible mappings with (E.A) property in fuzzy metric space by using the following function: Let is a set of all increasing and continuous functions such that for every 3.1 Example: [33] Let defined by We prove the following theorem 3.2 Theorem: Let (X, M, ) be a fuzzy metric space, where is a continuous t-norm. Further let A, B, S, T, P and Q be mappings from X into itself and satisfying the following condition

5 A Common Fixed Point Theorem For Occasionally Weakly Compatible Mappings 17 (3.2.1) For all x, y X, t>0 and for some 1 k 2. Suppose that the pair (P, AB) and (Q, ST) shares the property provided the pair (P, AB) and (Q, ST) are occasionally weakly compatible. Then P, Q, AB and ST have a unique common fixed point. Further if (A, B), (S, T), (A, P) and (S, Q) are commuting maps then A, B, S, T, P and Q have a unique common fixed point. Proof: Since (P, AB) and (Q, ST) satisfies common property, there exists a sequence and in X such that where.since, a point u X exists such that Abu=t. We assert that Pu=ABu. Let on the contrary Pu ABu, then there exists > 0 such that (3.2.2) To support the claim, let it be untrue. Then we have =...= 1, As. This shows that for all t > 0 which contradicts Pu ABu and hence (3.2.2) is proved. On using inequality (3.2.1), with, We get For all. As, it follows that as, we have

6 18 P.SrikanthRao and Veena Kulkarni, Which contradicts (3.2.2), we have Pu=ABu=t. Therefore, u is a coincidence point of the pair (P, AB) As, there exists a point such that STv=t. We show that Qv=STv. Let on the contrary Qv, then there exists > 0 such that (3.2.3) To support the claim, let it be untrue. Then we have =...= 1, As. This shows that for all t >0 which contradicts Qv STv and hence (3.2.3) is proved. On using inequality (3.2.1), with, We get For all. As, it follows that as, we have, Which contradicts (3.2.2), we have Qv=STv=t, which shows that v is a coincidence point of the pair (Q, ST) Hence t=pu=abu=qv=stv. u is the coincidence point of P and AB v is the coincidence point of Q and ST Since the pair (P, AB) are occasionally weakly compatible so by definition there exists a point such that Pu=ABu and P(AB)u=(AB)Pu Since the pair (Q, ST) are occasionally weakly compatible so by definition there exists a point such that Qv=STv and Q(ST)v=(ST)Qv Moreover, if there is another point z such that Pz= ABz, then, using (3.2.1) it follows that Pz=ABz=Qv=STv, or Pu=Pz and w=pu=abu is unique point of coincidence of P

7 A Common Fixed Point Theorem For Occasionally Weakly Compatible Mappings 19 and AB. By Lemma [13], w is the unique common fixed point of P and AB. i.e., w=pw=abw. Similarly there is a unique point such that z=qz=stz Uniqueness: Suppose that. Using inequality (3.2.1) with x=w, y=z, we get for some > 0 For all. As we have = which is a contradiction. Therefore z=w and z is a common fixed point. By the preceding argument it is clear that z is unique. z is the common fixed point of P, Q, AB and ST Finally we need to show that z is a common fixed point of A, B, P, Q, S and T Since (A, B), (A, P) are commutative Az=A(ABz)=A(BAz)=(AB)Az; Az=APz=PAz Bz=B(ABz)=(BA)Bz=(AB)Bz; Bz=BPz=PBz Which shows that Az, Bz are common fixed point of (AB, P) yielding then by Az=z=Bz=Pz=ABz in the view of uniqueness of common fixed point of the pairs (P, AB) Similarly using the commutativity of (S, T) and (S, Q) it can be shown that. Sz=z=Tz=Qz=Az=Bz=Pz. which shows that z is a common fixed point of A, B, P, Q, S and T. we can easily prove the uniqueness of z from (3.2.1) 3.3 Example: Let (X, M, ) be a fuzzy metric space, wherex= [3, 14), with t-norm is defined by for all and

8 20 P.SrikanthRao and Veena Kulkarni for all. Let the function defined by Define the self mappings P, Q, A, B, S and T by and Consider two sequences, property:. The pairs (P, AB) and (Q, ST) satisfy the AB(X) ST(X). Also, P(X)={3, 11} [3, 11)=ST(X) and Q(X)= {3, 5} [3, 5) {13}=AB(X). (P, AB) and (Q, ST) are OWC Hence, all the conditions of Theorem 3.2 are satisfied, and 3 is a unique common fixed point of the pairs (P, AB) and (Q, ST) which also remains a point of coincidence. Here, one may notice that the involved mappings are even discontinuous at their unique common fixed point 3. However, notice that the subspaces AB(X) and ST(X) are not closed subspaces of X. Taking T=B=I x identity self map in Theorem Corollary: Let (X, M, ) be a fuzzy metric space, where is a continuous t-norm. Further let A, S, P and Q be mappings from X into itself and satisfying the following condition (3.4.1) For all x, y X, t>0 and for some 1 k 2. Suppose that the pairs (P, A) and (Q, S) shares the property provided the pairs (P, A) and (Q, S) are occasionally weakly compatible. Then P, Q, A and S have a unique common fixed point. Taking P=Q and T=B=I x in Theorem 3.2 we get the following corollary. 3.5 Corollary: Let (X, M, ) be a fuzzy metric space, where is a continuous t-norm. Further let A, S and P be mappings from X into itself and satisfying the following condition

9 A Common Fixed Point Theorem For Occasionally Weakly Compatible Mappings 21 (3.5.1) For all x, y X, t>0 and for some 1 k 2. Suppose that the pairs (P, A) and (P, S) shares the property provided the pairs (P, A) and (P, S) are occasionally weakly compatible then P, A and S have a unique common fixed point. Taking A=S and T=B=I x in Theorem 3.2 we get the following corollary. 3.6 Corollary: Let (X, M, ) be a fuzzy metric space, where is a continuous t-norm. Further let A, P and Q be mappings from X into itself and satisfying the following condition (3.6.1) For all x, y X, t>0 and for some 1 k 2. Suppose that the pairs (P, A) and (Q, A) shares the property provided the pairs (P, A) and (Q, A) are occasionally weakly compatible then P, Q and A have a unique common fixed point. Taking P=Q and A=S and T=B=I x in theorem 3.2 we get the following corollary. 3.7 Corollary: Let (X, M, ) be a fuzzy metric space, where is a continuous t-norm. Further let A and P be mappings from X into itself and satisfying the following condition (3.7.1) For all x, y X, t>0 and for some 1 k 2. Suppose that the pair (P, A) shares the property provided the pair (P, A) are occasionally weakly compatible. Then P and A have a unique common fixed point.

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