28.04.2014 Views

Chapter 3 The Contraction Mapping Theorem In this chapter we ...

Chapter 3 The Contraction Mapping Theorem In this chapter we ...

Chapter 3 The Contraction Mapping Theorem In this chapter we ...

SHOW MORE
SHOW LESS

Create successful ePaper yourself

Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.

¢¡¤£¦¥¤§©¨<br />

! " ¢#$#$ %& '()+*<br />

<br />

,.-+/1032547680:9=@?BAC=D4E/19F/>=D9S03H©G34¤g^H


Ï!ÐtѹÒ:ÓqÔJÕdÖV×`Ø>ÕdÙjÓqÔÚK×mÛ`ÛFÙ»Ô@ÜbÏ%ÐtÑÌÓqÖVÑ>Ý<br />

¨´%w¢±u03254¦2d4¢25kjk5n:4E/>?89J/1=@GD2j-ß{2jpn3?1=C­:wj©pw¦à%HpiX=`/>25iX=@4¢9¹i9F;419J/>2d4Eg^³!25-3‘V­:wj©®–oAc2»/10X¥¤á$©<br />

Þ<br />

9MåqPRQqUVL`PBL8NJOlPŽQ1S0:9F/¹9¤6TH?8925H=@iœ4E/19F/>=t4¤/10:9J/79¤4E/>?1256`/<br />

6`Hp-!/>?89=°í3£©=@G©ïR;„H=@i4hg^Hp?i9F;:4419J/1254Eg^³%2j-3‘V­:wj©®–DAc2»/10¥”áð©2d6M4E;l92d6¤4>;:90ãiX=`/1?>2d6¤G3=`/>=@?>iX25-3=@Gãz!³â•„‘V¦:“T©t–báþ©0:=T-ã/103=¤i92d6f•lwr±u0%n:4@a!2jgoAC=76T9F-¤í:-:G(9F-%³XiX=`/>?12d6BHp-¤’þ4>n:680¤/>0:9F/u’<br />

6`Hp-!/>?8903=@-(AC=7H03==T£%2d4E/>=T-l6`=9F-:G°n3-32dv!n3=T-:=@414cHFg¦9bí3£©=@G°;lHp2j-!/@w|,Ì/<br />

i9®³¤0:90l9J/¤’è254B-3HF/B6TH?12d6T4ug^Hp?¤Ac032d680(Hp-3=h6@9F-M;3?1HJI0l9J/hšî2d4h9+6`Hp-!/>?89n32j/192d6•lw<br />

Œ ‚:† Œ¦¥ ~}¨§©>D‚„ƒZ…T†J‡3ˆF…t‰V‚„ƒ ¥ ‡¦‰Vƒ ,Ìg¦š ›p’þú’è254f96`Hp-p/1?19p6q/12jHp-°i9F;:;32j-:<br />

¢¤£<br />

H?1256c4>;:90l9J/¤ª&áš ª w<br />

ÍqÎ<br />

2d4h9Fkd4>He6T9Fk5k5=@Gâ9e6TH?8925H2d4Eg^³!25-3ã‘Ž­3w5©t–Ac2j/>0ã¥e¨ä©s2d4h6@9Fk5kj=tG<br />

šh‘Vª„–ráª<br />

-s/103=¹HF/>0:=T?u0:9F-lG¬a!/>03=72dG©=T-!/>2j/m³i9F;°H;:92d4Eí:=@4¹‘V­:wj©®–rAc2j/>0M¥¤á©


¥ ; HON J §<br />

K1L<br />

P<br />

¥ ;<br />

©!Q¥R<br />

•l‘VªF“Eª –`“<br />

§<br />

–¤254S5C9Fn:680%³=@i4C92d4>=f96T2j;:kj25-3=@4@a%2j-l6`k5n:G©25-3X;30%³%4>2d6T4@a<br />

6803=@ib2d4E/>?1³Hpz3k5=Ti4{2j-Xi9J/103=Ti9J/1256@4Tap9F-:Gh/>03=u;3?1H/>25=@4¦H=@i4<br />

9F?1=7HFg{2j-!/>?125-:4E2d6¹i9J/>0:=Ti9J/1256@9Fk25-!/>=T?1=@4E/@w<br />

G©³%-:9FiX2d6T9Fk¬4>³©4m/1=Ti 2d4cG©=`í:-:=@Gsz%³¤9b4E/19J/1=¹4>;:9p6`=B’a©Ac03Hp4>=B=@kj=@ib=@-!/14cG©=t4>6T?>25zl=f/>0:=<br />

—<br />

6T9=@k59F/>=@4/>03=u4E/19J/1=²ª&e|¯M’<br />

G©2db„=@?>=@-!/¦4E/19F/>=@4}/>0:=²4>³%4E/>=@i<br />

9J/¹/>25iX=gf¤/>H°/>03=X4m/89J/1=D9J/9°;3?>=@I%2jHpn:4c/12jiX=2d9Fz3k5=K2d4f6TH25-%n3H=T/>=


Ï!ÐtѹÒ:ÓqÔJÕdÖV×`Ø>ÕdÙjÓqÔÚK×mÛ`ÛFÙ»Ô@ÜbÏ%ÐtÑÌÓqÖVÑ>Ý<br />

Ac03=@?>=u/103=fG©H25I/>25z3k5=0:=T-D/>03=cG©³%-:9=@iä25iX;3kj25=@4B/>0l9J/¤/103=T?1=h2d4B9n:-325v!n3=D=@v!n325kj25z3?12jn:i<br />

9F-lG”/10:9J//103=s4>³%4E/>=@iþ9=9H+25-©í:-32j/m³y4E/19


Èq¾q½¿ÅŽÊŽ½¿Ç¤ÆoŽ jÈqËlÊ Š Ë ‹w‹‘ ½ ’ ‹ Ë ‹ ÉTÊ“8Æ Œ)”‹ ŬÈo!Ê Š‹ ŽJÆ8ËRÆo ‹ Ê ‹ Ë•3À3¼JÈqË@•S–v—˜Î —¢½¿Å„Ê Š‹ ÈqÉ Œ š<br />

¼3½¿¾®À@ÁtÀΉˆBŠ‹Œ<br />

‹ ‘ Ž


D Q´±Dáÿ¦_$ ª<br />

x¢=?>=@Ac?>2j/>=¹/>0:254c=tv!n:9J/12jHp-(9p4C/>03=7í3£©=tG¤;„HHpz3k5=Ti<br />

±<br />

ª4¸<br />

$<br />

±<br />

R<br />

“<br />

ª&<br />

±„wXx¢=bAc25k5kr;3?1HJI25H=pa©A²=6TH25G3=T?u/>0:=h925H=KiX=`/1?>2d6K4>;:92j/>0:ĩ<br />

HFg{/>0:==@v!n:9J/12jHp-<br />

±u03=K9p4>4>H©6`2d9J/>=tGs2j/>=@?19F/>25H6803=@iX=¹2d4C/>0:=T-<br />

á © ª<br />

é P3ª A<br />

6`Hp?>?1=@4>;„HH9bi9F;Mšç›l‘V¦3“1¹–r‘V¦3“1¹–Cp2jIp=T-(z%³<br />

šBª°á ©<br />

$ ‘V­:w º!–<br />

éµ<br />

ª·A<br />

±2d4u9Kí:£%=tG¤;„H=@HJI0:=D9®I?12567Hp-xhCw<br />

6TH?89=T=7Ac03=T/>03=@?cš<br />

±<br />

R ª<br />

±<br />

D R ª<br />

¾š¤ª@4QyšBª&D ¾ á ¿<br />

©<br />

P©ª/A<br />

é<br />

©<br />

é<br />

P©ª&D4A Q<br />

á ©<br />

©!Q<br />

¾ª Qyª D ¾ $<br />

é<br />

¿ ¿¿<br />

,Ì/ug^H0l9J/cšç6TH?89032d4c6`Hp-:G©2j/>25H0:=s4E=tvpn:=T-:6T=bªyáüš ª 6`H=@HJI0:=


­:wé3a%/>03=K6TH2j/E/1=T-925H=Tp?1925H0:=iX9=s/10:9J/<br />

›c÷ ±l“oÁ1ù°Ù÷ ±„“oÁ1ùfÝh<br />

2d4b9e6`Hp-p/12j-%n3Hpn:4hg^n3-:6q/12jHp-4>n:680<br />

/>0l9J/<br />

‘^ªo“E«3– ¾ •9F/>2d4mí:=t4‘V­:w ­!–qw<br />

¤¦¦<br />

xy=¹;3?1HJ<strong>In</strong>3kj/Cz%³4E0:HJAc2j-3h/>0:9F/@a%Ac03=@-y‘V­:wj©t¦p–|254C419J/>2d4Eí:=@G¬a!/103=fi9


Ï!ÐtѹÒ:ÓqÔJÕdÖV×`Ø>ÕdÙjÓqÔÚK×mÛ`ÛFÙ»Ô@ÜbÏ%ÐtÑÌÓqÖVÑ>Ý<br />

zla;:9=pwƒ¼eH=@iüé3w<br />

4>2j-:6T=H?1=T25-!/>=@?>;3?1=`/¦/103254|kj25iX2»/C9=T?12j=t4Twoxy=fG©=`í:-:=<br />

g^H


à ékA7èyAÂè D As$G$F$ Ä m¦‘^ªZ–<br />

J N<br />

,.-gV9p6q/@a¬‘Ž­3w5©@­p–¢?>=t9Fk5kj³b254C9Kp=THpib=T/>?1256B4E=@?>25=@4r/>0:9F/C2d4²9H=@kj³X6`Hp-!Ip=T?10¤?>=t4E;„=@6`/<br />

/>H°94En32j/19


šp=`/84©¨D2»/84¤25-32j/>2d9Fk}4E/19F/>=K9F-lG+9F;3;3?1Hp9p6803=@4u925n3i<br />

—B4þfr<br />

Ž‘Vª„–VA7ÿ©‘^ªZ–íü¦‘^ª„–Cá¦:“ ¦X¨)ªe¨ÿ©p“ ‘V­3w5©€p–<br />

Q!ü<br />

šF“ ü¦‘m©®–rá š $ ‘V­3w5©Gº!–<br />

ü¦‘V¦!–rá<br />

,Ìg/ÿb254¤9¤6`H25H03=@-ã‘V­3w5©€p–C254B=@9p4E³¤/>Hs4>H25H=@;3kd903=7-3Hp-303HpibHp=·tM25479°-3=TA<br />

2d4u/>0:=kj25-3=@9Hp;lH!4E2j/>25H³bIJ9Fk5n3=B;:?>Hpz3kj=@iü9p4|97í:£%=tG<br />

g^H=tG©03H=Tp?19³sIJ9Hpz3kj=@i<br />

<br />

Q!t „á9m„“<br />

<br />

<br />

Ê —<br />

Ï!ÐtѹÒ:ÓqÔJÕdÖV×`Ø>ÕdÙjÓqÔÚK×mÛ`ÛFÙ»Ô@ÜbÏ%ÐtÑÌÓqÖVÑ>Ý<br />

9F-lG°šVta:?>=t4E;„=@6`/>25I=@-!/>2d9Fk¢=@v!n:9F/>25H


á<br />

ß<br />

<br />

ß<br />

<br />

<br />

<br />

ß<br />

m¦‘ Þ –©• Þ<br />

<br />

<br />

ß<br />

<br />

ß<br />

<br />

Ac03H!4E=öp=T?1-3=Tkr2d4/103=&%¹?1=T=@-BÍ4g^n3-:6q/12jHp-wMxy=¤Ac2jk5k²4E/>n:G3³'%¹?1=T=@-BÍ4Kg^n3-:6`/>25H=t9J/>=@?<br />

G©=T/19F25k25-§5²0l9F;©/1=T?K©t¦3w<br />

z:a!AC=Bi9®³D?1=TAc?12j/>=h‘Ž­3w5©G­p–r925H=¹/>0:=2j-!/>=@25ïR0:92dG©=)M9F?1=7=!ã!|90:=2j-!/>=@=t4E;„=@6`/u/>H«s25-(/>032d4c=@v!n:9F/>25H


¢¤£ Œ ‚:† Œ¦¥ ~},+ ,ÌgƒÿD›:÷¦3“@©`ù„Ôhç254c6TH25-%n3HH+z„H=T-!/cAc2j/>0+±u03=THp?>=@i ­3w)€©w<br />

à%25-:6`=<br />

±u03=(-3H=t9Fö©4bG©HJAc-)Hp-3k5³â2»gSÿe2d4D-:=Tp9F/>25I2d4E=t425-hi9F-%³¹HF/103=T?{;3?1H2jiX;3k5=+i9J£©25ihn3i<br />

,Ìg%ÿ©‘^ª„–eµÄ¦¢2j-<br />

¦3a²4>Hy/>03=+4>H25H


l<br />

e<br />

e43<br />

25479¤6TH?8925H


N<br />

á<br />

á<br />

F F<br />

ÕF<br />

Õ<br />

Õ<br />

e<br />

3 e<br />

e<br />

3 e<br />

e<br />

e43<br />

m¦‘ Þ “wü¦‘ Þ –E–ƒQ´m¦‘ Þ “ut„‘ Þ –>–©• Þ F<br />

äGm¦‘ Þ “wü¦‘ Þ –E–Q´m¦‘ Þ “ut„‘ Þ –>–Fä7• Þ<br />

ãiä1ü¦‘ Þ –Q¬t„‘ Þ –Fä7• Þ<br />

254D9e6`Hp-p/1?19p6q/12jHp-¢Hp-ÀãX‘E÷ fí=pa<br />

,Ì/hg^Hpkjk5HJA¤47/10:9J/b2»gI@¨œ©…Uãba}/103=T-š<br />

fí=M"<br />

4E=@=+/>0:=yG©2d4>6Tn:414E25H25HïÌI®925H


“ ü{‘Ž¦p–ráêü<br />

Ac03=@?>=%±s›hÔhç2d4c96`H=tGã9J/ ü&¯Ëh wà%n3;3;„Hp4>=¤/10:9J/pm¦‘¦fq“uüZ–K254<br />

2d4h/103=M6`k5Hp4>=@Gâz:9


N<br />

á<br />

N<br />

á<br />

F F<br />

F<br />

Õ<br />

e<br />

3 e<br />

F<br />

F<br />

F<br />

F<br />

Õ<br />

e<br />

3 e<br />

e<br />

3 e<br />

—î4E25n3kj/¹2d4¹/>0:9F/2»g!m¦‘¦fq“wü¬–f25476TH25-%n3H0:=T-¢/>03=X=T£%2d4E/>=T-l6`=b/>25iX=m@¤H


µ¦¤9H25HIJ903=7g^Hp?>i<br />

K<br />

l<br />

l<br />

g^H=Tg^H


Ï!ÐtѹÒ:ÓqÔJÕdÖV×`Ø>ÕdÙjÓqÔÚK×mÛ`ÛFÙ»Ô@ÜbÏ%ÐtÑÌÓqÖVÑ>Ý<br />

‰ ‘çÿ%“Œ‡„–¹2d4K9(0yg^n3-:6`/>25H=<br />

€F¦Jù©9F-lG|¼e9


Y Œ †Jˆp‰VŠ Œ ~ûö ,Ìg{šÿ419J/>2d4Eí:=@47‘V­:wj©®–¢g^Hp?c¥f¨ ©03HJA/>0l9J/cA²=76T903=7G©2d4m/89F-:6T=<br />

K<br />

/103=2j-:2»/125903=7í3£©=tG¤;„H25H0:=7í3£©=@G°;lHp2j-!/¤ªMg^?1H0:=s4En3iæHFgC/>0:=s9Fz:4>H=XIJ9Fk5n3=t4HFgC/>03=¤Hb„ïÌG©2d9FpH4¹/10:9F-¢/103=¤9Fzl4EHpkjn©/1=<br />

á’ QC“Q_” Ac03=T?1=#’æ254hG©259

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!