GENERAL CONTRACTIVE MAPPING CONDITION OF FUZZY METRIC SPACES IN FIXED POINT THEOREM

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1 Joural of Mathematics ad Statistics (): 65-7, 4 ISSN: Sciece Publicatios doi:.3844/jmssp Published Olie () 4 ( GENERAL CONTRACTIVE MAPPING CONDITION OF FUZZY METRIC SPACES IN FIXED POINT THEOREM Ramasamy Moha Raj ad Veerachamy Malliga Devi Departmet of Mathematics, Faculty of Sciece ad Humaities, Aa Uiversity Tiruelveli Regio, Tiruelveli, Tamiladu, Idia Departmet of Mathematics, Faculty of Sciece ad Humaities, Aa Uiversity, Tuticori Campus, Tamil Nadu, Idia Received 3-4-9; Revised 4--4; Accepted 4--6 ABSTRACT I several ways, related fixed poit theorems o two or three metric spaces have bee demostrated. By applyig cotractive coditio of itegral type for class of weakly compatible maps i ucompleted ituitioistic fuzzy metric spaces without cosiderig ay cotiuous mappigs, i this paper, we verify some frequet fixed poit theorems for differet mappigs. Keywords: Fuzzy Metric Space, Fixed Poit Theorem, Baach Fixed Theorem, Mappig. INTRODUCTION I 9, Baach a polish mathematicia proved a theorem uder appropriate coditios ad showed the existece ad uiqueess of a fixed poit this result is called Baach fixed poit theorem. This theorem is also applied to prove the existece ad uiqueess of the solutios of differetial equatios. May authors have made differet geeralizatio of Baach fixed theorem. There are so may researches are available o this fixed poit theorems. Normally, fixed poit theory is classified ito 3 categories such as () Topological Fixed poit theory () Metric Fixed poit theory ad (3) Discrete fixed poit theory. I these three areas, the boudary lies ca be detected by employig the theorems such as () Brouwer s Fixed Poit Theorem () Baach s Fixed Poit Theorem (3) Tarski s Fixed Poit Theorem. The cocept of Fuzzy set as a ew way to represet vagueess i our everyday life (Zadeh, 965). However, whe the ucertaity is due to fuzziess rather tha radomess, as sometimes i the measuremet of a ordiary legth, it seems that the cocept of a fuzzy metric space is more suitable. We ca divide them ito followig two groups: The first group ivolves those map where X represets the totality of all fuzzy poits of a set ad satisfy some axioms which are aalogous to the ordiary metric axioms. Thus, i such a approach umerical distaces are set up betwee fuzzy objects. O the other had i secod group, we keep those results i which the distace betwee objects is fuzzy ad the objects themselves may or may ot be fuzzy (Maro et al., )... Prelimiaries Some prelimiary defiitios are give below. Defiitio. Zadeh (965) A fuzzy set A i X is a fuctio with domai X ad values i [, ]. Defiitio. Schweizer ad Sklar (96)A biary operatio *: [, ] [, ] [, ] is a cotiuous t-orms if * is satisfyig coditios: *is a commutative ad associative *is cotiuous a* = a for all a [, ] a*b c*d wheever a c ad b d ad a, b, c, d [, results i which a fuzzy metric o a set X is treated as a ] Correspodig Author: Veerachamy Malliga Devi, Departmet of Mathematics, Faculty of Sciece ad Humaities, Aa Uiversity (Tuticori Campus), Tamil Nadu, Idia Sciece Publicatios 65

2 Ramasamy Moha Raj ad Veerachamy Malliga Devi / Joural of Mathematics ad Statistics (): 65-7, 4 Defiitio.3 George ad Veeramai (994) a 3-tuple (X, M, *) is said to be a fuzzy metric space if X is a arbitrary set, * is a cotiuous t-orm ad M is a fuzzy set o X [, ] satisfyig the followig coditios, for all x, y, z X, such that t is i [, ]: (f) M(x, y, t) > (f) M(x, y, t) = if ad oly if x = y (f3) M(x, y, t) =, x, t) (f4) M(x, y, t) *, z, s) M(x, z, t + s) (f5) M(x, y, *): (, ) (, ] is cotiuous The M is called a fuzzy metric o X. The M(x, y, t) deotes the degree of earess betwee x ad y with respect to t. Defiitio.4 Let (X, M, *) is a fuzzy metric space: x i X is said to be coverget to a A sequece { } poit x X (deoted by ): Sciece Publicatios lim x = x) if, for all t >, lim M(x t) = + p A sequece {x } i X called a Cauchy sequece if, for all t > ad p > : lim M(x t) = + p A fuzzy metric space i which every Cauchy sequece is coverget is said to be complete Defiitio.5 Rhoades (988) a pair of self-mappigs (f, g) of a fuzzy metric space (X, M, *) is said to be: Weakly commutig if: M (fgx, gfx, t) M(fx, gx, t) x X & t > R-weakly commutig if there exists some R > such that M(fgx, gfx, t) M(fx, gx, t/r) x X ad t > Defiitio.6 Jugck ad Rhoades (8) two self-mappigs f ad g of a fuzzy metric space (X, M, *) are called compatible if: lim M(fgX,gfX = 66 wheever {X } is a sequece i X such that: lim fx = lim gx = x for some x i X Defiitio.7 Grabiec (988): Two self-maps f ad g of a fuzzy metric space (X, M, *) are called reciprocally cotiuous o X if: LimfgX = fx ad lim gfx = gx wheever {X } is a sequece i X such that: Defiitio.8 lim fx = lim gx = x for some x i X Let X be a set, f, g self maps of X. A poit x i X is called a coicidece poit of f ad gifffx = gx. We shall call w = fx = gx a poit of coicidece of f ad g. Defiitio.9 Jugck ad Rhoades (8) a pair of maps S ad T is called weakly compatible pair if they commute at coicidece poits. Defiitio. Two self-maps f ad g of a set X are occasioally weakly compatible (owc) iff there is a poit x i X which is a coicidece poit of f ad g at which f ad g commute... Literature Review Klim ad Wardowski (7) itroduced the cocept of cotractio for set-valued maps i metric spaces ad the coditios guarateeig the existece of a fixed poit for such a cotractio are established. Oe of our results essetially geeralizes the Nadler ad Feg-Liu theorems ad is differet from the Mizoguchi-Takahashi result. The secod result was differet from the Reich ad Mizoguchi-Takahashi results. The method which had used i the proofs of our results was ispired by Mizoguchi-Takahashi ad Fe-Liu sideas. Mishra et al. () proved some fixed poit theorems for weakly compatible maps i fuzzy metric space satisfyig itegral type iequality but without assumig the completeess of the space or cotiuity of the mappigs ivolved. Paper has exteded this cocept to fuzzy metric space ad established the existece of commo fixed poits for a pair of self-mappigs. The result obtaied i the fuzzy metric space by usig the

3 Ramasamy Moha Raj ad Veerachamy Malliga Devi / Joural of Mathematics ad Statistics (): 65-7, 4 otio of o-compatible maps or the property (E.A) were very iterestig. Paper has proved commo fixed poit theorems for weakly compatible maps i fuzzy metric space by usig the cocept of (E.A) property, however, without assumig either the completeess of the space or cotiuity of the mappigs ivolved. Sigh et al. () aimed was to prove some commo fixed poit theorems i (GV)-fuzzy metric spaces. To prove the results, the research had employed the idea of compatibility. Wherei coditios o completeess of the uderlyig space (or subspaces) together with coditios o cotiuity i respect of ay oe of the ivolved maps were relaxed. Research results substatially geeralized ad improved a multitude of relevat commo fixed poit theorems of the existig literature i metric as well as fuzzy metric spaces which iclude some relevat results. Bugajewski ad Kasprzak (9) proved a collectio of ew fixed poit theorems forso-called weakly F- cotractive mappigs. By aalogy, paper has itroduced a class of strogly F-expasive mappigs ad proved fixed poit theorems for such mappigs. Paper has also provided a few examples, which illustrated these results ad, as a applicatio, paper proved a existece ad uiqueess theorem for the geeralized Fredholm itegral equatio of the secod kid. At last, paper has applied the Möch fixed poit theorem to prove two results o the existece of approximate fixed poits of some cotiuous mappigs. Mishra ad Choudhary () preseted some commo fixed poit theorems for occasioally weakly compatible mappigs ifuzzy metric spaces uder various coditios. Du () discussed several characterizatios of MTfuctios. Usig the characterizatios of MT-fuctios, paper established some existece theorems for coicidece poit ad fixed poit i complete metric spaces. Results showed the ew geeralizatios of Beride Beride s fixed poit theorem ad Mizoguchi Takahashi s fixed poit theorem for oliear multivalued cotractive maps. Maro et al. () i this study, we prove commo fixed poit theorems i fuzzy metric spaces. We also discuss result related to R-weakly commutig type mappigs. Chauha (9) proved a commo fixed poit theorem for two pairs of weakly compatible mappigs i M-fuzzy metric spaces... Problem Defiitio Lemma Grabiec (988): For all x, y X, M (x, y, ) is odecreasig. Sciece Publicatios 67 Lemma Mishra et al. (994) ad Cho (997): Let {y } be a sequece i a fuzzy metric space (X, M, *) with the coditio (FM-6). If there exists a umber k (, ) such that +, y +, kt) M (y +, y, t). Lemma 3 Mishra et al. (994): If for all x, y X, t > ad for a umber k (, ) M(x, y, kt) = M(x, y, t). Theorem 4. Let (X, M, ) be a fuzzy metric space with cotiuous t- orm defied by t t t for all t [, ]. Let A, B, S, T, P ad Q be mappigs from X ito itself such that: P(X) AB(X) ad Q(X) ST(X) There exists a costat k (, ) such that: M(px,Qy,Kt) where, ϕ: R + R + is a Lebesque-itegrable mappig which is summable, oegative ad such that: ϕ (t) dt > for each > M(ABy,Qy, t)m(stx, Px, t), M(STx,Qy, t), α = mi M(ABy,Px,( α )t), M(ABy,STx, t) For all x, y, (, ) ad t > ad: If oe of P(X), AB(X), ST(X) or Q(X) is a complete subspace of X, the P ad ST have a coicidece poit Q ad AB have a coicidece poit AB = BA, QB = BQ, QA = AQ, PT = TP, ST = TS The pair {P, ST} is weakly compatible, the A, B, S, T, P ad Q have a uique commo fixed poit i X Proof By (a), sice P(X) AB(X), for ay poit x X, there exists a poit x X such that Px = ABx. Sice Q(X) ST(X), for this poit x we ca choose a poit x

4 Ramasamy Moha Raj ad Veerachamy Malliga Devi / Joural of Mathematics ad Statistics (): 65-7, 4 X such that Qx = STx ad so o. Iductively, we ca defie a sequecei {y } i X such that for =,,,: Y = Px = ABx ad Y = Qx = STx By (b), for all t> ad α = -q, with q (, ), we have: +,kt) M(Qx,Px +,kt) ϕ (t)dt = + +,kt) mi +, + + Cosequetly, it follows that for =,,., p =, :,kt) mi,,t /k p ) By otig that +, y +, t/k p ) as p, we have for =,, : M(Px,Px +,kt) M(x +,x + =, + +,kt) mi +,x = mi M(ABx,Qx,M(STx,Px, M(STx +,Qx, αt), M(ABx Px,( α)t), M(ABx,STx +, t), y,, y, = mi, y +, α t),, y +,( + q)t),, y +, y,, mi, y, +,qt),, y,, mi, y,qt) Sice the t-orm * is cotiuous ad M (x, y,.) is left cotiuous, lettig q i (a),we have:, y, +, x mi, y, t).,kt) mi, 3 Hece by Lemma, {y } is a Cauchy sequece. Now suppose ST(X) is complete. Note that the subsequece {y } is cotaied i ST(X) ad has a limit i ST(X). Call it z. Let u ST - z. The STu = z. We shall use the fact that the subsequece {y } also coverges to z. By (b), we have: Take α = : M(Pu,kt) M(Pu,Qx,kt) m(u,x+ ϕ (t)dt = M(ABx,Qx,M(STu,Pu, m(u, x, t) = mi M(STu,Qx, t), M(ABx, Pu, t), M(ABx,STu, t), y,m(z,pu,m(z = mi,pu,,z which implies that, as : m(u,x + mi (z,pu,,m(z,pu,,m = = M(z,Pu Similarly, we also have: 3,kt) mi, 3 Sciece Publicatios I geeral, we have for =,, : 68 M(Pu,z,kt) M(Pu,z Therefore, by Lemma.3, we have Pu = z. Sice STu = z thus Pu = z = STu, i.e., u is a coicidece poit of P ad ST. This proves (i).

5 Ramasamy Moha Raj ad Veerachamy Malliga Devi / Joural of Mathematics ad Statistics (): 65-7, 4 Sice P(X) AB(X), Pu = z implies that z AB(X). Let v AB-z. The ABv = z. By (b), we have: Take α = :,Qv,kt) M(Px,Qv,kt),v ϕ (t)dt = M(ABx,Qx, t), M(STz, Pz, t), + + m(z,x + = mi M(STz,Qx +,M(ABx,Pz, M(ABx,STz, t) m(y,m(pz,pz, + = mi Mpz, y, t,,pz,m(y,pz,v = M(ABv,Qv, t), M(STx, t)m(stx,qv, t), mi M(ABv,Px,M(ABv,STx Sciece Publicatios M(z,Qv,, = mi,qv, t), M(z,M(z, y which implies that, as : M(z, Qv, t),, + = = M(z,Qv, t) v mi M(z,Qu,, M(z,Qv,kt) M(z,Qv Therefore, by Lemma.3, we have Qv = z. Sice ABv = z thus Qv = z = ABv, i.e., v is a coicidece poit of Q ad AB. This proves (ii). The remaiig two cases pertai essetially to the previous cases. Ideed if P(X) or Q(X) is complete, the by (a) z P(X) AB(X) or z Q(X) ST(X). Thus (i) ad (ii) are completely established. Sice the pair {P, ST} is weakly compatible therefore P ad ST commute at their coicidece poit, i.e., P(STu) = (ST)Pu or Pz = STz. By (d), we have Q(ABv) = (AB)Qv or Qz = ABz. Now, we prove that Pz = z, by (b), we have: Take α = : M(Pz,kt) M(Pz,Qx,kt) m(z,x 69 Proceedig limit as, we have:,,m(pz,z,m = = M(Pz,z m(z,x + mi (z,pz,m(z,pz m(pz,z,kt) M(Pz,z Therefore, by Lemma.3, we have Pz = z so Pz = STz = z. By (b), we have: Take α = : m(y,qz,kt) M(Px,Qz,kt),z ϕ (t)dt =, M(ABz,Qz,M(STx,Px, + +, z, t) = mi M(STx,Qz, t), m(abz, Px +, t), M(ABz,STx +, t) M(Qz,Qz,, y, + + = mi +,Qz, t), M(Qz, y,m(qz, y Proceedig limit as, we have:,,m(z,qz, = = M(z,Qz.,z mi M(Qz,z,M(Qz,z M(z,Qz,kt) M(z,Qz,

6 Ramasamy Moha Raj ad Veerachamy Malliga Devi / Joural of Mathematics ad Statistics (): 65-7, 4 Therefore, by Lemma 3, we have Qz = z so Qz = ABz = z. By (b) ad usig (d), we have: Take α = : Sciece Publicatios M(z,Bz,kt) m(z,bz M(Pz,QBz,kt) ϕ (t)dt = ϕ(t), M(AB(Bz),Q(Bz), t), M(STz, Pz, t), m(z, Bz, t) = mi M(STz,Q(Bz), t), M(ABz, Pz, t), M (AB(Bz),STz { } { } = mi M(Bz, Bz, t), M(z, z, t), M(z, Bz, t), = mi,, M(z, Bz, t),, M(Bz, z, t) = M(z, Bz, t) M(z,Bz,kt) M(z,Bz. Therefore, by Lemma 3, we have Bz = z. Sice ABz = z, therefore Az = z. Agai by (b) ad usig (d), we have: M(Tz,z,kt) Take α = : m(tz,z M(P(Tz),Qz,kt) ϕ (t)dt =,, M(ABz, Qz, t), M(ST(Tz), P(Tz), t), m(tz, z, t) = mi M(ST(Tz)Qz, t), M(ABz, p(tz), t), M(ABz,ST(Tz), t) M(Qz,Qz, t), M(Tz,Tz, t), = mi M(Tz, z, t), M(z, Tz, t), M(z, Tz, t) = mi,, M(Tz, z, t), M(z, Tz, t), M(z, Tz, t) { } = M(Tz, z, t). M(Tz,z,kt) M(Tz,z. Therefore, by Lemma 3, we have Tz = z. Sice STz = z, therefore Sz = z. By combiig the above results, we have: Az = Bz = Sz = Tz = Pz = Qz = z, That is z is a commo fixed poit of A, B, S, T, P ad Q. The uiqueess of the commo fixed poit of A, B, S, T, P ad Q follows easily from (b). This completes the proof. 7 If we put P = Q i Theorem 4., we have the followig result. Corollary 4.: Let (X, M, ) be a fuzzy metric space with cotiuous t- orm defied by t t t for all t [, ]. Let A, B, S, T ad P be mappigs from X ito itself such that: P(X) AB(X) ad P(X) ST(X) There exists a costat k (, ) such that M(Px,Py,kt). where, ϕ: R + R + is a Lebesque-itegrable mappig which is summable, oegative ad such that: ϕ (t)dt > for each > M(ABy,Py,M(STx,Px,M(STx, αt),, y = mi M(ABy,Px, α )t),m(aby,stx For all x, y X, α (, ) ad t > ad: If oe of P(X), AB(X) or ST(X) is a complete subspace of X, the P ad ST have a coicidece poit P ad AB have a coicidece poit AB = BA, PB = BP, PA = AP, PT = TP, ST = TS ad The pair {P, ST} is weakly compatible, The A, B, S, T ad P have a uique commo fixed poit i X. If we put B = T = Ix (the idetity mappig o X) i Theorem 4., we have the followig result: Corollary 4. Let (X, M, ) be a fuzzy metric space with cotiuous t-orm defied by t tt for all t [, ]. Let A, S, P ad Q be mappigs from X ito itself such that: P(X) A(X) ad Q(X) S(X) There exists a costat k (, ) such that: M(Px,Qy,kt) where, ϕ: R + R + is a Lebesque-itegrable mappig which is summable, oegative ad such that: ϕ (t) dt > for each >

7 Ramasamy Moha Raj ad Veerachamy Malliga Devi / Joural of Mathematics ad Statistics (): 65-7, 4 M(Ay,Qy, t), M(Sx, Px, t), M(Sx,Qy, αt), = mi M(Ay,Px, α )t),m(ay,sx For all x, y X, α (, ) ad t > ad: If oe of P(X), A(X), S(X) or Q(X) is a complete subspace of X the P ad S have a coicidece poit ad Q ad A have a coicidece poit. QA = AQ ad The pair {P, S} is weakly compatible, the A, S, P ad Q have a uique commo fixed poit i X If we put A = S i Corollary 4., we have the followig result: Corollary 4.3 Let (X, M, ) be a fuzzy metric space with cotiuous defied by t t t for all t [, ]. Let A, P ad Q be mappigs from X such that: P(X) A(X) ad Q(X) A(X) There exists a costat k (, ) such that: Sciece Publicatios M(Px,Qy,kt) where, ϕ: R + R + is a Lebesque-itegrable mappig which is summa oegative ad such that: ϕ (t)dt > foreach > M(Ay,Qy, t), M(Sx, Px, t), M(Sx,Qy, αt), = mi M(Ay,Px, α )t),m(ay,sx For all x, y X, α (, ) ad t > ad: If oe of P(X), Q(X) or A(X) is a complete subspace of X, the P ad A have a coicidece poit ad Q ad A have a coicidece poit QA = AQ ad The pair {P, A} is weakly compatible, the A, P ad Q have a uique commo fixed poit i X 7 I Theorem 4., if we replace the coditio QA = AQ by weak compatibility of the pair {Q, AB} the we have the followig theorem. Theorem 4. Let (X, M, ) be a fuzzy metric space with cotiuous t- orm defied by t t t for all t [, ]. Let A, B, S, T, P ad Q be mappigs from X ito itself such that: P(X) AB(X) ad Q(X) ST(X) There exists a costat k (, ) such that: M(Px,Qy,kt) where, ϕ: R + R + is a Lebesque-itegrable mappig which is summable, oegative ad such that: = ϕ (t)dt > for each >, M(ABy,Qy, t), M(STx, Px, t), M(STx,Qy, αt), mi M(ABy,Px, α )t),m(aby,stx For all x, y X, α (, ) ad t > ad: If oe of P(X), AB(X), ST(X) or Q(X) is a complete subspace of X, the P ad ST have a coicidece poit Q ad AB have a coicidece poit AB = BA, QB = BQ, PT = TP, ST = TS The pairs {P, ST} ad {Q, AB} are weakly compatible, The A, B, S, T, P ad Q have a uique commo fixed poit i X. By usig Theorem 4., we have the followig theorem Theorem 4.3 Let (X, M, ) be a fuzzy metric space with cotiuous t-orm defied by t t t for all t [, ]. Let A, B, S, T ad Pi, for i =,,,, be mappigs from X ito itself such that: P (X) AB(X) ad Pi(X) ST(X), for i N There exists a costat k (, ) such that: M(P x,p i y,kt)

8 Ramasamy Moha Raj ad Veerachamy Malliga Devi / Joural of Mathematics ad Statistics (): 65-7, 4 where, ϕ: R + R + is a Lebesque-itegrable mappig which is summable, oegative ad such that: = Sciece Publicatios ϕ (t)dt > for each >, M(ABy,Pi y,m(stx,p x,m(stx,pi y, αt), mi M(ABy,P x, α )t),m(aby,stx For all x, y X, α (, ) ad t > ad: If oe of P (X), AB(X) or ST(X) is a complete subspace of X or alteratively, P i, for i N, are complete subspace of X, the P ad ST have a coicidece poit For i N, P i ad AB have a coicidece poit AB = BA, P i B = BP i (i N ), PT = TP, ST = TS The pairs {P, ST} ad {P i (i N ), AB} are weakly compatible, the A, B, S, T ad P i, for i =,,, have a uique commo fixed poit i X. CONCLUSION Some coditios ivolve liear ad oliear expressios (ratioal, irratioal ad of geeral type). Recetly, some fixed poit results for mappigs satisfyig a itegral type cotractive coditio. Fixed poit theorems for various geeralizatios of cotractio mappigs i probabilistic ad fuzzy metric space were described. I our proposed method, we proved some commo fixed poit theorems for six mappigs by usig cotractive coditio of itegral type for class of weakly compatible maps i ocomplete ituitioistic fuzzy metric spaces, without takig ay cotiuous mappig of itegral type. The theorems ad corollary proved that this research proved the fixed poit theorems i terms of mappigs i fuzzy metric spaces. 3. REFERENCES Bugajewski, D. ad P. Kasprzak, 9. Fixed poit theorems for weakly F -cotractive ad strogly F- expasivemappigs. J. Math. Aal. Appli., 359: Chauha, S.S., 9. Commo fixed poit theorem for two pairs of weakly compatible mappigs i M- fuzzy metric spaces. It. J. Math. Aal., 3: Cho, Y.J., 997. Fixed poit i fuzzy metric spaces. J. Fuzzy Mathem., 5: Du, W.S.,. O coicidece poit ad fixed poit theorems for oliear multivalued maps. Topol. Appli., 59: George, A. ad P. Veeramai, 994. O some results i fuzzy metric spaces. Fuzzy Sets Syst., 64: DOI:.6/65-4(94)96-7 Grabiec, M., 988. Fixed poit i fuzzy metric spaces. Fuzzy Sets Syst., 7: Jugck, G. ad B.E. Rhoades, 8. Fixed poit theorems for occasioally weakly compatible mappigs. Erratum, Fixed Poit Theory, 9: Klim, D. ad D. Wardowski, 7. Fixed poit theorems for set-valued cotractios i complete metric spaces. J. Math. Aal. Appli., 334: DOI:.6/j.jmaa.6.. Maro, S., S.S. Bhatia ad S. Kumar,. Commo fixed poit theorems i fuzzy metric spaces. Aals Fuzzy Mathem. Iform., 3: Mishra, M.K., P. Sharma ad D.B. Ojha,. Fixed poits theorem i fuzzy metric space for weakly compatible maps satisfyig itegral type iequality. It. J. Applied Eg. Res., : Mishra, R.K. ad S. Choudhary,. O fixed poit theorems i fuzzy metricspaces. IMACST, : -. Mishra, S.N., N. Sharma ad S.L. Sigh, 994. Commo fixed poits of maps o fuzzy metric spaces. It. J. Math. Math. Sci., 7: Rhoades, B.E., 988. Cotractive defiitios ad cotiuity. Cotemporary Math, 7: Schweizer, B. ad A. Sklar, 96. Statistical metric spaces. Pacific J. Math., : Sigh, D., M. Sharma, R. Sharma ad N. Sigh,. Some commo fixed poit theorems i fuzzy metric spaces. Adv. Fixed Poit Theory, : 9-7. Zadeh, L.A., 965. Fuzzy sets. Iform. Cotrol, 89: