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Bd32360363

  1. 1. Deepak Singh, M.S. Rathore, Krishnapal Singh Sisodia / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.360-363 A Common Fixed Point Theorem through Weak and Semi- Compatibility in Fuzzy Metric Space Deepak Singh, M.S. Rathore, Krishnapal Singh Sisodia *NITTTR, Bhopal (M.P.), India **Department of Mathematics, Govt. P. G. College Sehore (M.P.), India ***Department of Mathematics, SISTec-E, Ratibad, Bhopal (M.P.), IndiaAbstract In this paper, a common fixed point (fm5) M(x, y, .): (0, ) [0,1] is continuous.theorem for six self mappings has been (fm6) for all x, y X.established using the concept of semi- Then M is called a Fuzzy metric on X. The functioncompatibility and weak compatibility in Fuzzy M (x, y, t) denote the degree of nearness between xmetric space, which generalizes the result of and y with respect to t.Singh B.S., Jain A. and Masoodi A.A. [6]. Example 1: Let (X, d) be a metric space. Denote a b=ab for a, b [0, 1] and let Md be Fuzzy set onKeywords: Fuzzy metric space, common fixedpoint, t-norm, compatible map, semi-compatible, X2 (0, ) defined as follows:weak compatible map. Md (x, y, t) =2000 Mathematics Subject Classification: 47H10,54H25. Then (X, Md, ) is a Fuzzy metric space, we call this Fuzzy metric induced by a metric d the standardIntroduction: The concept of Fuzzy sets was intuitionistic Fuzzy metric.introduced by Zadeh [9]. Following the concept of Definition 3 [2]: Let (X, M, ) be a Fuzzy metricFuzzy sets, Fuzzy metric spaces have been space, thenintroduced by Kramosil and Michalek [4] and (a) A sequence {xn} in X is said to beGeorge and Veeramani [2] modified the notion of convergent to x in X if for each >0 andFuzzy metric spaces with the help of continuous t-norms. Vasuki [8] investigated some fixed point each t>0, there exists n 0 N such that Mtheorems in Fuzzy metric spaces for R-weakly (xn, x, t)>1- for all n n0.commuting mappings. Inspired by the results of B. (b) A sequence {xn} in X is said to be CauchySingh, A. Jain and A.A. Masoodi [6], in this paper, if for each >0 and each t>0, there existswe prove a common fixed point theorem for six self n0 N such that M (xn, xm, t)>1- for all n,maps under the condition of weak compatibility andsemi-compatibility in Fuzzy metric spaces. m n0. (c) A Fuzzy metric space in which everyPreliminaries: Cauchy sequence is convergent is said toDefinition 1: A binary operation : be complete.[0,1] [0,1] [0,1] is a continuous t-norm if is Proposition 1: In a Fuzzy metric space (X, M, ), ifsatisfying the following conditions: a a a for a [0, 1] then a b=min {a, b} for all a, b (a) is commutative and associative; [0, 1]. (b) is continuous; Definition 4 [7]: Two self mappings A and S of a (c) a 1=a for all a [0,1]; Fuzzy metric space (X, M, ) are called compatible (d) a b c d whenever a c and b d and if whenever a,b,c,d [0,1]. { } is a sequence in X such thatDefinition 2 [2]: A 3-tuple (X, M, ) is said to be a for some in X.Fuzzy metric space if X is an arbitrary set, is a Definition 5 [7]: Two self mappings A and S of acontinuous t-norm and M is a Fuzzy set onX2 (0, ) satisfying the following conditions: for Fuzzy metric space (X, M, ) are called semi-all x,y,z X, s,t>0, compatible if(fm1) M (x, y, t)>0; whenever { } is a sequence in X such that(fm2) M (x, y, t) =1 iff x=y; for some in X.(fm3) M (x, y, t) = M (y, x, t);(fm4) M (x, y, t) M(y, z, s) M(x, z, t+s); 360 | P a g e
  2. 2. Deepak Singh, M.S. Rathore, Krishnapal Singh Sisodia / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.360-363 Definition 6 [7]: Two self maps A and S of a Fuzzy metric space (X, M, ) are called weakly compatible (or coincidentally commuting) if they commute at their coincidence points, i.e. if Ax=Sx for some X then ASx=SAx. Proposition 2 [7]: In a Fuzzy metric space (X, By lemma 1 and lemma 2; we get M, ), limit of a sequence is unique. Proposition 3 [7]: In a Fuzzy metric space (X, Similarly; we have M, ), If (A, S) is a semi-compatible pair of self maps and S is continuous, then (A, S) is compatible Thus we have Remark 1: If self maps A and S of a Fuzzy metric space (X, M, ) are compatible then they are weakly Using lemma 3; we have compatible. { } is a Cauchy sequence in X. Lemma 1 [3]: Let (X, M, ) be a Fuzzy metric Since (X, M, ) is complete, { } converges to space. Then for all x, y єX, some point X. Also its subsequences converges is a non decreasing function. to the same point i.e. X. Lemma 2 [5]: Let (X, M, ) be a Fuzzy metric i.e. { } →z and { } →z ---------- space. If there exists k [0, 1] such that (1) { } →z and { } →z --------------- (2) then x=y. Case I: suppose P is continuous. Lemma 3 [1]: let {yn} be a sequence in a Fuzzy Step-2: since P is continuous and (P, AB) is semi- metric space (X, M, ) with the condition (fm6). If compatible pair, we have there exists k [0, 1] such that →Pz and for all t>0 →ABz. and n N, then {yn} is a Cauchy sequence in X. Since the limit in Fuzzy metric space is unique, we Main Results: get In this result, we are proving a theorem to obtain a . ---------------- (3) unique common fixed point for six self mappings Step-3: putting x=Px2n and y=x2n+1 in condition (v); using weak and semi-compatibility in a Fuzzy we have metric space. Theorem 1: Let (X, M, ) be a complete Fuzzy metric space and let A, B, S, T, P and Q be Taking n→∞ and using (1), (2) and (3); we get mappings from X into itself such that the following conditions are satisfied: } (i) P(X) ST(X), Q(X) AB(X); (ii) AB=BA, ST=TS, PB=BP, QT=TQ } (iii) Either AB or P is continuous; i.e. (iv) The pair (P, AB) is semi-compatible and Therefore by using lemma 2; we have (Q, ST) is weakly compatible. . (v) There exists such that for every Step-4: putting x=Bz and y= x2n+1 in condition (v); and t>0 we get. Then A, B, S, T, P and Q have a unique common fixed point. BP=PB, AB=BA; so we have Proof: Let x0 . By (i), there exists x1, x2 X such P(Bz)=B(Pz)=Bz and AB(Bz)=B(AB)z=Bz that Px0=STx1= y0 and Qx1=ABx2=y1. Inductively, Taking n→∞ and using (1), we get we can construct sequences {xn} and {yn} in X such that Px2n=STx2n+1=y2n and Qx2n+1=ABx2n+2=y2n+1 for n=0, 1, 2, ----. Therefore, by using lemma 2, we get Step-1: putting x=x2n and y= x2n+1 in (v), we have . And also we have 361 | P a g e
  3. 3. Deepak Singh, M.S. Rathore, Krishnapal Singh Sisodia / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.360-363 . Thus, . ------------ (4) Now, we prove .Step-5: since P(X) ST(X), there exists X such Step-8: putting and inthat (v); we get --------------- (5)Putting x=x2n and y= u in (v), we haveTaking n→∞ and using (2) and (5), we get Taking n→∞ and using (1), (9) and (10); we getTherefore by using lemma 2; we get Therefore, by using lemma 2, we get . .Hence Step-9: putting x=z and y= x2n+1 in condition (v); weSince (Q, ST) is weakly compatible, therefore we gethaveThus ----------- (6) Taking n→∞ and using (1); we getStep-6: Putting x=x2n and y= z in (v), we have Taking n→∞ and using (2) and (6); we get Therefore, by using lemma 2, we get . .Therefore by using lemma 2, we get Further, using step (4), we get .Step-7: putting x=x2n and y= Tz in (v), we have Thus, gives A And so . Also it follows from steps (5), (6) and (7) thatAs QT=TQ and ST=TS, we have .QTz=TQz=Tz and STTz=TSTz=T(STz)=T(Qz)=Tz Hence weTaking n→∞ and using (2); we get get . i.e. z is a common fixed point of A, B, P, Q, S and T in this case also. Uniqueness : Let w be another common fixed pointTherefore by using lemma 2; we get of A, B, P, Q, S and T, ThenNow implies .Hence -------------- (7) Putting x=z and y= w in condition (v); we getCombining (4) and (7); we getHence z is the common fixed point of A, B, S, T, Pand Q.Case II: suppose AB is continuous. Therefore, by using lemma 2, we getSince AB is continuous and (P, AB) is semi- .compatible, we have Therefore z is the unique common fixed point of self → ------------- (8) maps A, B, P, Q, S and T. ----------- (9) Remark 2: If we take B=T=IX (the identity map on ----------- (10) X) in the theorem 1, then condition (ii) is satisfied trivially and we get 362 | P a g e
  4. 4. Deepak Singh, M.S. Rathore, Krishnapal Singh Sisodia / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 2, March -April 2013, pp.360-363Corollary 1: Let (X, M, ) be a complete Fuzzymetric space and let A, S, P and Q be mappingsfrom X into itself such that the following conditionsare satisfied: (i) P(X) S(X), Q(X) A(X); (ii) Either A or P is continuous; (iii) The pair (P, A) is semi-compatible and (Q, S) is weakly compatible. (iv) There exists such that for every and t>0 . Then A, S, P and Q have a unique common fixed point.References: [1] Y.J. Cho, H.K. Pathak, S.M. Kang, J.S. Jung, Common Fixed points of compatible maps of type (β) on Fuzzy metric spaces, Fuzzy sets and systems, 93(1998), 99-111. [2] A. George, P. Veeramani, On some results in Fuzzy metric spaces, Fuzzy sets and systems, 64(1994), 395-399. [3] M. Grebiec, Fixed points in Fuzzy metric space, Fuzzy sets and systems, 27(1998), 385-389. [4] O. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika 11(1975), 326-334. [5] S. Sharma, Common Fixed point theorems in Fuzzy metric spaces, Fuzzy sets and systems, 127(2002), 345-352. [6] B. Singh, A. Jain and A.A. Masoodi, Semi- compatibility, weak compatibility and fixed point theorem in Fuzzy metric space, International Mathematical Forum, 5, 2010, no. 61, 3041-3051 [7] B. Singh and S. Jain, Semi-compatibility, compatibility and fixed point theorems in Fuzzy metric space, Journal of Chungecheong Math. Soc. 18(1), (2005), 1- 22. [8] R. Vasuki, Common Fixed points for R- weakly commuting maps in Fuzzy metric spaces, Indian J. Pure Appl. Math., 30(1999), 419-423. [9] L.A. Zadeh, Fuzzy sets, Inform and Control, 8(1965), 338-353. 363 | P a g e

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