BULETINUL Uiversităţii Petrol Gaze di Ploieşti Vol LXII No /00 60-65 Seria Mateatică - Iforatică - Fizică Coo Fixed Poits for Multifuctios Satisfyig a Polyoial Iequality Alexadru Petcu Uiversitatea Petrol-Gaze di Ploiești, Bd București 39, Ploiești e-ail: alexadrupetcu005@gailco Abstract Coo fixed poit theores i coplete etric spaces (X, d) are give for two or ore ultifuctios which satisfy polyoial iequalities usig oly the distace d, without usig the Hausdorff etric Keywords: ultifuctios, coo fixed poit Itroductio The paper of B Fisher [] cotais the followig result: Theore A Let ( X, d ) be a coplete etric space ad ST, : X X dsxty (, ) (, ) (, ) c d x Sx + d y Ty d ( x, Sx ) + d ( y, Ty ) two appigs such that () for all x, y fro X which verify the coditio d( x, Sx) + d( y, Ty) 0, where 0< c < The S ad T have a coo fixed poit that is there exists u X such that u = Su ad u = Tu Reark The appigs S ad T which verify theore A have the property: (a) Proof Let us assue that for ay coverget sequece ( x) 0 fro X with li x = x, x+ = Sx, x = Tx it results x = Sx, x = Tx d( x, Tx) 0 Based o coditio () we have d ( x, Sx) + d ( x, Tx) d ( x, x+ ) + d ( x, Tx) d( x+, Tx) = d( Sx, Tx) c = c, d( x, Sx ) + d( x, Tx) d( x, x ) + d( x, Tx) + fro where, for, it results d( x, Tx) cd( x, Tx) which is a cotradictio sice 0 < c < Therefore d( x, Tx ) = 0, that is x = Tx Aalogously we prove that x = Sx I the followig we will cosider appigs, called ultifuctios, defied o the etric space ( X, d ) with values i PX ( ), that is i the faily of oepty subsets of X
Coo Fixed Poits for Multifuctios Satisfyig a Polyoial Iequality 6 I [4] V Popa has proved coo fixed poit theores for ultifuctios which verify ratioal iequalities, which cotai the Hausdorff etric i their expressios ad which geeralize theore A I the preset paper we will preset other coo fixed poit theores for two or ore ultifuctios without usig the Hausdorff etric ad which geeralize ot oly theore A but also the theores obtaied by V Popa [4] for the case p = Coo Fixed Poit Theores for Multifuctios Fixed poit of the ultifuctio T : ( X, d) P( X) is ay eleet u X with the property u Tu We ote F( T ) the set of fixed poits of the ultifuctio T Lea Let ( X, d ) be a etric space ad ST, : X PX ( ) two ultifuctios such that ( )x X, ( )y Sx (or y Tx) there exists z Ty (respectively z Sy), the followig iequality occurs: ( cd ) ( yz, ) + d ( xyd, ) ( yz, ) cd ( xy, ) 0 where, 0 < c < ad FS ( ) φ The FT ( ) φ ad F( S) = F( T) Proof Let u F( S), that is u Su, it results that there exists z Tu ad () becoes ( cd ) ( uz, ) + d ( uud, ) ( uz, ) cd ( uu, ) 0 fro where we get ( cd ) ( uz, ) 0, that is duz (, ) = 0 It results z= u ad therefore u Tu which iplies F( S) F( T) Aalogously we prove that F( T) F( S) Therefore F( S) = F( T) Let V : X P( X) with ( X, d ) a etric space The followig property will be used further: (b) for ay coverget sequece ( x) 0 fro X with li x= x, x Vx (or x Vx ) it results x Vx Theore Let ( X, d ) be a coplete etric space ad ST, : X PX ( ) two ultifuctios such that ( )x X, ( ) y Sx (or y Tx) there exists z Ty (respectively z Sy) iequality () occurs, where, 0< c < If oe of the ultifuctios S, T verifies coditio (b) the S ad T have coo fixed poits ad F( S) = F( T) Proof Let x0 The there exists x3 X arbitrary fixed ad x Sx0 The there exists x Tx such that ( cd ) ( x, x) + d ( x, x) d ( x, x) cd ( x, x) 0 0 0 Sx such that ( c) d ( x, x ) + d ( x, x ) d ( x, x ) cd ( x, x ) 0 3 3 Cotiuig this reasoig we obtai a sequece x 0, x, x, x 3,, x, x, x with Sx, x Tx ad which verifies the iequality ( c) d ( x, x ) + d ( x, x ) d ( x, x ) cd ( x, x ) 0, ( ) (3) + + ()
6 Alexadru Petcu The first eber fro iequality (3) is a secod degree trioial i the variable with the discriiat d ( x, x + ) Δ= d ( x, x ) + 4( c) cd ( x, x ) = ( + 4c 4 c ) d ( x, x ) > 0 Iequality (3) occurs if d ( x, x + ) is betwee the roots of the trioial, that is 4c 4c 4c 4c ( c) ( c) d ( x, x) d ( x, x+ ) d ( x, x) We ote + + 4c 4c k = ( c) A siple calculatio shows that k < ad sice d( x, x + ) 0 it results 0 d ( x, x ) k d ( x, x ), + that is d( x, x ) kd( x, x ), ( ), fro where we deduce + d( x, x+ ) k d( x0, x ), ( ) A routie calculatio leads to k d( x, x+ p) d( x0, x),, p N, k which shows that ( x) 0 it is a Cauchy sequece ad sice the space X is coplete it results that ( x ) is coverget Let u = li x, u X 0 We have x Sx ad assuig that S verifies (b) it results that u With lea we deduce that u Tu ad F( S) = F( T) Su Corollary Theore geeralizes theore A of B Fisher [] Proof We assue that the coditios of theore A are true Eliiatig the deoiator, () becoes d( Sx, Ty) d( x, Tx) + d( Sx, Ty) d( y, Ty) cd ( x, Sx) c d ( y, Ty) 0 (4) We observe that (4) occurs for ay x, y X, eve if d( x, Sx) + d( y, Ty) = 0 We cosider y = Sx ad ote z = Ty Iequality (4) becoes ( cd ) ( yz, ) + d( xyd, ) ( yz, ) cd ( xy, ) 0, which is the sae with () for = ad ST, : X X Based o this reark occurs (a) which covers (b) i this particular case Theore A is this way prove I the particular case S = T : X P( X) fro theore it results Theore Let ( X, d ) be a coplete etric space ad T : X P( X) that ( )x X, ( )y Tx, there exists z Ty with the property a ultifuctio such
Coo Fixed Poits for Multifuctios Satisfyig a Polyoial Iequality 63 ( cd ) ( yz, ) + d ( xyd, ) ( yz, ) cd ( xy, ) 0, where, 0 < c < The T has a fixed poit Theore 3 Let ( X, d) be a coplete etric space ad ( T) a sequece of ultifuctios T : X P( X) such that for ay occurs the property ( ) x X, ( )y Tx (or y T x), there exists z T y (respectively z T y) which verify the coditio ( cd ) ( yz, ) + d ( xyd, ) ( yz, ) cd ( xy, ) 0, where, 0 < c < If oe of the ultifuctios T verifies (b) the the sequece ( T) has coo fixed poits ad F( T ) = F( T ), ( ) Theore 3 results fro theore ad lea Other Cosequeces of Theore I this sectio we will deduce theores, 3 ad 5 fro [4] (V Popa) for the particular case p =, like a cosequece of theore fro this paper Let ( X, d ) be a etric space, P ( X ) the faily of oepty subsets, closed ad bouded fro X ad the Hausdorff-Popeiu etric with { } H( A, B) = ax sup d( a, B),sup d( b, A) a A b B dab (, ) = if dab (, ), where AB, P ( X) We also ote b B { } δ ( AB, ) = sup dab (, ): a Ab, B Particularizig the well kow result (lea (V) [5]) which says that if AB, P ( X) ad k R, k >, the for ay a A there exists b B such that da (, b) kh ( AB, ), we obtai Lea Let k > ad the ultifuctios ST, : X P ( X) The for ay x X ad ay y Sx (or y Tx) there exists z Ty (respectively z Sy) such that d( y, z) kh( Sx, Ty) Theore 4 (Theore [4]) Let ( X, d ) be a coplete etric space ad two ultifuctios such that T, T : X P ( X) d ( x, Tx) + d ( y, T y) H ( Tx, Ty) c δ (, ) ( yt, y) x Tx (5) for ay x, y fro X for which δ ( xtx, ) ( yty, ) 0, (6) where, 0 < c < The T ad have coo fixed poits ad Proof Eliiatig the deoiator, (5) becoes T F T F T ( ) = ( ) ( δ δ ) ( ) H ( TxT, y) ( xtx, ) + ( yt, y) c d ( xtx, ) + d ( yt, y) (7) which occurs eve if coditio (6) is ot satisfied
64 Alexadru Petcu Iequality (7) is valid for ay x, y fro X ad i particular for y Tx Let k c < < For x X, y Tx with lea it results that there exists z T y such that ad fro here we have d( y, z) kh( Tx, T y) ( δ ) ( δ ) d ( y, z) ( xtx, ) ( yt, y) k H ( TxT, y) ( xtx, ) ( yt, y) ad ow with (7) we obtai or eve ore ( δ ) ( + ) d ( yz, ) ( xtx, ) ( yty, ) ck d ( xtx, ) d ( yty, ) ( ) ( ) d ( yz, ) d ( xy, ) + d ( yz, ) ck d ( xy, ) + d ( yz, ), fro where it results that ( )x X, ( )y Tx, there exists z T y such that ( ck ) d ( y, z) d ( x, y) d ( y, z) ck d ( x, y) 0 +, where, 0 < ck <, coditio which has the for of iequality () We prove ow that T verifies coditio (b) Let ( x) 0 be a coverget sequece fro X with li x = x X ad We have fro where with (7) we obtai or ore x Tx, x Tx dtxx (, ) HTxTx (, ) ( δ ) ( + ) d ( Txx, ) ( xtx, ) ( x, Tx ) c d ( xtx, ) d ( x, Tx ) ( + ) ( + ) d ( Tx, x ) d ( x, Tx) d ( x, x ) c d ( x, Tx) d ( x, x ) fro where, for, it results dtxx (, ) cdxtx (, ), that is dtxx (, ) = 0 Because Tx it is a closed set we deduce x Tx It results that the coditios of theore are satisfied ad together with lea results theore 4 Theore 5 (Theore 3 [4]) Let ( X, d ) be a coplete etric space ad ultifuctio such that the followig iequality occurs for all x, H ( Tx, Ty) c (, ) (, ) δ ( x, Tx) ( y, Ty) d x Tx + d y Ty y i X with δ ( xtx, ) ( yty, ) 0 The T has a fixed poit This theore results with theore 4 T : X P ( X) a
Coo Fixed Poits for Multifuctios Satisfyig a Polyoial Iequality 65 Theore 6 (Theore 5 [4]) Let ( X, d) be a coplete etric space ad ultifuctios T : X P ( X) such that the followig iequality occurs d ( x, Tx) + d ( y, T y) H ( Tx, T y) c (, ) ( yt, y) δ x Tx, ( ) ( T ) a sequece of for all x, 0< c < The the sequece ( ) y i X which verify the coditio δ ( xtx, ) ( yty, ) 0, where, has coo fixed poits ad T F( T ) = F( T ), ( ) This theore results with theore ad lea Note I paper [] fixed poit theores i etric spaces ( X, d ) are give for ultifuctios T : X P( X), called (d)-cotractive, without usig the Hausdorff etric, havig the property ( )x X, ( )y Tx, there exists z Ty such that d( y, z) α d( x, y), where 0 < α < See also [3] for related results o coo fixed poits theores for (d)-cotractive ultifuctios Refereces [] Fisher, B, Coo Fixed Poits ad Costat Mappigs Satisfyig a Ratioal Iequality, Matheatics Seiar Notes, 6, pp 9-35, 978 [] Petcu, Al, Teoree de puct fix petru ultifucții (d)-cotractive î spații etrice, Buletiul Uiversității Petrol-Gaze di Ploiești, Seria Mateatică, Iforatică, Fizică, LVII(), pp -7, 005 [3] Petcu, Al, Coo Fixed Poits for (d)-cotractive Multifuctios i Metric Spaces, Bulleti of Petroleu Gas Uiversity of Ploiesti, Matheatics, Iforatics, Physics Series, LX(), pp -4, 008 [4] Popa V, Coo fixed poits for ultifuctios satisfyig a ratioal iequality, Kobe Joural of Matheatics, (, pp 3-8, 985 [5] Rus, I A, Fixed Poits Theores for Multi-valued Mappigs i Coplete Metric Spaces, Matheatica Japoicae, 0, pp -4, 975 Rezuat Pucte fixe coue petru ultifucții care verifică o iegalitate polioială Se dau teoree de puct fix cou î spații etrice coplete ( X, d ) petru două sau ai ulte ultifucții care îdepliesc iegalități polioiale expriate uai cu distața etrica Hausdorff d, fără a utiliza