A Common Fixed Point Theorem Using Compatible Mappings of Type (A-1)

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1 Aals of Pure ad Applied Mathematics Vol. 4, No., 07, 55-6 ISSN: X (P), (olie) Published o 7 September 07 DOI: 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 (Telagaa), Idia Departmet of Mathematics, Uiversity Post Graduate College, Secuderabad Osmaia Uiversity, Hyderabad (Telagaa), Idia 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

2 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

3 { } 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 = 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

4 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)

5 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

6 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

7 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) R.P.Pat, A Commo fixed poit theorem uder a ew coditio, Idia J. of Pure ad App. Math., 30() (999) Jugck, Compatible mappigs ad commo fixed poits (), Iterat. J. Math. Ad Math. Sci., (988) G.Jugck ad B.E. Rhoades, Fixed poit for set valued fuctios without cotiuity, Idia J. Pure Appl. Math, 9(3) (998), S.Sessa, O a weak commutativity coditio i a fixed poit cosideratios, Publ. Ist Math. (Beograd), 3(46)(98) P.C.Lohai ad V.H.Badshah, Compatible mappigs ad commo fixed poit for four mappigs, Bull. Cal. Math. Soc., 90(998) G.Jugck, P.P.Murthy ad Y.J.Cho, Compatible mappigs of type (A) ad commo fixed poits, Math. Japa, 38 (993) 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) V.Sriivas ad V.Nagaraju, Commo fixed poit theorem o compatible mappigs of type (P), Ge. Math. Notes, () (04) 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) 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) 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)

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