Some Fixed Point Theorems using Weak Compatibility OWC in Fuzzy Metric Space

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1 Ieraioal Joural of Applied Egieerig Research ISSN Volume 13, Number 23 (2018) pp Research Idia Publicaios. hp:// Some Fixed Poi Theorems usig Weak Compaibiliy OWC i Fuzzy Meric Space Meeu Research Scholar, Dep. of Mahemaics, Baba Masah Uiversiy, Rohak ad Ass. Prof., Dep. of Mahemaics, A.I.J.H.M. College, Rohak, Haryaa, Idia. Viod Kumar Professor, Deparme of Mahemaics, Baba Masah Uiversiy, Rohak. Haryaa, Idia. Sushma Ass. Professor, Dep. of Mahemaics, Kaya Mahavidyalaya, Kharkhoda (Soepa), Haryaa, Idia. Absrac I his paper, a fixed poi heorem for six self mappigs is preseed by usig he cocep of weak compaible maps also preses some commo fixed poi heorems for occasioally weakly compaible mappig i fuzzy meric space. Keywords: Commo fixed poi, Fuzzy meric space, occasioally weakly compaible mappigs. 1. INTRODUCTION The fixed poi heory has bee sudied ad geeralized i differe spaces. Fuzzy se heory is oe of uceraiy approaches where i opological srucure are basic ools o develop mahemaical models compaible o cocree real life siuaio. Fuzzy se was defied by Zadeh [27]. Kramosil ad Michalek [15] iroduced fuzzy meric space, George ad Veermai [7] modified he oio of fuzzy meric spaces wih he help of coiuous orms. May researchers have obaied commo fixed poi heorem for mappig saisfyig differe ypes of commuaiviy codiios. Vasuki [26] proved fixed poi heorems for R weakly commuaig mappig. Pa [19, 20, 21] iroduced he ew cocep reciprocally coiuous mappigs ad esablished some commo fixed poi heorems. Balasubramaiam [5] have show ha Rhoades [23] ope problem o he exisece of coracive defiiio which geeraes a fixed poi bu does o force he mappig o be coiuous a he fixed poi, posses a affirmaive aswer. Rece lieraure o fixed poi i fuzzy meric space ca be viewed i [1, 2, 3, 10, 17]. Jai ad Sigh [29] proved a fixed poi heorem for six self maps i a fuzzy meric space. I his paper, a fixed poi heorem for six self maps has bee esablished usig he cocep of weak compaibiliy of pairs of self maps i fuzzy meric space. Also preses some commo fixed poi heorems for more geeral commuaive codiio i.e. occasioally weakly compaible mappigs i fuzzy meric space. For he sake of compleeess, we recall some defiiio ad kow resuls i fuzzy meric space. 2. PRELIMINARY NOTES Defiiio 2.1. A fuzzy se A i X is a fucio wih domai X ad values i [0, 1]. Defiiio 2.2. A biary operaio * : [0, 1] [0, 1] [0, 1] is a coiuous orms if * is saisfyig codiios (i) * is a commuaive ad associaive ; (ii) * is coiuous ; (iii) a * 1 = a for all a [0, 1] ; (iv) a * b c * d wheever a c ad b d, ad a, b, c, d [0, 1]. Defiiio 2.3. A 3 uple (X, M, *) is said o be a fuzzy meric space if X is a arbirary se, * is a coiuous orm ad M is a fuzzy se o X 2 (0, ) saisfyig he followig codiios, for all x, y, z X, s, > 0, (F1) M(x, y, ) > 0 ; (F2) M(x, y, ) = 1 if ad oly if x = y (F3) M(x, y, ) = M(y, x, ) ; (F4) (F5) M(x, y, ) * M(y, z, s) M(x, z, + s); M(x, y, ) : (0, ) (0, 1] is coiuous The M is called a fuzzy meric o X. The M(x, y, ) deoes he degree of earess bewee x ad y wih respec o. Example 2.1.(Iduced fuzzy meric) Le (X, d) be a meric space. Deoe a * b = ab for all a, b [0, 1] ad le M d be fuzzy ses o X 2 (0, ) defied as follows M d(x, y, ) d(x y) 16538

2 Ieraioal Joural of Applied Egieerig Research ISSN Volume 13, Number 23 (2018) pp Research Idia Publicaios. hp:// The (X, M d, *) is a fuzzy meric space. We call his fuzzy meric iduced by a meric d as he sadard iuiioisic fuzzy meric. Defiiio 2.4. Le (X, M, *) be a fuzzy meric space. The (a) (b) (c) a sequece {x } i X is said o coverges o x i X if for each > 0 ad each > 0, here exiss 0 N such ha M(x, x, ) > 1 for all 0. a sequece {x } i X is said o be Cauchy if for each > 0 ad each >0, here exiss 0 N such ha M(x, x m, ) > 1 for all, m 0. a fuzzy meric space i which every Cauchy sequece is coverge is said o be complee. Defiiio 2.5. A pair of self mappig (f, g) of a fuzzy meric space (X, M, *) is said o be (i) (ii) weakly commuig if M(fgx, gfx, ) M(fx, gx, ) for all x X ad > 0 R weakly commuig if here exiss some R>0 such ha M(fgx, gfx, ) M(fx, gx, /R) for all x X ad > 0. Defiiio 2.6. Two self mappig f ad g of a fuzzy meric space (X, M, *) are called compaible if lim M(fgx,gfx, ) 1 wheever {x } is a sequece i X such ha lim fx lim gx x for some x i X. Defiiio 2.7. Two self maps f ad g of a fuzzy meric space (X, M, *) are called reciprocally coiuous o X if lim fgx = fx ad lim gfx = gx wheever {x } is a sequece i X such ha lim fx = i X. lim gx = x for some x Defiiio 2.8. Le X be a se, f, g self maps of X. A poi x i X is called a coicidece poi of f ad g iff fx = gx. We shall call w = fx = gx a poi of coicidece of f ad g. Defiiio 2.9. A pair of maps S ad T is called weakly compaible pair if hey commue a coicidece pois i.e. if Sx = Tx for some x X he STx = TSx. Defiiio Two self maps f ad g of a se X are occasioally weakly compaible (OWC) iff here is a poi x i X which is a coicidece poi of f ad g a which f ad g commue. Al Thagafi ad Nasur Shahzad [4] show ha occasioally weakly is weakly compaible bu coverse is o rue. Example 2.2. Le R be he usual meric space. Defie S, T : R R by Sx = 2x ad Tx = x 2 for all x R. The Sx = Tx for x = 0, 2 bu STO = TSO ad ST2 TS2. S ad T are occasioally weakly compaible self maps bu o weakly compaible. Proposiio 2.1. Self mappig A ad S of a fuzzy meric space (X, M, *) are compaible. Proof. Suppose A p = S p for some p i X. Cosider a sequece {P } = P. Now {Ap } A p ad {Sp } S p(a p). As A ad S are compaible we have M(ASp, SAp, ) 1 for all > 0 as. Thus ASp = SAp ad we ge ha (A, S) is weakly compaible. The followig is a example of pair of self maps i a fuzzy meric space which are weakly compaible bu o compaible. Example 2.3. Le (X, M, *) be a fuzzy meric space where X = [0, 2], orm is defied by a * b = mi {a, b} for all a, b [0, 1] ad M(x, y, ) = e x y Defie self maps A ad S o X as follows ad Takig The Also A S x x 2 x if 0 x 1 2 if 1 x 2 x if 0 x 1 2 if 1 x 2 1 x 1, 1,2,3,... for all x, y X. x x < 1 ad 2 x > 2 for all. Ax, Sx 1 as 1 M(ASx, SAx, 2 ) = e 1 as Hece he pair (A, S) is o compaible. Also se of coicidece pois of A ad S is [1, 2]. Now for ay x [1, 2]. Ax = Sx = 2 ad AS(x) = A(2) = 2 = S(2) = SA(x). Thus A ad S are weakly compaible bu o compaible. From he above example, i is obvious ha he cocep of weak compaibiliy is more geeral ha ha of compaibiliy. Proposiio 2.2. I a fuzzy meric space a sequece is uique. (X, M, *) limi of Lemma 2.1. Le (X, M, *) be a fuzzy meric space. The for all x, y X, M(x, y, ) is a o decreasig fucio. Lemma 2.2. Le (X, M, *) be a fuzzy meric space. If here exiss q (0, 1) such ha for all x, y X, M(x, y, q) M(x, y, ), for all > 0, he x = y. Lemma 2.3. Le {x } be a sequece i a fuzzy meric space (X, M, *). If here exiss a umber q (0, 1) such ha M(x +2, x +1, q) M(x +1, x, ) for all > 0 ad N. The {x } is a Cauchy sequece i X. Lemma 2.4. Le X be a se, f, g owc self maps of X. If f ad g have a uique poi of coicidece, w = fx = gx, he w is he uique commo fixed poi of f ad g

3 Ieraioal Joural of Applied Egieerig Research ISSN Volume 13, Number 23 (2018) pp Research Idia Publicaios. hp:// Lemma 2.5. The oly orm * saisfyig r * r r for all r [0, 1] is he miimum orm, ha is a * b = mi {a, b} for all a, b [0, 1]. 3. MAIN RESULT Theorem 3.1. Le (X, M, *) be a complee fuzzy meric space ad le A, B, S, T, P ad Q be self mappigs from X io iself such ha he followig codiios are saisfied (a) P(X) ST(X), Q(X) AB(X) ; (b) AB = BA, ST = TS, PB = BP, QT = TQ; (c) eiher AB or P is coiuous (d) (P, AB) is compaible ad (Q, ST) is weakly compaible, (e) here exiss q (0, 1) such ha for every x, y X ad > 0 M(Px, Qy, q) mi {M(ABx, STy, ), M(Px, ABx, ),M(Qy, STy, ),M(Px, STy, ), M(Qx, ABy, )} Proof : Le x 0 X. From (a) here exiss x 1, x 2 X such ha Px 0 = STx 1 ad Qx 1 = ABx 2. Iducively, we ca cosruc sequece {x } ad {y } i X such ha Px 2 2 = STx 2 1 = y 2 1 ad Qx 2 1 = ABx 2 = y 2 for = 1, 2, 3, Sep 1 : Pu x = x 2 ad y = x 2+1 i (e), we ge (Px 2, Qx 2+1, q) mi {M(ABx 2, STx 2+1, ), M(Px 2, ABx 2, ), M(Qx 2+1, STx 2+1, ), M(Px 2, STx 2+1, ), M(Qx 2, ABx 2+1, )} = mi{m(y 2, y 2+1, ), M(y 2+1, y 2, ), M(y 2+2, y 2+1, ), M(y 2+1, y 2+1, ), M(y 2+1, y 2+1, )} mi{m(y 2, y 2+1, ), M(y 2+1, y 2+2, ) M(y 2+1, y 2+2, q) mi{m(y 2, y 2+1, ), This implies, M(y 2+1, y 2+2, ) M (y 2+1, y 2+2, q) M (y 2, y 2+1, ) Similarly, (y 2+2, y 2+3, q) M(y 2+1, y 2+2, ) Thus, M(y +1, y +2, q) M(y, y +1, ) for = 1, 2, 3, M(y, y +1, ) M(y, y +1, q ) M(y 2, y 1, as q 2 ). M(y1, y2, q ) 1 ad hece M(y, y +1, ) 1 as for ay > 0 for each > 0 ad > 0, we ca choose 0 N such ha M(y, y +1, ) > 1 for all > 0. For m, N, we suppose m >. The we have M(y, y m, ) mi {M(y, y +1, m ), M(y 1, y 2, ),...,M(y m 1, y m, )} m m mi {(1 ), (1 ),, (1 ) (m ) imes} (1 ) ad hece {y } is a Cauchy sequece i X. Sice (X, M, *) is complee {y } coverges o some poi z X. Also is subsequeces coverges o he same poi z X i.e. {Qx } z ad {STx } z (1) {Px } z ad {ABx } z (2) 2 2 Case I : Suppose AB is coiuous. Sice AB is coiuous, we have (AB) 2 x 2 ABz d ABPx 2 ABz. As (P, AB) is compaible pair, he PABx 2 ABz. Sep 2 : Pu x = ABx 2 ad y = x 2+1 i (e), we ge M(PABx 2, Qx 2+1, q) mi {M(ABABx 2, STx 2+1, ), M(PABx 2, ABABx 2, ), M(Qx 2+1, STx 2+1, ), M(PABx 2, STx 2+1, ), M(QABx 2, ABx 2+1, )} Takig, we ge M(ABz, z, q) mi {M(ABz, z, ), M(ABz, ABz, ),M(z, z, ), M(ABz, z, ), M(ABz, z, )} = M(ABz, z, ) i.e. M(ABz, z, q) M(ABz, z, ) Therefore, by usig Lemma 2.2, we ge ABz = z Sep 3 : Pu x = z ad y = x 2+1 i (e), we have M(Pz, Qx 2+1, q) mi {M(ABz, STx 2+1, ), (3) M(Pz, ABz, ), M(Qx 2+1, STx 2+1, ), M(Pz, STx 2+1, ), M(Qz, ABx 2+1, )} Takig ad usig equaio (1), we ge M(Pz, z, q) mi{m(z, z, ), M(Pz, z, ), M(z, z, ), M(Pz, z, ), M(z, z, )}= M(Pz, z, ) i.e. M(Pz, z, q) M(Pz, z, ) Therefore by usig Lemma 2.2, we ge Pz = z. Therefore ABz = Pz = z. Sep 4 : Puig x = Bz ad y = x 2+1 i codiio (e), we ge 16540

4 Ieraioal Joural of Applied Egieerig Research ISSN Volume 13, Number 23 (2018) pp Research Idia Publicaios. hp:// M(PBz, Qx 2+1,q) mi{m(abbz,stx 2+1, ), M(PBz, ABBz, ), M(Qx 2+1, STx 2+1, ), M(PBz, STx 2+1, ), M(QBz, ABx 2+1, )} As BP = PB, AB = BA, so we have P(Bz) = B(Pz) = Bz ad (AB)(Bz) = (BA)(Bz) = B(ABz) = Bz. Takig ad usig (1), we ge M(Bz, z, q) mi{m(bz, z, ), M(Bz, Bz, ), M(z, z, ),M(Bz, z, ) M(Bz, z, )} = M(Bz, z, ) i.e. M(Bz, z, q) M(Bz, z, ) Therefore, by usig Lemma 2.2, we ge Bz = z ad also we have ABz = Az = z. Therefore Az = Bz = Pz = z (4) Sep 5 : As P(X) ST(X), here exiss u X such ha z = Pz = STu, Puig x = x 2 ad y = u i (e), we ge M(Px 2, Qu, q) mi{m(abx 2, STu, ), M(Px 2, ABx 2,),M(Qu,STu,),M(Px 2,STu, ), M(Qx 2, ABu, )} Takig ad usig (1) ad (2), we ge M(z, Qu, q) mi{m(z, z, ), M(z, z, ), M(Qu, z, ),M(z, z, ), M(z, z, )}= M(Qu, z, ) i.e. M(z, Qu, q) M(z, Qu, ) Therefore by Lemma 2.2, we ge Qu = z, Hece STu = z = Qu. Sice (Q, ST) is weak compaible. Therefore, we have QSTu = STQu. Thus Qz = STz. Sep 6 : Puig x = x 2 ad y = z i (e), we ge M(Px 2, Qz, q) mi{m(abx 2, STz, ), M(Px 2, ABx 2, ),M(Qz,STz,),M(Px 2,STz, ), M(Qx 2, ABz, )} Takig ad usig (2) ad sep 5, we ge M(z, Qz, q) mi{m(z, Qz, ), M(z, z, ), M(Qz, Qz, ), M(z, Qz, ), M(z, Qz, )} = M(z, Qz, ) i.e. M(z, Qz, q) M(z, Qz, ) Therefore, by usig lemma 2.2, we ge Qz = z. Sep 7: Puig x = x 2 ad y = Tz i (e), we ge M(Px 2, QTz, q) mi{m(abx 2, STTz, ), M(Px 2, ABx 2,), M(QTz, STTz, ), M(Px 2, STTz, ), M(Qx 2, ABTz, )} As QT = TQ ad ST = TS, we have QTz = TQz = Tz ad ST(Tz) = T(STz) = TQz = Tz Takig, we ge M(z, Tz, q) mi{m(z, Tz, ), M(z, z, ), M(Tz, Tz, ),M(z, Tz, ), M(z, Tz, )} = M(z, Tz, ) i.e. M(z, Tz, q) M(z, Tz, ) Therefore by usig lemma 2.2, we ge Tz = z, ow STz = Tz = z implies Sz = z. Hece Sz = Tz = Qz = z Combiig (4) ad (5), we ge Az = Bz = Pz = Qz = Tz = Sz = z Hece, z is he commo fixed poi of A, B, S, T, P ad Q. (5) Case II: Suppose P is coiuous. As P is coiuous, P 2 x 2 Pz ad P(AB) x 2 Pz. As (P, AB) is compaible, we have (AB) Px 2 Pz. Sep 8: Puig x = Px 2 a y = x 2+1 i codiio (e), we have M(PPx 2, Qx 2+1, q) mi{m(abpx 2, STx 2+1, ) M(PPx 2, ABPx 2, ), M(Qx 2+1, STx 2+1, ),M(PPx 2, STx 2+1, ),M(QPx 2, ABx 2+1, )} Takig, we ge M(Pz, z, q) mi{m(pz, z, ), M(Pz, Pz, ), M(z, z, ), M(Pz, z, ), M(Pz, z, )}= M(Pz, z, ) i.e. M(Pz, z, q) M(Pz, z, ) Therefore by usig Lemma 2.2, we have Pz = z. Furher usig seps 5, 6, 7, we ge z = STz = Sz = Tz = z Sep 9: As Q(X) AB(X), here exiss w X, such ha z = Qz = ABw. Pu x = w ad y = x 2+1 i (e), we have M(Pw, Qx 2+1, q) mi{m(abw, STx 2+1, ), M(Pw, ABw, ), M(Qx 2+1,STx 2+1,), M(Pw, STx 2+1, ), M(Qw, ABx 2+1, )} Takig, we ge M(Pw, z, q) mi{m(z, z, ), M(Pw, z, ), M(z, z, ), M(Pw, z, ), M(z, z, )}= M(Pw, z, ) i.e. M(Pw, z, q) M(Pw, z, ) Therefore, by usig Lemma 2.2, we ge Pw = z Therefore, ABw = Pw = z. As (P, AB) is compaible. We have Pz = ABz. Also, from sep 4, we ge Bz = z. Thus Az = Bz = Pz = z ad we see ha z is he commo fixed poi of he six maps i his case also. Uiqueess : Le u be aoher commo fixed poi of A, B, S, T, P ad Q. The Au = Bu = Pu = Su = Tu = u. Pu x = z ad y = u, i (e), we ge M(Pz, Qu, q) mi{m(abz, STu, ), M(Pz, ABz, ), M(Qu, STu, ), M(Pz, STu, ), M(Qz, ABu, )} 16541

5 Ieraioal Joural of Applied Egieerig Research ISSN Volume 13, Number 23 (2018) pp Research Idia Publicaios. hp:// Takig, we ge M(z, u, q) mi{m(z, u, ), M(z, z, ), M(u, u, ), M(z, u, ), M(z, u, )} = M(z, u, ) i.e. M(z, u, q) M(z, u, ) Therefore, by usig Lemma 2.2, we ge z = u. Therefore z is he uique fixed poi of self maps A, B, S, T, P ad Q. Corollary 3.1. Le (X, M, *) be a complee fuzzy meric space ad le A, S, P ad Q be mappig from X io iself such ha he followig codiios are saisfied (a) P(X) S(X), Q(X) A(X) ; (b) (c) (d) (e) eiher A or P is coiuous (P, A) is compaible ad (Q, S) is weakly compaible; here exiss q (0, 1), such ha for every x, y X ad > 0 ; M(Px, Qy, q) mi{m(ax, Sy, ), M(Px, Ax, ), M(Qy, Sy, ), M(Px, Sy, ), M(Qx, Ay, )} The A, S, P ad Q have a uique fixed poi i X. Theorem 3.2. Le (X, M, *) be a complee fuzzy meric space ad le A, B, S, T, P ad Q be self mappigs of X. Le pair {A, S}, {B,T} ad {P, Q} be owc. If here exiss q (0, 1) such ha M(Px, Qy, q) mi{m(abx, STy, ), M(Px, ABx, ), M(Qy, STy, ), M(Px, STy, ), M(ABx, Qy, )} (1) for all x, y X ad for all > 0, he here exiss a uique poi w X such ha ABw = Pw = w ad a uique poi z X such ha STy = Qy = z. Moreover, z = w, so ha here is a uique commo fixed poi of A, B, S, T, P ad Q. Proof: Le he pair {A, S}, {B, T} ad {P, Q} be owc, so here are pois x, y X such ha ABx = Px ad STy = Qy. We claim ha ABx = STy. If o, by iequaliy (1) M(Px, Qy, q) mi{m(abx, STy, ), M(Px, ABx, ), M(Qy, STy, ), M(Px, STy, ), M(ABx, Qy, )} = mi{m(px, Qy, ), M(Px, Px, ), M(Qy, Qy, ), M(Px, Qy, ), M(Px, Qy, )} = M(Px, Qy, ) Therefore ABx = STy i.e. ABx = Px = STy = Qy. Suppose ha here is a aoher poi z such ha ABz = STz he by (1), we have ABz = Pz = STy = Qy so ABx = ABz ad w = ABx = STx is he uique poi of coicidece of A ad S. By lemma 2.4, w is he oly commo fixed poi of A ad S. Similarly here is a uique poi z X such ha z = Bz = Tz ad z = Pz = Qz Assume ha w z, we have M(w, z, q) = M(Pw, Qw, q) mi{m(abw, STz, ), M(Pw, ABw, ), M(Qz, STz, ), M(Pw, STz, ), M(ABw, Qz, )} = mi{m(w, z, ), M(w, w, ), M(z, z, ), M(w, z, ),M(w, z, )} = M(w, z, ) Therefore, we have z = w by Lemma 2.4 ad z is a commo fixed poi of A, B, S, T, P ad Q. The uiqueess of he fixed poi holds from (1). Theorem 3.3. Le (X, M, *) be a complee fuzzy meric space ad le A, B, S, T, P ad Q be self mappig of X. Le he pair {A, S}, {B, T} ad {P, Q} be owc. If here exiss q (0, 1) such ha M(Px, Qy, q) (M(ABx, STy, ), M(Px, ABx, ), M(Qy, STy, ),M(Px, STy, ), M(ABx, Qy, )) (1) for all x, y X ad : [0, 1] 5 [0, 1] such ha (, 1, 1,, ) > for all 0 < < 1, he here exis a uique commo fixed poi of A, B, S, T, P ad Q. Proof : le he pairs {A, S}, {B, T} ad {P, Q} are owc here are pois x, y X such ha ABx = Px ad STy = Qy. We claim ha ABx = STy. If o, by iequaliy (1) M(Px, Qy, q) (M(ABx, STy, ), M(Px, ABx, ), M(Qy, STy, ), M(Px, STy, ), M(ABx, Qy, ))= (M(Px, Qy, ), M(Px, Px, ), M(Qy, Qy, ),M(Px, Qy, ), M(Px, Qy, )) = (M(Px, Qy, ), 1, 1, M(Px, Qy, ), M(Px, Qy, ) > M(Px, Qy, ) a coradicio. Therefore ABx = STy, i.e. ABx = Px = STy = Qy. Suppose ha here is a aoher poi z such ha ABz = STz he by (1) we have ABz = Pz = STy = Qy, so ABx = ABz ad w = ABx = STx is he uique poi of coicidece of A ad T. By Lemma 2.4, w is a uique commo fixed poi of A ad S. Similarly here is a uique poi z X such ha z = Bz = Tz ad z = Pz = Qz. Thus z is a commo fixed poi of A, B, S, T, P ad Q. The uiqueess of he fixed poi holds from (1). Theorem 3.4. Le (X, M, *) be a complee fuzzy meric space ad le A, B, S, T, P ad Q be self mappigs of X. Le he pair {A, S}, {B, T} ad {P, Q} be owc. If here exiss q (0, 1) such ha M(Px, Qy, q) (mi{m(abx, STy, ), M(Px, ABx, ), M(Qy, STy, ), M(Px, STy, ), M(ABx, Qy, )}) (1) for all x, y X ad : [0, 1] [0, 1] such ha () > 1 for all 0 < < 1, he here exiss a uique commo fixed poi of A, B, S, T, P ad Q. Proof : The proof follows from Theorem

6 Ieraioal Joural of Applied Egieerig Research ISSN Volume 13, Number 23 (2018) pp Research Idia Publicaios. hp:// Theorem 3.5. Le (X, M, *) be a complee fuzzy meric space ad le A, B, S, T, P ad Q be self mappigs of X. Le he pair {A, S}, {B, T} ad {P, Q} are owc. If here exiss a poi q (0, 1) for all (x, y) X ad > 0. M(Px, Qy, q) M(ABx, STy,)*M(Px, ABx, ) *M(Qy, STy, )*M(Px, STy,)* M(ABx, Qy, ) The, here exiss a uique commo fixed poi of A, B, S, T, P ad Q. Proof : Le he pair {A, S}, {B, T} ad {P, Q} are owc, here are pois x, y X such ha ABx = Px ad STy = Qy. We claim ha ABx = STy, by iequaliy (1) M(Px, Qy, q) M(ABx, STy, ) * M(Px, ABx, ) * M(Qy, STy, )*M(Px, STy, ), M(ABx, Qy, ) = M(Px, Qy, ) * M(Px, Px, ) * M(Qy, Qy, ) * M(Px, Qy, ) * M(Px, Qy, ) M(Px, Qy, ) * 1 * 1 * M(Px, Qy, ) * M(Px, Qy, ) M(Px, Qy, ) Thus, we have ABx = STy, i.e. ABx = Px = STy = Qy, suppose ha here is a aoher poi z such ha ABz = STz he by (1) we have ABz = Pz = STy = Qy so ABx = ABz ad w = ABx = STx is he uique poi of coicidece of A ad S. Similarly here is a uique poi z X such ha z = Bz = Tz ad z = Pz = Qz. Thus z is a commo fixed poi of A, B, S, T, P ad Q. Corollary 3.2. Le (X, M, *) be a complee fuzzy meric space ad A, B, S, T, P ad Q be self mappigs of X. Le he pair {A, S}, {B, T} ad {P, Q} ad owc. If here exiss a poi q (0, 1) for all x, y X ad > 0. M(Px, Qy, q) M(ABx, STy,)*M(Px,ABx, ) * M(Qy, STy, )* (M(Px, STy, 2) * M(ABx, Qy, ) The, here exiss a uique commo fixed poi of A, B, S, T, P ad Q. Proof : We have M(Px, Qy, q) M(ABx, STy, ) * M(Px, ABx, ) * M(Qy, STy, )*M(Px, STy, 2) * M(ABx, Qy, ) M(ABx, STy, ) * M(Px, ABx, ) * M(Qy, STy, )* M(STy,ABx,) *M(ABx,Px,)* M(ABx,Qy, ) M(ABx, STy, ) * M(Px, ABx, ) * M(Qy, STy, )* M(ABx, Qy, ) Ad herefore, from above heorem A, B, S, T, P ad Q have a commo fixed poi. REFERENCES [1] C. T. Aage, J. N. Saluke, Commo Fixed Poi Theorems i Fuzzy Meric Spaces, 56(2), 2009, pp , Ieraioal Joural of Pure ad Applied Mahemaics. [2] C. T. Aage, J. N. Saluke, Some Fixed Poi Theorems i Fuzzy Meric Space, 56(3) 2009, pp , Ieraioal Joural of Pure ad Applied Mahemaics. [3] C. T. Aage, J. N. Saluke, O Fixed Poi Theorems i Fuzzy Meric Spaces Usig A Corol Fucio, Submied. [4] A. Al Thagafi ad Naseer Shahzad, Geeralized I Noexpasive Selfmaps ad Ivaria Approximaio, Eglish Series May, 2008, Vol. 24, No. 5, pp , Aca Mahemaica Siica. [5] P. Balasubramaiam, S. Muralisakar, R.P. Pa, Commo fixed pois of four mappigs i a fuzzy meric space, 10(2) (2002), , J. Fuzzy Mah.. [6] Y. J. Cho, H. K. Pahak. S.M. Kag, J. S. Jug, Commo fixed pois of compaible maps of ype (A) o fuzzy meric spaces, 93 (1998), Fuzzy Ses ad Sysems. [7] A. George, P. Veeramai, O some resuls i fuzzy meric spaces, 64 (1994), , Fuzzy Ses ad Sysems. [8] M. Grabiec, Fixed pois i fuzzy mericspace, 27(1988), Fuzzy Ses ad Sysems. [9] O. Hadzic, Commo fixed poi heorems for families of mappig i complee meric space, 29 (1984), , Mah. Japa. [10] Mohd. Imdad ad Javid Ali, Some commo fixed poi heorems i fuzzy meric space,11(20060, ,Mahemaical Commuicaios. [11] G. Jugck, Compaible mappigs ad commo fixed pois (2) (1988), , I. J. Mah. Mah. Sci. [12] G. Jugck ad B. E. Rhoades, Fixed Poi for Se Valued fucios wihou Coiuiy, 29(3) (1998), pp , Idia J. Pure Appl. Mah.. [13] G. Jugck ad B. E. Rhoades, Fixed Poi Theorems for Occasioally Weakly compaible Mappigs, Volume 7, No. 2, 2006, Fixed Poi Theory. [14] G. Jugck ad B. E. Rhoades, Fixed Poi Theorems for Occasioally Weakly compaible Mappigs, Erraum, Volume 9, No. 1, 2008, , Fixed Poi Theory. [15] O. Kramosil ad J. Michalek, Fuzzy meric ad saisical meric space, 11(1975), , Kybereika. [16] S. Kuukcu, A fixed poi heorem for coracio ype mappigs i Meger spaces, 4(6) (2007), , Am. J. Appl. Sci

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