Completion of a Dislocated Metric Space

Transcription

1 69 Chpter-3 Completio o Dislocte Metric Spce 3 Itrouctio: metric o X [] i We recll tht istce uctio o set X is si to be islocte i) y ) ii) 0 implies y iii) z) z or ll y z i X I is islocte metric o X the X ) is clle islocte metric spce is clle islocte qusimetric i symmetry coitio i) is roppe rom the bove eiitio We bbrevite islocte metric s metric islocte qusi metric s q metric I is q metric D is eie o the X D ) is islocte metric spce X X by D y )

2 By theorem 5 the topology 70 D iuce by D coicies with r li the ottio o chpter Further the clss B { V )/ X 0}is ope bse or D The presece o the trigle iequlity les Hussor property or some ice properties to X D) I prticulr X D) stisies properties C through C 5 Moreover iv) Cuchy sequeces i X D) Cuchy sequeces i X ) re ieticl v) Coverget sequeces i X D) re Cuchy sequeces hve uique limits X D) is complete i X ) is complete I the trigulr iequlity is elete rom the ioms o the it is iicult to eie the cocept o completio o the resultig istce spce I such morphous spce eve costt sequeces my il to coverge This relte iiculties compel us to reti the trigle iequlity i the iscussio o completeess We ormlly recor the eiitio o completeess: A metric spce X ) is complete i every Cuchy sequece i X ) coverges ' Deiitio: Let X ) Y ) be istce spces A mp : X Y is clle isoistce i or ll y X oe hs ) ) X Y re si to be isoistt i there eist isoistce : X Y oto

3 7 3 Completio: I wht ollows is islocte metric o oempty set X Lemm 3: is isolte poit o X i X {} or i 0 y Proo: Suppose is isolte poit o X The there eists r 0 such tht y r This implies X {} or i r 0 y Coversely suppose X {} or i 0 y I X {} the clerly is isolte poit o X I X {} the i r 0 y Hece is isolte poit o X which implies r r or ll y Corollry 3: I ) 0the is isolte poit o X Proo: I y the ) y ) so ) or ll y i X So { } X or i ) 0 y Lemm33: I R R re equivlece reltios o isjoit sets A A the R R R is equivlece reltio o A A A Proo: Routie hece omitte

4 7 Theorem 34: Let X ) be islocte metric spce The there eists complete islocte metric spce X ) isoistce T : X ) X ) such tht T X ) is ese i X Proo: Let I be the collectio o isolte poits o X J X I I be the collectio o sequeces i X which re ultimtely costt elemet lyig i I RI Let J eote the clss o Cuchy sequeces i J We eie reltios RJ respectively o I J s ollows: I ) y) re sequeces i I the ) RI y) i the ultimtely costt vlue o ) coicies with tht o y ) I ) y) re sequeces i J the ) RJ y) i lim y) 0 Veriictio tht J R I is equivlece reltio is esy Let us ow veriy tht R is equivlece reltio Suppose ) J 0 Sice ) is Cuchy sequece i J lim ) 0 hece ) RJ ) provig tht RJ is releive Suppose ) RJ y) or ) y) J The lim y) lim y ) 0 Hece RJ is symmetric I ) y ) z ) J ) R y ) y ) R z ) the there eist two J J itegers such tht y) i y z) i

5 73 Hece z) y) y z) i m{ } This proves ) RJ z) hece tht RJ is trsitive Let X I J The RRis equivlece reltio o X Let X eote X I X X cotiig the sequece ) eotes the equivlece clss i We eie : X X [0 ) s ollows: ) i y re respectively the y y y I ultimtely costt terms o ) y) lim ) i y y I y J evetully This limit eists sice y ) is Cuchy sequece is iepeet o the choice o the represettive sequece i the equivlece clss y I I ) J y) I the eie y y ) J y) J the eie y lim y) We ow check tht this limit eists is iepeet o the choice o represettive elemets rom the equivlece clsses y ote tht it ollows rom the trigle iequlity tht First we

6 y) m ym) m) y ym) Sice ) y ) re Cuchy sequeces give 0 there eists positive iteger 0 y y m ) or ll m 0 This implies tht m m such tht 74 m ) y ) y ) provig tht y is Cuchy sequece o rel umbers By the completeess o R this sequece coverges The eiitio o RJ mkes it obvious tht lim y) is iepeet o the choice o the represettive sequeces ) y) Veriictio tht respectively rom the clsses y is -metric o X : Clerly y ) 0 y ) y ) or y X Suppose y ) 0 Let ) y ) y We irst see tht either ) y) re both i I or both i J Suppose o the cotrry ) ) I y ) J Let be the ultimtely costt vlue o Now 0 y ) lim y) But 0 ) y ) Hece 0 ) lim y) 0 cotrry to the ct tht I

7 Suppose y I ) y ) y with b the ultimtely costt vlues o ) y) respectively 75 The y ) 0 b) 0 b ) y ) y Suppose y J ) y ) y y ) 0 lim y ) 0 ) y ) y Veriictio o the trigulr iequlity is routie Embeig o X i X : I X let ) be the costt sequece ) where the equivlece clss cotiig ) Deie T : X X by T ) It is cler tht T is isoistce We ow veriy tht T X ) is ese i Cse i): X Let X 0 ) I I this cse let be the ultimtely costt vlue o ) The by the eiitio o T T X ) The Thus TX) i this cse Cse ii): ) J Sice ) is Cuchy sequece there eists positive iteger 0 such tht

8 76 ) i m Let The by virtue o the ct tht m J ) 0 ) ˆ lim Hece T X ) is ese i X ) is complete: Let ) 0 X be Cuchy sequece i m 0 implies m Sice T X ) is ese i tht z ) Hece 0 X 0 There eists 0 such tht There is o hrm i ssumig tht X or ech positive iteger there eists m m m m z z ) z ) ) z ) m 3 i m Hece z is Cuchy sequece i T X ) Sice T is isoistce z ) is Cuchy sequece i X Moreover z z ) z z ) i m 0 Let z eote z m m z ) z ) z z ) z i X such

9 77 lim z z m) m i lim z ) 0 provig tht X ) is complete This completes the proo o theorem 34 Deiitio 35: Let X ) X ) be metric spces X ) be completio o X ) i is si to i) X ) is complete ii) there is isoistce T : X ) X ) such tht T X ) is ese i X Note: I X ) is complete metric spce the its completio is X ) itsel Lemm 36: Let X ) be - metric spce X ) be completio o X ) Let T : X Xbe isoistce embeig X i X with T X ) ese i X The poit y o Xis isolte poit i oly i y T) or some isolte poit o X Proo: Suppose y is isolte poit o X Suppose i possible y TX ) the sice T X ) is ese i X there eists sequece { T )} i T X ) such tht lim T ) 0 By lemm 3 it ollows tht y is ot isolte poit o X cotrictio

10 Hece y T) or some i X Now T is isolte poit o X hece tht o T X ) Sice X T X ) re isoistt is isolte poit o X Coversely suppose is isolte poit o X Sice T is isoistce T ) is isolte poit o T X ) Suppose to obti cotrictio T ) is ot isolte poit o X The or ech positive iteger k there eists elemet k i X such tht k 0 k T Sice k X either k T X) or there eists yk i T X ) such tht y ) T 0 k k k 78 Now Also yk T yk k ) k T k k y k sice y ) T k k k k Hece 0 yk T which by lemm 3 cotricts the ct tht T ) k is isolte poit o T X ) Theorem 37: Let X ) be metric spce X ) X ) be completios o X ) Ti : X ) Xi i ) i ) be isoistces such tht Ti X ) is ese i X i The there eists isoistce o to T : X ) X ) such tht ollowig igrm is commuttive Proo:

11 79 T Deiitio o T : I X is isolte poit o Xthe ) isolte poit o X hece T T is isolte poit o X is Deie T ) T T I X is ot isolte poit there eists sequece z ) i X such tht { Tz } coverges to i X ) Sice T is isoistce { T z } is coverget hece ) Cuchy sequece so { z } is Cuchy sequece i X Sice T is isoistce { T z } is Cuchy sequece i X ) Sice X ) is complete there eists z X such tht lim T z z) 0 This z is iepeet o the choice o the sequece { z } i X Deie T ) z By eiitio T T T T is isoistce: I y X T T T ) T T ) T T T ) T T ) T ) T ) T ) T ) So I y X X limt y limt y where y X the

12 80 T T lim limt T lim y limt lim T T ) y ) lim T y y ) ) whe X X y X or whe X y Xthe rgumet is similr Hece T is isoistce T is bijectio : Iterchgig the plces o X X we get i similr wy isoistce S : X X such tht T ST Sice ST T TT T We hve TST TT T A STT ST T Sice T X ) is ese i X T X) i X We get TS ietity o X ST is ietity o X Hece S T re bijectios

13 8 33 Completio o the metric ssocite with - metric: Pscl Hitzler [] itrouce - metric ssocite with metric use this metric i provig some ie poit theorems tht re eee i progrmmig lguges I this sectio we estblish tht the metric ssocite with the completio o metric o spce is the completio o the metric ssocite with the metric Deiitio 33: Let ) X be metric spce Deie o X X by i 0 i y y is metric o X is clle the metric ssocite with Clerly 0 y ) wheever y I s{ } r 0 X write B s r r r ) { y / s r} The V ) B ) { } V ) B ) { } r The collectio { V ) / Xr 0} { V ) / Xr 0} geerte the sme topology o X However coverget sequeces i X re ot ecessrily the sme sice costt sequeces re coverget sequeces with respect to while this hols wrt or with ) 0 oly r s r s r

14 As metioe i 4 eistece o poits with positive sel istce les to uplestess i the etesio o the cocept o cotiuity i metric spces s well This is eviet rom the ollowig Emple: Let be metric o set X which is ot metric so tht the set A { / ) 0} is oempty I is the metric ssocite with the the ietity mp i : X ) X ) is cotiuous i the usul sese But i A the costt sequece ) coverges i X ) while it oes ot coverge i X ) Deiitio: I X ) be metric spce Y ) be the metric spce ssocite with We cll : X Y sequetilly cotiuous i lim ) 0 lim ) 0 Propositio 33: lim ) 0 i either i) evetully ie there eists N such tht or N or ii) ) c be split ito subsequeces y ) z ) where y or every z or y lim z ) 0 Proo: Assumelim ) 0 The or y subsequece u ) o ) lim ) 0 I there oes ot eist N such tht u or ll N let u y ) be the possibly iite) subsequece o ) such tht y or ll Let z ) be the subsequece o ) tht remis ter eletig ech y Clerly z or y lim ) 0 Further z or every so z lim z ) 0 Coversely ssume tht the coitio hols 8

15 83 The lim ) 0 Deiitio: I s { } ) is sequece i X we sy tht ) is s -Cuchy sequece or simply s - Cuchy i ) is Cuchy sequece i X s ) Propositio 333: I sequece ) i X is - Cuchy the ) is - Cuchy Coversely i ) is - Cuchy is ot evetully costt the ) is - Cuchy Proo: Sice 0 m) m) - Cuchy - Cuchy Coversely suppose tht ) is - Cuchy Give 0 N ) such tht ) i N) m N) So i m N) N) m the m ) m I ) is ot evetully costt N) there eists m N) such tht m The ) ) ) m m ) m ) m Thus i ) is ot evetully costt the or ll m N) N) ) m Hece ) is - Cuchy

16 84 Emple 334: Let X 0 ) y the y 0 i i y y I ) is y sequece i 0 ) the ) is - Cuchy i 0 there eists N ) such tht m This implies tht lim 0 Coversely i lim 0 the 0 N ) such tht or m N) N) Hece m or m N) N) Costt sequeces re ot - Cuchy but - Cuchy Theorem 335: Let X ) be metric spce be the metric ssocite with o X X ) be the completio o X ) be the metric ssocite with o X The X ) is the completio o X ) I prticulr i X ) is complete metric spce the X ) is complete metric spce We prove tht i) X is ese i X ) ii) Every - Cuchy sequece i X is coverget Proo o i) Let X X The there eists sequece ) i X such tht lim ) 0 Sice X so tht implies tht X is ese i X ) lim ) 0 This

17 85 Proo o ii) Let ) be - Cuchy i X I ) is evetully costt the there eists N X such tht or N I this cse lim ) 0 or N hece ) is coverget Suppose ) is ot evetully costt The by propositio 333 ) is - Cuchy sequece Sice X ) is complete there eists X such tht lim ) 0 Sice 0 ) ) 0 lim ) 0 Hece ) coverget to This completes the proo o ii) 34 Fie poit theorems: Let X ) be metric spce : X X be mppig Write V ) z ) { / V ) 0} Clerly every poit o z ) is ie poit o but the coverse is ot ecessrily true We cll poits o z ) s coiciece poits o The set z ) is close subset o X Mthew s theorem [8] sttes tht cotrctio o complete metric spce hs uique ie poit The sme theorem hs bee justiie by lterte proo by Pscl Hitzler[8] We preset etesio o this theorem or coiciece poits

18 Theorem 34[]: Let X ) be complete metric spce : X X be cotrctio The there is uique coiciece poit or Proo: For y Cosequetly i Hece { X the sequece o itertes stisies 86 where is y cotrctive costt m m m m = } is Cuchy sequece i X ) I the ) lim lim So lim ) Sice ) Sice 0 ; lim ) 0 Hece 0 Uiqueess : I 0 the ) ) so tht ) ) ) ) )

19 87 ) 0 Hece Theorem 34[] : Let ) sequetilly cotiuous Assume tht X be metric spce : X X be ) ) m{ y ) } wheever 0 The hs uique coiciece poit wheever Cl ) is oempty or some X Proo: Recll ) { ) / 0} write V ) Z { / V ) 0} Sice is sequetilly cotiuous so is V I Z the V ) m{ ) } m{ V ) V } V V ) Wheever V ) 0 i e Z ) k I ) Z= the V V k k Hece V ) is coverget ) Let be cluster poit o ) i ) i N = lim i ) k =lim i k k ) Cl )

20 88 Sice V is sequetilly cotiuous k V ) = lim V i k ) ) Let = lim V i = V ) Also = lim V i ) = V ; k V V ) ) From ) 3) it ollows tht V ) =0 ie 0 I ) V ) 0 V the = ) = ) I ) 0 ) = ) <m{v )V ) )}= ) which is cotrictio Hece )=0 BE Rhoes [3] presete list o eiitios o cotrctive type coitios or sel mp o metric spce X ) estblishe implictios oimplictios mog them there by cilittig to check i y ew cotrctive coitio implies y oe o the coitios metioe i [3] so s to erive ie poit theorem Amog the coitios i [3] Seghl s coitio is sigiict s goo umber o Cotrctive coitios imply Seghl s coitio We ow cosier vliity o islocte versios o these coitios

21 Let X ) be islocte metric spce : X X be mppig y be y elemets o X Cosier the ollowig coitios Bch) : There eists umber 0 such tht or ech y X ) Rkotch) : There eists mootoe ecresig uctio : 0 ) [0) such tht or ech y X y 3 Eelstei) : For ech y X y ) ) 4 K) :There eists umber 0 y X y ) y ) 89 such tht or ech 5 Bichii): There eists umber h 0 h such tht or ech ) hm y ) y X y 6 For ech y X y ) m y ) 7 Reich): There eist oegtive umbers b c stisyig b c such tht or ech y X y ) b y ) c 8 Reich) : There eist mootoiclly ecresig uctios b c rom 0 ) to [0 ) stisyig t) b t) c t) such tht or ech y X y ) b y ) c

22 90 9 There eist oegtive uctios b c stisyig sup y b y c y} such tht or ech y X y yx ) t bt y ) ct where t y 0 Sehgl): For ech y X y ) m y ) y Theorem 343[]: I is sel mp o islocte metric spce X ) stisies y o the coitios ) through 9) the hs uique ie poit provie Cl ) is oempty or some X Proo: I [3] BE Rhoes prove tht whe is metric ) ) 3) 0) 4) 5) 6) 0) 4) 7) 8) 0) 5) 7) 9) 0) whe 0 is replce by y these implictios hol goo i metric spce s well sice y 0 i metric spce It ow ollows tht hs ie poit which is uique whe ) hs cluster poit or some

23 9 Emple 344[]: Deie y For y i R is islocte metric o R I 0 0) ) B I 0 0 B ) = i i Also 0 B ) i < Emple 345[]: Deie : R R by ) Every oegtive rel umber is ie poit o but 0 is the oly coiciece poit ) )= + =0 0 Thus 0 is the oly coiciece poit while ll oegtive rel umbers re ie poits Theorem 346[4] : Let X ) be complete -metric spce sel mp o X 0h I is sequetilly cotiuous or ll y with 0 ) ) h m{ ) y y )}) the hs uique coiciece poit Proo: Assume tht stisies ) For X y positive iteger write m) { ) )} [ m )] =Sup{ u v) /{ u v} m)} We irst prove the ollowig Lemm: For every positive iteger m there eists positive iteger k m such tht [ m)] k m

24 9 Proo: To prove this it is eough i we prove tht [ m )] m m where m m { ) } ) We prove this by usig iuctio Assume tht ) is true or m ie [ m )] m We hve to prove or m ie [ m )] m ) We hve [ ) m)] Also i m m{ ) m m } } i ) m m i m m { ) ) } 3) Hece [ m )] =Sup { ) / 0 i j m } i j =Sup {{ ) m m } i j { ) / 0 i j m }} m{ ) m m [ ) m]} m rom ) 3) Hece [ m )] Proo o the Theorem: I m This proves the lemm i m j m

25 93 i j i j ) ) [ m )] m i j i i h m{ ) ) ) i j j i j j ) ) ) ) } h[ m)] Also j j m { ) ) } I m re positive itegers such tht m the by ) m m ) ) h ) k h k ; k m - - ) or some 0 h h [ m )] ) m - +) by bove lemm) By the lemm k 0 k m [ m )] k Assume k ) h [ m)] h k k k k h

26 94 I k =0 [ m )] ) + ) ) +h ) h m Hece ) h m h h This is true or every m > Sice 0 h < lim Hece { m )} is Cuchy sequece i X ) h = 0 Sice X is complete z X so tht lim ) z We prove tht z z 0 0 z z z z z) z By sequetil cotiuity o lim z) z 0 Hece z z 0 hece z is coiciece poit o Suppose z z re coiciece poits o The z z )= z z 0 similrly z z)=0 I z z) 0 the by ) z z)= z ) z h m{ z z ) z z z z z z z z }

27 95 h z z) cotrictio Hece z z)=0hece z zthis completes the proo We ow prove ie poit theorem or sel mp o metric spce stisyig the logue o 9) i [3] Theorem 347[4]: Let X ) be complete metric spce : X X be sequetilly cotiuous mppig such tht there eist rel umbers 0 0 mi{ } stisyig 4 t lest oe o the ollowig or ech i ) ) ii iii y X ) ) [ + y ) ] ) ) [ ) + y ] The hs uique coiciece poit Proo: Puttig ) y i the bove = m { } we get Agi puttig y = ) i the bove i) ii) iii) yiels ) ) )

28 96 I h=m { } the 0 h ) h I m re positive itegers such tht m ) h h sice 0 h ; lim h 0 m the Hece { )} is Cuchy s sequece i X ) Sice X is complete z i X lim ) = z Sice is sequetilly cotiuous lim ) = z) i X ) Sice 0 z z z ) z It ollows tht z z =0 Hece z is coiciece poit Uiqueess : I z zre coiciece poits o the by hypothesis Either z z) z z) or 0 or z z) Sice 0 Hece z z 0 we must hve z z)=0 4 The metric versio or the cotrctive iequlity 0) i the moiie orm ) give below yiels the ollowig Theorem 348[4] : Let X ) be complete metric spce : X X be sequetilly cotiuous mppig Assume tht there eist o-egtive costts i stisyig < such tht or ech y X with y

29 97 ) ) ) y y y y y The hs uique coiciece poit Proo:Cosier ) ) ) = ) ) ) ) ) ) ) ) where I m the

30 98 m m ) ) m = ) Hece { )} is Cuchy s sequece i X ) hece coverget Let lim the ) lim sice is sequetilly cotiuous ) So = lim ) lim Sice 0 =0 Hece ) = Hece is coiciece poit or Uiqueess: I 0 ) = )

31 99 Cosier ) ) ) ) where 4 5 ) 0 Hece It is ot out o plce to preset here ew ie poit theorems or sel mps o complete metric spces erive by the uthor re publishe i vrious jourls A ew o theorems re liste without proo here uer Theorem 349[5]: Let X ) be complete -metric spce Let A B S T : X X be cotiuous mppigs stisyig I T X ) A X ) S X ) B X ) II The pirs S A) T B) re wekly comptible III S T m{ A B A S) By T} For ll y X where0 the A B S T hve uique commo ie poit Theorem 340[5]: Let X ) be complete metric spce Let A B S T : X X be cotiuous mppig stisyig I T X ) A X ) S X ) B X ) II The pirs S A) T B) re wekly comptible

32 III For ll A S) By T S T { } A B A B o yo X where 0 0 the A B S 4 T hve uique commo ie poit Theorem 34[6]: Let X ) A B S T : X X be cotiuous mppigs stisyig I T X ) A X ) S X ) B X ) be complete metric spce Let II The pirs S A) T B) re wekly comptible III 00 5 S T A B A S) 3 By T 4 A T By S) For ll y X where the A B S T hve uique 0 5 commo ie poit Theorem:34[6]: Let X ) be complete metric spce Let S T : X X be cotiuous mppigs stisyig S T T S T S) 3 Sy T 4 T T 5 Sy S) or ll y X where the S T hve uique ie poit