Điểm bất động của các ánh xạ co suy rộng trong không gian meetric suy rộng và không gian meetric nón suy rộng

❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦

❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤

◆❣✉②Ơ♥ ❱➝♥ ❚➞♠

➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣
tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭
❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣
▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝

◆❣❤Ö ❆♥ ✲ ✷✵✶✺


❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦

❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤

◆❣✉②Ơ♥ ❱➝♥ ❚➞♠

➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣
tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭
❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣
▲✉❐♥ ❱➝♥ ❚❤➵❝ ❙ü ❚♦➳♥ ❍ä❝
❈❤✉②➟♥ ♥❣➭♥❤✿

❚♦➳♥ ●✐➯✐ tÝ❝❤

▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷

❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝

P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥

◆❣❤Ư ❆♥ ✲ ✷✵✶✺


1
▼ơ❝ ▲ơ❝

❚r❛♥❣
▼ơ❝ ❧ơ❝

✶

▲ê✐ ♥ã✐ ➤➬✉

✷

❈❤➢➡♥❣ ✶✳

➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

♠➟tr✐❝ s✉② ré♥❣

✶✳✶

❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥

✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✶✳✷ ➜✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝
s✉② ré♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❈❤➢➡♥❣ ✷✳

➜✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

♠➟tr✐❝ ♥ã♥ s✉② ré♥❣

✷✳✶

✷✽

➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ➤Þ❛ ♣❤➢➡♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥
♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✷

✶✵

➜✐Ó♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉

✷✽

ϕ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

♥ã♥ s✉② ré♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✽

❑Õt ❧✉❐♥

✹✻

❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦

✹✼


2

❧ê✐ ♥ã✐ ➤➬✉

▲ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ ♠ét tr♦♥❣ ♥❤÷♥❣ ✈✃♥ ➤Ị ♥❣❤✐➟♥ ❝ø✉ q✉❛♥
trä♥❣ ❝đ❛ ❣✐➯✐ tÝ❝❤✳ ◆ã ❝ã r✃t ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ tr♦♥❣ ♥❣➭♥❤ t♦➳♥ ❤ä❝ ✈➭ ❝➳❝
♥❣➭♥❤ ❦ü t❤✉❐t✳ ❑Õt q✉➯ q✉❛♥ trä♥❣ ➤➬✉ t✐➟♥ ♣❤➯✐ ♥ã✐ ➤Õ♥ tr♦♥❣ ❧ý t❤✉②Õt
➤✐Ó♠ ❜✃t ➤é♥❣ ➤ã ❧➭ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ
❝ñ❛ ❇❛♥❛❝❤ ✭♥➝♠ ✶✾✾✷✮✳ ▼ét ❤➢í♥❣ ♥❤×♥ ❦❤➳❝ ✈Ị ♥❣❤✐➟♥ ❝ø✉ ể t ộ
ờ t ò t r ệ tì ể ❜✃t ➤é♥❣ ❝đ❛ ♠ét ➳♥❤ ①➵ ❧➭ ✈✃♥ ➤Ị ❝ã
♥❤✐Ị✉ ø♥❣ ❞ô♥❣ tr♦♥❣ ❣✐➯✐ tÝ❝❤ ♥❤✃t ❧➭ ❧ý t❤✉②Õt ❝➳❝ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥✱
♣❤➢➡♥❣ tr×♥❤ ➤➵♦ ❤➭♠ r✐➟♥❣✱ ♣❤➢➡♥❣ tr×♥❤ tÝ❝❤ ♣❤➞♥✳ ◆❣➢ê✐ t❛ ➤➲ t×♠ ❝➳❝❤
♠ë ré♥❣ ♥❣✉②➟♥ ❧ý ♥➭② ❝❤♦ ♥❤✐Ò✉ ➳♥❤ ①➵ ✈➭ ♥❤✐Ò✉ ❧♦➵✐ ❦❤➠♥❣ ❣✐❛♥ ❦❤➳❝ ♥❤❛✉✳
❑❛♥♥❛♥ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ♠ét ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ➳♥❤ ①➵ ❝♦ ♠➭
♥ã ❦❤➠♥❣ ➤ß✐ ❤á✐ tÝ♥❤ ❧✐➟♥ tơ❝ ❝đ❛ ➳♥❤ ①➵ ✭♥➝♠ ✶✾✻✽✮✳ ◆➝♠ ✷✵✵✹✱ ❇❡r✐♥❞❡ ➤➲
❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ❝➳❝ ➳♥❤ ①➵ ❝♦ ②Õ✉✱ ó ũ ò ợ ọ
✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ ❤➬✉
❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✳ ❙❛✉ ➤ã ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝ ❦❤➳❝ t✐Õ♣ tơ❝
♥❣❤✐➟♥ ❝ø✉ t❤❡♦ ❤➢í♥❣ ♥➭② ✈➭ t❤✉ ➤➢ỵ❝ ♥❤✐Ị✉ ❦Õt q✉➯ t❤ó ✈Þ✳

➜➷❝ ❜✐Ưt ✈➭♦ ♥➝♠ ✷✵✵✼✱ t❤❡♦ ❤➢í♥❣ ♠ë ré♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❍✉❛♥❣
▲♦♥❣✲●✉❛♥❣ ✈➭ ❩❤❛♥❣ ❳✐❛♥ ➤➲ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ư♠ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥
❜➺♥❣ ❝➳❝❤ t❤❛② ➤ỉ✐ t❐♣ sè t❤ù❝ tr♦♥❣ ➤Þ♥❤ ♥❣❤Ü❛ ♠➟tr✐❝ ❜ë✐ ♠ét ♥ã♥ ➤Þ♥❤
❤➢í♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥✳ ❍❛✐ t➳❝ ❣✐➯ ❝ị♥❣ ➤➲ ①➞② ❞ù♥❣ ❝➳❝ ❦❤➳✐
♥✐Ư♠ ✈Ị sù ❤é✐ tơ ❝đ❛ ❞➲②✱ tÝ♥❤ ➤➬② ➤đ ❝đ❛ ❦❤➠♥❣ ❣✐❛♥✱ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣
➤è✐ ✈í✐ ➳♥❤ ①➵ ❝♦ ✈➭ t❤✉ ➤➢ỵ❝ ữ ết q tú ị tr ớ
➤å♥❣ t❤ê✐ ❝ị♥❣ t❤✃② ➤➢ỵ❝ ♠ét sè ø♥❣ ❞ơ♥❣ ❝đ❛ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝
♥ã♥ tr♦♥❣ ❣✐➯✐ tÝ❝❤ ♣❤✐ t✉②Õ♥✱ tè✐ ➢✉ ✈Ð❝t➡✳ ❍✐Ư♥ ♥❛② ♥❣❤✐➟♥ ❝ø✉ ❝✃✉ tró❝ ❝đ❛
❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ➤❛♥❣ t❤✉ ❤ót sù q✉❛♥ t➞♠ ❝đ❛ ♠ét sè ♥❤➭ t♦➳♥ ❤ä❝
tr♦♥❣ ✈➭ ♥❣♦➭✐ ♥➢í❝✳ ◆❣➢ê✐ t❛ ❝ị♥❣ ➤➲ t×♠ ❝➳❝❤ ♠ë ré♥❣ ❦❤➳✐ ♥✐Ư♠ ❦❤➠♥❣


3

❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ❜➺♥❣ ❝➳❝❤ t❤❛② t❤Õ ❜✃t ➤➻♥❣ t❤ø❝ t t
tú ì ữ t ể r❛ ❦❤➳✐ ♥✐Ö♠ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ✈➭
♥❣❤✐➟♥ ❝ø✉ ❝➳❝ tÝ♥❤ ❝❤✃t ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr➟♥ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♥➭②✳
➜Ĩ t❐♣ ❞➢ỵt ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝✱ ❝❤ó♥❣ t➠✐ t✐Õ♣ ❝❐♥ ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉
♥❤➺♠ t×♠ ❤✐Ĩ✉ ❝➳❝ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣
❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✳ ❚r➟♥ ❝➡ së
❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣
t➠✐ ➤➲ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✿ ✧➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣

❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✧✳
▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣
❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱
❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝♦✱ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣✱ ➳♥❤ ①➵ ❝♦ ➤Þ❛ ♣❤➢➡♥❣✱ ➳♥❤ ①➵ ❦✐Ĩ✉

ϕ✲❝♦✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝
s✉② ré♥❣ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱✳✳✳

❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣

❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ơ❝ ✶ ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉
♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❧✉❐♥ ✈➝♥✱ ❣å♠✿ ❦❤➠♥❣ ❣✐❛♥
♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ➳♥❤ ①➵ ❝♦✱
➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ♠ét sè ➤Þ♥❤ ❧ý ✈➭ ❤Ư q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦✳
▼ơ❝ ✷ ❝❤ó♥❣ t trì ột số ị ý ệ q ✈➭ ✈Ý ❞ơ ✈Ị ➤✐Ĩ♠ ❜✃t
➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳
❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣

❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ơ❝ ✶ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐
♥✐Ư♠ ✈Ị✿ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱ ➳♥❤ ①➵ ❝♦
➤Þ❛ ♣❤➢➡♥❣✱ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ✈Ị ❝➳❝ ➳♥❤ ①➵ ❝♦ ➤Þ❛ ♣❤➢➡♥❣ tr♦♥❣
❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ✈➭ ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳ ▼ô❝ ✷ ❣✐í✐ t❤✐Ư✉
❦❤➳✐ ♥✐Ư♠ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉

ϕ✲❝♦✱ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý✱ ❤Ư q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t


4
➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉

ϕ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✳

▲✉❐♥ ✈➝♥ ♥➭② ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ ❞➢í✐ sù ❤➢í♥❣
❞➱♥ t❐♥ t×♥❤ ❝❤✉ ➤➳♦ ❝ñ❛ t❤➬② ❣✐➳♦ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ t➳❝ ❣✐➯ ①✐♥ ❜➭② tá
sù ❜✐Õt ➡♥ s➞✉ s➽❝ tí✐ ❚❤➬②✳ ◆❤➞♥ ❞Þ♣ ♥➭② t➳❝ ❣✐➯ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❇❛♥
❝❤đ ♥❤✐Ư♠ ❦❤♦❛ ❚♦➳♥✱ P❤ß♥❣ ➤➭♦ t➵♦ ❙❛✉ ➤➵✐ ❤ä❝✱ q✉ý t❤➬②✱ ❝➠ ❣✐➳♦ tr♦♥❣
❜é ♠➠♥ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ❦❤♦❛ ❙➢ ♣❤➵♠ ❚♦➳♥ ❤ä❝✱ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ➤➲ t❐♥
t×♥❤ ❣✐ó♣ ➤ì t➳❝ ❣✐➯ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ◆❤➞♥

➤➞②✱ t➳❝ ❣✐➯ ❝ò♥❣ ①✐♥ ❝➯♠ ➡♥ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ ❝❛♦ ❤ä❝ ❦❤ã❛ ✷✶ ●✐➯✐ tÝ❝❤ t➵✐
tr➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✳ ❈✉è✐ ❝ï♥❣✱ t➳❝ ❣✐➯ ①✐♥ ❝❤➞♥ t❤➭♥❤ ì ố
ẹ ị tt ❝➯ ❜➵♥ ❜❒ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ ❣✐ó♣ ➤ì t➳❝ ❣✐➯ ❤♦➭♥ t❤➭♥❤
♥❤✐Ư♠ ✈ơ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣✳
▼➷❝ ❞ï ➤➲ tÝ❝❤ ❝ù❝ ➤➬✉ t➢ ✈➭ ❝ã ♥❤✐Ò✉ ❝è ❣➽♥❣ tr♦♥❣ ♥❣❤✐➟♥ ❝ø✉✱ t❤ù❝
❤✐Ư♥ ➤Ị t➭✐✱ s♦♥❣ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr ỏ ữ s sót
ợ ữ ý ❦✐Õ♥ ➤ã♥❣ ❣ã♣ ❝ñ❛ q✉ý ❚❤➬②✱ ❈➠ ✈➭ ❜➵♥ ➤ä❝ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝
❤♦➭♥ t❤✐Ư♥✳

❱✐♥❤✱ ♥❣➭② ✵✽ t❤➳♥❣ ✵✽ ♥➝♠ ✷✵✶✺

◆❣✉②Ô♥ ❱➝♥ ❚➞♠


5

❝❤➢➡♥❣ ✶

➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣
tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣
1.1

❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥

P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ q✉❛ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝
tr×♥❤ ❜➭② ❧✉❐♥ ✈➝♥✱ ❝➳❝ ♠è✐ q✉❛♥ ❤Ư ❣✐÷❛ ❝➳❝ ❦❤➳✐ ♥✐Ư♠✱ ❦Õt q✉➯ tr➟♥ ✈➭ ❝❤♦
♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳

1.1.1

➜Þ♥❤ ♥❣❤Ü❛✳

♠ét ♠➟tr✐❝ tr➟♥

✭❬✶❪✮ ❈❤♦ t ợ

X = d : X ì X → R ➤➢ỵ❝ ❣ä✐ ❧➭

X ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥

✭✶✮

d(x, y) ≥ 0 ✈í✐ ♠ä✐ x, y ∈ X ✈➭ d(x, y) = 0 ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x = y ✳

✭✷✮

d(x, y) = d(y, x) ✈í✐ ♠ä✐ x, y ∈ X ✳

✭✸✮

d(x, y) ≤ d(x, z) + d(z, y) ✈í✐ ♠ä✐ x, y, z ∈ X ✳

❚❐♣

X ❝ï♥❣ ✈í✐ ♠ét ♠➟tr✐❝ d tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

✈➭ ❦Ý ❤✐Ö✉ ❧➭

(X, d) ❤❛② ➤➡♥ ❣✐➯♥ ❧➭ X ✳ ❙è d (x, y) ❣ä✐ ❧➭ ❦❤♦➯♥❣ ❝➳❝❤ tõ ➤✐Ĩ♠ x

➤Õ♥ ➤✐Ĩ♠ y ✳

1.1.2

➜Þ♥❤ ♥❣❤Ü❛✳

✭❬✶❪✮ ❈❤♦ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

(X, d) ✈➭ (Y, ρ)✳

➳♥❤ ①➵

f : (X, d) → (Y, ρ) ➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ ❝♦ ♥Õ✉ tå♥ t➵✐ α ∈ [0, 1) s❛♦ ❝❤♦
ρ[f (x) , f (y)] ≤ αd (x, y) ,

1.1.3

➜Þ♥❤ ❧ý✳

✈í✐ ♠ä✐

✭❬✶❪✮ ✭◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦✮ ●✐➯ sö

x, y ∈ X.

(X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

➤➬② ➤ñ✱ f : X → X ❧➭ ➳♥❤ ①➵ ❝♦ tõ X ✈➭♦ ❝❤Ý♥❤ ♥ã✳ ❑❤✐ ➤ã tå♥ t➵✐ ❞✉② ♥❤✃t
➤✐Ó♠ x∗ ∈ X ✳
➜✐Ó♠

①➵

f✳

x∗ ∈ X ❝ã tÝ♥❤ ❝❤✃t f (x∗ ) = x∗ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤


6
▼ë ré♥❣ ◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤✱ P✳ ◆✳ tt r
t ợ ết q s

1.1.4

ị ý

❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ T

:X→X

❧➭ ♠ét tù ➳♥❤ ①➵ t❤á❛ ♠➲♥ ❜✃t ➤➻♥❣ t❤ø❝✿

ψ (d (T x, T y)) ≤ ψ (d (x, y)) − ϕ (d (x, y)) ✈í✐ ♠ä✐ x, y ∈ X,
tr♦♥❣ ➤ã ψ, ϕ : [0, +∞) → [0, +∞) ❧➭ ❝➳❝ ❤➭♠ ❧✐➟♥ tơ❝✱ ➤➡♥ ➤✐Ư✉ ❦❤➠♥❣ ❣✐➯♠
✈➭ ψ(t) = ϕ(t) = 0 ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ t ❂ ✵✳ ❑❤✐ ➤ã T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣
t

1.1.5

ị ĩ

t ợ

ột tr s rộ tr

X = d : X ì X R ợ ❣ä✐ ❧➭

X ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥

✭✶✮

d(x, y) = 0 ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x = y ❀

✭✷✮

d(x, y) = d(y, x) ✈í✐ ♠ä✐ x, y ∈ X ❀

✭✸✮

d(x, y) ≤ d(x, w) + d(w, z) + d(z, y) ✈í✐ ♠ä✐ x, y ∈ X ✈➭ ✈í✐ ♠ä✐ ❝➷♣ ➤✐Ĩ♠
♣❤➞♥ ❜✐Ưt

❚❐♣

w, z ∈ X \ {x, y}✳

X ❝ï♥❣ ✈í✐ ♠ét ♠➟tr✐❝ s✉② ré♥❣ d tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣

❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❦Ý ❤✐Ö✉ ❧➭ (X, d) ❤❛② ➤➡♥ ❣✐➯♥ ❧➭ X ✳
➜✐Ị✉ ❦✐Ư♥ ✭✸✮ ➤➢ỵ❝ ❣ä✐ ❧➭ ❜✃t ➤➻♥❣ t❤ø❝ tø ❣✐➳❝✳

1.1.6

◆❤❐♥ ①Ðt✳

✭❬✷❪✮ ●✐➯ sö

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ❱í✐

x ∈ X ✈➭ ε > 0 t❛ ❦ý ❤✐Ö✉ B(x, ε) = {y ∈ X : d(x, y) < ε}✳ ❑❤✐ ➤ã ❤ä B = {B(x, r) :
x ∈ X, r > 0} ❧❐♣ t❤➭♥❤ ♠ét ❝➡ së ❝ñ❛ ♠ét t➠♣➠ τd tr➟♥ X ✳
1.1.7

❱Ý ❞ô✳

s❛♦ ❝❤♦

✭❬✶✷❪✮ ❳Ðt

X = {t, 2t, 3t, 4t, 5t} ✈í✐ t > 0 ❧➭ ❤➺♥❣ sè✳ ❈❤♦ sè γ ∈ X

γ > 0✳ ❚❛ ①➳❝ ➤Þ♥❤ ❤➭♠ d : X × X → R ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝

✭❛✮

d(x, x) = 0 ✈í✐ ♠ä✐ x ∈ X ✳

✭❜✮

d(x, y) = d(y, x) ✈í✐ ♠ä✐ x, y ∈ X ✳

7

✭❝✮

d(t, 2t) = 3γ ✳

✭❞✮

d(t, 3t) = d(2t, 3t) = γ ✳

✭❡✮

d(t, 4t) = d(2t, 4t) = d(3t, 4t) = 2γ ✳

✭❢✮

d(t, 5t) = d(2t, 5t) = d(3t, 5t) = d(4t, 5t) = 23 γ ✳

❑❤✐ ➤ã ❞Ơ ❞➭♥❣ ❦✐Ĩ♠ tr❛ ➤➢ỵ❝ r➺♥❣
ré♥❣✱ ♥❤➢♥❣

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉②

(X, d) ❦❤➠♥❣ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ✈× t❛ ❝ã
d(t, 2t) = 3γ > γ + γ = d(t, 3t) + d(3t, 2t).

1.1.8

❱Ý ❞ô✳

✭❬✶✷❪✮ ❳Ðt

X =

1
n

: n = 1, 2, . . .

∪ {0, 2}✳ ❚❛ ①➳❝ ➤Þ♥❤ ❤➭♠

d : X × X → R+ ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝


0 ♥Õ✉



 1 ♥Õ✉
n
d(x, y) =
1

♥Õ✉

n




1 ♥Õ✉

x = y,
x ∈ {0, 2} ✈➭ y = n1 ,
x=

y ∈ {0, 2}

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱

(X, d) ❦❤➠♥❣ ❧➭ ❦❤➠♥❣ tr ì t ó
d

1.1.9



x, y tộ trờ ợ ò .

ó ễ tử t r


1
n

ị ĩ

1 1
1
1

1 1
,
= 1 > + = d , 0 + d 0, .
2 3
2 3
2
3

✭❬✺❪✮ ❈❤♦

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ❞➲②

{xn } ⊂ X ➤➢ỵ❝ ❣ä✐ ❧➭ ❤é✐ tơ ✈Ị ➤✐Ĩ♠ x ∈ X ♥Õ✉ ✈í✐ ♠ä✐ ε > 0 tå♥ t➵✐ n0 ∈ N∗
s❛♦ ❝❤♦ ✈í✐ ♠ä✐

n ≥ n0 t❛ ❝ã d (xn , x) < ε✳ ▲ó❝ ➤ã t❛ ❦Ý ❤✐Ư✉ ❧➭ lim xn = x ❤❛②
n→+∞

xn → x ❦❤✐ n → +∞✳

1.1.10

➜Þ♥❤ ♥❣❤Ü❛✳

✭❬✺❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❞➲②

{xn } ⊂ X ✳ ❚❛ ♥ã✐ r➺♥❣ {xn } ❧➭ ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣
(X, d) ♥Õ✉ ✈í✐ ♠ä✐ ε > 0✱ tå♥ t➵✐ nε ∈ N∗ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n > m ≥ nε ✱ t❛ ❝ã
d(xn , xm ) < ε✳

8

1.1.11

ị ĩ

tr s rộ

(X, d) ợ ọ ❧➭ ➤➬②

➤đ ♥Õ✉ ♠ä✐ ❞➲② ❈❛✉❝❤② tr♦♥❣ (X, d) ➤Ị✉ ❤é✐ tơ tr♦♥❣ ♥ã✳
❚➢➡♥❣ tù ♥❤➢ tr➢ê♥❣ ❤ỵ♣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ♥❣➢ê✐ t❛ ➤➲ t❤✉ ➤➢ỵ❝ ❝➳❝
❦Õt q✉➯ s❛✉✳

1.1.12

▼Ư♥❤ ➤Ị✳

✭❬✶✷❪✮ ◆Õ✉

{xn } ❧➭ ❞➲② ❤é✐ tô tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉②

ré♥❣✱ t❤× ♥ã ❧➭ ❞➲② ❈❛✉❝❤②✳

1.1.13

▼Ư♥❤ ➤Ị✳

✭❬✶✷❪✮ ◆Õ✉

{xn } ❧➭ ♠ét ❞➲② ❤é✐ tơ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

s✉② ré♥❣✱ t❤× ❣✐í✐ ❤➵♥ ❝đ❛ ♥ã ❧➭ ❞✉② ♥❤✃t✳

1.1.14

▼Ư♥❤ ➤Ị✳

✭❬✶✷❪✮ ◆Õ✉

{xn } ❧➭ ❞➲② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣

X ♠➭ ♥ã ❤é✐ tơ ✈Ị ➤✐Ĩ♠ x ∈ X ✱ t❤× ♠ä✐ ❞➲② ❝♦♥ {xnk } ❝đ❛ ♥ã ❝ị♥❣ ❤é✐ tơ ✈Ị
➤✐Ĩ♠ x✳

✭❬✸❪✮ ●✐➯ sư

1.1.15

➜Þ♥❤ ♥❣❤Ü❛✳

♠➟tr✐❝

(X, d) ✈➭♦ ❝❤Ý♥❤ ♥ã✳

➜✐Ĩ♠
①➵

T, f : X → X ❧➭ ❝➳❝ ➳♥❤ ①➵ tõ ❦❤➠♥❣ ❣✐❛♥

y X ợ ọ trị t ❝♦✐♥❝✐❞❡♥❝❡✮ ❝ñ❛ ❤❛✐ ➳♥❤

T ✈➭ f tr➟♥ X ♥Õ✉ tå♥ t➵✐ x ∈ X s❛♦ ❝❤♦ y = f (x) = T (x)✳ ❑❤✐ ➤ã ➤✐Ĩ♠ x ∈ X

➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ✭❝♦✐♥❝✐❞❡♥❝❡ ♣♦✐♥t✮ ❝đ❛ ❤❛✐ ➳♥❤ ①➵
❈➷♣ ➳♥❤ ①➵

(T, f ) ➤➢ỵ❝ ❣ä✐ ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ ♥Õ✉ T ✈➭ f ❣✐❛♦ ❤♦➳♥ ✈í✐

♥❤❛✉ t➵✐ ❝➳❝ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝❤ó♥❣✱ ♥❣❤Ü❛ ❧➭
➤✐Ĩ♠

T ✈➭ f ✳

T f (x) = f T (x) t➵✐ ❝➳❝

x ∈ X ♠➭ T (x) = f (x)✳

1.1.16

➜Þ♥❤ ♥❣❤Ü❛✳

ré♥❣ ✈➭

✭❬✶✶❪✮ ●✐➯ sư

X = φ✳ ◆Õ✉ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉②

(X, ) ❧➭ ♠ét t❐♣ ❤ỵ♣ s➽♣ t❤ø tù ❜é ♣❤❐♥ ớ q ệ tứ tự

tì

(X, d, ) ợ ọ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ t❤ø tù✳ ❑❤✐ ➤ã✱ ❤❛✐ ♣❤➬♥ tư
x, y ∈ X ➤➢ỵ❝ ❣ä✐ ❧➭ s♦ s ợ ế x
1.1.17

ị ĩ



T, f : X X ✳

✭❬✶✶❪✮ ❈❤♦

y ❤❛② y

x✳

(X, ) ❧➭ t❐♣ ❤ỵ♣ s➽♣ t❤ø tù ❜é ♣❤❐♥ ✈➭ ❤❛✐


9

➳♥❤ ①➵ T
❦❤✐

f (x)

1.1.18

➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ➳♥❤ ①➵

f ✲❦❤➠♥❣ ❣✐➯♠ ♥Õ✉ T x

T y ❦❤✐ ✈➭ ❝❤Ø

f (y) , ✈í✐ ♠ä✐ x, y ∈ X ✳

➜Þ♥❤ ♥❣❤Ü❛✳

✭❬✻❪✮ ●✐➯ sư

X ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ t➠♣➠✱ f : X → R✳ ❚❛ ♥ã✐

r➺♥❣✿
✭❛✮ ❍➭♠

f ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ t➵✐ x0 ∈ X ♥Õ✉ ✈í✐ ♠ä✐ sè t❤ù❝

α ∈ R ♠➭ f x0 < α t❤× tå♥ t➵✐ ❧➞♥ ❝❐♥ ♠ë U ❝ñ❛ x0 tr♦♥❣ X s❛♦ ❝❤♦ f x < α
✈í✐ ♠ä✐
✭❜✮ ❍➭♠

x ∈ U✱

f ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐ t➵✐ x0 ∈ X ♥Õ✉ ✈í✐ ♠ä✐ sè t❤ù❝

α ∈ R ♠➭ f x0 > α t❤× tå♥ t➵✐ ❧➞♥ ❝❐♥ ♠ë V ❝đ❛ x0 tr♦♥❣ X s❛♦ ❝❤♦ f x > α
✈í✐ ♠ä✐

✭❝✮ ❍➭♠
tr➟♥

x∈V✱

f ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ ✭t➢➡♥❣ ø♥❣ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐✮

X ♥Õ✉ f ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ ✭t➢➡♥❣ ø♥❣ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐✮ t➵✐

♠ä✐ ➤✐Ĩ♠ t❤✉é❝

1.1.19

➜Þ♥❤ ♥❣❤Ü❛✳

X✳

✭❬✹❪✮ ❈❤♦

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ ➳♥❤ ①➵ T :

X X ớ ỗ số n N t❛ ❦Ý ❤✐Ö✉ O(x, n) = {x, T x, . . . , T n x}✳ ❚❐♣ ❤ỵ♣
O(x, ∞) = {x, T x, . . . , T n x, . . .} ➤➢ỵ❝ ❣ä✐ ❧➭ q✉ü ➤➵♦ ❝đ❛ T t➵✐ x✳
❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ X ➤➢ỵ❝ ❣ä✐ ❧➭ T ✲q✉ü ủ ế ỗ

{xn } X ♠➭ {xn } ⊂ O(x, ∞) ✈í✐ x ∈ X ✱ t❤× xn → z ∈ X ✳
❘â r➭♥❣✱ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ❜✃t ❦ú ❧➭ ♠ét ❦❤➠♥❣

T qỹ

ủ ề ợ ú

1.1.20

ị ❧ý✳

✭❬✷❪✮ ❈❤♦

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ T : X → X ❧➭

♠ét ➳♥❤ ①➵ s❛♦ ❝❤♦

ρ(T x, T y) ≤ qρ(x, y), ✈í✐ ♠ä✐ x, y ∈ X,
tr♦♥❣ ➤ã q ∈ [0, 1)✳ ◆Õ✉ X ❧➭ ❦❤➠♥❣ ❣✐❛♥ T ✲q✉ü ➤➵♦ ➤➬② ➤đ✱ t❤× T ❝ã ♠ét
➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t tr♦♥❣ X ✳


10

1.2

➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣
❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣

P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵
❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❝➳❝ ❝➳❝ ❤Ư q✉➯ ❝đ❛ ❝❤ó♥❣ ➤å♥❣ t❤ê✐
❝❤♦ ♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ➤Þ♥❤ ❧ý ➤➲ ➤➢❛ r❛ ë tr➟♥✳

1.2.1

❇ỉ ➤Ị✳

✭❬✼❪✮ ❈❤♦

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ xi ∈ X ✱

xi−1 = xi ✱ 1 ≤ i ≤ n✱ n ≥ 3✱ x0 = x✱ xn = y ✳ ❑❤✐ ➤ã✱ ❤♦➷❝
n

d(x, y) ≤

d(xi−1 , xi ),
i=1

❤♦➷❝
n−2

d(x, y) ≤

d(xi−1 , xi ) + d(xn−2 , y).
i=1

❈❤ø♥❣ ♠✐♥❤✳ ❚❤❡♦ ❜✃t ➤➻♥❣ t❤ø❝ tø ❣✐➳❝✱ ➤å♥❣ t❤ê✐ sö ❞ơ♥❣ ♣❤Ð♣ q✉②
♥➵♣ t❤❡♦

k ✱ ✈í✐ ♠ä✐ k ∈ N ✈➭ ✈í✐ ♠ä✐ ti ∈ X, 0 ≤ i ≤ 2k + 3✱ ✈í✐ ti = ti+1 ✱ t❛ ❝ã
n−2

d(t0 , t2k+3 ) ≤

d(ti−1 , ti )

(1.1)

i=1

❚❛ ❝❤♦
◆Õ✉

xi ∈ X, xi−1 = xi ✈í✐ ♠ä✐ i = 1, ..., n, n ≥ 3, x0 = x, xn = y ✳ ❚❛ ❝ã

n − 3 ❧➭ sè ❝❤➼♥ t❤× tå♥ t➵✐ k ∈ N s❛♦ ❝❤♦ n = 2k + 3✳ ❉♦ ➤ã ♥❤ê ✭✶✳✶✮ t❛ ❝ã
n

d(x, y) ≤

d(xi−1 , xi ).
i=1

◆Õ✉

n − 3 ❧➭ sè ❧❰ t❤× tå♥ t➵✐ k ∈ N s❛♦ ❝❤♦ n − 1 = 2k + 3✳ ❱× t❤Õ ♥❤ê ✭✶✳✶✮ t❛ ❝ã
n−2

d(x, y) ≤

d(xi−1 , xi ) + d(xn−2 , y).
i=1

1.2.2

➜Þ♥❤ ♥❣❤Ü❛✳

✭❬✺❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣

(X, d)✳

➳♥❤ ①➵ f

:

X → X ➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ ❝♦ ♥Õ✉ tå♥ t➵✐ α ∈ [0, 1) s❛♦ ❝❤♦
d[f (x) , f (y)] ≤ αd (x, y) ,

✈í✐ ♠ä✐

x, y ∈ X.

❑❤✐ ➤ã t❛ ❝ã ➤➢ỵ❝ ◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ❝➳❝ ❦❤➠♥❣
❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳


11

1.2.3

➜Þ♥❤ ❧ý✳

✭❬✺❪✮ ❈❤♦

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤ñ ✈➭

T : X → X ❧➭ ♠ét ➳♥❤ ①➵ ❝♦✳ ❑❤✐ ➤ã✱ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t tr♦♥❣
X✳
❈❤ø♥❣ ♠✐♥❤✳ ❈❤♦ x ∈ X ❧➭ ♠ét ➤✐Ó♠ ❜✃t ❦ú✳ ➜➷t x0 = x, x1 = T x0 , x2 =

T x1 , ..., xn = T xn−1 , ... ❚õ ➤➞②✱ t❛ ❝ã t❤Ó ①➞② ❞ù♥❣ ➤➢ỵ❝ ♠ét ❞➲② ❝➳❝ ➤✐Ĩ♠ tr♦♥❣
X ♥❤➢ s❛✉✿ xn+1 = T xn = T n+1 x, n = 0, 1, ...
◆Õ✉ tå♥ t➵✐

n0 ∈ N s❛♦ ❝❤♦ xn0 +1 = xn0 t❤× t❛ ❝ã T xn0 = xn0 ✈➭ xn0 ❧➭

➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛

T ✳ ❇ë✐ ✈❐② ❦❤➠♥❣ ♠✃t tÝ♥❤ tỉ♥❣ q✉➳t✱ t❛ ❝ã t❤Ĩ ❣✐➯ sư r➺♥❣

xn = xn+1 ✈í✐ ♠ä✐ n ∈ N✳
❱×

T ❧➭ ➳♥❤ ①➵ ❝♦ ớ ỗ n N t ó
d(xn , xn+1 ) = d(T xn−1 , T xn )
≤ αd(xn−1 , xn ) ≤ α2 d(xn−2 , xn−1 ) ≤ ... ≤ αn d(x, x1 ),

tr♦♥❣ ➤ã

α ∈ [0, 1)✳ ❚õ ➤ã t❤❡♦ ❇ỉ ➤Ị ✶✳✷✳✶ ✈í✐ ♠ä✐ m ≥ n + 3 t❛ ❝ã ❤♦➷❝

d(T n x, T m x) ≤ d(T n x, T n+1 x) + d(T n+1 x, T n+2 x) + ... + d(T m−1 x, T m x),
❤♦➷❝

d(T n x, T m x) ≤ d(T n x, T n+1 x) + d(T n+1 x, T n+2 x) + ... + d(T m−2 x, T m x).
❙✉② r❛ ❤♦➷❝
m−1

d(T n x, T m x) ≤

αk d(x, x1 )

(1.2)

k=n

❤♦➷❝
m−3

d(T n x, T m x) ≤

αk d(x, x1 ) + d(T m−2 x, T m x)
k=n
m−3

(1.3)
αk d(x, x1 ) + αm−2 d(x, x2 ).

≤
k=n

❚õ ✭✶✳✸✮ ✈í✐

m ≥ n + 3✱ t❛ ❝ã

m−2

d(T n x, T m x) ≤

αk M.

(1.4)

k=n

❱í✐

M = max{d(x, x1 ), d(x, x2 )}✳ ❱× ✈❐② tõ ✭✶✳✷✮ ✈➭ ✭✶✳✹✮ t❛ ❝ã
m−1
n

m

αk M,

d(T x, T x) ≤
k=n

(1.5)


12
✈í✐ ♠ä✐

m, n ∈ N, m ≥ n + 3✳ ❱× α ∈ [0, 1) ♥➟♥ tõ ✭✶✳✺✮ t❛ s✉② r❛ {xn } ❧➭ ♠ét ❞➲②

❈❛✉❝❤②✳ ❱×

X ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤ñ ♥➟♥ tå♥ t➵✐ u ∈ X s❛♦ ❝❤♦

xn → u ❦❤✐ n → ∞✳ ❉♦ T ❧➭ ➳♥❤ ①➵ ❝♦ ♥➟♥ t❛ s✉② r❛ T ❧✐➟♥ tơ❝✳ ❱× t❤Õ✱ t❛ ❝ã
u = lim xn = lim T xn−1 = T u.
n→∞

❉♦ ➤ã

n→∞

u ❧➭ ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❝đ❛ T ✳

➜Ĩ ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❞✉② ♥❤✃t✱ t❛ ❣✐➯ sư r➺♥❣ tå♥ t➵✐
❝ị♥❣ ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛

w ∈ X s❛♦ ❝❤♦ w

T ✳ ❑❤✐ ➤ã✱ ✈× T ❝♦ ♥➟♥ t❛ ❝ã

d(w, u) = d(T w, T u) ≤ αd(w, u).
❱×
❱❐②

0 ≤ α < 1 ♥➟♥ tõ ❜✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ❝ï♥❣ ♥➭② t❛ s✉② r❛ d(w, u) = 0✳

w = u✳

1.2.4

➜Þ♥❤ ❧ý✳

✭❬✶✷❪✮ ❈❤♦

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤ñ

✈➭ T : X → X ❧➭ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥

d(T x, T y) ≤ α[d(x, T x) + d(y, T x) + d(y, T y) + d(x, y)],
✈í✐ ♠ä✐ x, y ∈ X ✈➭ 0 < α <

1
✳
4

(1.6)

❑❤✐ ➤ã✱ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t

tr♦♥❣ X ✳
❈❤ø♥❣ ♠✐♥❤✳ ▲✃② ➤✐Ó♠ tï② ý x0 ∈ X ✳ ❚❛ ①➳❝ ➤Þ♥❤ ♠ét ❞➲② {xn } ♥❤➢ s❛✉✿
❈❤ä♥

x1 ∈ X s❛♦ ❝❤♦ x1 = T x0 ✱ x2 ∈ X s❛♦ ❝❤♦ x2 = T x1 ✳ ❚✐Õ♣ tơ❝ q✉➳ tr×♥❤ tr

ớ ỗ

n 2 t xn+1 X s ❝❤♦ xn+1 = T xn ✳

◆Õ✉ tå♥ t➵✐ sè
❝ã

n0 ∈ N s❛♦ ❝❤♦ xn0 +1 = xn0 ✱ t❤× t❛ ❝ã T xn0 = xn0 ✈➭ ❦❤✐ ➤ã t❛

xn0 ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✳
❇➞② ❣✐ê t❛ ❣✐➯ sư r➺♥❣ ✈í✐ ♠ä✐

♥❤ê ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✶✳✻✮✱ ✈í✐

n ∈ N t❛ ❝ã xn+1 = xn ✳ ❑❤✐ ➤ã ✈í✐ n ≥ 1

x = xn−1 ✈➭ y = xn t❛ ❝ã

d(xn , xn+1 ) = d(T xn−1 , T xn ) ≤ α[d(xn−1 , T xn−1 ) + d(xn , T xn−1 ) + d(xn , T xn ) + d(xn−1 , xn )]
≤ α[d(xn−1 , xn ) + d(xn , xn ) + d(xn , xn+1 ) + d(xn−1 , xn )].
❙✉② r❛

d(xn , xn+1 ) − αd(xn , xn+1 ) ≤ αd(xn−1 , xn ) + αd(xn−1 , xn ),


13
❤❛②

(1 − α)d(xn , xn+1 ) ≤ 2αd(xn−1 , xn ).
❚õ ➤➞② t❛ ❝ã

d(xn , xn+1 ) ≤
✈í✐

r=

2α
1−α

t❤á❛ ♠➲♥

2α
d(xn−1 , xn ) = r.d(xn−1 , xn ),
1−α

0 < r < 1✳ ◆❤ê ♣❤Ð♣ tr✉② ❤å✐ ✈í✐ ♠ä✐ n ∈ N∗ t❛ t❤✉ ➤➢ỵ❝
d(xn , xn+1 ) ≤ rn d(x0 , x1 ).

❇➞② ❣✐ê t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣
✈í✐

{xn } ❧➭ ♠ét ❞➲② ❈❛✉❝❤② tr♦♥❣ X ✳ ❚❤❐t ✈❐②✱

m > n✱ ✈× xn+1 = xn ✈í✐ ♠ä✐ n ∈ N✱ ♥➟♥ ♥❤ê ❜✃t ➤➻♥❣ t❤ø❝ tø ❣✐➳❝ t❛ ❝ã
d(xn , xm ) ≤ d(xn , xn+1 ) + d(xn+1 , xn+2 ) + d(xn+2 , xm )
≤ rn d(x0 , x1 ) + rn+1 d(x0 , x1 ) + d(xn+2 , xm )
= (rn + rn+1 )d(x0 , x1 ) + d(xn+2 , xm )
..................................................................
≤ (rn + rn+1 + ... + rm−2 )d(x0 , x1 ) + d(xm−1 , xm )
≤ (rn + rn+1 + ... + rm−2 + rm−1 )d(x0 , x1 )
rn
d(x0 , x1 ).
≤

1−r

❈❤♦ n, m
❙✉② r❛

→ ∞ tr♦♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥✱ t❛ ➤➢ỵ❝ d(xn , xm ) → 0 ❦❤✐ n, m → ∞✳

{xn } ❧➭ ♠ét ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤đ✱ ❞♦

➤ã tå♥ t➵✐ ♠ét ➤✐Ĩ♠

u ∈ X s❛♦ ❝❤♦ xn → u✳

▲➵✐ ♥❤ê ❜✃t ➤➻♥❣ t❤ø❝ tø ❣✐➳❝ t❛ ❝ã

d(u, T u) ≤ d(u, xn ) + d(xn , xn+1 ) + d(xn+1 , T u)
≤ d(u, xn ) + d(xn , xn+1 ) + d(T xn , T u).
❱× t❤Õ ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✶✳✻✮ t❛ ❝ã

d(u, T u) ≤ d(u, xn ) + d(xn , xn+1 ) + α[d(xn , T xn ) + d(u, T xn ) + d(u, T u) + d(xn , u)]
≤ d(u, xn ) + d(xn , xn+1 ) + αd(xn , T xn ) + αd(u, T xn ) + αd(u, T u) + αd(xn , u)
≤ d(u, xn ) + d(xn , xn+1 ) + αd(xn , xn+1 ) + αd(u, xn+1 ) + αd(u, T u) + αd(xn , u).


14

❚õ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ s✉② r❛

(1 − α)d(u, T u) ≤ d(u, xn ) + d(xn , xn+1 ) + αd(xn , xn+1 ) + αd(u, xn+1 ) + αd(xn , u)
❈❤♦

n → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ t❤✉ ➤➢ỵ❝ (1 − α)d(u, T u) ≤ 0✳ ❱×

0 < α < 41 ✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❛ s✉② r❛ d(u, T u) = 0 ✈➭ T u = u✳ ❉♦ ➤ã u ❧➭
♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛

T tr♦♥❣ X ✳

❇➞② ❣✐ê t❛ sÏ ❝❤Ø r❛ r➺♥❣
❜✃t ➤é♥❣ ❝ñ❛

u ❧➭ ❞✉② ♥❤✃t✳ ❚❤❐t ✈❐②✱ ❣✐➯ sư u, v ❧➭ ❤❛✐ ➤✐Ĩ♠

T ✳ ❑❤✐ ➤ã d(u, v) = d(T u, T v)✳ ◆❤ê ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✶✳✻✮ t❛ ❝ã
d(u, v) ≤ α[d(u, T u) + d(v, T u) + d(v, T v) + d(u, v)]
≤ α[d(u, u) + d(v, u) + d(v, v) + d(u, v)
≤ 2αd(u, v).

❱×

0 < α < 41 ✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ s✉② r❛ d(u, v) = 0 ✈➭ u = v ✳
❱❐②

1.2.5

T ❝ã ❞✉② ♥❤✃t ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣✳

➜Þ♥❤ ❧ý✳

✭❬✶✷❪✮ ❈❤♦

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤ñ

✈➭ T : X → X ❧➭ ➳♥❤ ①➵ tõ X ✈➭♦ ❝❤Ý♥❤ ♥ã t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ ✭✶✳✻✮

d(T x, T y) ≤ α[d(x, T x) + d(y, T x) + d(y, T y) + d(x, y)],
✈í✐ ♠ä✐ x, y ∈ X ✱ 0 < α <

1
✳
4

◆Õ✉ {T n x} ❝ã ♠ét ❞➲② ❝♦♥ {T nk x} t❤á❛ ♠➲♥

lim T nk x = u ∈ X ✳ ❑❤✐ ➤ã u ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ❝đ❛ T tr♦♥❣ X ✳

k→∞

❈❤ø♥❣ ♠✐♥❤✳ ❚õ ❣✐➯ t❤✐Õt t❛ ❞Ô ❞➭♥❣ t❤✃② r➺♥❣ ♥Õ✉ tå♥ t➵✐ k ∈ N s❛♦ ❝❤♦

T nk x = T nk +1 x✱ t❤× t❛ ❝ã T nk x = u ✈➭ ❧➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝đ❛ T ✳
❇➞② ❣✐ê t❛ ❣✐➯ t❤✐Õt r➺♥❣ ✈í✐ ♠ä✐

k ∈ N t❛ ❝ã T nk x = T nk +1 x✳ ❑❤✐ ➤ã ♥❤ê

❜✃t ➤➻♥❣ t❤ø❝ tø ❣✐➳❝ t❛ ❝ã

d(u, T u) ≤ d(u, T nk x) + d(T nk x, T nk +1 x) + d(T nk +1 x, T u).
▼➷t ❦❤➳❝ ➳♣ ❞ơ♥❣ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✶✳✻✮ ✈í✐

x = T nk x ✈➭ y = u t❛ ❝ã

d(T nk +1 x, T u) = d(T T nk x, T u)
≤ α[d(T nk x, T nk +1 x) + d(u, T nk +1 x) + d(u, T u) + d(T nk x, u)].

(1.7)


15

❚õ ❜✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ♥➭② ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✼✮ t❛ s✉② r❛

d(u, T u) ≤ d(u, T nk x) + d(T nk x, T nk +1 x)
+ α[d(T nk x, T nk +1 x) + d(u, T nk +1 x) + d(u, T u) + d(T nk x, u)].
▲✃② ❣✐í✐ ❤➵♥ ❤❛✐ ✈Õ ❝đ❛ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ ❦❤✐
❉♦ ➤ã t❛ ❝ã

k → ∞✱ t❛ ➤➢ỵ❝ d(u, T u) = 0✳

T u = u✳ ❚Ý♥❤ ❞✉② ♥❤✃t ❝ñ❛ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T tr♦♥❣ X ➤➢ỵ❝

❝❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù ♥❤➢ ➜Þ♥❤ ❧ý ✶✳✷✳✸✳

1.2.6

❇ỉ ➤Ị✳

✭❬✶✵❪✮ ●✐➯ sư

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ T :

X → X ❧➭ ♠ét ➳♥❤ ①➵ s❛♦ ❝❤♦ ✈í✐ ♠ét sè α ∈ (0, 21 ) ♥➭♦ ➤ã✱ t❛ ❝ã
d(T x, T y) ≤ α[d(x, T x) + d(y, T y)] ớ ọ x, y X.

(1.8)

ó ớ ỗ n ∈ N✱ t❛ ❝ã

d(T n x, T n+1 x) ≤

α
1−α

n

d(x, T x) ✈í✐ ♠ä✐ x ∈ X.

(1.9)

❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✾✮ ❜➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ q✉②
♥➵♣✳
❱í✐

n = 1✱ tõ ✭✶✳✽✮✱ t❤❛② y = T x t❛ ❝ã
d(T x, T 2 x) ≤ α[d(x, T x) + d(T x, T 2 x)].

❚õ ➤ã s✉② r❛

d(T x, T 2 x) ≤
❇➞② ❣✐ê ❣✐➯ sư ✭✶✳✾✮ ➤ó♥❣ ✈í✐

n

d(T x, T

α
d(x, T x).
1−α

n✱ ♥❣❤Ü❛ ❧➭ t❛ ❝ã
n+1

x) ≤

α
1−α

n

d(x, T x).

❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✾✮ ➤ó♥❣ ✈í✐
✭✶✳✽✮✱ t❤❛②

n + 1✳ ❚❤❐t ✈❐②✱ tõ

x = T n x ✈➭ y = T n+1 x✱ t❛ ➤➢ỵ❝
d(T n+1 x, T n+2 x) ≤ αd(T n x, T n+1 x) + αd(T n+1 x, T n+2 x).

❚õ ➤➞② t❛ s✉② r❛

d(T n+1 x, T n+2 x) ≤

α
d(T n x, T n+1 x).
1−α


16
❉♦ ➤ã

d(T

n+1

x, T

n+2

α
d(T n x, T n+1 x) ≤
x) ≤
1−α

α
1−α

n+1

d(x, T x).

ổ ề ợ ứ

1.2.7

ị ý



(X, d) ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤ñ ✈➭

T : X → X ❧➭ ♠ét ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥
d(T x, T y) ≤ λ[d(x, T x) + d(y, T y)],

(1.10)

✈í✐ ♠ä✐ x, y ∈ X ✈➭ λ ∈ [0, 12 )✳ ❑❤✐ ➤ã✱ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t tr♦♥❣

X✳
❈❤ø♥❣ ♠✐♥❤✳ ▲✃② ➤✐Ó♠ x0 ∈ X ❜✃t ❦ú✳ ➜➷t x1 = T x0 ✱ ♥Õ✉ x1 = x0 t❤×

x0 = T x0 ✱ ➤✐Ị✉ ♥➭② ❝ã ♥❣❤Ü❛ r➺♥❣ x0 ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✳ ●✐➯ sö r➺♥❣
x1 = x0 ✳ ➜➷t x2 = T x1 ✳ ❇➺♥❣ ❝➳❝❤ ♥➭②✱ t❛ ❝ã t❤Ó ①➳❝ ị ợ ột
ể ủ

X s
xn+1 = T xn = T n+1 x0 , xn = xn+1 , n = 0, 1, 2, ...

❙ư ❞ơ♥❣ ❝ï♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ❝❤ø♥❣ ♠✐♥❤ ♥❤➢ tr♦♥❣ ❇ỉ ➤Ị ✶✳✷✳✻✱ t❛ ➤➢ỵ❝

d(xn , xn+1 ) = d(T n x0 , T n+1 x0 ) ≤

❚❛ ❝ị♥❣ ❝ã t❤Ĩ ❣✐➯ sư r➺♥❣
♥❣➢ỵ❝ ❧➵✐

λ
λ
d(T n−1 x0 , T n x0 ) =
d(xn−1 , xn ).
1−λ
1−λ

x0 ❦❤➠♥❣ ♣❤➯✐ ❧➭ ♠ét ➤✐Ĩ♠ t✉➬♥ ❤♦➭♥✱ ✈× ♥Õ✉

x0 ❧➭ ➤✐Ĩ♠ t✉➬♥ ❤♦➭♥ t❤× sÏ tå♥ t➵✐ n ∈ N s❛♦ ❝❤♦ xn = x0 ✳ ❑❤✐ ➤ã✱ t❛

❝ã

d(x0 , T x0 ) = d(xn , T xn ) = d(T n x0 , T n+1 x0 ) ≤
➜➷t

r=

λ
✳
1−λ

❑❤✐ ➤ã✱ ✈×

➤➞② t❛ s✉② r❛ d(x0 , T x0 )

λ

1−λ

n

d(x0 , T x0 ).

λ ∈ [0, 21 )✱ ♥➟♥ t❛ ❝ã r < 1 ✈➭ (1 − rn )d(x0 , T x0 ) ≤ 0✳ ❚õ

= 0 ✈➭ x0 = T x0 ✱ ❤❛② x0 ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✳ ❉♦

➤ã tr♦♥❣ ❝➳❝ ♣❤➬♥ t✐Õ♣ t❤❡♦ ❝ñ❛ ❝❤ø♥❣ ♠✐♥❤✱ t❛ ❝ã t❤Ĩ ❣✐➯ sư r➺♥❣
✈í✐

n = 1, 2, 3, ...

❇➞② ❣✐ê tõ ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✶✵✮ t❛ s✉② r❛

d(T n x0 , T n+m x0 ) ≤ λ[d(T n−1 x0 , T n x0 ) + d(T n+m−1 x0 , T n+m x0 )]
≤ λ[rn−1 d(x0 , T x0 ) + rn+m−1 d(x0 , T x0 )].

T n x0 = x0


17

❱×

r < 1 ♥➟♥ tõ ❜✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ❝ï♥❣ ♥➭②✱ t❛ s✉② r❛ d(xn , xn+m ) → 0 ❦❤✐

n → ∞✱ ✈í✐ ♠ä✐ m ∈ N✳ ❙✉② r❛ {xn } ❧➭ ♠ét ❞➲② ❈❛✉❝❤② tr♦♥❣ X ✳ ❱× X ❧➭ ➤➬②

➤ñ ♥➟♥ tå♥ t➵✐

u ∈ X s❛♦ ❝❤♦ xn → u✳ ◆❤ê ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✶✵✮✱ t❛ ❝ã

d(T u, u) ≤ d(T u, T n x0 ) + d(T n x0 , T n+1 x0 ) + d(T n+1 x0 , u)
≤ λ[d(u, T u) + d(T n−1 x0 , T n x0 )] + rn d(x0 , T x0 ) + d(T n+1 x0 , u).
❚õ ➤➞② t❛ s✉② r❛

rn
1
d(x0 , T x0 ) +
d(T n+1 x0 , u)
1−λ
1−λ
rn
1
≤ rn d(x0 , T x0 ) +
d(x0 , T x0 ) +
d(T n+1 x0 , u).
1−λ
1−λ

d(T u, u) ≤ rd(T n−1 x0 , T n x0 ) +

❈❤♦

n → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ ✈➭ sư ❞ơ♥❣ d(an , y) → d(a, y) ✈➭ d(x, an ) →

d(x, a) ❦❤✐ n → ∞✱ ✈í✐ {an } ❧➭ ♠ét ❞➲② tr♦♥❣ X ♠➭ an → a ∈ X ❦❤✐ n → ∞ ✈➭
r < 1✱ t❛ ❝ã d(T u, u) = 0✱ s✉② r❛ u = T u✳

❈✉è✐ ❝ï♥❣✱ t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣
❣✐➯ sö r➺♥❣ tå♥ t➵✐ ♠ét ➤✐Ó♠

T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❚❛

v ❦❤➳❝ tr♦♥❣ X s❛♦ ❝❤♦ v = T v ✳ ❑❤✐ ➤ã✱ ♥❤ê ➤✐Ị✉

❦✐Ư♥ ✭✶✳✶✵✮✱ t❛ ❝ã

d(v, u) = d(T v, T u) ≤ r[d(v, T v) + d(u, T v)] ≤ r[d(v, v) + d(u, u)] = 0.
❱× t❤Õ t❛ ❝ã

1.2.8

❱Ý ❞ô✳

d(v, u) = 0✱ ❤❛② v = u✳

❈❤♦

X = {1, 2, 3, 4}✳ ❚❛ ①➳❝ ➤Þ♥❤ d : X × X → R ♥❤➢ s❛✉

d(1, 1) = d(2, 2) = d(3, 3) = d(4, 4) = 0
d(1, 2) = d(2, 1) = 3
d(2, 3) = d(3, 2) = d(1, 3) = d(3, 1) = 1
d(1, 4) = d(4, 1) = d(2, 4) = d(4, 2) = d(3, 4) = d(4, 3) = 4.
❘â r➭♥❣✱ ❦❤✐ ➤ã ❝ã t❤Ó ❦✐Ó♠ tr❛ ➤➢ỵ❝
ré♥❣ ➤➬② ➤đ✱ ♥❤➢♥❣

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉②

(X, d) ❦❤➠♥❣ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ✈× ♥ã ❦❤➠♥❣ t❤á❛ ♠➲♥

➤✐Ị✉ ❦✐Ư♥ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✱ ❞♦

d(1, 2) = 3 > 2 = 1 + 1 = d(1, 3) + d(3, 2).


18
❇➞② ❣✐ê t❛ ①Ðt ➳♥❤ ①➵

T : X → X ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝
Tx =

❑❤✐ ➤ã

3 ♥Õ✉ x = 4,
1 ♥Õ✉ x = 4.

d(T1 , T2 ) = d(T1 , T3 ) = d(T2 , T3 ) = 0 ✈➭ tr♦♥❣ ❝➳❝ tr➢ê♥❣ ❤ỵ♣ ❦❤➳❝ t❛ ❝ã

d(T x, T y) = 1✳ ▼➷t ❦❤➳❝ t❛ ❧✉➠♥ ❝ã d(x, T x) + d(y, T y) ≥ 4✳ ❉♦ ➤ã ♥Õ✉ t❛ ❝❤ä♥
λ=

1
3

t❤× t❛ ❝ã

d(T x, T y) ≤ λ[d(x, T x) + d(y, T y)]✱ ✈í✐ ♠ä✐ x, y ∈ X ì tt

ề ệ ủ ị ý ợ tỏ ó ụ ị ý ✶✳✷✳✼
t❛ s✉② r❛

1.2.9

T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t x = 3✳

❑ý ❤✐Ư✉✳

❱í✐ ♠ä✐

❈❤♦

X ❧➭ ♠ét t❐♣ ❤ỵ♣ ❦❤➳❝ rỗ T : X X ột ①➵✳

x, y ∈ X ✱ t❛ ❦ý ❤✐Ö✉
M (x, y) = max{d(x, y), d(x, T y), d(y, T y)}.

(1.11)

❚❛ ❝ò♥❣ sÏ ❦ý ❤✐Ö✉

Ψ = {ψ| ψ : [0, ∞) → [0, ∞), ❧✐➟♥ tô❝✱ ❦❤➠♥❣ ❣✐➯♠ ✈➭ ψ(t) = 0 ⇔ t = 0},
✈➭

Φ = {φ| φ : [0, ∞) → [0, ∞), ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐, φ(t) > 0 ✈í✐ ♠ä✐ t > 0 ✈➭ φ(0) = 0}.
◆Õ✉

1.2.10

ψ ∈ tì ợ ọ t ổ

ị ĩ

X ột t ợ rỗ

ợ ❣ä✐ ❧➭ ❝ã ➤✐Ó♠ t✉➬♥ ❤♦➭♥ ♥Õ✉ tå♥ t➵✐ ♠ét ➤✐Ĩ♠
✈í✐ sè ♥➭♦ ➤ã

➳♥❤ ①➵ T : X → X

u ∈ X s❛♦ ❝❤♦ u = T p u

p ≥ 1✳ ▲ó❝ ➤ã t❛ ❝ị♥❣ ♥ã✐ ➤✐Ĩ♠ u ∈ X ❧➭ ➤✐Ĩ♠ t✉➬♥ ❤♦➭♥ ❝đ❛ T ✳

❉Ơ t❤✃② r➺♥❣ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛

T ❧➭ ➤✐Ĩ♠ t✉➬♥ ❤♦➭♥ ❝đ❛ T ✈➭ ♥Õ✉

p = 1 t❤× ➤✐Ĩ♠ t✉➬♥ ❤♦➭♥ u ❝đ❛ T ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✳
1.2.11

➜Þ♥❤ ❧ý✳

✭❬✶✶❪✮ ❈❤♦

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤ñ

✈➭ ❍❛✉s❞♦r❢❢✳ ●✐➯ sö r➺♥❣ T : X → X ❧➭ ➳♥❤ ①➵ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X ✱ t❛

❝ã

ψ(d(T x, T y)) ≤ ψ(M (x, y)) − φ(M (x, y)),

(1.12)


19
tr♦♥❣ ➤ã ψ ∈ Ψ✱ φ ∈ Φ ✈➭ M (x, y) ợ ị ở tứ (1.11) ➤ã✱
tå♥ t➵✐ ♠ét ➤✐Ó♠ ❞✉② ♥❤✃t u ∈ X s❛♦ ❝❤♦ u = T u✳
❈❤ø♥❣ ♠✐♥❤✳ ❚r➢í❝ ❤Õt ❞Ơ t❤✃② r➺♥❣ M (x, y) = 0 ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x = y ✈➭

x ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✳
▲✃② ➤✐Ó♠ tï② ý

x0 ∈ X ✳ ❇➺♥❣ q✉② ♥➵♣ t❛ ①➞② ❞ù♥❣ ♠ét ❞➲② {xn } s❛♦ ❝❤♦

x1 = T x0 , x2 = T x1 = T 2 x0 , . . .
xn+1 = T xn = T n+1 x0 ✈í✐ ♠ä✐ n ≥ 0.

(1.13)

❇➢í❝ ✶✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣

lim d(xn , xn+1 ) = 0.

n→∞

❚❤❛②

(1.14)

x = xn ✈➭ y = xn−1 tr♦♥❣ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✶✳✶✷✮✱ ➤å♥❣ t❤ê✐ sư ❞ơ♥❣ ❝➳❝ tÝ♥❤

❝❤✃t ❝đ❛ ❤➭♠

ψ, φ t❛ ➤➢ỵ❝
ψ(d(xn+1 , xn )) = ψ(d(T xn , T xn−1 ))
≤ ψ(M (xn , xn−1 )) − φ(M (xn , xn−1 ))

(1.15)

≤ ψ(M (xn , xn−1 )).
➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦

d(xn+1 , xn ) ≤ M (xn , xn−1 ) ✈í✐ ♠ä✐ n ≥ 1.

(1.16)

❈❤ó ý r➺♥❣

M (xn , xn−1 ) = max{d(xn , xn−1 ), d(xn , T xn ), d(xn−1 , T xn−1 )}
(1.17)
= max{d(xn , xn−1 ), d(xn , xn+1 )}.
◆Õ✉ tå♥ t➵✐ sè

n ≥ 1 s❛♦ ❝❤♦ d(xn−1 , xn ) < d(xn , xn+1 ) t❤× t❛ ❝ã M (xn , xn−1 ) =

d(xn , xn+1 ) > 0 ✈➭ ♥❤ê tÝ♥❤ ❝❤✃t ❝ñ❛ ❤➭♠ φ t❛ s✉② r❛ φ(d(xn+1 , xn )) > 0✳ ❑❤✐ ➤ã
✭✶✳✶✺✮ trë t❤➭♥❤

0 < ψ(d(xn+1 , xn )) ≤ ψ(d(xn+1 , xn )) − φ(d(xn+1 , xn )) < ψ(d(xn+1 , xn )).
➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❱× t❤Õ ✈í✐ ♠ä✐

(1.18)

n ≥ 1✱ t❛ ❝ã

d(xn+1 , xn ) ≤ d(xn−1 , xn ) = M (xn−1 , xn ).

(1.19)


20
❚õ ✭✶✳✶✾✮ t❛ s✉② r❛ r➺♥❣ ❞➲②
❝❤➷♥ ❞➢í✐✳ ❉♦ ➤ã tå♥ t➵✐ sè

{d(xn , xn+1 )} ❧➭ ❞➲② ➤➡♥ ➤✐Ö✉ ❦❤➠♥❣ t➝♥❣ ✈➭ ❜Þ

r ≥ 0 s❛♦ ❝❤♦

lim d(xn , xn+1 ) = lim M (xn−1 , xn ) = r.

n→∞

❇➺♥❣ ❝➳❝❤ ❧✃②

n→∞

(1.20)

lim sup ✷ ✈Õ tr♦♥❣ ❝➠♥❣ t❤ø❝ ✭✶✳✶✺✮✱ sư ❞ơ♥❣ tÝ♥❤ ❝❤✃t ❝đ❛ ❣✐í✐
n→∞

❤➵♥ tr➟♥✱ tÝ♥❤ ❧✐➟♥ tơ❝

ψ ✈➭ tÝ♥❤ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐ ❝đ❛ φ✱ t❛ t❤✉ ➤➢ỵ❝ ψ(r) ≤

ψ(r) − φ(r)✳ ➜✐Ị✉ ♥➭② s✉② r❛ φ(r) = 0 ✈➭ ♥❤ê tÝ♥❤ ❝❤✃t ❝ñ❛ ❤➭♠ φ t❛ ❝ã r = 0✳
◆❤➢ ✈❐② ✭✶✳✶✹✮ ➤➲ ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤✳

❇➢í❝ ✷✳ ❈❤ó♥❣ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣

lim d(xn , xn+2 ) = 0.

n→∞

(1.21)

❚❤❐t ✈❐②✱ ♥❤ê ✭✶✳✶✷✮ t❛ ❝ã

ψ(d(xn+2 , xn )) = ψ(d(T xn+1 , T xn−1 ))
≤ ψ(M (xn+1 , xn−1 )) − φ(M (xn+1 , xn−1 ))

(1.22)

≤ ψ(M (xn+1 , xn−1 )).
➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦

d(xn+2 , xn ) ≤ M (xn+1 , xn−1 ) ✈í✐ ♠ä✐ n ≥ 1,

(1.23)

tr♦♥❣ ➤ã

M (xn+1 , xn−1 ) = max{d(xn+1 , xn−1 ), d(xn+1 , T xn+1 ), d(xn−1 , T xn−1 )}
= max{d(xn+1 , xn−1 ), d(xn+1 , xn+2 ), d(xn−1 , xn )}

(1.24)

= max{d(xn+1 , xn−1 ), d(xn−1 , xn )}.
➜➷t

αn = d(xn+2 , xn ) ✈➭ βn = d(xn , xn+1 )✳ ❱× t❤Õ✱ ♥❤ê ✭✶✳✷✷✮ t❛ ❝ã t❤Ĩ ✈✐Õt
ψ(αn ) ≤ ψ(max{αn−1 , βn−1 }) − φ(max{αn−1 , βn−1 }) ✈í✐ ♠ä✐ n ≥ 1.

❚õ ➤ã ♥❤ê tÝ♥❤ ❝❤✃t ❝ñ❛ ❤➭♠

(1.25)

ψ t❛ s✉② r❛ r➺♥❣

αn ≤ max{αn−1 , βn−1 }.
▼➷t ❦❤➳❝ ♥❤➢ ➤➲ ❝❤Ø r❛ ë tr➟♥ t❛ ❝ã ❞➲②

(1.26)

{d(xn , xn+1 )} = {βn } ❧➭ ❞➲② ➤➡♥ ➤✐Ö✉

❦❤➠♥❣ t➝♥❣✱ ❞♦ ➤ã t❛ ❝ã

βn ≤ βn−1 ≤ max{αn−1 , βn−1 }.

(1.27)


21
❚õ ✭✶✳✷✻✮ ✈➭ ✭✶✳✷✼✮ t❛ ❝ã

max{αn , βn } ≤ max{αn−1 , βn−1 } ✈í✐ ♠ä✐ n ≥ 1.
❱× t❤Õ✱ ❞➲②

{max{αn , βn }} ❧➭ ❞➲② ➤➡♥ ➤✐Ö✉ ❦❤➠♥❣ t➝♥❣ ❝➳❝ sè ❦❤➠♥❣ ➞♠✱ ♥➟♥

♥ã ❤é✐ tơ ✈Ị ♠ét sè

t ≥ 0✳ ●✐➯ sö r➺♥❣ t > 0✳ ❑❤✐ ➤ã✱ tõ ✭✶✳✶✹✮ ❤✐Ó♥ ♥❤✐➟♥ t❛ ❝ã

lim sup αn = lim sup max{αn , βn } = lim max{αn , βn } = t.

n→∞

▲✃②

(1.28)

n→∞

n→∞

(1.29)

lim sup tr♦♥❣ ✭✶✳✷✺✮ rå✐ sư ❞ơ♥❣ ✭✶✳✷✾✮ ✈➭ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ψ ✈➭ φ✱ t❛ ➤➢ỵ❝
n→∞

ψ(t) = ψ( lim sup αn ) = lim sup ψ(αn )
n→∞

≤ lim sup ψ(max{αn−1 , βn−1 }) − lim inf φ(max{αn−1 , βn−1 })
n→∞

n→∞

(1.30)

≤ ψ( lim max{αn−1 , βn−1 }) − φ( lim max{αn−1 , βn−1 })
n→∞

n→∞

= ψ(t) − φ(t).
❚õ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❛ s✉② r❛

φ(t) = 0✱ ❞♦ ➤ã t = 0✳ ➜✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥✳

❇ë✐ ✈❐② tõ ✭✶✳✷✾✮ t❛ ❝ã

lim sup αn = 0,

n→∞

✈➭ ✈× t❤Õ t❛ ❝ã

(1.31)

lim αn = 0✳ ❉♦ ➤ã ✭✶✳✷✶✮ ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤✳

n→∞

❇➢í❝ ✸✳ ❚❛ sÏ ❝❤Ø r❛ r➺♥❣ T ❝ã ♠ét ➤✐Ĩ♠ t✉➬♥ ❤♦➭♥✳
●✐➯ sư ♥❣➢ỵ❝ ❧➵✐ r➺♥❣

T ❦❤➠♥❣ ❝ã ➤✐Ĩ♠ t✉➬♥ ❤♦➭♥✳ ❑❤✐ ➤ã {xn } ❧➭ ♠ét

❞➲② ❝➳❝ ➤✐Ó♠ ♣❤➞♥ ❜✐Ưt✱ ♥❣❤Ü❛ ❧➭
tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♥➭②

{xn } ❧➭ ♠ét ❞➲② ❈❛✉❝❤②✳ ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐ {xn } ❦❤➠♥❣

❧➭ ❞➲② ❈❛✉❝❤②✳ ❑❤✐ ➤ã tå♥ t➵✐ sè
❝➳❝ sè ♥❣✉②➟♥

xm = xn ✈í✐ ♠ä✐ m = n✳ ❚❛ sÏ ❝❤Ø r❛ r➺♥❣

ε > 0 s ớ ỗ số k ❝ã tå♥ t➵✐

m(k) > n(k) > k ✱ s❛♦ ❝❤♦
d(xn(k) , xm(k) ) > .

ớ ỗ số

(1.32)

k t ❝ã t❤Ó ❧✃② m(k) ❧➭ sè ♥❣✉②➟♥ ❞➢➡♥❣ ❜Ð ♥❤✃t ❧í♥ ❤➡♥

n(k) ✈➭ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✶✳✸✷✮ s❛♦ ❝❤♦
d(xn(k) , xm(k)−1 ) ≤ ε.

(1.33)


22
❇➞② ❣✐ê ✈×

{xn } ❧➭ ♠ét ❞➲② ❝➳❝ ➤✐Ĩ♠ ♣❤➞♥ ❜✐Ưt ♥➟♥ t❛ sư ❞ơ♥❣ ✭✶✳✸✷✮✱ ✭✶✳✸✸✮ ✈➭

❜✃t ➤➻♥❣ t❤ø❝ ì ữ t ể t ợ

< d(xm(k) , xn(k) )
≤ d(xm(k) , xm(k)−2 ) + d(xm(k)−2 , xm(k)−1 ) + d(xm(k)−1 , xn(k) )

(1.34)

≤ d(xm(k) , xm(k)−2 ) + d(xm(k)−2 , xm(k)−1 ) + ε.
❑❤✐ ➤ã✱ ♥❤ê ✭✶✳✶✹✮ ✈➭ ✭✶✳✷✶✮ t❛ s✉② r❛

lim d(xn(k) , xm(k) ) = ε.

(1.35)

k→∞

▲➵✐ ờ t tứ ì ữ t t ó

d(xm(k) , xn(k) ) ≤ d(xm(k) , xm(k)−1 ) + d(xm(k)−1 , xn(k)−1 ) + d(xn(k)−1 , xn(k) )
(1.36)
d(xm(k)−1 , xn(k)−1 ) ≤ d(xm(k)−1 , xm(k) ) + d(xm(k) , xn(k) ) + d(xn(k) , xn(k)−1 ).
❈❤♦

k → ∞ tr♦♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥✱ sư ❞ơ♥❣ ✭✶✳✶✹✮ ✈➭ ✭✶✳✸✺✮ t❛ ❝ã
lim d(xm(k)−1 , xn(k)−1 ) = ε.

(1.37)

k→∞

❱× ✈❐② tõ ✭✶✳✶✹✮ ✈➭ ✭✶✳✸✼✮ t❛ t❤✉ ➤➢ỵ❝

M (xm(k)−1 , xn(k)−1 ) = max{d(xm(k)−1 , xn(k)−1 ), d(xm(k)−1 , xm(k) )+d(xn(k)−1 , xn(k) )} → ε,
(1.38)
❦❤✐

k → ∞✳
❇➞② ❣✐ê ❜➺♥❣ ❝➳❝❤ ➳♣ ❞ơ♥❣ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✶✳✶✷✮ ✈í✐

x = xm(k)−1 ✈➭ y =

xn(k)−1 t❛ t❤✉ ➤➢ỵ❝
ψ(d(xm(k) , xn(k) )) = ψ(T xm(k)−1 , T xn(k)−1 )

(1.39)

≤ ψ(M (xm(k)−1 , xn(k)−1 )) − φ(M (xm(k)−1 , xn(k)−1 )).
❈❤♦

k → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ ✈➭ sư ❞ơ♥❣ ✭✶✳✸✺✮ ✈➭ ✭✶✳✸✽✮ t❛ ➤➢ỵ❝
ψ(ε) ≤ ψ(ε) − φ(ε).

❚õ ❜✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ❝ï♥❣ ♥➭② t❛ s✉② r❛
✈❐②

(1.40)

φ(ε) = 0✱ ❞♦ ➤ã ε = 0✱ ♠➞✉ t❤✉➱♥✳ ❱×

{xn } ❧➭ ♠ét ❞➲② ❈❛✉❝❤②✳ ❱× (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤ñ


23
♥➟♥ tå♥ t➵✐
✈í✐

u ∈ X s❛♦ ❝❤♦ xn → u ❦❤✐ n → ∞✳ ▲➵✐ ➳♣ ❞ơ♥❣ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✶✳✶✷✮

x = xn ✈➭ y = u✱ t❛ ❝ã
ψ(d(xn+1 , T u)) = ψ(d(T xn , T u))
(1.41)
≤ ψ(M (xn , u) − φ(M (xn , u)) ≤ ψ(M (xn , u)).

◆❤ê tÝ♥❤ ❝❤✃t ❝ñ❛ ❤➭♠

ψ t❛ s✉② r❛
d(xn+1 , T u) ≤ M (xn , u),

(1.42)

M (xn , u) = max{d(xn , u), d(xn , xn+1 ), d(u, T u)}.

(1.43)

tr♦♥❣ ➤ã

❱×

lim d(xn , u) = limn→∞ d(xn , xn+1 ) = 0✱ tõ ➤➻♥❣ t❤ø❝ tr➟♥ t❛ t❤✉ ➤➢ỵ❝

n→∞

lim M (xn , u) = d(u, T u).

(1.44)

lim sup d(xn+1 , T u) ≤ d(u, T u).

(1.45)

n→∞

❚õ ➤ã s✉② r❛
n→∞

❇➞② ❣✐ê t❛ ①Ðt ❤❛✐ tr➢ê♥❣ ❤ỵ♣ s❛✉✿

❚r➢ê♥❣ ❤ỵ♣ ✶✳ ◆Õ✉ xn = u ✈➭ xn = T u ✈í✐ ♠ä✐ n 2 tì ờ t
tứ ì ữ t

d(u, T u) ≤ d(u, xn ) + d(xn , xn+1 ) + d(xn+1 , T u),

(1.46)

✈➭ sư ❞ơ♥❣ ✭✶✳✶✹✮✱ t❛ t❤✉ ➤➢ỵ❝

d(u, T u) ≤ lim sup d(xn+1 , T u).

(1.47)

lim sup d(xn+1 , T u) = d(u, T u).

(1.48)

n→∞

❚õ ✭✶✳✹✺✮ ✈➭ ✭✶✳✹✼✮ t❛ ❝ã

n→∞

▲✃② ❣✐í✐ ❤➵♥

lim sup tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✹✶✮ ✈➭ sư ❞ơ♥❣ ✭✶✳✹✹✮✱ ✭✶✳✹✽✮ ❝ï♥❣
n→∞

❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛

ψ ✈➭ φ✱ t❛ ➤➢ỵ❝
ψ(d(u, T u)) ≤ ψ(d(u, T u)) − φ(d(u, T u)).

(1.49)