Aals of Pure ad Applied Mathematics Vol. 4, No., 07, 55-6 ISSN: 79-087X (P), 79-0888(olie) Published o 7 September 07 www.researchmathsci.org DOI: http://dx.doi.org/0.457/apam.v4a8 Aals of A Commo Fixed Poit Theorem Usig Compatible Mappigs of Type (A-) V.Nagaraju ad Bathii Raju Departmet of Mathematics, Uiversity College of Egieerig (Autoomous) Osmaia Uiversity, Hyderabad-500007 (Telagaa), Idia Departmet of Mathematics, Uiversity Post Graduate College, Secuderabad Osmaia Uiversity, Hyderabad-500003 (Telagaa), Idia Email: viswaag007@gmail.com, bathiiraju07@gmail.com Correspodig author Received 9 July 07; accepted 3 August 07 Abstract. I this paper we preset a commo fixed poit theorem i a metric space usig the weaker coditios such as compatible mappigs of type (A-) ad associated sequece which geeralizes the result of P.C.Lohai & V.H.Badshah. Keywords: Fixed poit, self maps, compatible mappigs of type (A-), associated sequece. AMS Mathematics Subject Classificatio (00): 54H5, 47H0. Itroductio I 976, Jugck proved some commo fixed poit theorems for commutig maps which geeralize the Baach cotractio priciple. Further these results were geeralized ad exteded i various ways by several authors. O the other had Sessa [5] itroduced the cocept of weak commutativity ad proved a commo fixed poit theorem for weakly commutig maps. I 986, Jugck [] itroduced the cocept of compatible maps which is more geeral tha that of weakly commutig maps. I 993, Jugck ad Cho [7] itroduced the cocept of compatible mappigs of type (A) by geeralizig the weakly uiformly cotractio maps. Afterwards Pathak ad Kha [0] itroduced the cocepts of A-compatibility ad S-compatibility by splittig the defiitio of compatible mappigs of type (A). I 007, Pathak et.al [8] reamed A- compatibility ad S-compatibility as compatible mappigs of type (A-) ad compatible mappigs of type (A-) respectively. The purpose of this paper is to prove a commo fixed poit theorem for four self maps i a metric space usig compatible mappigs of type (A-). 55
V.Nagaraju ad Bathii Raju. Defiitios ad prelimiaries Defiitio.. [] Two self maps S ad T of a metric space (X,d) are said to be compatible mappigs if lim d( STx, TSx ) = 0 wheever < x > is a sequece i X such that lim Sx = limtx = t for some t X. Defiitio.. [7] Two self maps S ad T of a metric space (X,d) are said to be compatible mappigs of type (A) if lim d( STx, TTx ) = 0 ad lim d( TSx, SSx ) = 0 wheever < > is a sequece i X such that lim Sx = limtx = t, for some t X. x Defiitio.3. [8] Two self maps S ad T of a metric space (X,d) are said to be compatible mappigs of type(a-) if lim d( TSx, SSx ) = 0 wheever { x } is a sequece i X such that lim Sx = limtx = t, for somet X. Defiitio.4. [9] Suppose P, Q, S ad T are self maps of a metric space ( X, d) such that S( X ) Q( X ) ad T ( X ) P( X ). The for ay arbitrary x0 Sx X,we have Sx0 S( X ) Q( X ) so that there is a x X such that = Qx ad for this x, there is a poit x X such that Tx = Px ad so o. 0 Repeatig this process to obtai a sequece + + + + < x > i X such that y = Sx = Qx ad y = Px = Tx for 0. We shall call this sequece < x > a associated sequece of x 0 relative to the four self maps P,Q,S ad T. Lemma.5. Let P, Q, S ad T be self mappigs from a metric space ( X, d) ito itself satisfyig S( X ) Q( X ) ad T ( X ) P( X ) (.5.) ad d( Qy, Ty)[ + d( Px, Sx)] d( Sx, Ty) α + βd( Px, Qy) [ + d( Px, Qy)] (.5.) for all x, y i X where α, β 0, α + β <. Further if X is complete, the for ay x0 sequece{x } relative to four self maps, the sequece X ad for ay of its associated 56
{ } A Commo Fixed Poit Theorem Usig Compatible Mappigs of Type (A-) y = { Sx0, Tx, Sx, Tx3,... Sx, Tx +,...} coverges to some poit i X. Proof: From (.4) ad (.5.), we have d( y, y ) = d( Sx, Tx ) + + d( Qx, Tx )[ + d( Px, Sx )] α [ + d( Px, Qy )] + + + + d( y, y )[ + d( y, y )] α [ + d( y, y )] + = +, + βd( Px, Qy ) βd( y, y ) = αd( y y ) + β d( y, y ) implies that + ( α ) d( y, y ) β d( y, y ) so that + β d( y, y ) d( y, y) = hd( y, y), where h + ( α) = β α. That is, d( y y ) h( y, y ). (.5.3), + Similarly, we ca prove that d( y+, y+ ) hd( y, y+ ). (.5.4) Hece, from (.5.3) ad (.5.4), we get d( y, y ) hd( y, y ) h d( y, y )... h d( y, y ). (.5.5) + 0 Now for ay positive iteger p, we have d( y, y ) d( y, y ) + d( y, y ) +... + d( y, y ) + p + + + + p + p h d( y, y ) + h d( y, y ) +... + h d( y, y ) + + p 0 0 0 = h + h + + h d y y + + p (... ) ( 0, ) = h + h + h + + h d y y p (... ) ( 0, ) h < d( y0, y) 0 as,sice h<. h Thus the sequece { y } is a Cauchy sequece i X. Sice X is a complete metric space, the sequece { y } coverges to some poit z i X. 57
V.Nagaraju ad Bathii Raju Remark.6. The coverse of the above Lemma is ot true. That is, if P,Q,S ad T are self maps of a metric space ( X, d) satisfyig (.5.), (.5.) ad eve if for ay x0 i X ad for ay of its associated sequece coverges, the the metric space ( X, d) eed ot be complete. Example.7. Let ( 0,] S,T,P ad Q o X by x if 0 < x Sx = Tx = if < x X = with d( x, y) = x y for x, y X. Defie the self maps ad 3x if 0 < x Px = Qx = 3x if < x The S( X ) = T( X ) =, while P( X ) = Q( X ) =,. Clearly S( X ) Q( X ) ad T ( X ) P( X ). It is also easy to see that the sequece Sx0, Tx, Sx, Tx3,... Sx, Tx +,... coverges to. Also the iequality (.5.) ca easily be verified for appropriate values of α, β 0, α + β <. Note that (X, d) is ot complete. Now we geeralize the result of P.C.Lohai ad V.H.Badshah [6] i the followig form. 3. Mai result We ow state our mai theorem as follows. Theorem 3.. Let P,Q,S ad T are self maps of a metric space ( X, d) satisfyig S( X ) Q( X ) ad T ( X ) P( X ) (3..) d( Qy, Ty)[ + d( Px, Sx)] d( Sx, Ty) α + βd( Px, Qy) [ + d( Px, Qy)] (3..) for all x,y i X whereα,β 0,α+β<. P ad Q are cotiuous ad (3..3) the pairs (S,P) ad (T,Q) are compatible mappigs of type (A-) o X. (3..4) Further if there is poit x0 X ad a associated sequece { x } of x0 relative to four self maps P, Q, S ad T such that the sequece Sx, Tx, Sx, Tx... Sx, Tx +... coverges to some poit z X, (3..5) 0 3 58
A Commo Fixed Poit Theorem Usig Compatible Mappigs of Type (A-) the z is a uique commo fixed poit of S,P,Q ad T. Proof: By (3..5), we have Sx z, Qx z, Tx z ad Px z as + + + (3..6) Suppose the pair (S,P) is compatible mappigs of type(a-). The we have lim PSx = lim SSx (3..7) Sice P is cotiuous, PPx, PSx Pz as (3..8) Now from (3..7) ad (3..8), we get SSx Pz as. (3..9) Suppose the pair (T,Q) is compatible mappigs of type(a-). The we have lim QTx+ = limttx+. (3..0) Sice Q is cotiuous, QQx, QTx+ Qz as. (3..) Now from (3..0) ad (3..), we get TTx+ Qz as. (3..) We shall ow prove that Pz = Qz = Sz = Tz = z. To prove Pz = Qz, put x = Sx ad y = Tx + i (3..), we get d( QTx, TTx )[ + d( PSx, SSx )] d( SSx, TTx ) α + βd( PSx, QTx ). + + + + [ + d( PSx, QTx+ )] Lettig ad usig (3..8), (3..9), (3..0), (3..) ad (3..) i the above iequality, we get d( Qz, Qz)[ + d( Pz, Pz)] d( Pz, Qz) α + βd( Pz, Qz) [ + d( Pz, Qz)] = β d( Pz, Qz) so that ( β ) d( Pz, Qz) 0. Sice β 0, α + β <,we have d( Pz, Qz ) = 0 which implies Pz = Qz. To prove Sz = Qz, put x = z ad y = Tx + i (3..), we get d( QTx+, TTx+ )[ + d( Pz, Sz)] d( Sz, TTx+ ) α + βd( Pz, QTx+ ). [ + d( Pz, QTx )] + Lettig ad usig (3..) ad (3..) i the above iequality, we get d( Qz, Qz)[ + d( Pz, Sz)] d( Sz, Qz) α + β d( Pz, Qz) [ + d( Pz, Qz)] which implies d( Sz, Qz) 0, sice Pz = Qz. Hece d( Sz, Qz ) = 0 which implies Sz = Qz. 59
Therefore Pz = Sz = Qz. V.Nagaraju ad Bathii Raju To prove Sz = Tz, put x = z ad y = z i (3..), we get d( Qz, Tz)[ + d( Pz, Sz)] d( Sz, Tz) α + β d( Pz, Qz) [ + d( Pz, Qz)] d( Sz, Tz)[ + d( Sz, Sz)] = α + β d( Pz, Pz), sice Pz = Qz = Sz. [ + d( Pz, Pz)] d( Sz, Tz) αd( Sz, Tz) so that ( α) d( Sz, Tz) 0. Siceα 0, α + β <, we have d( Sz, Tz ) = 0 which implies Sz = Tz. Therefore Sz = Pz = Qz = Tz. Fially to prove Tz = z, put x = x ad y = z i (3..), we get d( Qz, Tz)[ + d( Px, Sx )] d( Sx, Tz) α + β d( Px, Qz) [ + d( Px, Qx )] Lettig ad usig (3..6) i the above iequality, we get d( Tz, Tz)[ + d( z, z)] d( z, Tz) α + β d( z, Tz) [ + d( z, z)] = β d( z, Tz) so that ( β ) d( z, Tz) 0. Siceα 0, α + β <, we have d( z, Tz ) = 0 which impliestz = z. Therefore Sz = Pz = Qz = Tz = z, showig that z is a commo fixed poit of P,Q,S ad T. Uiqueess: Let z ad w be two commo fixed poits of P,Q,S ad T. The we have z = Sz = Pz = Qz = Tz ad w = Sw = Pw = Qw = Tw. Put x = z ad y = w i (3..), we get d( w, w)[ + d( z, z)] d( z, w) α + βd( z, w) [ + d( z, w)] = β d( z, w) < d(z,w), a cotradictio. Thus we have d( z, w ) = 0which implies that z = w. Hece z is a uique commo fixed poit of S,P,Q ad T. 60
A Commo Fixed Poit Theorem Usig Compatible Mappigs of Type (A-) Remark 3.. From the example (.7), clearly the pairs (S,P) ad (Q,T) are compatible mappigs of type(a-) ad P,Q are cotiuous. Also, if we take x = + for, the the sequece Sx0, Tx, Sx, Tx3... Sx, Tx +... coverges to X.Moreover, the ratioal iequality holds for the values of α, β 0, α + β <.It may be oted that is the uique commo fixed poit of P, Q, S ad T. REFERENCES. G.Jugck, Compatible mappigs ad commo fixed poits, Iterat. J. Math. Math. Sci., 9 (986) 77-778.. R.P.Pat, A Commo fixed poit theorem uder a ew coditio, Idia J. of Pure ad App. Math., 30() (999) 47-5. 3. Jugck, Compatible mappigs ad commo fixed poits (), Iterat. J. Math. Ad Math. Sci., (988) 85-88. 4. G.Jugck ad B.E. Rhoades, Fixed poit for set valued fuctios without cotiuity, Idia J. Pure Appl. Math, 9(3) (998), 7-38. 5. S.Sessa, O a weak commutativity coditio i a fixed poit cosideratios, Publ. Ist Math. (Beograd), 3(46)(98) 49-53. 6. P.C.Lohai ad V.H.Badshah, Compatible mappigs ad commo fixed poit for four mappigs, Bull. Cal. Math. Soc., 90(998) 30-308. 7. G.Jugck, P.P.Murthy ad Y.J.Cho, Compatible mappigs of type (A) ad commo fixed poits, Math. Japa, 38 (993) 38-39. 8. M.S.Kha, H.K.Pathak ad George Rey, Compatible mappigs of type (A-) ad type(a-) ad commo fixed poits i fuzzy metric spaces, Iteratioal Math. Forum, ()(007) 55-54. 9. V.Sriivas ad V.Nagaraju, Commo fixed poit theorem o compatible mappigs of type (P), Ge. Math. Notes, () (04) 87-94. 0. H.K.Pathak ad M.S.Kha, A compariso of various types of compatible maps ad commo fixed poits, Idia J. Pure Appl. Math, 8(4) (997) 477-485.. V.Sriivas ad S.Ravi, Geeratio of a commo fixed poit theorem usig A- compatible ad B-compatible mappigs of type(e), Global Joural of Pure ad Applied Mathematics, 3(6) (07) 735-744.. K.Jha, R.P.Pat ad K.B.Maadhar, A commo fixed poit theorem for reciprocally cotiuous compatible mappigs i metric space, Aals of Pure ad Applied Mathematics, 5() (04) 0-4. 6
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