Sự tồn tại điểm bất động của ánh xạ co trong không gian b mêtric nón

❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦

❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤

❚r➬♥ ❚❤Þ ❍➵♥❤

❙ù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦
tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥
▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝

◆❣❤Ö ❆♥ ✲ ✷✵✶✻


❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦

❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤

❚r➬♥ ❚❤Þ ❍➵♥❤

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

❚♦➳♥ ●✐➯✐ tÝ❝❤

▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷

❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝
P●❙✳ ❚❙✳ ➜✐♥❤ ❍✉② ❍♦➭♥❣

◆❣❤Ö ❆♥ ✲ ✷✵✶✻


1
▼ô❝ ▲ô❝

❚r❛♥❣
▼ô❝ ❧ô❝

✶

▲ê✐ ♥ã✐ ➤➬✉

✷

❈❤➢➡♥❣ ✶✳

❑❤➠♥❣ ❣✐❛♥

b✲♠➟tr✐❝ ♥ã♥

✺

✶✳✶✳ ▼ét sè ❦✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✶✳✷✳ ◆ã♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✽

✶✳✸✳ ❑❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷

❈❤➢➡♥❣ ✷✳

❙ù tå♥ t➵✐ ➤✐Ó♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

b✲♠➟tr✐❝ ♥ã♥

✶✼

✷✳✶✳ ▼ét sè ❦Õt q✉➯ ✈Ò sù tå♥ t➵✐ ➤✐Ó♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

b✲♠➟tr✐❝

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

✶✼

✷✳✷✳ ▼ét sè ❦Õt q✉➯ ✈Ị sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ tr♦♥❣
❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥

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

✷✹

❑Õt ❧✉❐♥

✸✾

❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦

✹✵


2
❧ê✐ ♥ã✐ ➤➬✉

▲ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ ♠ét tr♦♥❣ ữ ề ợ ề
t ọ q t ♥❣❤✐➟♥ ❝ø✉✳ ◆❣➢ê✐ t❛ ➤➲ t×♠ t❤✃② sù ø♥❣ ❞ơ♥❣ ➤❛ ❞➵♥❣
❝đ❛ ❧ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ t♦➳♥ ❤ä❝ ✈➭ ♥❤✐Ị✉ ♥❣➭♥❤ ❦ü t❤✉❐t ❦❤➳❝✳ ❙ù
♣❤➳t tr✐Ĩ♥ ❝đ❛ ❧ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❣➽♥ ❧✐Ị♥ ✈í✐ t➟♥ t✉ỉ✐ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝
❧í♥ ♥❤➢ ❇r♦✉✇❡r✱ ❇❛♥❛❝❤✱ ❙❝❤❛✉❞❡r✱ ❑❛❦✉t❛♥✐✱ ✳✳✳
❑Õt q✉➯ q✉❛♥ trä♥❣ ➤➬✉ t✐➟♥ ♣❤➯✐ ❦Ó ➤Õ♥ tr♦♥❣ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭
♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ❝ñ❛ ❇❛♥❛❝❤ ✭✶✾✾✷✮✳ ◆❣➢ê✐
t❛ ➤➲ ♠ë ré♥❣ ♥❣✉②➟♥ ❧ý ♥➭② ❝❤♦ ♥❤✐Ò✉ ❧♦➵✐ ➳♥❤ ①➵ ✈➭ ♥❤✐Ị✉ ❧♦➵✐ ❦❤➠♥❣ ❣✐❛♥
❦❤➳❝ ♥❤❛✉✳ ▼ét tr♦♥❣ ♥❤÷♥❣ ❤➢í♥❣ ♠ë ré♥❣ ➤ã ❧➭ t❤❛② ➤ỉ✐ ➤✐Ị✉ ❦✐Ư♥ tr♦♥❣
➤Þ♥❤ ♥❣❤Ü❛ ♠➟tr✐❝✱ tõ ➤ã t❤✉ ➤➢ỵ❝ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ré♥❣ ❤➡♥ ❧í♣ ❦❤➠♥❣ ❣✐❛♥
♠➟tr✐❝✳ ❙❛✉ ➤ã✱ ♥❣➢ê✐ t❛ ♥❣❤✐➟♥ ❝ø✉ sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❧í♣ ❝➳❝
❦❤➠♥❣ ❣✐❛♥ ✈õ❛ ➤Þ♥❤ ♥❣❤Ü❛✳✳✳
◆➝♠ ✷✵✵✼✱ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝ ❚r✉♥❣ ◗✉è❝✿ ❍✉❛♥❣ ▲♦♥❣ ✲ ●✉❛♥❣ ✈➭ ❩❤❛♥❣
❳✐❛♥ ✭❬✻❪✮ ➤➲ t❤❛② ❣✐➯ t❤✐Õt ❤➭♠ tr trị tr t ợ số tự
➞♠ ❜ë✐ ♥❤❐♥ ❣✐➳ trÞ tr♦♥❣ ♠ét ♥ã♥ ➤Þ♥❤ ❤➢í♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛✲
♥❛❝❤ ✈➭ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ö♠ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ❙❛✉ ➤ã✱ ♥❤✐Ò✉ ♥❤➭
t♦➳♥ ❤ä❝ ➤➲ ♥❣❤✐➟♥ ❝ø✉ ✈➭ ➤➵t ➤➢ỵ❝ ♥❤✐Ị✉ ❦Õt q✉➯ ✈Ị sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t
➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ◆❤÷♥❣ ♥❣➢ê✐ t❤✉ ➤➢ỵ❝ ♥❤✐Ị✉ ❦Õt q✉➯
t❤❡♦ ❤➢í♥❣ ♥➭② ❧➭✿ ❏✳ ❙✳ ❯♠❡✱ ❘✳ ❆✳ ❙t♦❧t❡♥❜❡❣✱ ❈✳ ❙✳ ❲♦♥❣✱ ❍✳ ▲✳ ●✉❛♥❣ ✈➭
❩✳ ❳✐❛♥ ✳✳✳ ❑❤➳✐ ♥✐Ư♠ ❦❤➠♥❣ ❣✐❛♥

b✲♠➟tr✐❝ ➤➢ỵ❝ ➤➢❛ r❛ ✈➭ ♥❣❤✐➟♥ ❝ø✉ ❜ë✐ ❙✳

❈③❡r✇✐❦ ✭❬✺❪✮✳ ❚r♦♥❣ ❬✼❪✱ ◆✳ ❍✉ss❛✐♥ ✈➭ ❝➳❝ ❝é♥❣ sù ➤➲ ♠ë ré♥❣ ❧í♣ ❦❤➠♥❣
❣✐❛♥ b✲♠➟tr✐❝ ✈➭ ♠➟tr✐❝ ♥ã♥ ❜➺♥❣ ❝➳❝❤ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ö♠ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝
♥ã♥ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè tÝ♥❤ ❝❤✃t t➠♣➠ ✈➭ sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣
❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♥➭②✳
➜Ĩ t❐♣ ❞➢ỵt ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝ ✈➭ ❧Ü♥❤ ❤é✐ ✈Ị ❧ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣✱


3
❝❤ó♥❣ t➠✐ t×♠ ❤✐Ĩ✉✱ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ ✈➭
sù tå♥ t➵✐ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥✳ ❱×
t❤Õ ❝❤ó♥❣ t➠✐ ❝❤ä♥ ➤Ị t➭✐ ♥❣❤✐➟♥ ❝ø✉ ❧➭✿ ✧❙ù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤

①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥✧✳
❱í✐ ♠ơ❝ ➤Ý❝❤ ➤ã✱ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❝❤✐❛ ❧➭♠ ❤❛✐ ❝❤➢➡♥❣✳
❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị ❑❤➠♥❣ ❣✐❛♥

b✲♠➟tr✐❝

♥ã♥✳ ❚r♦♥❣ ♠ơ❝ ✶✱ ❝❤ó♥❣

t➠✐ tr×♥❤ ❜➭② ♠ét sè ❦❤➳✐ ♥✐Ư♠ ✈➭ ❦Õt q✉➯ ❝➡ ❜➯♥ ✈Ò ❦❤➠♥❣ ❣✐❛♥ t➠♣➠✱ ❦❤➠♥❣
❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ➳♥❤ ①➵ ❧✐➟♥ tô❝✳✳✳ ❝➬♥ tr♦♥❣ ❧✉❐♥ ụ
trì ị ĩ í ụ ột sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝ñ❛ ♥ã♥ tr♦♥❣ ❦❤➠♥❣
❣✐❛♥ ❇❛♥❛❝❤✳ ụ trì ị ĩ í ụ ột sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥
❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥✳
❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ❙ù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

b✲

♠➟tr✐❝ ♥ã♥✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ơ❝ ✶ ❝❤ó♥❣ t➠✐ trì ột số ị ý ề sự

tồ t ể ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ❇❛♥❛❝❤✱ ❈❤❛tt❡r❥❡❛✱ ❑❛♥♥❛♥✱✳✳✳
tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

b✲♠➟tr✐❝

➤➲ ➤➢ỵ❝ ❣✐í✐ t❤✐Ư✉ tr♦♥❣ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❬✾❪✳

▼ơ❝ ✷ ❝❤ó♥❣ t➠✐ ➤➢❛ r❛ ♠ét ✈➭✐ ❦Õt q✉➯ ♠í✐ ✈Ị sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣
tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

b✲ ♠➟tr✐❝ ♥ã♥✱ ➤ã ❧➭ ❝➳❝ ➜Þ♥❤ ❧ý ✷✳✷✳✶✱ ✷✳✷✳✸✱ ✷✳✷✳✽✱ ✷✳✷✳✶✵ ✈➭

❝➳❝ ❍Ö q✉➯ ✷✳✷✳✹✱ ✷✳✷✳✺✱ ✷✳✷✳✼✱ ✷✳✷✳✾ ✈➭ ✷✳✷✳✶✷✳ ❈➳❝ ❦Õt q✉➯ ♥➭② ❧➭ ♠ë ré♥❣ ❝ñ❛
♠ét sè ❦Õt q✉➯ tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✹✱ ✽✱ ✾✱ ✶✵❪✳
▲✉❐♥ ✈➝♥ ➤➢ỵ❝ t❤ù❝ ❤✐Ư♥ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥
t❐♥ t×♥❤ ✈➭ ♥❣❤✐➟♠ ❦❤➽❝ ❝đ❛ P●❙✳❚❙✳ ➜✐♥❤ ❍✉② ❍♦➭♥❣✳ ❚➳❝ ❣✐➯ ①✐♥ ❜➭② tá
❧ß♥❣ ❜✐Õt ➡♥ s➞✉ s➽❝ ❝đ❛ ♠×♥❤ ➤Õ♥ ❚❤➬②✳
❚➳❝ ❣✐➯ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❇❛♥ ❈❤đ ♥❤✐Ư♠ ❑❤♦❛ ❙❛✉ ➤➵✐ ❤ä❝✱ ❇❛♥
❈❤đ ♥❤✐Ư♠ ❑❤♦❛ ❚♦➳♥ ✲ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✳
❚➳❝ ❣✐➯ ①✐♥ ➤➢ỵ❝ ❝➯♠ ➡♥ q✉ý ❚❤➬② ❣✐➳♦✱ ❈➠ ❣✐➳♦ ❚ỉ ●✐➯✐ tÝ❝❤ tr♦♥❣ ❑❤♦❛
❚♦➳♥ ✲ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ➤➲ ♥❤✐Ưt t×♥❤ ❣✐➯♥❣ ❞➵② ✈➭ ❣✐ó♣ ➤ì t➳❝ ❣✐➯ tr♦♥❣
s✉èt t❤ê✐ ❣✐❛♥ ❤ä❝ t❐♣✳


4
❈✉è✐ ❝ï♥❣ ①✐♥ ❝➯♠ ➡♥ ❣✐❛ ➤×♥❤✱ ➤å♥❣ ♥❣❤✐Ư♣✱ ❜➵♥ ❜❒✱ ➤➷❝ ❜✐Ưt ❧➭ ❝➳❝ ❜➵♥

tr♦♥❣ ❧í♣ ❈❛♦ ❤ä❝ ✷✷ ✲ ❈❤✉②➟♥ ♥❣➭♥❤ ●✐➯✐ tÝ❝❤ ➤➲ ❝é♥❣ t➳❝✱ ❣✐ó♣ ➤ì ✈➭ ➤é♥❣
✈✐➟♥ t➳❝ ❣✐➯ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ♥❣❤✐➟♥ ❝ø✉✳
▼➷❝ ❞ï ➤➲ ❝ã ♥❤✐Ò✉ ❝è ❣➽♥❣ ♥❤➢♥❣ ❞♦ ❝ß♥ ❤➵♥ ❝❤Õ ✈Ị ♠➷t ❦✐Õ♥ t❤ø❝ ✈➭
t❤ê✐ ❣✐❛♥ ♥➟♥ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ t❤✐Õ✉ sãt✳ ❑Ý♥❤ ♠♦♥❣ q✉ý
❚❤➬② ❈➠ ✈➭ ❜➵♥ ❜❒ ➤ã♥❣ ❣ã♣ ý ❦✐Õ♥ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥ ❤➡♥✳

❱✐♥❤✱ t❤➳♥❣ ✽ ♥➝♠ ✷✵✶✻

❚r➬♥ ❚❤Þ ❍➵♥❤


5




btr ó

trì ị ĩ í ❞ơ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝đ❛
♥ã♥ ✈➭ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥✱ ❧➭♠ ❝➡ së ❝❤♦ ❈❤➢➡♥❣ ✷✳

1.1

▼ét số ế tứ ị

ụ trì ột số ❦❤➳✐ ♥✐Ư♠ ✈➭ ❦Õt q✉➯ ❝➡ ❜➯♥ ✈Ị ❦❤➠♥❣ ❣✐❛♥
♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥

b✲♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥✱✳✳✳

❝➬♥ ❞ï♥❣ tr♦♥❣ ❧✉❐♥

✈➝♥✳ ❈➳❝ ❦Õt q✉➯ ♥➭② ➤➢ỵ❝ trÝ❝❤ r❛ tõ ❝➳❝ t➭✐ ❧✐Ư✉ t

1.1.1


ị ĩ



X

sử

d ợ ọ tr tr X

t rỗ

ế ớ ọ

d : X × X → R✳

x, y, z ∈ X

❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉

➤➢ỵ❝ t❤á❛ ♠➲♥
✭✐✮

✭✐✐✮

✭✐✐✐✮

d(x, y) ≥ 0 ✈➭ d(x, y) = 0 ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ x = y ❀
d(x, y) = d(y, x)❀
d(x, z) ≤ d(x, y) + d(y, z)✳

❚❐♣

X

❝ï♥❣ ✈í✐ ♠ét ♠➟tr✐❝

➤➢ỵ❝ ❦Ý ❤✐Ư✉ ❜ë✐

1.1.2

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

(X, d) ❤♦➷❝ X ✳

➜Þ♥❤ ĩ



sử

X

t rỗ

s 1



X ì X → [0, +∞) ➤➢ỵ❝ ❣ä✐ ❧➭ b✲♠➟tr✐❝ ♥Õ✉ ✈í✐ ♠ä✐ x, y, z ∈ X ✱ t❛ ❝ã
✭✐✮

✭✐✐✮

d(x, y) = 0 ⇔ x = y ❀
d(x, y) = d(y, x)❀

d :


6
d(x, y) ≤ s[d(x, z) + d(z, y)]

✭✐✐✐✮

X

❚❐♣

❝ï♥❣ ✈í✐ ♠ét

✭❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✮✳

b✲♠➟tr✐❝ tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣

✈í✐ t❤❛♠ sè s✱ ♥ã✐ ❣ä♥ ❧➭ ❦❤➠♥❣ ❣✐❛♥

❣✐❛♥

b✲♠➟tr✐❝

b✲♠➟tr✐❝ ✈➭ ➤➢ỵ❝ ❦Ý ❤✐Ư✉ ❜ë✐ (X, d) ❤♦➷❝

X✳
1.1.3

❱Ý ❞ơ✳

✶✮ ●✐➯ sư (X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ d

: X×X → [0, +∞)

❧➭ ❤➭♠ ➤➢ỵ❝ ❝❤♦ ❜ë✐

d(x, y) = (p(x, y))2 , ∀x, y ∈ X.
❑❤✐ ➤ã✱

d ❧➭ b✲♠➟tr✐❝ ✈í✐ s = 2✳

✷✮ ●✐➯ sö

X = R ✈➭ tr➟♥ R t❛ ①Ðt tr t tờ ị

d : R ì R → [0, +∞) ❜ë✐
d(x, y) = |x − y|2 , ∀x, y ∈ R.
❑❤✐ ➤ã✱

d ❧➭ ♠➟tr✐❝ ✈í✐ s = 2

✭t❤❡♦ ✶✮✮ ♥❤➢♥❣

d ❦❤➠♥❣ ❧➭ ♠➟tr✐❝ tr➟♥ R ✈×

d(1, −2) = 9 > 5 = d(1, 0) + d(0, −2).
1.1.4

➜Þ♥❤ ♥❣❤Ü❛✳

✭❬✺❪✮✳

●✐➯ sư

{xn }

❧➭ ❞➲② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

b✲♠➟tr✐❝

(X, d)✳
❉➲②
❤✐Ư✉ ❜ë✐

n0

{xn } ➤➢ỵ❝ ❣ä✐ ❧➭ b✲❤é✐ tơ ✭♥ã✐ ❣ä♥ ❧➭ ❤é✐ tơ✮ tí✐ x ∈ X

xn → x ❤♦➷❝ limn→∞ xn = x ♥Õ✉ ✈í✐ ♠ä✐ > 0✱ tå♥ t➵✐ sè tù ♥❤✐➟♥

s❛♦ ❝❤♦

❦❤✐

✈➭ ➤➢ỵ❝ ❦Ý

d(xn , x) <

✈í✐ ♠ä✐

n ≥ n0 ✳

◆ã✐ ❝➳❝❤ ❦❤➳❝✱

xn → x ❦❤✐ ✈➭ ❝❤Ø

d(xn , x) → 0 ❦❤✐ n → ∞✳
❉➲②

{xn } ➤➢ỵ❝ ❣ä✐ ❧➭ ❞➲② ❈❛✉❝❤② ♥Õ✉ ✈í✐ ♠ä✐ > 0✱ tå♥ t➵✐ sè tù ♥❤✐➟♥

n0 s❛♦ ❝❤♦ d(xn , xm ) <
❑❤➠♥❣ ❣✐❛♥
➤Ị✉ ❤é✐ tơ✳

✈í✐ ♠ä✐

n, m ≥ n0 ✳

b✲♠➟tr✐❝ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➬② ➤ñ ♥Õ✉ ♠ä✐ ❞➲② ❈❛✉❝❤② tr♦♥❣ ♥ã


7
1.1.5

➜Þ♥❤ ♥❣❤Ü❛✳

K = C✳

❍➭♠

●✐➯ sư

E

❧➭ ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ tr➟♥ tr➢ê♥❣

p : E → R ➤➢ỵ❝ ❣ä✐ ❧➭ ❝❤✉➮♥ tr➟♥ E

K = R ❤♦➷❝

♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉

❦✐Ư♥ s❛✉
✭✐✮

✭✐✐✮

✭✐✐✐✮

❙è
❝đ❛

p(x) ≥ 0, ∀x ∈ E ✈➭ p(x) = 0 ⇔ x = 0❀
p(λx) = |λ|p(x), ∀x ∈ E, ∀λ ∈ K❀
p(x + y) ≤ p(x) + p(y), ∀x, y ∈ E ✳
p(x) ➤➢ỵ❝ ❣ä✐ ❧➭ ❝❤✉➮♥ ❝đ❛ ✈❡❝t➡ x ∈ E ✳

x ❧➭ x

✳ ❑❤➠♥❣ ❣✐❛♥ ✈❡❝t➡

❚❛ t❤➢ê♥❣ ❦Ý ❤✐Ö✉ ❝❤✉➮♥

E ❝ï♥❣ ớ ột ị tr ó ợ

ọ ị

1.1.6

ệ ề

ế

E

ị tì t❤ø❝

d(x, y) = x − y , ∀x, y ∈ E,
①➳❝ ➤Þ♥❤ ♠ét ♠➟tr✐❝ tr➟♥

E✳

❚❛ ❣ä✐ ♠➟tr✐❝ ♥➭② ❧➭ ♠➟tr✐❝ s✐♥❤ ❜ë✐ ❝❤✉➮♥ ❤❛② ♠➟tr✐❝ ❝❤✉➮♥✳

▼ét ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥ ✈➭ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ t❤❡♦ ♠➟tr✐❝
s✐♥❤ ❜ë✐ ợ ọ

1.1.7

ị ý

ế


é ộ

E

ị tì

x x , x ∈ E ❀

(x, y) → x + y, ∀(x, y) ∈ E × E ❀

✈➭ ♣❤Ð♣ ♥❤➞♥ ✈í✐ ✈➠ ❤➢í♥❣✿

(λ, x) → λx, ∀(λ, x) ∈ K × E

❧➭ ❝➳❝ ➳♥❤

①➵ ❧✐➟♥ tơ❝✳

E ❧➭ ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥✳ ❑❤✐ ➤ã✱ ớ ỗ a E

1.1.8

ị ý

ỗ

K, = 0 ❝➳❝ ➳♥❤ ①➵

●✐➯ sö

x → x + a, x → λx, ∀x ∈ E
❧➭ ❝➳❝ ♣❤Ð♣ ➤å♥❣ ♣❤➠✐

E

❧➟♥

E✳

8
1.1.9

t ợ

ị ĩ



ệ

X





ột q ệ tr➟♥

➤➢ỵ❝ ❣ä✐ ❧➭ q✉❛♥ ❤Ư t❤ø tù ❜é ♣❤❐♥ tr➟♥

X

X✳

♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝

➤✐Ị✉ ❦✐Ư♥ s❛✉
✭✐✮

x ≤ x ✈í✐ ♠ä✐ x ∈ X ❀

✭✐✐✮ ❚õ

✭✐✐✐✮

x ≤ y ✈➭ y ≤ x s✉② r❛ x = y ✈í✐ ♠ä✐ x, y ∈ X ❀

x ≤ y ❀ y ≤ z s✉② r❛ x ≤ z ✈í✐ ♠ä✐ x, y, z ∈ X ✳

❚❐♣ ❤ỵ♣

X ❝ï♥❣ ✈í✐ ♠ét t❤ø tù ❜é ♣❤❐♥ tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ s➽♣ t❤ø

tù ❜é ♣❤❐♥ ✈➭ ❦Ý ❤✐Ư✉

(X, ≤) ❤♦➷❝ X ✳

1.1.10

●✐➯ sư ✧≤✧ ❧➭ ♠ét q✉❛♥ ❤Ư ❤❛✐ ♥❣➠✐ tr➟♥

➜Þ♥❤ ♥❣❤Ü❛✳

✭✐✮ P❤➬♥ tư

X ✈➭ A ⊆ X ✳

x ∈ X ➤➢ỵ❝ ❣ä✐ ❧➭ ❝❐♥ tr➟♥ ✭t➢➡♥❣ ø♥❣ ❝❐♥ ❞➢í✐✮ ❝đ❛ A ♥Õ✉

a ≤ x ✭t➢➡♥❣ ø♥❣ x ≤ a✮ ✈í✐ ♠ä✐ ♣❤➬♥ tư a ∈ A❀
✭✐✐✮ P❤➬♥ tư

❝đ❛

A

x ∈ X ➤➢ỵ❝ ❣ä✐ ❧➭ ❝❐♥ tr➟♥ ➤ó♥❣ ✭t➢➡♥❣ ø♥❣ ❝❐♥ ❞➢í✐ ➤ó♥❣✮

♥Õ✉

x

❧➭ ♠ét ❝❐♥ tr➟♥ ✭t➢➡♥❣ ø♥❣ ❝❐♥ ❞➢í✐✮ ❝đ❛

❝ị♥❣ ❧➭ ♠ét ❝❐♥ tr➟♥ ✭t➢➡♥❣ ø♥❣ ❝❐♥ ❞➢í✐✮ ❝đ❛

✈➭ ♥Õ✉

A t❤× x ≤ y

y

✭t➢➡♥❣

y ≤ x✮✳ ❑❤✐ ➤ã✱ t❛ ❦Ý ❤✐Ö✉ x = sup A ✭t➢➡♥❣ ø♥❣ x = inf A✮✳

ø♥❣

1.2

A

◆ã♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤

1.2.1

➜Þ♥❤ ♥❣❤Ü❛✳

R✳ ▼ét t❐♣ ❝♦♥ P
✭✐✮

P

❝đ❛

❧➭ ➤ã♥❣✱

✭✐✐✮ ❱í✐

✭❬✹❪✮✳ ❈❤♦

E

❧➭ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ tr➟♥ tr➢ê♥❣ sè t❤ù❝

E ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ã♥ tr♦♥❣ E ♥Õ✉

P = ∅✱ P = {0}❀

a, b ∈ R✱ a, b ≥ 0 ✈➭ x, y ∈ P

✭✐✐✐✮ ◆Õ✉

x∈P

✈➭

−x ∈ P

t❤×

x = 0✳

t❤×

ax + by ∈ P ❀


9
1.2.2

❚r♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❝➳❝ sè t❤ù❝

R ✈í✐ ❝❤✉➮♥ t❤➠♥❣

P = {x ∈ R : x ≥ 0} ❧➭ ♠ét ♥ã♥✳

t❤➢ê♥❣✱ t❐♣

2)

1)

❱Ý ❞ô✳

E = R2 ✱ P = {(x, y) ∈ E : x, y ≥ 0} ⊂ R2 ✳ ❑❤✐ ➤ã P

●✐➯ sư

t❤á❛ ♠➲♥

❜❛ ➤✐Ị✉ ❦✐Ư♥

P

✭✐✮

❧➭ ➤ã♥❣✱

P = ∅✱ P = {0}❀

(x, y), (u, v) ∈ P

✭✐✐✮ ❱í✐ ♠ä✐

✈➭ ♠ä✐

a, b ∈ R✱ a, b ≥ 0✱ t❛ ❝ã a(x, y) +

b(u, v) ∈ P ❀
(x, y) ∈ P

✭✐✐✐✮ ❱í✐

❱❐②

3)

P

●✐➯ sư

❚❛ ➤➲ ❜✐Õt

✈➭

(−x, −y) ∈ P ✱ t❛ ❝ã (x, y) = (0, 0)✳

❧➭ ♠ét ♥ã♥ tr➟♥

C[a,b]

E✳

❧➭ t❐♣ t✃t ❝➯ ❝➳❝ ❤➭♠ sè ♥❤❐♥ ❣✐➳ trÞ t❤ù❝ ❧✐➟♥ tơ❝ tr➟♥

[a, b]✳

C[a,b] ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈í✐ ❝❤✉➮♥

f = sup |f (x)|, ∀f ∈ C[a,b] .
x∈[a,b]

❚r➟♥

C[a,b]

❝ã q✉❛♥ ❤Ö t❤ø tù ộ t tờ



ợ ị ở

f, g C[a,b] ✱
f ≤ g ⇔ f (x) ≤ g(x), ∀x ∈ [a, b].
➜➷t

P = {f ∈ C[a,b] : 0 ≤ f }✳ ❑❤✐ ➤ã P
P

✭✐✮

❧➭ ➤ã♥❣✱

✭✐✐✮ ❱í✐ ♠ä✐

t❤á❛ ♠➲♥ ❜❛ ➤✐Ị✉ ❦✐Ö♥

P = ∅✱ P = {0}❀

a, b ∈ R✱ a, b ≥ 0 ✈➭ ♠ä✐ f, g ∈ P ✱ t❛ ❝ã
0 ≤ af (x) + bg(y), ∀x ∈ [a, b].

❉♦ ➤ã

af + bg ∈ P ❀

✭✐✐✐✮ ◆Õ✉

f ∈P

−f ∈ P ✱ t❤× f = 0✳

❱❐②

P

❧➭ ♠ét ♥ã♥ tr➟♥

❈❤♦

P

❧➭ ♠ét ♥ã♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤

q✉❛♥ ❤Ö t❤ø tù ”
❚❛ ✈✐Õt
➤ã

✈➭

E✳

≤ ” ①➳❝ ➤Þ♥❤ ❜ë✐ P

♥❤➢ s❛✉✿

❧➭ ♣❤➬♥ tr♦♥❣ ❝đ❛

P✳

❚r➟♥

E

t❛ ➤Þ♥❤ ♥❣❤Ü❛

x ≤ y ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ y−x ∈ P ✳

x < y ♥Õ✉ x ≤ y ✈➭ x = y ❀ ✈➭ ✈✐Õt x

intP

E✳

y ♥Õ✉ y − x ∈ intP ✱ tr♦♥❣


10
1.2.3

ó

ị ĩ

P





P

ột ó tr

ợ ọ ♥ã♥ ❝❤✉➮♥ t➽❝ ♥Õ✉ tå♥ t➵✐ sè t❤ù❝

x, y ∈ E

✈➭

0 ≤ x ≤ y ✱ t❛ ❝ã x ≤ K y

K > 0 s❛♦ ❝❤♦ ✈í✐ ♠ä✐

✳ ❙è t❤ù❝ ❞➢➡♥❣

♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ♥➭② ➤➢ỵ❝ ❣ä✐ ❧➭ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ ❝đ❛

1.2.4

✭❬✹❪✮✳ ●✐➯ sư

❇ỉ ➤Ị✳

P

✭✐✮ ◆Õ✉

✭✐✐✮ ◆Õ✉

✭✐✐✐✮ ◆Õ✉

✭✐✈✮

a

c❀

a ≤ b ✈➭ b

c t❤× a

c❀

c2

d t❤× a + c

✭✈✐✐✐✮ ◆Õ✉

✭✐①✮ ◆Õ✉

✭①✮ ◆Õ✉
t❤×

b + d❀

δ > 0 ✈➭ x ∈ intP ✱ tå♥ t➵✐ 0 < γ < 1 s❛♦ ❝❤♦ γx < δ ❀
c1 ∈ intP

✈➭

c2 ∈ P ✱ tå♥ t➵✐ d ∈ intP

c1 , c2 ∈ intP ✱ tå♥ t➵✐ e ∈ intP

a∈P

s❛♦ ❝❤♦

c1

d ✈➭

✈➭

a ≤ x ✈í✐ ♠ä✐ x ∈ intP

s❛♦ ❝❤♦

t❤×

e

c1 ✈➭ e

c2 ❀

a = 0❀

a ≤ λa ✈í✐ a ∈ P ✱ 0 < λ < 1 tì a = 0
0 xn yn ớ ỗ n ∈ N ✈➭ limn→∞ xn = x✱ limn→∞ yn = y

0 ≤ x ≤ y✳

❈❤ø♥❣ ♠✐♥❤✳

(i)

❱× ♣❤Ð♣ ❝é♥❣ ❧✐➟♥ tơ❝ ♥➟♥ intP +intP

c t❤× b − a ∈ intP

✈➭

intP + intP ⊂ intP ✳ ❱❐② a
➜Ó ý r➺♥❣ intP

+P =

⊂ intP ✳ ◆Õ✉ a

c − b ∈ intP ✳

❙✉② r❛

b

c − b ∈ intP ✳ ❙✉② r❛ c − a = c − b + b − a ∈
c✳

x∈P (x + intP ) ❧➭ t❐♣ ♠ë ✈➭ P ❧➭ ♥ã♥ ♥➟♥ s✉② r❛

x + intP ⊂ P ✳ ❉♦ ➤ã P + intP ⊂ intP ✳ ◆Õ✉ a ≤ b b


E a, b, c

d

ớ ỗ

(ii)

P

intP intP

ớ ỗ

b

b c

ỏ t tỏ

số tự ó

c tì a

ớ ỗ





b b

a

K

❧➭ ♥ã♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤

E ✱ {xn }, {yn } ❧➭ ❝➳❝ ❞➲② tr♦♥❣ E

E✳

c t❤× b − a ∈ P

c − a = c − b + b − a ∈ intP + P ⊂ intP

❤❛②


11
c − a ∈ intP ✳ ❱❐② a
(iii)

❚❛ ❝ã

a

b

✈➭

b − a + d − c ∈ intP
(iv)
(v)

c✳
c

d

❤❛②

♥➟♥

b − a ∈ intP

❑❤✐ ➤ã ✈í✐

δ
n x t❤á❛ ♠➲♥

γ=

❈❤ä♥

s✉② r❛

b + d✳

α intP ⊂ intP ✳

δ > 0 ✈➭ x ∈ intP ✱ ❝❤ä♥ sè tù ♥❤✐➟♥ n > 1 s❛♦ ❝❤♦

γx ≤ γ
(vi)

d − c ∈ intP ✱

(b + d) − (a + c) ∈ intP ✱ ❞♦ ➤ã a + c

❱× é ớ tụ
ớ ỗ

n x

< 1

0 < γ < 1 ✈➭
x ≤

δ
n x

x ≤

δ
< δ.
n

δ > 0 s❛♦ ❝❤♦ c1 + B(0, δ) ⊂ intP ✱ tr♦♥❣ ➤ã B(0, δ) = {x ∈ E :

x < δ}✳ ❉♦ tÝ♥❤ ❤ót ❝đ❛ B(0, δ) ♥➟♥ tå♥ t➵✐ m > 1 s❛♦ ❝❤♦ c2 ∈ mB(0, δ)✱
s✉② r❛

−c2 ∈ mB(0, δ) ✈➭ mc1 − c2 ∈ intP ✳

➜➷t

d = mc1 − c2 ✳

❑❤✐ ➤ã✱

d

t❤á❛ ♠➲♥ ✭✈✐✮✳

(vii)

> 0 s❛♦ ❝❤♦ c1 +B(0, δ ) ⊂ intP ✱ c2 +B(0, δ ) ⊂ intP

❈❤ä♥ δ

tr♦♥❣ ➤ã

B(0, δ ) = {x ∈ E : x < δ }✳ ❉♦ tÝ♥❤ ❤ót ❝ñ❛ B(0, δ ) ♥➟♥ tå♥ t➵✐ m > 0
s❛♦ ❝❤♦

c1 ∈ mB(0, δ )✱ c2 ∈ mB(0, δ )✱

s✉② r❛

−c1 ∈ mB(0, δ )✱ −c2 ∈

mB(0, δ ) ✈➭ mc1 −c1 ∈ intP ✱ mc2 −c2 ∈ intP ✳ ➜➷t e = mc1 −c1 +mc2 −c2 ✳
e t❤á❛ ♠➲♥ ✭✈✐✐✮✳

❑❤✐ ➤ã✱

(viii)
➤ã

x
n

x
n −a

●✐➯ sư

−a ∈ P

❱×

❱×

❧➭ ♥ã♥ ♥➟♥

a ≤ λa ♥➟♥ λa − a ∈ P
❚õ ➤ã s✉② r❛

−a ∈ P ✳ ❱× P
❚❛ ❝ã xn

▼➷t ❦❤➳❝✱

x
n

❧➭ ♥ã♥ ♥➟♥

−a =

❤❛②

1
1−λ a

x
n

=
✈➭

P

n = 1, 2, ...✱ ❞♦

→ 0 ♥➟♥

➤ã♥❣ tr♦♥❣

a(λ − 1) ∈ P ✳
∈ P✱

x
n

→ 0✳

❉♦ ➤ã

E ♥➟♥ −a ∈ P ✳

❤❛②

❉♦

0 < λ < 1 ♥➟♥

−a ∈ P ✳

◆❤➢ ✈❐②✱

a

✈➭

a = 0✳

≤ yn s✉② r❛ yn −xn ∈ P ✳ ❉♦ P

➤ã♥❣ ♥➟♥ limn→∞ (yn −xn )
♥➟♥

∈ P✳

limn→∞ (yn − xn ) = y − x✳

y − x ∈ P ✱ ❞♦ ➤ã x ≤ y ✳ ❍♦➭♥ t♦➭♥ t➢➡♥❣ tù ♥❤➢ tr➟♥✱ t❛ ❝❤ø♥❣

♠✐♥❤ ➤➢ỵ❝ tõ

❇ỉ ➤Ị✳

❧➭ ❞➲② tr♦♥❣

x
n ✈í✐ ♠ä✐

a≤

a = 0✳

limn→∞ xn = x✱ limn→∞ yn = y

❚õ ➤ã s✉② r❛

1.2.5

n = 1, 2, ...✳

a ✈➭ −a ∈ P ✳ ❱× P

1 − λ > 0✳
(x)

✈í✐ ♠ä✐

❚õ ❣✐➯ t❤✐Õt s✉② r❛

→ −a✳ ▼➷t ❦❤➳❝✱ ✈× ❞➲② { nx − a} ⊂ P

◆❤➢ ✈❐②

(ix)

x ∈ intP ✳

0 ≤ xn s✉② r❛ 0 ≤ x✳ ❱❐② 0 ≤ x ≤ y ✳
✭❬✹❪✮✳ ●✐➯ sö

P✳

P

❑❤✐ ➤ã✱ ♥Õ✉

❧➭ ♥ã♥ tr♦♥❣

E



{xn }

xn 0 tì ớ ỗ c ∈ intP ✱ tå♥ t➵✐ n0 ∈ N


12
s❛♦ ❝❤♦

xn

c ✈í✐ ♠ä✐ n ≥ n0 ✳

❈❤ø♥❣ ♠✐♥❤✳

✈×

intP

{xn } ❧➭ ❞➲② tr♦♥❣ P

●✐➯ sư

xn → 0✳

❱í✐ ♠ä✐

c ∈ intP ✱

δ > 0 s❛♦ ❝❤♦ c + BE (0, δ) ⊂ intP ✱ tr♦♥❣ ➤ã

❧➭ t❐♣ ♠ë ♥➟♥ tå♥ t➵✐

BE (0, δ) ❧➭ ❤×♥❤ ❝➬✉ ♠ë t➞♠ 0✱ ❜➳♥ ❦Ý♥❤ δ
x < δ t❤× c − x ∈ intP ✳

✈➭

❱í✐

tr♦♥❣

E✳

❉♦ ➤ã✱ ♥Õ✉

x∈E

♠➭

δ > 0 ①➳❝ ➤Þ♥❤ ♥❤➢ tr➟♥✱ tå♥ t➵✐ n0 ∈ N s❛♦

❝❤♦

x < δ, ∀n ≥ n0
❙✉② r❛

c − xn ∈ intP

1.3

❑❤➠♥❣ ❣✐❛♥

✈í✐ ♠ä✐

n ≥ n0 ✳ ❉♦ ➤ã✱ xn

c ✈í✐ ♠ä✐ n ≥ n0 ✳

b✲♠➟tr✐❝ ♥ã♥

▼ơ❝ ♥➭② trì ị ĩ í ụ ột số tí ❝❤✃t ❝➡ ❜➯♥ ❝đ❛ ❦❤➠♥❣
❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥✳
❚õ ➤➞② ✈Ị s❛✉✱ t❛ ❧✉➠♥ ❣✐➯ t❤✐Õt
❝đ❛

E

❜ë✐

P✳

✈í✐

1.3.1
❍➭♠

intP = ∅✱ ≤

➜Þ♥❤ ♥❣❤Ü❛✳

E

❧➭ ❦❤➠♥❣ ❣✐❛♥ tự

P

ó

E

ợ ị

sử X t rỗ d

: X ìX E



d ➤➢ỵ❝ ❣ä✐ ❧➭ b✲♠➟tr✐❝

❧➭ ❤❛✐ q✉❛♥ ❤Ư t❤ø tù tr➟♥

♥ã♥ tr➟♥

X

♥Õ✉ tå♥ t➵✐

s ≥ 1 s❛♦ ❝❤♦ ✈í✐ ♠ä✐

x, y, z ∈ X ✱ t❛ ❝ã
✶✮

d(x, y) ∈ P

✷✮

d(x, y) = d(y, x)❀

✸✮

d(x, y) ≤ s[d(x, z) + d(z, y)]✳

❚❐♣

X

✭tø❝

0 ≤ d(x, y)✮ ✈➭ d(x, y) = 0 ⇔ x = y ❀

❝ï♥❣ ✈í✐ ♠ét b✲♠➟tr✐❝ ♥ã♥

d tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝

♥ã♥ ✈í✐ t❤❛♠ sè

s ✈➭ ➤➢ỵ❝ ❦Ý ❤✐Ư✉ ❜ë✐ (X, d)✳

1.3.2

❚r♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✸✳✶✱ ♥Õ✉

❈❤ó ý✳

s = 1

tì t ợ ị

ĩ tr ó ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ◆ã✐ ❝➳❝❤ ❦❤➳❝✱ ❦❤➠♥❣ ❣✐❛♥


13
♠➟tr✐❝ ♥ã♥ ❧➭ tr➢ê♥❣ ❤ỵ♣ ➤➷❝ ❜✐Ưt ❝đ❛ ❦❤➠♥❣ ❣✐❛♥

b✲♠➟tr✐❝

♥ã♥ ❦❤✐

s = 1✳

❚å♥ t➵✐ ♥❤÷♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ ♠➭ ❦❤➠♥❣ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳
❚r♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✸✳✶✱ ♥Õ✉ ❧✃②
♥❣❤Ü❛ ❦❤➠♥❣ ❣✐❛♥

1.3.3

❱Ý ❞ô✳

E = R ✈➭ P = [0, ) tì t ợ ị

btr



E = R2 P = {(x, y) ∈ E : x, y ≥ 0}✱ X = R ✈➭

d : X × X → E ❧➭ ❤➭♠ ①➳❝ ➤Þ♥❤ ❜ë✐

d(x, y) = (|x − y|β , α|x − y|β ), ∀(x, y) ∈ X × X,
tr♦♥❣ ➤ã
❑❤✐ ➤ã✱

α ✈➭ β ❧➭ ❤❛✐ ❤➺♥❣ sè✱ α ≥ 0✱ β > 1✳

(X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ ✈í✐ t❤❛♠ sè s ≥ 2β > 1 ♥❤➢♥❣

(X, d) ❦❤➠♥❣ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳
❈❤ø♥❣ ♠✐♥❤✳

➜Ó ❝❤ø♥❣ ♠✐♥❤

(X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥✱

❧➢ỵt ❦✐Ĩ♠ tr❛ ❜❛ ➤✐Ị✉ ❦✐Ư♥ b✲♠➟tr✐❝ ♥ã♥ ❝đ❛ ❤➭♠
✶✮ ❍✐Ĩ♥ ♥❤✐➟♥

t❛ sÏ ❧➬♥

d✳ ❚❛ ❝ã

|x − y|β ≥ 0 ✈➭ α|x − y|β ≥ 0 ✈í✐ ♠ä✐ x, y ∈ R, α ≥

0, β > 1✳ ❉♦ ➤ã✱ d(x, y) ∈ P, ∀(x, y) X ì X
ữ

d(x, y) = (|x − y|β , α|x − y|β ) = (0, 0)
⇔

|x − y|β = 0
β

α|x − y| = 0

⇔ |x − y|β = 0 ⇔ x = y.

✷✮ ◆❤ê tÝ♥❤ ❝❤✃t ❝đ❛ ❣✐➳ trÞ t✉②Ưt ➤è✐

|x − y| = |y − x|, ∀x, y ∈ R✱ t❛

❞Ơ ❞➭♥❣ s✉② r❛ ➤➢ỵ❝

d(x, y) = (|x − y|β , α|x − y|β ) = (|y − x|β , α|y − x|β ) = d(y, x),
✈í✐ ♠ä✐

x, y ∈ R, α ≥ 0, β > 1✳

✸✮ ❱í✐ ♠ä✐

x, y, z ∈ R✱ t❛ ❝ã

d(x, y) ≤ s[d(x, z) + d(z, y)]
⇔ sd(x, z) + sd(z, y) − d(x, y) ∈ P
⇔ s(|x − z|β , α|x − z|β ) + s(|z − y|β , α|z − y|β ) − (|x − y|β , α|x − y|β ) ∈ P


14
⇔ (s|x − z|β + s|z − y|β − |x − y|β , α[s|x − z|β + s|z − y|β − |x − y|β ]) ∈ P
⇔ s|x − z|β + s|z − y|β ≥ |x − y|β .

s ≥ 2β

▼➷t ❦❤➳❝✱ ✈×

♥➟♥

|x − y|β ≤ (|x − y| + |y − z|)β ≤ (2 max{|x − y|, |y − z|})β
2β (max{|x − y|, |y − z|})β ≤ 2β (|x − z|β + |z − y|β ) ≤ s(|x − z|β + |z − y|β ).
β
β
❉♦ ➤ã s|x−z| +s|z−y|
✈í✐ ♠ä✐
❱❐②

≥ |x−y|β ✳ ◆❤➢ ✈❐②✱ d(x, y) ≤ s[d(x, z)+d(z, y)]✱

s ≥ 2β > 1✳

(X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ ✈í✐ t❤❛♠ sè s ≥ 2β > 1✳

❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣

s = 1, x = 4, y = 0, z = 1, β = 2 ✈➭ α = 1✱ t❛ ❝ã
|4 − 1|2 + |1 − 0|2 < |4 − 0|2 .

❚õ ➤ã s✉② r❛

d(4, 0) > d(4, 1) + d(1, 0)✳

❉♦ ➤ã

(X, d) ❦❤➠♥❣ ❧➭ ❦❤➠♥❣ ❣✐❛♥

♠➟tr✐❝ ♥ã♥✳

1.3.4
✈➭

➜Þ♥❤ ♥❣❤Ü❛✳

✭❬✼❪✮✳ ●✐➯ sö

(X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥✱ x ∈ X

{xn } ❧➭ ❞➲② tr♦♥❣ X ✳
✶✮ ❉➲②

❤♦➷❝

{xn } ➤➢ỵ❝ ❣ä✐ ❧➭ ❤é✐ tơ tí✐ x ✈➭ ➤➢ỵ❝ ❦Ý ❤✐Ư✉ ❜ë✐ limn→∞ xn = x
xn → x ♥Õ✉ ✈í✐ ♠ä✐ c ∈ intP ✱ tå♥ t➵✐ sè tù ♥❤✐➟♥ nc s❛♦ ❝❤♦
d(x, xn )

✷✮ ❉➲②

c

✈í✐ ♠ä✐

n ≥ nc ;

{xn } ➤➢ỵ❝ ❣ä✐ ❧➭ ❞➲② ❈❛✉❝❤② ♥Õ✉ ✈í✐ ♠ä✐ c ∈ intP ✱ tå♥ t➵✐ sè tù

♥❤✐➟♥

nc s❛♦ ❝❤♦
d(xn , xm )
b

✸✮ ❑❤➠♥❣ ❣✐❛♥ ✲♠➟tr✐❝ ♥ã♥

tr♦♥❣

X

➤Ị✉ ❤é✐ tơ✳

c

✈í✐ ♠ä✐

m, n ≥ nc ;

(X, d) ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➬② ➤đ ♥Õ✉ ♠ä✐ ❞➲② ❈❛✉❝❤②


15
1.3.5

●✐➯ sư

❇ỉ ➤Ị✳

{xn }

❧➭ ❞➲② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

b✲♠➟tr✐❝ (X, d)

✈➭

xn → x ∈ X ✳ ❑❤✐ ➤ã✱ t❛ ❝ã
✶✮

{xn } ❧➭ ❞➲② ❈❛✉❝❤②❀

✷✮

x ❧➭ ❞✉② ♥❤✃t❀
y∈X

✸✮ ❱í✐ ♠ä✐

✈➭ ✈í✐ ♠ä✐

c ∈ intP ✱ tå♥ t➵✐ sè tù ♥❤✐➟♥ n0 s❛♦ ❝❤♦

1
d(x, y) − c ≤ d(xn , y) ≤ sd(x, y) + c, n n0 .
s

ứ

1)

ớ ỗ

c
2s ớ ♠ä✐

d(xn , x)

c ∈ intP ✱ ✈× xn → x ♥➟♥ tå♥ t➵✐ nc ∈ N s❛♦ ❝❤♦

n ≥ nc ✳ ❉♦ ➤ã✱ ✈í✐ ♠ä✐ n ✈➭ m ≥ nc ✱ t❛ ❝ã

|d(xn , xm )| ≤ s[d(xn , x) + d(xm , x)]
❉♦ ➤ã✱

2)

c.

{xn } ❧➭ ❞➲② ❈❛✉❝❤②✳

●✐➯ sö

xn → x ✈➭ xn → y ✳ ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐ c ∈ intP ✱ tå♥ t➵✐ nc ∈ N s❛♦

❝❤♦ ✈í✐ ♠ä✐

n ≥ nc ✱ t❛ ❝ã
d(xn , x)

c
, d(xn , y)
2s

c
.
2s

❉♦ ➤ã

d(x, y) ≤ s[d(x, xn ) + d(xn , y)]
❑Õt ❤ỵ♣ ✈í✐ ❇ỉ ➤Ị ✶✳✷✳✹ s✉② r❛

3)

❱í✐ ỗ

c.

d(x, y) = 0 tứ x = y

y X ✱ t❤❡♦ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✱ t❛ ❝ã
d(x, y) ≤ s[d(x, xn ) + d(xn , y)], ∀n = 1, 2, ...

❚õ ➤ã s✉② r❛ ✈í✐ ♠ä✐

n = 1, 2, ...✱ t❛ ❝ã

1
d(x, y) − d(x, xn ) ≤ d(xn , y) ≤ sd(xn , x) + sd(x, y).
s
▼➷t ❦❤➳❝✱ ✈× xn

c
s ✈í✐ ♠ä✐

→ x ♥➟♥ ✈í✐ ♠ä✐ c ∈ intP ✱ tå♥ t➵✐ n0 ∈ N s❛♦ ❝❤♦ d(xn , x)

n ≥ n0 ✳ ❉♦ ➤ã✱ ✈í✐ ♠ä✐ n ≥ n0 ✱ t❛ ❝ã

1
1
c
d(x, y)−c ≤ d(x, y)−
s
s
s

1
d(x, y)−d(x, xn ) ≤ d(xn , y) ≤ c+sd(x, y).
s


16
◆❤➢ ✈❐②

1

d(x, y) − c ≤ d(xn , y) ≤ sd(x, y) + c, ∀n ≥ n0 .
s


17
❝❤➢➡♥❣ ✷

❙ù tå♥ t➵✐ ➤✐Ó♠ ❜✃t ➤é♥❣
tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

b✲♠➟tr✐❝ ♥ã♥

❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ➤➬✉ t✐➟♥✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ❦Õt q✉➯ ✈Ị sù
tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ➤➲ ❝ã tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠
❦❤➯♦✳ ❙❛✉ ➤ã✱ ❝❤ó♥❣ t➠✐ ♠ë ré♥❣ ❝➳❝ ❦Õt q✉➯ ♥➭② ❝❤♦ ❦❤➠♥❣ ❣✐❛♥

b✲♠➟tr✐❝

♥ã♥✳

2.1

▼ét sè ❦Õt q✉➯ ✈Ị sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣
❦❤➠♥❣ ❣✐❛♥

b✲♠➟tr✐❝

❚r♦♥❣ ♠ơ❝ ♥➭②✱ ❝❤ó♥❣ t❛ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ✈Ị sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t
➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ❇❛♥❛❝❤✱ ❈❤❛tt❡r❥❡❛✱ ❑❛♥♥❛♥✱✳✳✳ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

b✲♠➟tr✐❝ ➤➲ ➤➢ỵ❝ ❣✐í✐ t❤✐Ư✉ tr♦♥❣ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❬✾❪✳
2.1.1

➜Þ♥❤ ♥❣❤Ü❛✳

●✐➯ sư

(X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ✈➭ f : X → X ✳

➳♥❤ ①➵ f ➤➢ỵ❝ ❣ä✐ ❧➭ ❝♦ ❦✐Ó✉ ❇❛♥❛❝❤ ♥Õ✉ tå♥ t➵✐ α ∈ [0, 1) s❛♦ ❝❤♦
d(f x, f y) ≤ αd(x, y), ∀x, y ∈ X
❑❤✐ ➤ã✱

2.1.2

α ➤➢ỵ❝ ❣ä✐ ❧➭ ❤➺♥❣ sè ❝♦ ❝đ❛ f ✳
❈❤ó ý✳

t❤❡♦ ♥❣❤Ü❛✱ tõ
❈❤ø♥❣ ♠✐♥❤✳

❞➲② tr♦♥❣

◆Õ✉

f :X →X

❧➭ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ❇❛♥❛❝❤ t❤×

{xn } ❧➭ ❞➲② tr♦♥❣ X

●✐➯ sö

f

✈➭

xn → x ∈ X

❦Ð♦ t❤❡♦

0 ≤ d(f xn , f x) ≤ αd(xn , x) → 0 ❦❤✐ n → ∞.

❧✐➟♥ tô❝

f xn → f x✳

❧➭ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ❇❛♥❛❝❤ ✈í✐ ❤➺♥❣ sè

X ✱ xn → x ∈ X ✳ ❑❤✐ ➤ã✱ t❛ ❝ã

f

α ✈➭ {xn } ❧➭


18
❉♦ ➤ã✱

d(f xn , f x) → 0 ❦❤✐ n → ∞ tø❝ f xn → f x✳

2.1.3

➜Þ♥❤ ❧ý✳

X→X
f

✭❬✾❪✮✳ ●✐➯ sư

(X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ➤➬② ➤đ ✈➭ f :

❧➭ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ❇❛♥❛❝❤ ✈í✐ ❤➺♥❣ sè ❝♦

❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t

❈❤ø♥❣ ♠✐♥❤✳ ▲✃②

x0 ∈ X

α✳ ❑❤✐ ➤ã✱ ♥Õ✉ α <

x∗ ✈➭ f n x0 → x∗ ✈í✐ ♠ä✐ x0 ∈ X ✳

✈➭ ①➞② ❞ù♥❣ ❞➲②

{xn } tr♦♥❣ X

❜ë✐

xn+1 = f xn = f n+1 x0 ∀n = 0, 1, ....

❱×

f

1
s t❤×

❧➭ ➳♥❤ ①➵ ❝♦ ✈í✐ ❤➺♥❣ sè ❝♦

(2.1)

α ∈ [0, 1) ♥➟♥ t❛ ❝ã

d(xn , xn+1 ) = d(f xn−1 , f xn ) ≤ αd(xn−1 , xn )
= αd(f xn−2 , f xn−1 ) ≤ α2 d(xn−2 , xn−1 )
≤ ... ≤ αn d(x0 , x1 ) ∀n = 1, 2, ....

(2.2)

❙ư ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝ ✈➭ ✭✷✳✷✮ t❛ ❝ã

d(xn , xn+p ) ≤ s[d(xn , xn+1 ) + d(xn+1 , xn+p )]
≤ sd(xn , xn+1 ) + s2 [d(xn+1 , xn+2 ) + d(xn+2 , xn+p )]
≤ ... ≤ sd(xn , xn+1 ) + s2 d(xn+1 , xn+2 ) + ...+
+ sp−1 [d(xn+p−2 , xn+p−1 ) + d(xn+p−1 , xn+p )]
≤ sαn d(x0 , x1 ) + s2 αn+1 d(x0 , x1 ) + ...+
+ sp−1 αn+p−2 d(x0 , x1 ) + sp−1 αn+p−1 d(x0 , x1 )
≤ (sαn + s2 αn+1 + ... + sp−1 αn+p−2 + sp−1 αn+p−1 )d(x0 , x1 )
1 − (sα)p
sαn

= sαn
d(x0 , x1 ) <
d(x0 , x1 )
1 − sα
1 − sα
(2.3)
✈í✐ ♠ä✐
❉♦

n = 1, 2, ... ✈➭ ♠ä✐ p = 0, 1, ... ✭✈× sα < 1✮✳

α ∈ [0, 1) ♥➟♥

sαn
1−sα d(x0 , x1 )

→ 0 ❦❤✐ n → ∞✳ ❑Õt ❤ỵ♣ ✈í✐ ✭✷✳✸✮ s✉② r❛

lim d(xn , xn+p ) = 0 ∀p = 0, 1, ...

n→∞
➜✐Ò✉ ♥➭② ❝❤ø♥❣ tá

{xn } ❧➭ ❞➲② ❈❛✉❝❤② tr♦♥❣ (X, d)✳ ❱× (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥

b✲♠➟tr✐❝ ➤➬② ➤đ ♥➟♥ tå♥ t➵✐ x∗ ✱ s❛♦ ❝❤♦ f n x0 = xn → x∗ ✳


19
❇➞② ❣✐ê✱ t❛ ❝❤ø♥❣ ♠✐♥❤

t❤❡♦ ❈❤ó ý ✷✳✶✳✷✱

f

x∗ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ f ✳ ❱× f

❧✐➟♥ tơ❝✳ ❉♦ ➤ã✱

❧➭ ➳♥❤ ①➵ ❝♦ ♥➟♥

f xn → f x∗ tø❝ ❧➭ xn+1 → f x∗ ✳ ▼➷t ❦❤➳❝✱

xn+1 → x∗ ✳ ❉♦ ➤ã✱ f x∗ = x∗ ✳ ❱❐② x∗ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ f ✳
●✐➯ sư

y∈X

❝ị♥❣ ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛

f

tø❝ ❧➭

f y = y✳

❑❤✐ ➤ã✱

t❛ ❝ã

d(x∗ , y) = d(f x∗ , f y) ≤ αd(x∗ , y).

❱×

α ∈ [0, 1) ♥➟♥ d(x∗ , y) = 0✱ tø❝ ❧➭ x∗ = y ✳

❱❐② ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛

f

❧➭

❞✉② ♥❤✃t✳

➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ❇❛♥❛❝❤ ❧➭ ❧✐➟♥ tơ❝✳ ❉♦ ➤ã ♠ét ✈✃♥ ➤Ò ➤➷t r❛ ♠ét ❝➳❝❤ tù
♥❤✐➟♥ ❧➭ ❝ã t❤Ĩ ➤➢❛ r❛ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝♦ s❛♦ ❝❤♦ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉
❦✐Ư♥ ❝♦ ♥➭② sÏ ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ♥❤➢♥❣ ❦❤➠♥❣ ❧✐➟♥ tơ❝✳ ➜Ĩ ❣✐➯✐ q✉②Õt ✈✃♥ ➤Ò
♥➭②✱ tr♦♥❣ ❬✸❪ ✈➭ ❬✼❪✱ ❑❛♥♥❛♥ ✈➭ ❈❤❛tt❡r❥❡❛ ➤➲ ➤➢❛ r❛ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ❝♦ ❦✐Ĩ✉
❑❛♥♥❛♥ ✈➭ ❝♦ ❦✐Ĩ✉ ❈❤❛tt❡r❥❡❛ s❛✉ ➤➞②✳

2.1.4

➜Þ♥❤ ♥❣❤Ü❛✳

✶✮ ✭❬✼❪✮

➳♥❤ ①➵ f

●✐➯ sư

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

➤➢ỵ❝ ❣ä✐ ❧➭ ❝♦ ❦✐Ó✉ ❑❛♥♥❛♥ ♥Õ✉ tå♥ t➵✐

α ∈ [0, 21 ) s❛♦

❝❤♦

d(f x, f y) ≤ α[d(x, f x) + d(y, f y)], ∀x, y ∈ X.
✷✮ ✭❬✸❪✮

➳♥❤ ①➵ f ➤➢ỵ❝ ❣ä✐ ❧➭ ❝♦ ❦✐Ĩ✉ ❈❤❛tt❡r❥❡❛ ♥Õ✉ tå♥ t➵✐ α ∈ [0, 12 ) s❛♦

❝❤♦

d(f x, f y) ≤ α[d(x, f y) + d(y, f x)], ∀x, y ∈ X.
◆Õ✉

(X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ t❤× ❑❛♥♥❛♥ ✭❬✼❪✮ ➤➲ ❝❤ø♥❣ ♠✐♥❤

♠ä✐ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ❑❛♥♥❛♥ tr➟♥

X ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ❝ß♥ ❈❤❛tt❡r❥❡❛

✭❬✸❪✮ ➤➲ ❝❤ø♥❣ tá ♠ä✐ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ❈❤❛tt❡r❥❡❛ tr➟♥

X

❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉②

♥❤✃t✳
➜Ĩ ♠ë ré♥❣ ❝➳❝ ❦Õt q✉➯ tr➟♥ ➤➞② ❝đ❛ ❑❛♥♥❛♥ ✈➭ ❈❤❛tt❡r❥❡❛ ❝❤♦ tr➢ê♥❣

❤ỵ♣ ❦❤➠♥❣ ❣✐❛♥
➤➞②✳

b✲♠➟tr✐❝✱

▼✳ ❑✐r ✈➭ ❍✳ ❑✐③✐❧t✉♥❝ ➤➲ ➤➢❛ r❛ ❤❛✐ ➤Þ♥❤ ❧ý s❛✉


20
2.1.5
➤đ ✈í✐

➜Þ♥❤ ❧ý✳

✭❬✾❪ ❚❤❡♦r❡♠

s ≥ 1 ✈➭ f : X → X

2✮✳ ●✐➯ sö (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ➤➬②

❧➭ ➳♥❤ ①➵ s❛♦ ❝❤♦ tå♥ t➵✐

µ ∈ [0, 21 ) t❤á❛ ♠➲♥

d(f x, f y) ≤ µ[d(x, T x) + d(y, T y)] ∀x, y ∈ X.
f

❑❤✐ ➤ã✱

❝ã ❞✉② ♥❤✃t ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣

x∗ ∈ X

✈➭

(2.4)

f n x0 → x∗

✈í✐ ♠ä✐

x0 ∈ X ✳
❈❤ø♥❣ ♠✐♥❤✳ ▲✃②

x0 ∈ X

✈➭ ①➞② ❞ù♥❣ ❞➲②

{xn } tr♦♥❣ X

❜ë✐

xn+1 = f xn = f n+1 x0 ∀n = 0, 1, ...
❚❤❡♦ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✹✮✱ t❛ ❝ã

d(xn , xn+1 ) = d(f xn−1 , f xn ) ≤ µ[d(xn−1 , xn ) + d(xn , xn+1 )]
✈í✐ ♠ä✐

n = 1, 2, ... ❉♦ ➤ã
d(xn , xn+1 ) ≤

µ
d(xn−1 , xn ) ∀n = 1, 2, ...
1à

(2.5)

ừ s r

d(xn , xn+1 )
ì

à [0, 21 ) ♥➟♥

µ
1−µ

n

µ
1−µ

< 1✳ ❉♦ ➤ã✱ f

d(x0 , x1 ) ∀n = 1, 2, ...

❧➭ ➳♥❤ ①➵ ❝♦✳ ❚✐Õ♣ t❤❡♦✱ ❜➺♥❣ ❝➳❝❤ ❝❤ø♥❣

♠✐♥❤ t➢➡♥❣ tù ♥❤➢ tr♦♥❣ ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ✷✳✶✳✸ ✭❬✾❪ ❚❤❡♦r❡♠
❧✉❐♥ ➤➢ỵ❝

{xn } ❧➭ ❞➲② ❈❛✉❝❤②✳

X

❧➭ ➤➬② ➤đ ♥➟♥ tå♥ t➵✐

❱×

(2.6)

x∗ ∈ X

s❛♦ ❝❤♦

1✮

t❛ ❦Õt

xn → x∗ ✳ ❙ư ❞ơ♥❣ ❜✃t ➤➻♥❣

t❤ø❝ t❛♠ ❣✐➳❝ ✈➭ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✹✮ t❛ ❝ã

d(x∗ , f x∗ ) ≤ s[d(x∗ , xn ) + d(xn , f x∗ )]
= sd(x∗ , xn ) + sd(f xn−1 , f x∗ )
≤ sd(x∗ , xn ) + sµ[d(xn−1 , xn ) + d(x∗ , f x∗ )].

(2.7)

s

sµ
d(x∗ , xn ) +
d(xn−1 , xn )
1 − sµ
1 − sµ

(2.8)

❉♦ ➤ã

0 ≤ d(x∗ , f x∗ ) ≤


21
✈í✐ ♠ä✐

n = 1, 2, ... ❱× xn → x∗

♥➟♥ s✉② r❛ ✈Õ ♣❤➯✐ ❝đ❛ ✭✷✳✽✮ ❞➬♥ tí✐

0 ❦❤✐

n → ∞✳ ❑Õt ❤ỵ♣ ✈í✐ ✭✷✳✽✮ s✉② r❛ d(f x∗ , x∗ ) = 0 tø❝ x∗ = f x∗ ✳ ◆❤➢ ✈❐②✱ x∗
❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛
●✐➯ sư

y∈X

f✳

❝ị♥❣ ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛

f ✳ ❑❤✐ ➤ã✱ t❤❡♦ ➤✐Ị✉ ❦✐Ư♥

✭✷✳✹✮ t❛ ❝ã

0 ≤ d(x∗ , y) = d(f x∗ , f y) ≤ µ[d(x∗ , f x∗ ) + d(y, f y)] = 0.
❉♦ ➤ã✱

d(x∗ , y) = 0 tø❝ x∗ = y ✳ ❱❐② ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ f

2.1.6

➜Þ♥❤ ❧ý✳

➤đ✱

f :X →X

✭❬✾❪ ❚❤❡♦r❡♠

3✮✳

●✐➯ sư

❧➭ ❞✉② ♥❤✃t✳

(X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ➤➬②

❧➭ ➳♥❤ ①➵ s❛♦ ❝❤♦ tå♥ t➵✐

λ ∈ [0, 12 ) t❤á❛ ♠➲♥ λs ∈ [0, 12 )

✈➭

d(f x, f y) ≤ λ[d(x, f y) + d(y, f x)] ∀x, y ∈ X.
❑❤✐ ➤ã✱

f

❝ã ❞✉② ♥❤✃t ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣

x∗ ∈ X

✈➭

f n x0 → x∗

(2.9)
✈í✐ ♠ä✐

x0 ∈ X ✳
●✐➯ sư

❈❤ø♥❣ ♠✐♥❤✳

{xn } ⊂ X

❧➭ ❞➲② ➤➢ỵ❝ ①➞② ❞ù♥❣ ♥❤➢ tr♦♥❣ ➜Þ♥❤ ❧ý

✷✳✶✳✺✳ ❚❤❡♦ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✾✮ t❛ ❝ã

d(xn , xn+1 ) = d(f xn−1 , f xn ) ≤ λ[d(xn−1 , xn+1 ) + d(xn , xn )]
≤ sλ[d(xn−1 , xn ) + d(xn , xn+1 )] ∀n = 1, 2, ...
❉♦ ➤ã✱ t❛ ❝ã

d(xn , xn+1 ) ≤

λs
d(xn−1 , xn ) ∀n = 1, 2, ...
1 − λs

(2.10)

❚õ ✭✷✳✶✵✮ s✉② r❛

d(xn , xn+1 ) ≤
❱× λs

λs
1 − λs

n

d(x0 , x1 ) ∀n = 1, 2, ...

(2.11)

λs
∈ [0, 21 ) ♥➟♥ 1−λs

< 1✳ ❉♦ ➤ã ❜➺♥❣ ❝➳❝❤ ❝❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù ♥❤➢ tr♦♥❣

❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ✷✳✶✳✸✱ t ết ợ {xn } ì (X, d) ➤➬②
➤ñ ♥➟♥ tå♥ t➵✐

x∗ ∈ X

s❛♦ ❝❤♦

xn → x∗ ✳


22
∗
❇➞② ❣✐ê t❛ ❝❤ø♥❣ tá x ❧➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝đ❛ f ✳ ❚❛ ❝ã✱ ✈í✐ ♠ä✐ n

= 1, 2, ...

d(x∗ , f x∗ ) ≤ sd(x∗ , xn+1 ) + sd(xn+1 , f x∗ )
= sd(x∗ , xn+1 ) + sd(f xn , f x∗ )
≤ sd(x∗ , xn+1 ) + sλ[d(xn , f x∗ ) + d(x∗ , xn+1 )]
= sd(x∗ , xn+1 ) + sλd(x∗ , xn+1 ) + sλd(xn , f x∗ )
❚r♦♥❣ ✭✷✳✶✷✮ ❝❤♦

n → ∞ t❛ ➤➢ỵ❝
d(x∗ , f x∗ ) ≤ sλd(x∗ , f x∗ )

❚õ

(2.12)

(2.13)

λs ∈ [0, 21 ) s✉② r❛ d(x∗ , f x∗ ) = 0 tø❝ ❧➭ x∗ = f x∗ ✳ ❉♦ ➤ã✱ x∗ ❧➭ ➤✐Ĩ♠ ❜✃t

➤é♥❣ ❝đ❛

f✳

●✐➯ sư

y∈X

❝ị♥❣ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛

f ✳ ❑❤✐ ➤ã✱ t❤❡♦ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✾✮

t❛ ❝ã

d(x∗ , y) = d(f x∗ , f y) ≤ λ[d(y, f x∗ ) + d(f y, x∗ )]
= λ[d(y, x∗ ) + d(y, x∗ )] = 2λd(x∗ , y).
λ<

❑Õt ❤ỵ♣ ✈í✐

f

1
2 s✉② r❛

d(x∗ , y) = 0 tø❝ ❧➭ x∗ = y ✳ ❱❐② ➤✐Ó♠ ❜✃t ➤é♥❣ ❝đ❛

❧➭ ❞✉② ♥❤✃t✳

2.1.7

◆❤❐♥ ①Ðt✳

❚r♦♥❣ ✈✐Ư❝ ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ✷✳✶✳✺ ✈➭ ➜Þ♥❤ ❧ý ✷✳✶✳✻ ✭tø❝

❚❤❡♦r❡♠ 2 ✈➭ ❚❤❡♦r❡♠ 3 tr♦♥❣ ❬✾❪✮✱ ❝➳❝ t➳❝ ❣✐➯ ▼✳ ❑✐r ✈➭ ❍✳ ❑✐③✐❧t✉♥❝ ➤➲ ♣❤➵♠
❝➳❝ s❛✐ ❧➬♠ s❛✉ ➤➞②
✶✮ ◆Õ✉

1 − sµ ≤ 0 tì từ s r ợ

ế btr

d tụ tì từ s ợ ✭✷✳✶✸✮✳

❚r♦♥❣ ❬✼❪✱ ◆✳ ❍✉ss❛✐♥ ✈➭ ❝➳❝ ❝é♥❣ sù ➤➲ ➤➢❛ r❛ ✈Ý ❞ơ ❝❤ø♥❣ tá tå♥ t➵✐ ♥❤÷♥❣

b✲♠➟tr✐❝ ❦❤➠♥❣ ❧✐➟♥ tơ❝✳
◆❤➢ ✈❐②✱ tõ ✭✷✳✶✷✮ ❦❤➠♥❣ s✉② r❛ ➤➢ỵ❝ ✭✷✳✶✸✮✱ ♠➭ t❛ ❝❤ø♥❣ ♠✐♥❤ ♥❤➢ s❛✉✳
❚❤❡♦ ✭✷✳✶✷✮ t❛ ❝ã

d(x∗ , f x∗ ) ≤ sd(x∗ , xn+1 ) + sλd(x∗ , xn+1 ) + sλd(xn , f x∗ )
= sd(x∗ , xn+1 ) + sλd(x∗ , xn+1 ) + s2 λ[d(xn , x∗ ) + d(x∗ , f x∗ )].

23
◆Õ✉

1 − s2 λ > 0 t❤× ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② s✉② r❛
s(1 + λ)
s2 λ
∗
0 ≤ d(x , f x ) ≤
d(x , xn+1 ) +
d(xn , x∗ ) → 0.
2
2
1−s λ
1−s λ
∗

❉♦ ➤ã✱

∗

d(x∗ , f x∗ ) = 0 tø❝ ❧➭ f x∗ = x∗ ✳

✸✮ ❱í✐ ❝➳❝ tết tr ị ý tì ❜➺♥❣ ❝➳❝❤ ❝❤ø♥❣
♠✐♥❤ t➢➡♥❣ tù ♥❤➢ ❈❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ✷✳✶✳✸ ✭tø❝ ❚❤❡♦r❡♠ 1 tr♦♥❣ ❬✾❪✮ ❦❤➠♥❣
❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝

{xn }

❧➭ ❞➲② ❈❛✉❝❤②✱ ♠➭ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤ ❜➺♥❣ ♣❤➢➡♥❣

♣❤➳♣ ❦❤➳❝ ♥❤➢ s❛✉
➜è✐ ✈í✐ ➜Þ♥❤ ❧ý ✷✳✶✳✺✱ sư ❞ơ♥❣ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✹✮ ✈➭ ✭✷✳✻✮ ts ❝ã

d(xn , xm ) = d(f xn−1 , f xm−1 ) ≤ µ[d(xn−1 , xn ) + d(xm−1 , xm )]
≤ µd(x0 , x1 ) (
❦❤✐

µ m−1
µ n−1
)
+(
)
→0
1−µ
1−µ

n, m → ∞✳ ❉♦ ➤ã✱ {xn } ❧➭ ❞➲② ❈❛✉❝❤②✳
➜è✐ ✈í✐ ➜Þ♥❤ ❧ý ✷✳✶✳✻✱ sư ❞ơ♥❣ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✾✮✱ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝ ✈➭

✭✷✳✶✶✮✱ t❛ ❝ã

d(xn , xm ) = d(f xn−1 , f xm−1 ) ≤ λ[d(xn−1 , xm ) + d(xm−1 , xn )]
≤ sλ[d(xn−1 , xn ) + d(xn , xm )] + sλ[d(xm−1 , xm ) + d(xm , xn )].
❚õ ➤ã s✉② r❛

sλ
[d(xn−1 , xn ) + d(xm−1 , xm )]
1 − 2sλ
sλ
sλ n−1

sλ m−1
≤
d(x0 , x1 ) (
)
+(
)
→ 0.
1 − 2sλ
1 − sλ
1 − sλ

d(xn , xm ) ≤

❦❤✐

n, m → ∞✳ ❉♦ ➤ã✱ {xn } ❧➭ ❞➲② ❈❛✉❝❤②✳
◆❤➢ ✈❐②✱ tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✺ ❝➬♥ ❜ỉ s✉♥❣ t❤➟♠ ➤✐Ị✉ ❦✐Ư♥

tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✻ ❝➬♥ ❜ỉ s✉♥❣ t❤➟♠ ➤✐Ị✉ ❦✐Ư♥

s 2 µ < 1

sà < 1 ò