Điểm bất động của các ánh xạ co suy rộng trong không gian g mêtric

❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦

❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤

◆❣✉②Ơ♥ ❍♦➭♥❣ ❍➢♥❣

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

G✲♠➟tr✐❝

▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝

◆❣❤Ö ❆♥ ✲ ✷✵✶✻


❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦
❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤

◆❣✉②Ơ♥ ❍♦➭♥❣ ❍➢♥❣

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

G✲♠➟tr✐❝

▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝
❈❤✉②➟♥ ♥❣➭♥❤✿

❚♦➳♥ ●✐➯✐ tÝ❝❤

▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷

❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝
P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥

◆❣❤Ö ❆♥ ✲ ✷✵✶✻


▼ơ❝ ▲ơ❝

❚r❛♥❣
▼ơ❝ ❧ơ❝

✐

▼ë ➤➬✉

✐✐

❈❤➢➡♥❣ ✶✳

➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉

Φ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲

♠➟tr✐❝

✶

✶✳✶

❈➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✷

➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉

❈❤➢➡♥❣ ✷✳

Φ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝

✶✹

➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵
❣✐❛♥

✷✳✷

✻

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

G✲♠➟tr✐❝
✷✳✶

✶

G✲♠➟tr✐❝

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

➜✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵
❣✐❛♥

G✲♠➟tr✐❝

G✲❝♦ ②Õ✉ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣
α✲❝♦

✶✹

②Õ✉ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣

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

✷✸

❑Õt ❧✉❐♥

✸✷

❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦

✸✸

✐


▼ë ➤➬✉
✶✳ ▲ý ❞♦ ❝❤ä♥ ➤Ò t➭✐

❚r♦♥❣ ✈➭✐ t❤❐♣ ❦û ❣➬♥ ➤➞②✱ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ♠➟tr✐❝ ➤➲ trë t❤➭♥❤
♠ét ❧Ü♥❤ ✈ù❝ ♥❣❤✐➟♥ ❝ø✉ q✉❛♥ trä♥❣ tr♦♥❣ ❦❤♦❛ ❤ä❝ t❤✉➬♥ tó② ✈➭ ❦❤♦❛ ❤ä❝ ø♥❣
❞ơ♥❣✳ ❚r♦♥❣ t❤ù❝ tÕ✱ ♥ã ➤➲ trë t❤➭♥❤ ♠ét tr♦♥❣ ♥❤÷♥❣ ❝➠♥❣ ❝ơ ❝èt ②Õ✉ ♥❤✃t
tr♦♥❣ ❣✐➯✐ tÝ❝❤ ❤➭♠ ♣❤✐ t✉②Õ♥✱ tè✐ ➢✉ ❤ã❛✱ t♦➳♥ ❤ä❝✱ ❝➳❝ ♠➠ ❤×♥❤ t♦➳♥ ❤ä❝✱ ❦✐♥❤
tÕ ✈➭ ② ❤ä❝✳ ❈➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ị♥❣ ➤ã♥❣ ♠ét
✈❛✐ trß q✉❛♥ trä♥❣ tr♦♥❣ ✈✐Ư❝ ①➞② ❞ù♥❣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ tr♦♥❣ t♦➳♥ ❤ä❝ ➤Ó ❣✐➯✐
q✉②Õt ❝➳❝ ✈✃♥ ➤Ị tr♦♥❣ t♦➳♥ ❤ä❝ ø♥❣ ❞ơ♥❣ ✈➭ ❦❤♦❛ ❤ä❝✳ ❱× ✈❐②✱ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥
♠➟tr✐❝ ✈➭ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣ ➤➲ ❧➠✐ ❝✉è♥ ♠ét sè ❧➢ỵ♥❣ ❧í♥ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝
❝ị♥❣ ❧➭ ➤✐Ị✉ ❞Ơ ❤✐Ĩ✉✳ ▼ét sè ♠ë ré♥❣ ❝đ❛ ❦❤➳✐ ♥✐Ư♠ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➲ ➤➢ỵ❝
➤Ị ①✉✃t ❜ë✐ ♠ét sè t➳❝ ❣✐➯✳ ◆➝♠ ✶✾✾✼✱ ❨✳ ■✳ ❆❧❜❡r ✈➭ ❙✳ ●✉❡rr❡ ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐
♥✐Ư♠ ➳♥❤ ①➵

α✲❝♦ ②Õ✉ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➭ t❤✐Õt ❧❐♣ ♠ét ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t

➤é♥❣ ❝❤♦ ❧í♣ ➳♥❤ ①➵ ➤ã✳ ❙❛✉ ➤ã ♥➝♠ ✷✵✵✶✱ ❇✳ ❊✳ ❘❤♦❛❞❡s ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣
❦❤➳✐ ♥✐Ư♠ ➳♥❤ ①➵ ❝♦ ế t ợ ột ị ý ể t ộ tr♦♥❣ ❦❤➠♥❣
❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✳ ❙❛✉ ➤ã✱ ♥❤✐Ị✉ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵
t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ♥➭♦ ➤ã ➤➲ ➤➢ỵ❝ ♥❣❤✐➟♥ ❝ø✉ ❜ë✐ ♥❤✐Ò✉ t➳❝ ❣✐➯ ♥❤➢✿ ■✳
❇❡❣ ✈➭ ▼✳ ❆❜❜❛s ✭✷✵✵✻✮✱ P✳ ◆✳ ❉✉tt❛ ✈➭ ❇✳ ❙✳ ❈❤♦✉❞❤✉r② ✭✷✵✵✽✮✱ ❲✳ ❙❤❛t❛♥❛✇✐
✭✷✵✶✵✮✱ ❍✳ ❆②❞✐ ✈➭ ❝➳❝ ❝é♥❣ sù ✭✷✵✵✶✮ ✈➭ ❖✳ ❩❤❛♥❣ ✈➭ ❨✳ ❙♦♥❣ ✭✷✵✵✾✮✳ ◆➝♠ ✶✾✻✻✱
❙✳ ●❛❤❧❡r ➤➲ ❣✐í✐ t❤✐Ư✉ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈Ị ❦❤➠♥❣ ❣✐❛♥
❉❤❛❣❡ ➤➲ ❣✐í✐ t❤✐Ư✉ ❝➳❝ ❦❤➳✐ ♥✐Ö♠ ❦❤➠♥❣ ❣✐❛♥

2✲♠➟tr✐❝ ✈➭ ♥➝♠ ✶✾✾✷ ❇✳ ❈✳

D✲♠➟tr✐❝✳

❙❛✉ ➤ã✱ ♥➝♠ ✷✵✵✻ ❩✳

▼✉st❛❢❛ ✈➭ ❇✳ ❙✐♠s ➤➲ ❝❤Ø r❛ r➺♥❣ ❤➬✉ ❤Õt ❝➳❝ ❦Õt q✉➯ ❧✐➟♥ q✉❛♥ ➤Õ♥ ❦❤➠♥❣ ❣✐❛♥

D✲♠➟tr✐❝ ❝ñ❛ ❇✳ ❈✳ ❉❤❛❣❡ ❧➭ ❦❤➠♥❣ ❝ß♥ ➤ó♥❣ ✈➭ ❤ä ➤➲ ❣✐í✐ t❤✐Ư✉ ♠ét ❦❤➳✐ ♥✐Ư♠
♠í✐ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝ ✈➭ ♥❣❤✐➟♥

❝ø✉ ♥❤✐Ị✉ ❦Õt q✉➯ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤♦ ❝➳❝ tù ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝

✈í✐ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ♥➭♦ ➤ã✳ ❙❛✉ ➤ã✱ ♥❤✐Ị✉ t➳❝ ❣✐➯ ➤➲ ♥❣❤✐➟♥ ❝ø✉ ✈Ị ❝➳❝ ➤✐Ĩ♠ ❜✃t
➤é♥❣ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉②

✐✐


ré♥❣✳ ●➬♥ ➤➞②✱ ❍✳ ❆②❞✐ ✈➭ ❝é♥❣ sù ➤➲ t❤✐Õt ❧❐♣ ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ ➤✐Ĩ♠ ❜✃t
➤é♥❣ ❝❤✉♥❣ ❝❤♦ ❤❛✐ tù ➳♥❤ ①➵
❝➳❝❤ ❣✐➯ sö r➺♥❣

f

f

❧➭ ♠ét ➳♥❤ ①➵

✈➭

g tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ X

❜➺♥❣

G✲❝♦ ②Õ✉ s✉② ré♥❣ ❦✐Ĩ✉ A ✈➭ B ➤è✐ ✈í✐ g ✳

❚r➟♥ ❝➡ së ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ P●❙✳❚❙ ❚r➬♥ ❱➝♥
➣♥ ❝❤ó♥❣ t➠✐ ➤➲ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐ ♥❣❤✐➟♥ ❝ø✉

❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

✧➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵

G✲♠➟tr✐❝✧✳

✷✳ ➜è✐ t➢ỵ♥❣ ✈➭ ♣❤➵♠ ✈✐ ♥❣❤✐➟♥ ❝ø✉
✲ ➜è✐ t➢ỵ♥❣ ✈➭ ♣❤➵♠ ✈✐ ♥❣❤✐➟♥ ❝ø✉ ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥
❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝

G✲♠➟tr✐❝✱

❞➲②

G✲

➤➬② ➤đ✱ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝

➳♥❤ ①➵ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ●✲♠➟tr✐❝✳
✲ P❤➵♠ ✈✐ ♥❣❤✐➟♥ ❝ø✉ ❧➭ ❝➳❝ tÝ♥❤ ❝❤✃t ✈➭ ♠è✐ q✉❛♥ ệ ữ ố tợ
tr ị ý ề ể ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉

❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵
❝➳❝ ➳♥❤ ①➵

φ✲❝♦✱ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠

G✲❝♦ ②Õ✉ s✉② ré♥❣✱ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛

α✲❝♦ ②Õ✉ s✉② ré♥❣ ✈➭ ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ❦Õt q✉➯ ➤ã✳

✸✳ P❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉
✲ ❉ï♥❣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❣✐➯✐ tÝ❝❤✱ t➠♣➠✱ ❣✐➯✐ tÝ❝❤ ❤➭♠✳
✲ ❙ư ❞ơ♥❣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ ❧ý t❤✉②Õt✱ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ t➭✐ ❧✐Ư✉
✈➭ sư ❞ơ♥❣ ♠ét sè ❦ü t❤✉❐t ❝❤ø♥❣ ♠✐♥❤ ♠í✐ ➤Ĩ ❣✐➯✐ q✉②Õt ❝➳❝ ✈✃♥ ➤Ị ➤➷t r❛✳
✲ ❉ù❛ ✈➭♦ ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦✱ ❜➺♥❣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ ♣❤➞♥
tÝ❝❤ tỉ♥❣ ❤ỵ♣✱ s♦ s➳♥❤ ✱ ❦❤➳✐ q✉➳t ❤♦➳✳✳✳ ➤Ĩ tr×♥❤ ❜➭② ♠ét ❝➳❝❤ ❤Ư t❤è♥❣ ❝➳❝ ❦✐Õ♥
t❤ø❝ ❧✐➟♥ q✉❛♥ ➤Õ♥ ❝➳❝ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉
➳♥❤ ①➵

Φ✲❝♦✱

G✲❝♦ ②Õ✉ s✉② ré♥❣✱ ➳♥❤ ①➵ α✲❝♦ ②Õ✉ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳

✹✳ ▼ơ❝ ➤Ý❝❤ ♥❣❤✐➟♥ ❝ø✉
▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝✱

❞➲②

G✲❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ✱ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝

➳♥❤ ①➵ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ●✲♠➟tr✐❝✱ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤

✐✐✐


①➵ ❦✐Ĩ✉

Φ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝

➳♥❤ ①➵

G✲❝♦ ②Õ✉ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t

➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵

G✲❝♦ α✲②Õ✉ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱✳✳✳ ✈➭ ❝❤♦

❝➳❝ ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳

✺✳ ◆é✐ ❞✉♥❣ ♥❣❤✐➟♥ ❝ø✉
✲ ◆❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉
❦❤➠♥❣ ❣✐❛♥

φ✲❝♦ tr♦♥❣

G✲♠➟tr✐❝✳

✲ ◆❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵
tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝✳ ❈➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳

✲ ◆❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵
tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

G✲❝♦ ②Õ✉ s✉② ré♥❣
α✲❝♦ ②Õ✉ s✉② ré♥❣

G✲♠➟tr✐❝✳ ❈➳❝ ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳

✻✳ ❈✃✉ tró❝ ❧✉❐♥ ✈➝♥
▲✉❐♥ ✈➝♥ ❣å♠ ✷ ❝❤➢➡♥❣✳
❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉
❦❤➠♥❣ ❣✐❛♥

Φ✲❝♦

tr♦♥❣

G✲♠➟tr✐❝✳ ▼ơ❝ ✶ ♥❤➺♠ ❣✐í✐ t❤✐Ư✉ ♠ét sè ❦❤➳✐ ♥✐Ư♠ ✈➭ ♠ét sè tÝ♥❤

❝❤✃t ❝đ❛ ❝❤ó♥❣ ❝➬♥ ❞ï♥❣ ❝❤♦ ❝➳❝ tr×♥❤ ❜➭② ✈Ị s❛✉✳ ▼ơ❝ ✷ ♥❤➺♠ tr×♥❤ ❜➭② ♠ét
sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉

Φ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝

➤➬② ➤đ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳
❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ②Õ✉ s✉② ré♥❣

G✲♠➟tr✐❝✳ ▼ơ❝ ✶ ♥❤➺♠ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t

➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ G✲❝♦ ②Õ✉ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ ✈➭

tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ▼ơ❝ ✷ trì ột số ị ý ể t ộ ❝ñ❛
❝➳❝ ➳♥❤ ①➵

α✲❝♦ ②Õ✉ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ ✈➭ ❝➳❝ ✈Ý ❞ơ

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

✐✈


❚❍P❚ ◗✉❛♥❣ ❚r✉♥❣✱ tØ♥❤ ❚➞② ◆✐♥❤ ➤➲ ❣✐ó♣ ➤ì✱ t➵♦ ề ệ t ợ t
tr q trì ọ t❐♣ ✈➭ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ◆❤➞♥ ➤➞② t➳❝ ❣✐➯ ①✐♥ ❝➳♠
➡♥ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ ❝❛♦ ❤ä❝ ●✐➯✐ ❚Ý❝❤ ❦❤♦➳ ✷✷ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❙➭✐ ●ß♥✳ ❈✉è✐
❝ï♥❣ t➳❝ ❣✐➯ ①✐♥ ❣ë✐ ❧ê✐ ❝➳♠ ➡♥ ➤Õ♥ ❇❛ ♠Ñ✱ ❝➳❝ tr ì t ề
ệ t ợ ❣✐ó♣ t➳❝ ❣✐➯ ❤♦➭♥ t❤➭♥❤ ♥❤✐Ư♠ ✈ơ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣✳
▼➷❝ ❞ï ➤➲ tÝ❝❤ ❝ù❝ ➤➬✉ t➢ ✈➭ ❝ã ♥❤✐Ị✉ ❝è ❣➽♥❣ tr♦♥❣ ♥❣❤✐➟♥ ❝ø✉✱ t❤ù❝ ❤✐Ư♥
➤Ị t➭✐✱ s♦♥❣ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ s❛✐ sãt✳ ❚➳❝ ❣✐➯ ợ
ữ ý ế ó ó ủ qý ❈➠ ✈➭ ❜➵♥ ➤ä❝ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥
t❤✐Ư♥✳

❱✐♥❤✱ ♥❣➭② ✷✵ t❤➳♥❣ ✽ ♥➝♠ ✷✵✶✻

◆❣✉②Ô♥ ❍♦➭♥❣ ❍➢♥❣

✈


❝❤➢➡♥❣ ✶

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

Φ✲❝♦

G✲♠➟tr✐❝

❈➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥

✶✳✶

P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ q✉❛ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ö❝ tr×♥❤
❜➭② ❝đ❛ ❧✉❐♥ ✈➝♥✳ ◆é✐ ❞✉♥❣ ❣å♠✿ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈Ị ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝✱ ❞➲②

G✲❤é✐ tơ✱ ❞➲② G✲❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛
❝❤ó♥❣ ❝➬♥ ❞ï♥❣ trì ề s

ị ĩ

ột tr tr

❈❤♦ t❐♣ ❤ỵ♣

X

X = φ✱ ➳♥❤ ①➵ d : X × X → R ➤➢ỵ❝ ❣ä✐

♥Õ✉ t❤á❛ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥

✶✮

d(x, y) ≥ 0 ✈í✐ ♠ä✐ x, y ∈ X

✷✮

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 ✳

d(x, y) = 0 ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x = y ✳

d tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ ❦Ý
❤✐Ö✉ ❧➭ (X, d) ❤❛② ➤➡♥ ❣✐➯♥ ❧➭ X ✳ ❙è d (x, y) ❣ä✐ ❧➭ ❦❤♦➯♥❣ ❝➳❝❤ ❣✐÷❛ ➤✐Ĩ♠ x ✈➭
➤✐Ĩ♠ y ✳
❚❐♣

X

✈➭

✶✳✶✳✷

❝ï♥❣ ớ ột tr

ị ĩ



X ột t rỗ ✈➭ G : X × X × X → R+

❧➭ ♠ét ❤➭♠ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ò✉ s❛✉
✭✶✮

G(x, y, z) = 0 ♥Õ✉ x = y = z ✱

✭✷✮

0 < G(x, x, y) ✈í✐ ♠ä✐ x, y ∈ X

✭✸✮

G(x, x, y) ≤ G(x, y, z)✱ ✈í✐ ♠ä✐ x, y, z ∈ X

✭✹✮

G(x, y, z) = G(x, z, y) = G(y, z, x) = ..., ✭➤è✐ ①ø♥❣ ë ❝➯ ✸ ❜✐Õ♥✮✱

✭✺✮

G(x, y, z) ≤ G(x, a, a) + G(a, y, z), ✈í✐ ♠ä✐ x, y, z, a ∈ X ✱ ✭❜✃t ➤➻♥❣

✈í✐

x = y✱
✈í✐

z = y✱

t❤ø❝ t❛♠ ❣✐➳❝✮✳

G ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ♠➟tr✐❝ s✉② ré♥❣✱ ❤❛② ❣ä♥ ❤➡♥ ❧➭ ♠ét G✲♠➟tr✐❝
tr➟♥ X ✱ ✈➭ ❝➷♣ (X, G) ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳

❑❤✐ ➤ã✱ ❤➭♠

✶


(X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ {xn } ❧➭ ♠ét
❞➲② ❝➳❝ ➤✐Ĩ♠ ❝đ❛ X ✳ ➜✐Ĩ♠ x ∈ X ➤➢ỵ❝ ❣ä✐ ❧➭ ❣✐í✐ ❤➵♥ ❝đ❛ ❞➲② {xn } ♥Õ✉
lim G(x, xn , xm ) = 0✳ ▲ó❝ ➤ã t❛ ♥ã✐ r➺♥❣ ❞➲② {xn } ❧➭ G✲❤é✐ tơ ✈Ị x✳
➜Þ♥❤ ♥❣❤Ü❛✳

✶✳✶✳✸

✭❬✼❪✮ ❈❤♦

n,m→∞

xn → x tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ (X, G) ♥Õ✉ ✈í✐ ♠ä✐ ε > 0✱
tå♥ t➵✐ k ∈ N s❛♦ ❝❤♦ G(x, xn , xm ) < ε ✈í✐ ♠ä✐ m, n ≥ k ✳
◆❤➢ ✈❐②✱

✶✳✶✳✹

✭❬✼❪✮ ❈❤♦

▼Ư♥❤ ➤Ị✳

{xn } ⊆ X

✈➭ ➤✐Ĩ♠

(X, G)

❧➭ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝✳

❑❤✐ ➤ã✱ ✈í✐ ❞➲②

x ∈ X ✱ ❝➳❝ ❦❤➻♥❣ ➤Þ♥❤ s❛✉ ➤➞② ❧➭ t➢➡♥❣ ➤➢➡♥❣

{xn } ❧➭ G✲❤é✐ tơ ✈Ị x❀

✭✶✮

❉➲②

✭✷✮

G(xn , xn , x) → 0✱ ❦❤✐ n → ∞❀

✭✸✮

G(xn , x, x) → 0✱ ❦❤✐ n → ∞❀

✭✹✮

G(xm , xn , x) → 0✱ ❦❤✐ m, n → ∞✳

(X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❉➲② {xn } ⊆
X ➤➢ỵ❝ ❣ä✐ ❧➭ G✲❈❛✉❝❤② ♥Õ✉ ✈í✐ ♠ä✐ ε > 0 tå♥ t➵✐ sè tù ♥❤✐➟♥ N ∈ N s❛♦
❝❤♦ G(xn , xm , xl ) < ε ✈í✐ ♠ä✐ n, m, l ≥ N ✱ ♥❣❤Ü❛ ❧➭ G(xn , xm , xl ) → 0 ❦❤✐
n, m, l → ∞✳
✶✳✶✳✺

➜Þ♥❤ ♥❣❤Ü❛✳

✭❬✼❪✮ ❈❤♦

G✲♠➟tr✐❝ (X, G) ➤➢ỵ❝ ❣ä✐ ❧➭ G✲➤➬② ➤đ ✭❤♦➷❝
❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ủ ế ỗ G tr (X, G) Gộ tơ
tr♦♥❣ (X, G)✳

✶✳✶✳✻

➜Þ♥❤ ♥❣❤Ü❛✳

✶✳✶✳✼

▼Ư♥❤ ➤Ị✳

✭❬✼❪✮ ❑❤➠♥❣ ❣✐❛♥

✭❬✼❪✮

❈❤♦

(X, G)

❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝✳

❑❤✐ ➤ã✱ ❝➳❝

❦❤➻♥❣ ➤Þ♥❤ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣

{xn } ❧➭ G✲❈❛✉❝❤②❀

✭✶✮

❉➲②

✭✷✮

❱í✐ ♠ä✐

ε > 0✱

tå♥ t➵✐ sè

k ∈N

s❛♦ ❝❤♦

n, m ≥ k ❀
✭✸✮

G(gn , gm , gm ) → 0 ❦❤✐ m, n → ∞✳

✷

G(xn , xm , xm ) < ε

✈í✐ ♠ä✐


(X, G) ✈➭ (X , G ) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭
➳♥❤ ①➵ f : (X, G) → (X , G )✳ ❑❤✐ ➤ã f ➤➢ỵ❝ ❣ä✐ ❧➭ G✲❧✐➟♥ tơ❝ t➵✐ ➤✐Ĩ♠ a ∈ X
❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ✈í✐ ♠ä✐ ε > 0✱ tå♥ t➵✐ sè δ > 0 s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X ♠➭
G(a, x, y) < δ t❛ ❝ã G (f (a), f (x), f (y)) < ε✳
➳♥❤ ①➵ f : (X, G) → (X , G ) ➤➢ỵ❝ ❣ä✐ ❧➭ G✲❧✐➟♥ tơ❝ tr➟♥ X ❦❤✐ ✈➭ ❝❤Ø ❦❤✐
♥ã ❧➭ G✲❧✐➟♥ tô❝ t➵✐ ♠ä✐ ➤✐Ĩ♠ a ∈ X ✳
➜Þ♥❤ ♥❣❤Ü❛✳

✶✳✶✳✽

✭❬✼❪✮ ❈❤♦

(X, G) ✈➭ (X , G ) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❑❤✐
➤ã✱ ➳♥❤ ①➵ f : X → X ❧➭ G✲❧✐➟♥ tô❝ t➵✐ x ∈ X ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ♥ã ❧➭ G✲❧✐➟♥
tô❝ t❤❡♦ ❞➲② t➵✐ x✱ ♥❣❤Ü❛ ❧➭ ✈í✐ ♠ä✐ ❞➲② {xn } ⊂ X ❧➭ G✲❤é✐ tô ➤Õ♥ x✱ t❛ ❝ã
❞➲② ❣✐➳ trÞ {f (xn )} ❧➭ G✲❤é✐ tơ ➤Õ♥ f (x)✳
▼Ư♥❤ ➤Ị✳

✶✳✶✳✾

✭❬✼❪✮ ❈❤♦

▼Ư♥❤ ➤Ị✳

✶✳✶✳✶✵

✭❬✼❪✮ ❈❤♦

(X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳

❑❤✐ ➤ã✱ ❤➭♠

G(x, y, z) ❧✐➟♥ tô❝ ➤å♥❣ t❤ê✐ t❤❡♦ t✃t ❝➯ ✸ ❜✐Õ♥ ❝ñ❛ ♥ã✳
(R, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ tr t tờ
ị GS : R ì R ì R → [0, +∞) ❝❤♦ ❜ë✐
❱Ý ❞ô✳

✶✳✶✳✶✶

✭❬✼❪✮

✶✮ ❈❤♦

Gs (x, y, z) = d (x, y) + d (y, z) + d (x, z)

❚❛ ①➳❝

✭✶✳✶✮

x, y, z ∈ R✳ ❑❤✐ ➤ã✱ râ r➭♥❣ (R, Gs ) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳
✷✮ ❈❤♦ X = {a, b}✳ ❍➭♠ G tr➟♥ X × X ì X [0, +) ị ở

ớ ọ

G (a, a, a) = G (b, b, b) = 0
G (a, a, b) = 1, G (a, b, b) = 2.

✭✶✳✷✮

G ❧➟♥ t♦➭♥ ❜é X × X × X ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ tÝ♥❤ ➤è✐ ①ø♥❣ ❝đ❛ ❝➳❝
❜✐Õ♥ sè✳ ❑❤✐ ➤ã✱ râ r➭♥❣ r➺♥❣ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳
❚❛ ♠ë ré♥❣

✶✳✶✳✶✷

♠ä✐

▼Ư♥❤ ➤Ị✳

x, y, z

✈➭

✭❬✼❪✮ ❈❤♦

(X, G)

❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥

a ∈ X ✱ t❛ ❝ã

G(x, y, z) = 0✱ t❤× x = y = z ✱

✭✶✮

◆Õ✉

✭✷✮

G(x, y, z) ≤ G(x, x, y) + G(x, x, z)✱

✭✸✮

G(x, y, y) ≤ 2G(y, x, x)✱
✸

G✲♠➟tr✐❝✳

❑❤✐ ➤ã✱ ✈í✐


✭✹✮

G(x, y, z) ≤ G(x, a, z) + G(a, y, z)✱

✭✺✮

2
G(x, y, z) ≤ [G(x, y, a) + G(x, a, z) + G(a, y, z)]✱
3

✭✻✮

G(x, y, z) ≤ G(x, a, a) + G(y, a, a) + G(z, a, a)✳

✶✳✶✳✶✸

X

➜Þ♥❤ ♥❣❤Ü❛✳

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ➳♥❤ ①➵ T : X →
②Õ✉ ♥Õ✉ ✈í✐ ♠ä✐ x, y ∈ X ❜✃t ➤➻♥❣ t❤ø❝ s❛✉ ➤➞②

✭❬✷❪✮ ❈❤♦

➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵

C ✲❝♦

➤ó♥❣

1
d(T x, T y) ≤ [d(x, T y) + d(y, T x)] − φ(d(x, T y), d(y, T x)),
2
φ : [0, +∞) × [0, +∞) → [0, +∞)
φ (x, y) = 0 ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x = y = 0✳
tr♦♥❣ ➤ã

❧➭ ♠ét ❤➭♠ sè ❧✐➟♥ tô❝ s❛♦ ❝❤♦

(X, G) ❧➭ ♠ét G✲❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ➳♥❤ ①➵ f :
X → X ➤➢ỵ❝ ❣ä✐ ❧➭ G✲❝♦ ②Õ✉ ♥Õ✉ ✈í✐ ♠ä✐ x, y, z ∈ X ❜✃t ➤➻♥❣ t❤ø❝ s❛✉ ➤➞②
✶✳✶✳✶✹

➜Þ♥❤ ♥❣❤Ü❛✳

✭❬✷❪✮ ❈❤♦

➤ó♥❣

G (f x, f y, f z) ≤

1
3 [G (x, f y, f y) + G (y, f z, f z) + G (z, f x, f x)]

−φ (G (x, f y, f y) , G (y, f z, f z) , G (z, f x, f x)) ,

φ : [0, +∞)3 → [0, +∞) ❧➭ ♠ét ❤➭♠ ❧✐➟♥ tơ❝ ✈í✐ φ (t, s, u) = 0 ♥Õ✉
✈➭ ❝❤Ø ♥Õ✉ t = s = u = 0✳
tr♦♥❣ ➤ã

✶✳✶✳✶✺

➜Þ♥❤ ♥❣❤Ü❛✳

X →X

(X, G) ❧➭ ♠ét G✲❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ➳♥❤ ①➵ f :
❦✐Ó✉ G✲❝♦ ②Õ✉ ♥Õ✉ ✈í✐ ♠ä✐ x, y, z ∈ X ❜✃t ➤➻♥❣

✭❬✷❪✮ ❈❤♦

➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵

t❤ø❝ s❛✉ ➤➞② ➤ó♥❣

G (f x, f y, f z) ≤

1
3 [G (x, x, f y) + G (y, y, f z) + G (z, z, f x)]

−φ (G (x, x, f y) , G (y, y, f z) , G (z, z, f x)) ,
φ : [0, +∞)3 → [0, +∞) ❧➭ ♠ét ❤➭♠ ❧✐➟♥ tơ❝ ✈í✐ φ (t, s, u) = 0 ♥Õ✉
✈➭ ❝❤Ø ♥Õ✉ t = s = u = 0✳
tr♦♥❣ ➤ã

✹

ị ĩ





số

: [0, +) [0, +)

ợ ọ ❧➭ ❤➭♠

t❤❛② ➤ỉ✐ ❦❤♦➯♥❣ ❝➳❝❤ ♥Õ✉ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➞② t❤á❛ ♠➲♥

✭✶✮

ψ ❧➭ ❤➭♠ ❧✐➟♥ tô❝ ✈➭ t➝♥❣✳

✭✷✮

ψ (t) = 0 ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ t = 0✳

✶✳✶✳✶✼

➜Þ♥❤ ♥❣❤Ü❛✳

✭❬✼❪✮ ❑❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝ (X, G) ➤➢ỵ❝ ❣ä✐ ❧➭ ➤è✐ ①ø♥❣ ♥Õ✉

G (x, x, y) = G (x, y, y)
✈í✐ ♠ä✐

x, y ∈ X ✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♥❣➢ỵ❝ ❧➵✐ ❦❤➠♥❣ ❣✐❛♥ X ợ ọ

ố ứ

ét

tr

ớ ỗ

Gtr tr t X ✱ ❝ã ♠ét ♠➟tr✐❝ dG ❧✐➟♥ ❦Õt ✈í✐ G✲

G ➤➢ỵ❝ ❝❤♦ ❜ë✐
dG (x, y) = G (x, x, y) + G (x, y, y) ,

◆Õ✉

G

❧➭ ➤è✐ ①ø♥❣ t❤× ❤✐Ĩ♥ ♥❤✐➟♥ t❛ ❝ã

tr♦♥❣ tr➢ê♥❣ ❤ỵ♣

✈í✐ ♠ä✐

x, y ∈ X.

dG (x, y) = 2G (x, x, y)✱

♥❤➢♥❣

G ❦❤➠♥❣ ❧➭ G✲♠➟tr✐❝ ➤è✐ ①ø♥❣ t❤× t❛ ❝ã

3
G (x, y, y) ≤ dG (x, y) ≤ 2G (x, y, y) , ✈í✐ ♠ä✐ x, y ∈ X.
2
✶✳✶✳✶✽

➜Þ♥❤ ♥❣❤Ü❛✳

✭❬✸❪✮ ●✐➯ sư

f, g : X → X

❧➭ ❝➳❝ ➳♥❤ ①➵ tõ t❐♣

X

✈➭♦ ❝❤Ý♥❤

♥ã✳

ω = f x = gx ớ x X tì x ợ ❣ä✐ ❧➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❝đ❛ f ✈➭
g ✈➭ ω ợ ọ trị trù ủ f g ✳
❈➷♣ ❝➳❝ ➳♥❤ ①➵ f, g ➤➢ỵ❝ ❣ä✐ ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ ♥Õ✉ ❝❤ó♥❣ ❣✐❛♦ ❤♦➳♥ t➵✐

❝➳❝ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❝đ❛ ❝❤ó♥❣✱ ♥❣❤Ü❛ ❧➭ gf x = f gx t➵✐ ♠ä✐ ➤✐Ĩ♠ x ∈ X ♠➭
f x = gx✳
◆Õ✉

✶✳✶✳✶✾

➜Þ♥❤ ❧ý✳

f, g : X → X

❧➭ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ ❝đ❛

g ❝ã ♠ét ❣✐➳ trÞ trï♥❣ ♥❤❛✉ ❞✉② ♥❤✃t ω = f x = gx✱ t❤× ω ❧➭
➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t ❝ñ❛ f ✈➭ g ✳

t❐♣

X ✳ ◆Õ✉ f

✭❬✸❪✮ ●✐➯ sö

✈➭

✺


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

✶✳✷

❣✐❛♥

G✲♠➟tr✐❝

Φ ❧➭ t❐♣ ❤ỵ♣ t✃t ❝➯ ❝➳❝ ❤➭♠ φ : [0, +∞) → [0, +∞) s❛♦ ❝❤♦ φ ❧➭
n
❤➭♠ ❦❤➠♥❣ ❣✐➯♠ ✈í✐ lim φ (t) = 0 ✈í✐ ♠ä✐ t (0, +) ế tì
ý ệ

n

ợ ❣ä✐ ❧➭ ♠ét

Φ✲➳♥❤ ①➵✳ ◆Õ✉ φ ❧➭ ♠ét Φ✲➳♥❤ ①➵✱ t❤× ❞Ơ ❞➭♥❣ t❤✃② r➺♥❣

✶✮

φ (t) < t ✈í✐ ♠ä✐ t ∈ (0, +∞)✳

✷✮

φ (0) = 0

❚r♦♥❣ ❝➯ ♠ô❝ ♥➭② t❛ sÏ ❦ý ❤✐Ư✉

φ ❧➭ ❝➳❝ Φ✲➳♥❤ ①➵✳

❇➞② ❣✐ê✱ ❝❤ó♥❣ t❛ ❣✐í✐

t❤✐Ư✉ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❦Õt q✉➯ ➤➬✉ t✐➟♥✳

①➵

✭❬✶✵❪✮ ❈❤♦

➜Þ♥❤ ❧ý✳

✶✳✷✳✶

T :X→X

X

❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝ ➤➬② ➤đ✳

●✐➯ sư ➳♥❤

t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥

G(T (x), T (y), T (z)) ≤ φ(G(x, y, z))
x, y, z ∈ X ✳
G✲❧✐➟♥ tơ❝ t➵✐ u✳
✈í✐ ♠ä✐

❑❤✐ ➤ã✱

T

❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t

✭✶✳✸✮

u ∈ X

✈➭

❈❤ø♥❣ ♠✐♥❤✳ ❚r➢í❝ ❤Õt t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ tå♥ t➵✐ ➤✐Ó♠ ❜✃t ➤é♥❣

T

❧➭

u∈X

T ✳ ❈❤ä♥ x0 ∈ X ✳ ➜➷t xn = T (xn−1 ), n ∈ N✳ ◆Õ✉ tå♥ t➵✐ n0 ∈ X s❛♦ ❝❤♦
xn0 = xn0 −1 ✱ t❤× xn0 −1 ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✳ ❇➞② ❣✐ê ❣✐➯ sư r➺♥❣ xn = xn−1 ✱
✈í✐ ♠ä✐ n ∈ N✳ ❚❛ sÏ ❝❤ø♥❣ tá r➺♥❣ ❞➲② (xn ) ❧➭ ♠ét G tr X
t ớ ỗ n N✱ t❛ ❝ã

❝ñ❛

G(xn , xn+1 , xn+1 ) = G(T (xn−1 ), T (xn ), T (xn ))
≤ φ(G(xn−1 , xn , xn ))
≤ φ2 (G(xn−2 , xn−1 , xn−1 ))

✭✶✳✹✮

..........................................

≤ φn (G(x0 , x1 , x1 )).
❱í✐ ❜✃t ❦ú
t➵✐ sè

ε > 0 ❝❤♦ tr➢í❝✱ ✈× lim φn G((x0 , x1 , x1 )) = 0 ✈➭ φ (ε) < ε ♥➟♥ tå♥
n→∞

k0 ∈ N s❛♦ ❝❤♦
φn (G (x0 , x1 , x1 )) < ε − φ (ε) ,
✻

✈í✐ ♠ä✐

n ≥ k0 .

✭✶✳✺✮


❉♦ ➤ã✱ t❛ ❝ã

G (xn , xn+1 , xn+1 ) < ε − φ (ε) ,
❇➞② ❣✐ê t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ✈í✐ ♠ä✐

G (xn , xm , xm ) < ε,

✈í✐ ♠ä✐

n ≥ k0 .

✭✶✳✻✮

m, n ∈ N✱ ♠➭ m > n t❛ ❝ã
✈í✐ ♠ä✐

m ≥ n ≥ k0 .

✭✶✳✼✮

m✳
❚❤❐t ✈❐②✱ ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✻✮ ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ ε − φ (ε) < ε
t❛ ❝ã ♥❣❛② ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✼✮ ➤ó♥❣ ✈í✐ m = n + 1✳ ❇➞② ❣✐ê ❣✐➯ sö ❜✃t ➤➻♥❣ t❤ø❝
✭✶✳✼✮ ➤ó♥❣ ✈í✐ m = k ✳ ❱í✐ m = k + 1 t❛ ❝ã
▼✉è♥ t❤Õ t❛ ❝❤ø♥❣ tá ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✼✮ ❜➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ q✉② ♥➵♣ t❤❡♦

G (xn , xk+1 , xk+1 ) ≤ G (xn , xn+1 , xn+1 ) + G (xn+1 , xk+1 , xk+1 )
< ε − φ (ε) + φ (G (xn , xk , xk ))

✭✶✳✽✮

< ε − φ (ε) + φ (ε) = ε.
m✱ t❛ ❦Õt ❧✉❐♥ r➺♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✼✮ ➤ó♥❣
✈í✐ ♠ä✐ m ≥ n ≥ k0 ✳ ❉♦ ➤ã (xn ) ❧➭ ❞➲② G✲❈❛✉❝❤② ✈➭ ✈× X ➤➬② ➤đ✱ ♥➟♥ (xn ) ❧➭
G✲❤é✐ tơ tí✐ u ∈ X ✳ ❱í✐ n ∈ N✱ t❛ ❝ã
❇➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ q✉② ♥➵♣ tr➟♥

G (u, u, T (u)) ≤ G (u, u, xn+1 ) + G (xn+1 , xn+1 , T (u))
≤ G (u, u, xn+1 ) + φ (G (xn , xn , u))

✭✶✳✾✮

≤ G(u, u, xn+1 ) + G(xn , xn , u).
n → ∞ ✈➭ sư ❞ơ♥❣ ❣✐➯ t❤✐Õt r➺♥❣ G ❧✐➟♥ tơ❝ t❤❡♦ ❝➳❝ ❜✐Õ♥ ❝đ❛ ♥ã✱ t❛ ♥❤❐♥
➤➢ỵ❝ G (u, u, T (u)) = 0✳ ❱× t❤Õ✱ t❛ ❝ã T (u) = u✳ ❉♦ ➤ã✱ u ❧➭ ➤✐Ó♠ ❜✃t ➤é♥❣
❝đ❛ T ✳
❇➞② ❣✐ê ❣✐➯ sư v ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦❤➳❝ ❝đ❛ T ♠➭ u = v ✳ ❱× φ ❧➭ ♠ét
Φ✲➳♥❤ ①➵✱ ♥➟♥ t❛ ❝ã
❈❤♦

G (u, u, v) ≤ G (T (u) , T (u) , T (v))
≤ φ (G (u, u, v))

✭✶✳✶✵✮

< G (u, u, v) .
➜✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❉♦ ➤ã t❛ ❝ã

u = v ✈➭ ✈× t❤Õ T

♥❤✃t✳

✼

❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉②


G✲❧✐➟♥ tơ❝ t➵✐ u ∈ X ✱ t❛ ❣✐➯ sư (yn ) ❧➭ ❞➲② ❜✃t ❦ú
tr♦♥❣ X ♠➭ (yn ) Gộ tụ tớ u ó ớ ỗ n ∈ N t❛ ❝ã
➜Ô ❝❤ø♥❣ ♠✐♥❤ r➺♥❣

T

❧➭

G (u, u, T (yn )) = G (T (u) , T (u) , T (yn ))
≤ φ (G (u, u, yn ))

✭✶✳✶✶✮

< G (u, u, yn ) .
❈❤♦

n → ∞ t❛ ♥❤❐♥ ➤➢ỵ❝ lim G (u, u, T (yn )) = 0✳
n→∞

G✲❤é✐ tô tí✐ u = T (u)✳ ❉♦ ➤ã✱ T

❧➭

❱× t❤Õ✱ ❞➲②

(T (yn )) ❧➭

G✲❤é✐ tơ t➵✐ u✳

❇➺♥❣ ❝➳❝❤ ➳♣ ❞ơ♥❣ ➜Þ♥❤ ❧ý ✶✳✷✳✶ t❛ t❤✉ ➤➢ỵ❝ ❝➳❝ ❦Õt q✉➯ s❛✉✳

G✲♠➟tr✐❝ ➤➬② ➤đ✳ ●✐➯ sư r➺♥❣
➳♥❤ ①➵ T : X → X tỏ ề ệ ớ ỗ m N t ❝ã

✶✳✷✳✷

❍Ö q✉➯✳

✭❬✶✵❪✮ ❈❤♦

X

❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥

G(T m (x), T m (y), T m (z)) ≤ φ(G(x, y, z))
✈í✐ ♠ä✐

x, y, z ∈ X ✳ ❑❤✐ ➤ã✱ T

❈❤ø♥❣ ♠✐♥❤✳

Tm

❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t

➳♣ ❞ơ♥❣ ➜Þ♥❤ ❧ý ✶✳✷✳✶ ❝❤♦ ➳♥❤ ①➵

❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t

T m✱

✭✶✳✶✷✮

u ∈ X✳

t❛ s✉② r❛ r➺♥❣ ➳♥❤ ①➵

u ∈ X ✳ ❑❤✐ ➤ã✱ tõ ❝➳❝ ➤➻♥❣ t❤ø❝

T (u) = T (T m (u)) = T m+1 (u) = T m (T (u))

✭✶✳✶✸✮

T (u) ❝ò♥❣ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ T m ✳ ◆❤➢♥❣ ✈× u ∈ X ❧➭
m
➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ❝đ❛ T ✱ t❛ s✉② r❛ T (u) = u✳ ❉♦ ➤ã✱ u ❝ị♥❣ ❧➭ ➤✐Ĩ♠
❜✃t ➤é♥❣ ❞✉② ♥❤✃t ❝đ❛ T ✳

t❛ s✉② r❛ r➺♥❣

✶✳✷✳✸

①➵

❍Ö q✉➯✳

T :X→X

✭❬✶✵❪✮ ❈❤♦

X

❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝ ➤➬② ➤đ✳

●✐➯ sư ➳♥❤

t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥

G (T (x) , T (y) , T (y)) ≤ φ (G (x, y, y))
x, y ∈ X ✳ ❑❤✐ ➤ã✱ T
tơ❝ t➵✐ u✳

✈í✐ ♠ä✐

❈❤ø♥❣ ♠✐♥❤✳

❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t

u∈X

✭✶✳✶✹✮
✈➭

T

❧➭

G✲❧✐➟♥

➳♣ ❞ơ♥❣ trù❝ t✐Õ♣ ➜Þ♥❤ ❧ý ✶✳✷✳✶✱ ❜➺♥❣ ❝➳❝❤ t❤❛② z = y✳

✽

X ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ✳ ●✐➯ sư ❝ã sè
k ∈ [0, 1) s❛♦ ❝❤♦ ➳♥❤ ①➵ T : X → X t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥
✶✳✷✳✹

❍Ư q✉➯✳

✭❬✶✵❪✮ ❈❤♦

G (T (x) , T (y) , T (z)) ≤ kG (x, y, z) ,
x, y, z ∈ X ✳
G✲❧✐➟♥ tô❝ t➵✐ u✳
✈í✐ ♠ä✐

❑❤✐ ➤ã✱

T

❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t

✭✶✳✶✺✮

u ∈ X

✈➭

T

❧➭

φ : [0, +∞) → [0, +∞) ❝❤♦ ❜ë✐ φ (ω) =
kω ✈í✐ ♠ä✐ ω ∈ [0, +∞)✳ ❑❤✐ ➤ã✱ ❞Ơ t❤✃② r➺♥❣ φ ❧➭ ♠ét ❤➭♠ ❦❤➠♥❣ ❣✐➯♠ ✈í✐
lim φn (t) = 0✱ ✈í✐ ♠ä✐ t > 0✳ ❱× t❤Õ tõ ✭✶✳✶✺✮
❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ①➳❝ ➤Þ♥❤ ➳♥❤ ①➵

n→+∞

G (T (x) , T (y) , T (z)) ≤ φ (G (x, y, z)) ,

✈í✐ ♠ä✐

x, y, z ∈ X,

✭✶✳✶✻✮

❦Õt ❧✉❐♥ ủ ệ q ợ s trự tế từ ị ý ✶✳✷✳✶✳

✶✳✷✳✺

①➵

❍Ư q✉➯✳

T :X→X

✭❬✶✵❪✮ ❈❤♦

X

❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝ ➤➬② ➤đ✳

●✐➯ sư ➳♥❤

t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥

G (T (x) , T (y) , T (z)) ≤
x, y, z ∈ X ✳
G✲❧✐➟♥ tô❝ t➵✐ u✳
✈í✐ ♠ä✐

❑❤✐ ➤ã✱

T

G (x, y, z)
,
1 + G (x, y, z)

❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t

❈❤ø♥❣ ♠✐♥❤✳ ❇➺♥❣ ❝➳❝❤ ①➳❝ ➤Þ♥❤ ❤➭♠ sè

✭✶✳✶✼✮

u ∈ X

✈➭

T

❧➭

φ : [0, +∞) → [0, +∞) ❝❤♦ ❜ë✐

ω

✈í✐ ♠ä✐ ω ∈ [0, +∞)✳ ❑❤✐ ➤ã✱ ❞Ơ t❤✃② r➺♥❣ φ ❧➭ ♠ét
1+ω
n
❤➭♠ ❦❤➠♥❣ ❣✐➯♠ ✈í✐ lim φ (t) = 0✱ ✈í✐ ♠ä✐ t > 0✳ ❱× t❤Õ t❛ ❝ã
❝➠♥❣ t❤ø❝

φ (ω) =

n→∞

G (T (x) , T (y) , T (z)) ≤ φ (G (x, y, z)) ,

✈í✐ ♠ä✐

x, y, z ∈ X,

✭✶✳✶✽✮

❦Õt ❧✉❐♥ ❝đ❛ ❤Ư q✉➯ ợ s trự tế từ ị ý

ị ý

T :XX

❈❤♦

X

❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝ ➤➬② ➤đ✳

●✐➯ sư ➳♥❤

t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ö♥

G(T (x), T (y), T (z)) ≤ φ(max{G(x, y, z)), G(x, T (x), T (x)),
G(y, T (y), T (y)), G(T (x), y, z)}),
✾

✭✶✳✶✾✮


x, y, z ∈ X ✳
G✲❧✐➟♥ tơ❝ t➵✐ u✳
✈í✐ ♠ä✐

❑❤✐ ➤ã✱

T

❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t

u ∈ X

✈➭

❈❤ø♥❣ ♠✐♥❤✳ ❚r➢í❝ ❤Õt t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ tå♥ t➵✐ ➤✐Ó♠ ❜✃t ➤é♥❣

T

❧➭

u∈X

T ✳ ▲✃② x0 ∈ X ✳ ➜➷t xn = T (xn−1 ), n ∈ N✳ ◆Õ✉ tå♥ t➵✐ n0 ∈ N s❛♦ ❝❤♦
xn0 = xn0 −1 ✱ t❤× xn0 −1 ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✳ ❇➞② ❣✐ê ❣✐➯ sư r➺♥❣ xn = xn−1 ✱
✈í✐ ♠ä✐ n ∈ N✳ ❚❛ sÏ ❝❤ø♥❣ tá r➺♥❣ ❞➲② (xn ) ❧➭ ♠ét G tr X
t ớ ỗ n N✱ t❛ ❝ã

❝ñ❛

G(xn , xn+1 , xn+1 ) = G(T (xn−1 ), T (xn ), T (xn ))
≤ φ(max{G(xn−1 , xn , xn ), G(xn−1 , xn , xn ),
G(xn , xn+1 , xn+1 ), G(xn , xn , xn )}).

✭✶✳✷✵✮
✭✶✳✷✶✮

◆Õ✉

max{G(xn−1 , xn , xn ), G(xn , xn+1 , xn+1 ), G(xn , xn , xn )} = G(xn , xn+1 , xn+1 ),
✭✶✳✷✷✮
t❤×

G (xn , xn+1 , xn+1 ) ≤ φ (G (xn , xn+1 , xn+1 )) < G (xn , xn+1 , xn+1 ) .

✭✶✳✷✸✮

➜✐Ò✉ ♥➭② ❞➱♥ ➤Õ♥ ♠➞✉ t❤✉➱♥✳ ❱× t❤Õ✱ ❝❤Ø ①➯② r❛ tr➢ê♥❣ ❤ỵ♣

♠❛① {G (xn−1 , xn , xn ) , G (xn , xn+1 , xn+1 ) , G (xn , xn , xn )}

= G (xn−1 , xn , xn ) .
✭✶✳✷✹✮

❉♦ ➤ã✱ tõ ❝➠♥❣ t❤ø❝ ✭✶✳✷✵✮ t❛ ♥❤❐♥ ➤➢ỵ❝

G (xn , xn+1 , xn+1 ) ≤ φ (G (xn−1 , xn , xn )) .
ì tế ớ ỗ



n N t❛ ❝ã

G (xn , xn+1 , xn+1 ) = G (T (xn−1 ) , T (xn ) , T (xn ))
≤ φ (G (xn−1 , xn , xn ))
≤ φ2 (G (xn−2 , xn−1 , xn−1 ))

... .............................
≤ φn (G(x0 , x1 , x1 )).
❇➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ❧❐♣ ❧✉❐♥ t➢➡♥❣ tù ♥❤➢ tr♦♥❣ ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ✶✳✷✳✶
t❛ ❝❤Ø r❛ r➺♥❣

(xn ) ❧➭ ♠ét ❞➲② G✲❈❛✉❝❤②✳
✶✵

❱×

X

❧➭ ❦❤➠♥❣ ❣✐❛♥

G✲➤➬② ➤ñ✱

♥➟♥


t❛ s✉② r❛ r➺♥❣ ❞➲②

(xn ) ❧➭ G✲❤é✐ tô ➤Õ♥ ột tử ó u X

ớ ỗ

n ∈ N✱ t❛ ❝ã
G(u, u, T (u)) ≤ G(u, u, xn ) + G(xn , xn , T (u)) ≤ G(u, u, xn )

✭✶✳✷✻✮

+ φ(max{G(xn−1 , xn−1 , u), G(xn−1 , xn , xn ), G(xn , xn−1 , u)}).
❇➞② ❣✐ê✱ t❛ ①Ðt ❝➳❝ tr➢ê♥❣ ❤ỵ♣ s❛✉✿
❚r➢ê♥❣ ❤ỵ♣ ✶✳ ◆Õ✉

max{G(xn−1 , xn−1 , u), G(xn−1 , xn , xn ), G(xn , xn−1 , u)} = G(xn−1 , xn , xn ),
t❤× t❛ ❝ã

G(u, u, T (u)) < G(u, u, xn ) + G(xn−1 , xn , xn ).

✭✶✳✷✼✮

n → ∞ tr♦♥❣ ❝➠♥❣ t❤ø❝ ✭✶✳✷✼✮✱ t❛ s✉② r❛ r➺♥❣ G(u, u, T (u)) = 0 ✈➭ ❞♦ ➤ã
T (u) = u✳

❈❤♦

❚r➢ê♥❣ ❤ỵ♣ ✷✳ ◆Õ✉

max{G(xn−1 , xn−1 , u), G(xn−1 , xn , xn ), G(xn , xn−1 , u)} = G(xn−1 , xn−1 , u),
t❤× t❛ ❝ã

G(u, u, T (u)) < G(u, u, xn ) + G(xn−1 , xn−1 , u).

✭✶✳✷✽✮

n → ∞ tr♦♥❣ ❝➠♥❣ t❤ø❝ ✭✶✳✷✽✮✱ t❛ s✉② r❛ r➺♥❣ G(u, u, T (u)) = 0 ✈➭ ❞♦ ➤ã
T (u) = u✳

❈❤♦

❚r➢ê♥❣ ❤ỵ♣ ✸✳ ◆Õ✉

max{G(xn−1 , xn−1 , u), G(xn−1 , xn , xn ), G(xn , xn−1 , u)} = G(xn , xn−1 , u),
t❤× t❛ ❝ã

G(u, u, T (u)) < G(u, u, xn ) + G(xn , xn−1 , u)

✭✶✳✷✾✮

≤ G(u, u, xn ) + G(xn , xn−1 , xn−1 ) + G(xn−1 , xn−1 , u).
n → ∞✱ tr♦♥❣ ❝➠♥❣ t❤ø❝ ✭✶✳✷✾✮✱ t❛ s✉② r❛ r➺♥❣ G(u, u, T (u)) = 0 ✈➭ ❞♦ ➤ã
T (u) = u✳ ❚r♦♥❣ t✃t ❝➯ tr➢ê♥❣ ❤ỵ♣✱ ❝❤ó♥❣ t❛ ➤Ị✉ ❦Õt ❧✉❐♥ r➺♥❣ u ❧➭ ♠ét ➤✐Ĩ♠
❜✃t ➤é♥❣ ❝đ❛ T ✳
❈❤♦

✶✶


v ∈ X ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦❤➳❝ ❝đ❛ T
❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ❤➭♠ φ t❛ ❝ã
❇➞② ❣✐ê✱ ❣✐➯ sư

♠➭

u = v ✳ ❑❤✐ ➤ã✱ ♥❤ê

G(u, v, v) ≤ φ(max{G(u, v, v), G(u, u, u), G(v, v, v), G(u, v, v)})
= φ(G(u, v, v)) < G(u, v, v).
G(u, v, v) = 0 ✈➭ ❞♦ ➤ã u = v ✳
➜Ó ❝❤ø♥❣ tá r➺♥❣ T ❧➭ G✲❧✐➟♥ tô❝ t➵✐ u✱ t❛ ❣✐➯ sö r➺♥❣ (yn )

tr♦♥❣ X s❛♦ ❝❤♦ (yn ) ❧➭ G✲❤é✐ tơ ➤Õ♥ u✳ ❑❤✐ ➤ã✱ t❛ ❝ã

➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❱× t❤Õ✱

❧➭ ❞➲② ❜✃t ❦ú

G(u, u, T (yn )) ≤ φ(max{G(u, u, yn ), G(u, u, u), G(u, u, u), G(u, u, yn )})
= φ(G(u, u, yn )) < G(u, u, yn ).

✭✶✳✸✵✮

n → ∞ tr♦♥❣ ❝➠♥❣ t❤ø❝ ✭✶✳✸✵✮✱ t❛ ❦Õt ❧✉❐♥ r➺♥❣ ❞➲② (T (yn )) ❧➭ G✲❤é✐ tô
➤Õ♥ T (u) = u✳ ❉♦ ➤ã✱ T ❧➭ G✲❧✐➟♥ tơ❝ t➵✐ u✳
❈❤♦

❚õ ➜Þ♥❤ ❧ý ✶✳✷✳✻ t❛ ❝ã ❝➳❝ ❦Õt q✉➯ s❛✉✳

X ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✳ ●✐➯ sö ❝ã sè
k ∈ [0, 1) s❛♦ ❝❤♦ ➳♥❤ ①➵ T : X → X t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥
✶✳✷✳✼

❍Ư q✉➯✳

✭❬✶✵❪✮ ❈❤♦

G(T (x), T (y), T (z)) ≤ k. max{G(x, y, z), G(x, T (x), T (x)), G(y, T (y), T (y)),
G(T (x), y, z)},

✈í✐ ♠ä✐

x, y, z ∈ X.

✭✶✳✸✶✮

u ∈ X ✈➭ T ❧➭ G✲❧✐➟♥ tô❝ t➵✐ u✳
❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ①➳❝ ➤Þ♥❤ ♥❣❤Ü❛ ❤➭♠ sè φ : [0, +∞) → [0, +∞) ❝❤♦ ❜ë✐
❝➠♥❣ t❤ø❝ φ (ω) = kω ✈í✐ ♠ä✐ ω ∈ [0, +∞)✳ ❑❤✐ ➤ã✱ ❞Ô t❤✃② r➺♥❣ φ ❧➭ ♠ét ❤➭♠
n
❦❤➠♥❣ ❣✐➯♠ ✈í✐ lim φ (t) = 0 ✈í✐ ♠ä✐ t > 0✳ ❱×
❑❤✐ ➤ã✱

T

❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t

n→∞

G(T (x), T (y), T (z)) ≤ φ(max{G(x, y, z), G(x, T (x), T (x)), G(y, T (y), T (y)),
G(T (x), y, z)}),
✈í✐ ♠ä✐

✶✳✷✳✽

x, y, z ∈ X ✱ ❦Õt ❧✉❐♥ ❝đ❛ ❤Ư q✉➯ ➤➢ỵ❝ s✉② trù❝ t✐Õ♣ tõ ➜Þ♥❤ ❧ý ✶✳✷✳✻✳

❍Ư q✉➯✳

➳♥❤ ①➵

✭✶✳✸✷✮

✭❬✶✵❪✮ ❈❤♦

T :X→X

X

❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝ ➤➬② ➤đ✳ ●✐➯ sư r➺♥❣

t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥

G(T (x), T (y), T (y)) ≤ φ(max{G(x, y, y), G(x, T (x), T (x)), G(y, T (y), T (y)),
G(T (x), y, y)}),
✶✷

✭✶✳✸✸✮


x, y ∈ X ✳
G✲❧✐➟♥ tơ❝ t➵✐ u✳

✈í✐ ♠ä✐

❑❤✐ ➤ã

T

❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t

❈❤ø♥❣ ♠✐♥❤✳ ❇➺♥❣ ❝➳❝❤ t❤❛②

z = y

❝❤ø♥❣ ♠✐♥❤✳

✶✸

u∈X

✈➭

T

❧➭

tr♦♥❣ ➜Þ♥❤ ❧ý ✶✳✷✳✻✱ t❛ ❝ã ➤✐Ị✉ ❝➬♥


❝❤➢➡♥❣ ✷

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

✷✳✶

tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝

G✲❝♦ ②Õ✉ s✉② ré♥❣

G✲♠➟tr✐❝

P❤➬♥ ♥➭②✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵

G✲❝♦ ②Õ✉ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✱ ♠ét sè ❤Ư q✉➯ ✈➭ ❝➳❝ ✈Ý
❞ơ ♠✐♥❤ ❤♦➵✳

(X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ❝➳❝ ➳♥❤ ①➵
f, g : X → X ✳ ❚❛ ♥ã✐ r➺♥❣ f ❧➭ ➳♥❤ ①➵ G✲❝♦ ②Õ✉ s✉② ré♥❣ ❦✐Ĩ✉ A ➤è✐ ✈í✐ g ♥Õ✉
✈í✐ ♠ä✐ x, y, z ∈ X ❜✃t ➤➻♥❣ t❤ø❝ s❛✉ ➤➞② ➤ó♥❣

✷✳✶✳✶

➜Þ♥❤ ♥❣❤Ü❛✳

✭❬✷❪✮ ❈❤♦

1
ψ(G(f x, f y, f z)) ≤ψ( (G(gx, f y, f y) + G(gy, f z, f z) + G(gz, f x, f x)))
3
− φ(G(gx, f y, f y), G(gy, f z, f z), G(gz, f x, f x)),
✭✷✳✶✮
tr♦♥❣ ➤ã
✭✶✮
✭✷✮

ψ ❧➭ ❤➭♠ t❤❛② ➤æ✐ ❦❤♦➯♥❣ ❝➳❝❤✳
φ : [0, ∞)3 → [0, ∞) ❧➭ ♠ét ❤➭♠ ❧✐➟♥ tơ❝ ✈í✐ φ(t, s, u) = 0 ♥Õ✉ ✈➭ ❝❤Ø
♥Õ✉ t = s = u = 0✳

G✲♠➟tr✐❝ ✈➭ ❝➳❝ ➳♥❤ ①➵
f, g : X → X s❛♦ ❝❤♦ f ❧➭ ➳♥❤ ①➵ G✲❝♦ ②Õ✉ s✉② ré♥❣ ❦✐Ó✉ A ➤è✐ ✈í✐ g ✳ ●✐➯
sư f (X) ⊆ g(X)✱ g(X) ❧➭ t❐♣ ❝♦♥ ➤➬② ➤ñ ❝ñ❛ (X, G) ✈➭ ❝➷♣ {f, g} ❧➭ t➢➡♥❣
t❤Ý❝❤ ②Õ✉✳ ❑❤✐ ➤ã✱ f ✈➭ g ❝ã ❝❤✉♥❣ ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳
❈❤ø♥❣ ♠✐♥❤✳ ❚õ ❣✐➯ t❤✐Õt f (X) ⊆ g(X)✱ t❛ ❝ã t❤Ó ①➞② ❞ù♥❣ ♠ét ❞➲② {xn }
tr♦♥❣ X s❛♦ ❝❤♦ gxn+1 = f xn ✈í✐ ♠ä✐ n ∈ N✳ ◆Õ✉ ❝ã sè tù ♥❤✐➟♥ n ≥ 1 ♥➭♦ ➤ã
s❛♦ ❝❤♦ gxn+1 = gxn t❤× t❛ ❝ã gxn = f xn ✳ ❑❤✐ ➤ã✱ f ✈➭ g ❝ã ➤✐Ĩ♠ trï♥❣ ♥❤❛✉✳
❉♦ ➤ã✱ ♥❤ê ➜Þ♥❤ ❧ý ✶✳✶✳✶✾ t❛ s✉② r❛ f ✈➭ g ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣✳ ❇ë✐ ✈❐②✱ t❛
❣✐➯ sö r➺♥❣ gxn+1 = gxn ✈í✐ ♠ä✐ n ∈ N✳
✷✳✶✳✷

➜Þ♥❤ ❧ý✳

✭❬✷❪✮ ❈❤♦

(X, G)

❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥

✶✹


ớ ỗ

n N sử ụ t t❤ø❝ ✭✷✳✶✮ ✈➭ ✭✺✮ tr♦♥❣ ➜Þ♥❤

♥❣❤Ü❛ ✶✳✶✳✷✱ t❛ ❝ã

ψ(G(gxn , gxn+1 , gxn+1 )) = ψ(G(f xn−1 , f xn , f xn ))
≤ ψ( 13 G(gxn−1 , f xn , f xn ) + G(gxn , f xn , f xn ) + G(gxn , f xn−1 , f xn−1 ))
−φ(G(gxn−1 , f xn , f xn ), G(gxn , f xn , f xn ), G(gxn , f xn−1 , f xn−1 ))
= ψ( 13 (G(gxn−1 , gxn+1 , gxn+1 ) + G(gxn , gxn+1 , gxn+1 ) + G(gxn , gxn , gxn )))
−φ(G(gxn−1 , gxn+1 , gxn+1 ) + G(gxn , gxn+1 , gxn+1 ) + G(gxn , gxn , gxn ))
≤ ψ( 31 (G(gxn−1 , gxn+1 , gxn+1 ) + G(gxn , gxn+1 , gxn+1 )))
≤ ψ( 31 G(gxn−1 , gxn , gxn ) + 23 G(gxn , gxn+1 , gxn+1 )).
✭✷✳✷✮
❱×

ψ ❧➭ ❤➭♠ t➝♥❣✱ ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ❝➠♥❣ t❤ø❝ ✭✷✳✷✮✱ t❛ ❝ã

1
G(gxn , gxn+1 , gxn+1 ) ≤ (G(gxn−1 , gxn+1 , gxn+1 ) + G(gxn , gxn+1 , gxn+1 ))
3
1
2
≤ G(gxn−1 , gxn , gxn ) + G(gxn , gxn+1 , gxn+1 ).
3
3
✭✷✳✸✮
❑❤✐ ➤ã t❛ ❞Ô ❞➭♥❣ s✉② r❛ r➺♥❣

G(gxn , gxn+1 , gxn+1 ) ≤ G(gxn−1 , gxn , gxn )
n ≥ 1✳
❱× ✈❐② {G(gxn , gxn+1 , gxn+1 ), n ∈ N} ❧➭ ♠ét ❞➲② ❤➭♠ ❦❤➠♥❣ t➝♥❣✳
tå♥ t➵✐ r ≥ 0 s❛♦ ❝❤♦

✭✷✳✹✮

✈í✐ ♠ä✐

❉♦ ➤ã✱

lim G(gxn , gxn+1 , gxn+1 ) = r.

n→∞
❈❤♦

✭✷✳✺✮

n → ∞ tr♦♥❣ ❝➠♥❣ t❤ø❝ ✭✷✳✸✮✱ t❛ ❝ã
r≤

1
1
2
1
lim G(gxn−1 , gxn+1 , gxn+1 ) + r ≤ r + r = r,
3 n→∞
3
3
3

➤✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ r➺♥❣

lim G(gxn−1 , gxn+1 , gxn+1 ) = 2r.

n→∞

✶✺

✭✷✳✻✮


❙ư ❞ơ♥❣ ❝➠♥❣ t❤ø❝ ✭✷✳✷✮ ♠ét ❧➬♥ ♥÷❛ t❛ ❝ã

1
ψ(G(gxn , gxn+1 , gxn+1 )) ≤ ψ( (G(gxn−1 , gxn+1 , gxn+1 )+
3
+ G(gxn , gxn+1 , gxn+1 )))
− φ(G(gxn−1 , gxn+1 , gxn+1 ), G(gxn , gxn+1 , gxn+1 ),
G(gxn , gxn , gxn )).
❈❤♦

n → ∞✱ ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ➤➻♥❣ t❤ø❝ ✭✷✳✺✮✱ ✭✷✳✻✮ ✈➭ tÝ♥❤ ❧✐➟♥ tơ❝ ❝đ❛ ψ ✈➭

φ✱ t❛ t❤✉ ➤➢ỵ❝
ψ(r) ≤ ψ(r) − φ(2r, r, 0),
✈× t❤Õ t❛ ❝ã

φ(2r, r, 0) = 0✳ ◆❤ê tÝ♥❤ ❝❤✃t ❝ñ❛ ❤➭♠ φ✱ t❛ s✉② r❛ r➺♥❣ r = 0✱ ♥❣❤Ü❛

❧➭ t❛ ❝ã

lim G (gxn , gxn+1 , gxn+1 ) = 0.

n→∞

❚✐Õ♣ t❤❡♦✱ t❛ ❝❤Ø r❛ r➺♥❣
❦❤➠♥❣ ❧➭ ❞➲②

✭✷✳✼✮

{gxn } ❧➭ ❞➲② G✲❈❛✉❝❤②✳ ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐ r➺♥❣ {gxn }

G✲❈❛✉❝❤②✱ ♥❣❤Ü❛ ❧➭
lim G(gxm , gxn , gxn ) = 0.

m,n→∞

ε > 0 ♠➭ ✈í✐ ♥ã t❛ ❝ã t❤Ĩ tì ợ {gxm(i) }
{gxn(i) } ủ {xn } s❛♦ ❝❤♦ n(i) ❧➭ ❝❤Ø sè ♥❤á ♥❤✃t s❛♦ ❝❤♦ ✈í✐

❑❤✐ ➤ã✱ tå♥ t➵✐

n(i) > m(i) > i,

t❛ ❝ã

G gxm(i) , gxn(i) , gxn(i) ≥ ε.

✈➭

✭✷✳✽✮

➜✐Ò✉ ♥➭② ❝ã ♥❣❤Ü❛ ❧➭

G(gxm(i) , gxn(i)−1 , gxn(i)−1 ) < ε.

✭✷✳✾✮

❇➞② ❣✐ê✱ ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ✭✺✮ tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✷✱ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✽✮✱
✭✷✳✾✮ ✈➭ sư ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✸✮ tr♦♥❣ ▼Ư♥❤ ➤Ị ✶✳✶✳✶✷✱ t❛ ❝ã

ε ≤ G(gxm(i) , gxn(i) , gxn(i) )
≤ G(gxm(i) , gxm(i)+1 , gxm(i)+1 ) + G(gxm(i)+1 , gxn(i) , gxn(i) )
≤ G(gxm(i) , gxm(i)+1 , gxm(i)+1 ) + G(gxm(i)+1 , gxn(i)−1 , gxn(i)−1 )+
+G(gxn(i)−1 , gxn(i) , gxn(i) )
≤ 3G(gxm(i) , gxm(i)+1 , gxm(i)+1 ) + G(gxm(i) , gxn(i)−1 , gxn(i)−1 )+
+ G(gxn(i)−1 , gxn(i) , gxn(i) )
≤ 3G(gxm(i) , gxm(i)+1 , gxm(i)+1 ) + ε + G(gxn(i)−1 , gxn(i) , gxn(i) ).
✶✻


❈❤♦

i → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ ✈➭ sö ❞ơ♥❣ ➤➻♥❣ t❤ø❝ ✭✷✳✼✮✱ t❛ ♥❤❐♥ ➤➢ỵ❝
lim G(gxm(i) , gxn(i) , gxn(i) ) = lim G(gxm(i)+1 , gxn(i) , gxn(i) )

i→∞

i→∞

= lim G(gxm(i) , gxn(i)−1 , gxn(i)−1 )
i→∞

= ε.

✭✷✳✶✵✮

❉♦ ➤ã✱ tõ ❝➠♥❣ t❤ø❝ ✭✷✳✶✮ t❛ ❝ã

ψ(G(gxm(i)+1 , gxn(i) , gxn(i) )) = ψ(G(f xm(i) , f xn(i)−1 , f xn(i)−1 ))
≤ ψ( 31 G(gxm(i) , f xn(i)−1 , f xn(i)−1 ) + G(gxn(i)−1 , f xn(i)−1 , f xn(i)−1 )+
+G(gxn(i)−1 , f xm(i) , f xm(i) )) − φ(G(gxm(i) , f xn(i)−1 , f xn(i)−1 ),
G(gxn(i)−1 , f xn(i)−1 , f xn(i)−1 ), G(gxn(i)−1 , f xm(i) , f xm(i) ))
= ψ( 31 G(gxm(i) , gxn(i) , gxn(i) ) + G(gxn(i)−1 , gxn(i) , gxn(i) )+
+G(gxn(i)−1 , gxm(i)+1 , gxm(i)+1 )) − φ(G(gxm(i) , gxn(i) , gxn(i) ),
G(gxn(i)−1 , gxn(i) , gxn(i) ), G(gxn(i)−1 , gxm(i)+1 , gxm(i)+1 ))
≤ ψ( 31 G(gxm(i) , gxn(i) , gxn(i) ) + G(gxn(i)−1 , gxn(i) , gxn(i) )+
+G(gxn(i)−1 , gxm(i)+1 , gxm(i)+1 )).
✭✷✳✶✶✮
▼ét ❧➬♥ ữ ì

t t ợ

1
G(gxm(i)+1 , gxn(i) , gxn(i) ) ≤ (G(gxm(i) , gxn(i) , gxn(i) )+
3
+ G(gxn(i)−1 , gxn(i) , gxn(i) ) + G(gxn(i)−1 , gxm(i)+1 , gxm(i)+1 )).
❉♦ ➤ã✱ ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✺✮ tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✷ ✈➭ ▼Ư♥❤ ➤Ị
✶✳✶✳✶✷✱ t❛ ❝ã

1
G(gxm(i)+1 , gxn(i) , gxn(i) ) ≤ (G(gxm(i) , gxn(i) , gxn(i) )+
3
+ G(gxn(i)−1 , gxn(i) , gxn(i) ) + G(gxn(i)−1 , gxm(i)+1 , gxm(i)+1 ))
1

≤ (G(gxm(i) , gxn(i) , gxn(i) ) + G(gxn(i)−1 , gxn(i) , gxn(i) )+
3
+ 2G(gxn(i)−1 , gxn(i)−1 , gxm(i)+1 ))
1
≤ (G(gxm(i) , gxn(i) , gxn(i) ) + G(gxn(i)−1 , gxn(i) , gxn(i) )+
3
+ 2G(gxn(i)−1 , gxn(i)−1 , gxm(i) ) + 2G(gxm(i) , gxm(i) , gxm(i)+1 )).
✶✼


❈❤♦

i → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥✱ ✈➭ sö ❞ô♥❣ ❝➠♥❣ t❤ø❝ ✭✷✳✼✮ ✈➭ ✭✷✳✶✵✮✱ t❛ ❝ã
lim G(gxn(i)−1 , gxm(i)+1 , gxm(i)+1 ) = 2ε.

i→∞

✭✷✳✶✷✮

i → ∞ tr♦♥❣ ❝➠♥❣ t❤ø❝ ✭✷✳✶✶✮ ✈➭ sư ❞ơ♥❣ ❝➠♥❣ t❤ø❝ ✭✷✳✼✮✱ ✭✷✳✶✵✮✱
✭✷✳✶✷✮ ✈➭ tÝ♥❤ ❧✐➟♥ tơ❝ ❝đ❛ ψ ✈➭ φ✱ t❛ ❝ã
❇➞② ❣✐ê✱ ❝❤♦

1
(ε + 0 + 2ε) − φ(ε, 0, 2ε).
3
❉♦ ➤ã✱ t ợ (, 0, 2) = 0 ì tế ♥❤ê ❝➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ ❤➭♠ φ✱ t❛ s✉②
r❛ ε = 0✳ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈× ε > 0✳ ❱× t❤Õ✱ {gxn } ❧➭ ❞➲② G✲❈❛✉❝❤② tr♦♥❣
g(X)✳ ❱× (g(X), G) ❧➭ ➤➬② ➤ñ✱ ♥➟♥ tå♥ t➵✐ t, u ∈ X s❛♦ ❝❤♦ {gxn } ❤é✐ tô ➤Õ♥
t = gu✱ ♥❣❤Ü❛ ❧➭

ψ(ε) ≤ ψ

lim G(gxn , gxn , gu) = lim G(gxn , gu, gu) = 0.

n→∞
❱×

n→∞

✭✷✳✶✸✮

G ❧➭ ❧✐➟♥ tơ❝ t❤❡♦ ❝➳❝ ❜✐Õ♥ ❝ñ❛ ♥ã✱ ♥➟♥ t❛ ❝ã
lim G(gxn , gxn , f u) = G(gu, gu, f u),

✭✷✳✶✹✮

lim G(gxn , f u, f u) = G(gu, f u, f u).

✭✷✳✶✺✮

n→∞
✈➭

n→∞
❇➞② ❣✐ê✱ t❛ ❝❤Ø r❛ r➺♥❣

f u = t✳ ◆❤ê ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✶✮✱ t❛ ❝ã

ψ(G(gxn+1 , gxn+1 , f u)) = ψ(G(f xn , f xn , f u)
1

≤ ψ( (G(gxn , f xn , f xn ) + G(gxn , f u, f u) + G(gu, f xn , f xn )
3
− φ(G(gxn , f xn , f xn ) + G(gxn , f u, f u) + G(gu, f xn , f xn ))
1
= ψ( (G(gxn , gxn+1 , gxn+1 ) + G(gxn , f u, f u) + G(gu, gxn+1 , gxn+1 )))
3
− φ(G(gxn , gxn+1 , gxn+1 ) + G(gxn , f u, f u) + G(gu, gxn+1 , gxn+1 )).
❈❤♦
✈➭

n → ∞✱ sư ❞ơ♥❣ ❝➳❝ ❝➠♥❣ t❤ø❝ ✭✷✳✼✮✱ ✭✷✳✶✸✮ − ✭✷✳✶✺✮ ✈➭ tÝ♥❤ ❧✐➟♥ tơ❝ ❝đ❛ ψ

φ✱ t❛ ❝ã

ψ(G(gu, gu, f u)) ≤ ψ

1
G(gu, f u, f u) − φ(0, G(gu, f u, f u), 0).
3

❚õ ❝➠♥❣ t❤ø❝ ✭✸✮ tr♦♥❣ ▼Ư♥❤ ➤Ị ✶✳✶✳✶✷✱ t❛ ❝ã

G(gu, f u, f u) ≤ 2G(gu, gu, f u)
✶✽

✭✷✳✶✻✮