Iteatioal Joual of Matheatics Teds ad Techology (IJMTT) Volue 47 Nube July 07 Coe Metic Saces, Coe Rectagula Metic Saces ad Coo Fixed Poit Theoes M. Sivastava; S.C. Ghosh Deatet of Matheatics, D.A.V. College Kau, U.P. ABSTRACT: A coo fixed oit theoe fo a ai of weakly coatible aigs is oved i a coe etic ace. We also oved a fixed it theoe i coe ectagula etic sace by usig atioal tye cotactive coditio. Itoductio: The study of coo fixed it of aig satisfyig cetai cotactive coditios has bee take a iotat ole i eset eseach activity. I 976, Jugek [5] oved a coo fixed oit theoe fo coutig aigs, geealized the faous Baach cotactio icile. Sessa [0] itoduced the otio of weakly coutig aigs. Also, Jugek [6] itoduced the otio of coatible aigs i ode to geealize the cocet of weak coutativity. Agai, Pat [9] Defied R-Weakly coutig as ad established soe coo fixed it theoe, assuig the cotiuity of at least oe of the aigs. Jugek ad Rhoades [7] defied a ai of self aigs to be weakly coatible if they coute at thei coicidece its. The alyig these cocets, seveal authos have obtaied coicidece oit esults fo vaious classes of aigs i a etic sace. O the othe had Huag ad Zhag [4] have itoduced the cocet of coe etic sace, whee the set of eal ubes is elaced by a odeed Baach sace, ad they have established soe fixed oit theoes fo cotactive tye aigs i a oal coe etic sace. O the othe had Huag ad Zhag [4] have itoduced the cocet of coe etic saces, whee the set of eal ubes is elaced by a odeed Baach sace, ad they have established soe fixed oit theoes fo cotactive tye aigs i colete coe etic saces. The study of fixed oit theoes i such saces is followed by soe othe atheaticias. Followig the idea of Baciai [3], Aga, Ashad ad Beg [] exteded the otio of coe etic saces by elacig the tiagula iequality by a ectagula iequality. The ai of this ae is to establish a coo fixed oit theoe fo a ai of weakly coatible aigs i a coe etic sace without exloitig the otio of the cotiuity ad we also oved a fixed oit theoe i coe ectagula etic sace by usig atioal tye cotactio aig. Let E be a eal Baach sace. A subset P of E is called a coe if, ad oly if (a) P is called, o ety ad {0} (b) a,br ad a,b 0 ad x,yp ilies ax + byp (c) P( P) {0} Give a coe PE, we defie a atial odeig with esect to P by x y x R, A coe is oal if thee is a ube K 0 such that fo all x, y E 0 x y ilies. x K y y if ad oly if, the iequality x The least ositive ube satisfyig the above iequality is called the oal costat of P, while y stads fo yx it P (iteio of P ). ISSN: 3-5373 htt://www.ijttjoual.og Page 5
Iteatioal Joual of Matheatics Teds ad Techology (IJMTT) Volue 47 Nube July 07 DEFINITION.. Let X be a o ety. Suose that the aig d : X X E satisfies. 0 d( x, y) fo all x, y X ad d( x, y) 0 if ad oly if x y. d( x, y) d( y, x) fo all x, y X. 3. d( x, y) d( x, z) d( z, y) fo all x, yz X.. The d is called a coe etic o X ad ( Xd, ) is called a coe etic sace. The cocet of a coe etic sace is oe geeal tha that of a etic sace. DEFINITION.. Let ( Xd, ) be a coe etic sace we say that { x } is a Cauchy sequece if fo eegy C i E with c 0, thee is N such that fo all N, d( x, x) C fo soe fixed x i X. A coe etic sace X is said to be colete if eegy Cauchy sequece i X is coveget i X. It is kow that { x } coveges to x X if, ad oly if d( x, x) 0 as. Also the liit of a coveget sequece is uique ovided P is a oal coe with oal costatk. DEFINITION.3. Let f ad g be self aigs of a set X. If w fx gx fo soe x i X. the x is called a coicidece oit f ad g, ad w is called a oit of coicidece of f ad g. DEFINITION.4. Two self aigs f ad g of a set X ae said to be weakly coatible if they coute at thei coicidece oits, that is if fu gu fo soe u X, the fgu gfu. DEFINITION.. Let X be a o ety set, suose the aig d : X X E satisfies sace.. 0 d( x, y) fo all x, y X, x y ad d( x, y) 0 if ad oly if x y. d( x, y) d( y, x) fo all x, y X.. 3. d( x, y) d( x, u) d( u, v) d( v, y) fo all x, y X ad fo all distict oits geeal. u, v X /{ x, y} (ectagula oety) The d is called a coe ectagula etic o X, ad ( Xd, ) is called a coe ectagula etic Note that ay coe etic sace is a coe ectagula etic sace but the covese is ot tue is ISSN: 3-5373 htt://www.ijttjoual.og Page 6
Iteatioal Joual of Matheatics Teds ad Techology (IJMTT) Volue 47 Nube July 07 DEFINITION.. Let ( Xd, ) be a coe ectagula etic sace let { x } be a sequece i X ad x X fo evey C E, c 0 thee is N such that fo all N, d( x, x) C, the { } coveget to x ad x is the liit of { x }. We deote this by x.. If x is said to be LEMMA.3. Let ( Xd, ) be a coe ectagula etic sace, P be a oal coe. Let { x } be a sequece i X. The x x as d( x, x) 0 as. Note that if ( Xd, ) is a coe etic sace ad { x } is coveget sequece i X. The the liit of { } uique. x is DEFINITION.4. Let ( xd, ) be a coe ectagula etic sace, { x } be a sequece i X. If fo ay C E with 0 C, thee is N such that fo all, N, ( x, x ) C, the { } sequece i X. x is called a Cauchy DEFINITION.5. Let ( Xd, ) be a coe ectagula etic sace ad P be a oal coe. Let { x } be a sequece i X. The { x } is a Cauchy sequece of ad oly if d( x, x) 0 as,. DEFINITION.6. Let ( Xd, ) be a coe ectagula etic sace. If evey Cauchy sequece is coveget i X, the X is called a colete coe ectagula etic sace. THEOREM :- Let ( Xd, ) be a coe etic sace ad P be a oal coe with oal costat k. Suose that the aigs f, g : X X satisfy the cotactive coditio d( fx, fy) [ d( gx, gy) d( fx, fy) d( fx, gy) d( fy, gx) d( fy, gy) d( fx, gx)] [0, 6) is a costat. If the age of g cotais the age of f ad gx ( ) is colete subsace of X. The f ad g have a uique coicidece oit i X. Moeove if f ad g ae weakly coatible Whee the f ad g have a uique coo fixed oit. x be a abitay oit i X. The sice ( ) ( ) Poof. Let 0 f X g X, we choose a oit f ( x0) g( x) cotiuig this ocess, havig chose x i X we obtai f ( x) g( x ). The x i X such that x i X such that ISSN: 3-5373 htt://www.ijttjoual.og Page 7
Iteatioal Joual of Matheatics Teds ad Techology (IJMTT) Volue 47 Nube July 07 d( gx, gx ) d( fx, fx ) [ d( gx, gx ) d( fx, fx ) d( fx, gx ) d( fx, gx ) d( fx, gx ) d( fx, gx )] [ d( gx, gx ) d( gx, gx ) d( gx, gx ) d( gx, gx )] [3 d( gx, gx ) 3 d( gx, gx )] 3 d( gx, gx ) 3 d( gx, gx ) hd( gx, gx ) Now fo 3 h 3 we get d( gx, gx ) d( gx, gx ) d( gx, gx ) d( gx, gx ) ( h h h ) d( gx, gx) h d( gx, gx0) h Usig the oality of coe P ilies that d( gx, gx ) 0 as, h d( gx, gx) K d( gx, gx0) h { gx } is a Cauchy sequece i X sice gx ( ) is colete subsace of X so thee exists q i gx ( ) such that ISSN: 3-5373 htt://www.ijttjoual.og Page 8
Iteatioal Joual of Matheatics Teds ad Techology (IJMTT) Volue 47 Nube July 07 g( ) q d( gx, f ) [ d( gx, g ) d( gx, f ) d( gx, f ) d( f, g ) d( fx, g ) d( fx, gx )] [ d( gx, g) d( gx, f ) d( gx, g ) d( gx, g )] Usig oality of coe 3 [ d ( gx, g )] 3 d( gx, f ) K d( gx, g 0 d( gx, f ) 0 as as. The uiqueess of a liit i a coe etic sace ilies that f ( O g( ) Agai we show f ad g have a uique oit of coicidece, if ossible assue that thee exists aothe oit t i X such that f ( t) g( t) d( gt, g) d( ft, f) [ d( gt, g) d( ft, f) d( ft, g) d( f, gt) d( ft, gt) d( f, g)] [ d( gt, g) d( ft, f) d( g, gt) d( g, gt)] d( gt, g) [ d( g, f)] 0 This show f ad g both have sae fixed oit. THEOREM. Let ( Xd, ) be a colete coe ectagula etic sace. P be a oal coe with oal costat K. Suose a aig f : X X satisfyig cotactive coditio d( x, fx) d( y, fy) d( fx, fy) d( x, fx) d( y, fy) d( x, y) xy X, d( x, y) ISSN: 3-5373 htt://www.ijttjoual.og Page 9
Iteatioal Joual of Matheatics Teds ad Techology (IJMTT) Volue 47 Nube July 07 whee [0, ) 5. The, (i) f has a uique fixed oit i X (ii) Fo ay x X Now fo x X we have d Tx T x (, ) d( Tx, TTx) d Tx T x, the iteative sequece T ( x) coveges to the fixed oit. d( Tx, T x) d( x, Tx) (, ) d( x, Tx) d T x T x d( x, Tx) d( Tx, T x) d( x, Tx) d( Tx, T x) d( x, Tx) d( x, Tx) 3 (, ) d( TTx, TT x) d T x T x 3 (, ) d( Tx, T x) d( x, Tx) 3 d( Tx, Tx ) d( Tx, Tx ) 3 d( Tx, T x) dd( T x, T x) d( Tx, T x) d( Tx, Tx ) 3 3 d( T x, T x) d( Tx, T x) d( T x, T x) d( Tx, T x) Thus i geeal, if is a ositive itege, the ( T x, T x) d( x, Tx) whee k [0, 5) We divide the oof ito two case.. k d( x, Tx) ISSN: 3-5373 htt://www.ijttjoual.og Page 0
Iteatioal Joual of Matheatics Teds ad Techology (IJMTT) Volue 47 Nube July 07 FIRST CASE: Let i.e. T x T y y whee, T x fo soe,, y N. Let. The T ( T x) T x T x. Now sice, we have, d y Ty d T y T y (, ) (, ) Sice [0,) d( y, Ty). we obtai d( y, Ty) P ad d( y, Ty) P which ilies that d( y, Ty) 0, i.e., Ty y. SECOND CASE: Assue that T x T x fo all, N,. Clealy, we have Ad (, ) (, ) (, ) d T x T x d x Tx d x Tx d T x t x d T x T x d T x T x (, ) ( (, ) (, )) ( d) x, Tx) d( x, Tx)) d( x, Tx) d( x, Tx) Now if, d( x, Tx). is odd the witig, ad usig the fact that T x T x fo, N, we ca easily show that d T x T x d T x T x d T x T x d T x T x (, ) ( (, ) (, ) (, ) d x Tx d x Tx d x Tx (, ) (, ) (, ) Agai if d( x, Tx). is eve the witig ad usig the sae aguets as befoe, we ca get ISSN: 3-5373 htt://www.ijttjoual.og Page
Iteatioal Joual of Matheatics Teds ad Techology (IJMTT) Volue 47 Nube July 07 d T x T x d T x T x d T x T x d T x T x d T x T x 3 3 4 (, ) (, ) (, ) (, ) (, ) d x Tx d x Tx d x Tx d x Tx 3 (, ) (, ) (, ) (, ) d( x, Tx). Thus cobiig all the cases we have Hece, we get d( T xt x) d( x, Tx),, N. of X, thee is * Tx fo ay d( T x, T x) d( x, Tx),, N. Sice k d( x, Tx) 0 as,( T x) is a Cauchy sequece. By the coleteess * x X such that We shall ow show that N. We have T x d x Tx d x T x d T x Tx * x as. * * (, ) (, ) (, ) Tx x. Without ay loss of geeality, we ca assue that T x x, d x T x d T x T d T x T x d x Tx * (, ) (, ) ( (, ) (, )) This ilies that Hece, d x Tx d x T x d T x T x * (, ) ( (, ) ( ) (, )).. k * d( x, Tx ) ( d( x, T x) ( ) d( T x, T x) ) 0 as So we obtai d( Tx, x ) 0, i.e., x Tx. ISSN: 3-5373 htt://www.ijttjoual.og Page
Iteatioal Joual of Matheatics Teds ad Techology (IJMTT) Volue 47 Nube July 07 Now, if * y is aothe fixed oit of T, the d x y d Tx Ty d x Tx d y Ty (, ) (, ) ( (, ) ( )) 0 which ilies that d( x, y ) 0 i.e., x y. REFERENCES: [] Abbas M. ad Jugck G., Coo fixed oit esults fo ocoutig aigs without cotiuity i coe etic saces, Joual of Matheatical Aalysis ad Alicatios, 34 (008), 46 40. [] Aza A., Ashad M. ad Beg I., Baach cotactio icile o coe ectagula etic saces, Al. Aal. Discete Math., (to ae),,. [3] Baciai A., A fixed oit theoe of Baach-Caccioli tye o a class of geealized etic saces, Publ. Math. Debece, 57 (000), 3 37. [4] Huag L.G. ad Zhag X., Coe etic saces ad fixed oit theoes of cotactive aigs, Joual of Matheatical Aalysis ad Alicaios, 33(), (007), 468 476. [5] Jugck G., Coutig aigs ad fixed oits, Ae. Math. Mothly., 73 (976), 6 63. [6] Jugck G., Coatibleaigs ad coo fixed oits, Iteatioal Joual of Matheatics ad Matheatical Seieces, 9 (986), 77 779. [7] Jugck G. ad Rhoades B.E., Fixed Poits fo set valued fuctios without cotiuity, Idia J. Pue Al. Math., 9 (998), 7 38. [8] Pat R.P., Coo fixed oits of o-coutatig aigs, J. Math. Aal. Al. 88 (994), 436-440. [9] Pat R.P., Pat V. ad Jha K., A ote o coo fixed oit ude stict cotactive coditios, J. Math Aal. Al. 7 (00), 879-880. [0] Sessa S., O a weak coutativity coditio of aigs i fixed oit cosideatios, Publ. Ist. Math. (Beogad)(N.S.), 3 (98), 49 53. [] Kaa R., Soe esults o fixed oits, Bull. Cal. Math. Soc. 60 (968), 7-76. ISSN: 3-5373 htt://www.ijttjoual.og Page 3