Điểm bất động bộ đôi của các ánh xạ co suy rộng trong không gian meetric thứ tự

❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦

❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤

◆❣✉②Ơ♥ ❍♦➭♥❣ ❉ị

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

◆❣❤Ö ❆♥ ✲ ✷✵✶✺


❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦

❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤

◆❣✉②Ơ♥ ❍♦➭♥❣ ❉ị

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

❚♦➳♥ ●✐➯✐ tÝ❝❤

▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷

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

◆❣❤Ö ❆♥ ✲ ✷✵✶✺


▼ơ❝ ▲ơ❝

❚r❛♥❣
▼ơ❝ ❧ơ❝

✶

▲ê✐ ♥ã✐ ➤➬✉

✷

❈❤➢➡♥❣ ✶✳

➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ♣❤✐ t✉②Õ♥ ❦❤➠♥❣ ❝ã

tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù

✺

✶✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥

✺

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

✶✳✷ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ♣❤✐ t✉②Õ♥ ❦❤➠♥❣ ❝ã tÝ♥❤
❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❈❤➢➡♥❣ ✷✳

➜✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉

❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù
❄❄

✶✶

✷✷

➜✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉
❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❄❄

✷✳✷ ➜✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥
❝♦ ❦✐Ĩ✉ ❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✼

❑Õt ❧✉❐♥

✸✸

❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦

✸✹

✶

❧ê✐ ♥ã✐ ➤➬✉

▲ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ ♠ét tr♦♥❣ ♥❤÷♥❣ ❝❤đ ➤Ị ♥❣❤✐➟♥ ❝ø✉ q✉❛♥
trä♥❣ ❝đ❛ ❣✐➯✐ tÝ❝❤✳ ◆ã ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ tr♦♥❣ t♦➳♥ ❤ä❝ ✈➭ ❝➳❝ ♥❣➭♥❤ ❦ü
t❤✉❐t✳ ❈➳❝ ❦Õt q✉➯ q✉❛♥ trä♥❣ ➤➬✉ t✐➟♥ ♣❤➯✐ ❦Ó ➤Õ♥ tr♦♥❣ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t
➤é♥❣ ❧➭ ♥❣✉②➟♥ ❧ý ➤✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❇r♦✇❡r ✈➭♦ ♥➝♠ ✶✾✶✷ ✈➭ ♥❣✉②➟♥ ❧ý ➳♥❤
①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ❝ñ❛ ❇❛♥❛❝❤ ✈➭♦ ♥➝♠ ✶✾✷✷✳ ◆❣✉②➟♥
❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ➤➲ trë t❤➭♥❤ ♠ét ❝➠♥❣ ❝ơ ♣❤ỉ ❞ơ♥❣ ➤Ĩ ❣✐➯✐ q✉②Õt ❝➳❝
❜➭✐ t♦➳♥ ✈Ị sù tå♥ t➵✐ tr♦♥❣ ♥❤✐Ị✉ ❝❤✉②➟♥ ♥❣➭♥❤ ❝đ❛ ●✐➯✐ tÝ❝❤ t♦➳♥ ❤ä❝ ✈➭ ❝ã
♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ tr♦♥❣ ❝➳❝ ♥❣➭♥❤ ❦❤♦❛ ❤ä❝ ❦❤➳❝✳ ❱× t❤Õ ➤➲ ❝ã ♠ét sè ❧í♥ ❝➳❝
♠ë ré♥❣ ❝đ❛ ♥❣✉②➟♥ ❧ý ❝➡ ❜➯♥ ♥➭② ❝❤♦ ❝➳❝ ❧í♣ ➳♥❤ ①➵ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ❦❤➳❝
♥❤❛✉✱ ❜➺♥❣ ❝➳❝❤ ➤✐Ị✉ ❝❤Ø♥❤ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝➡ ❜➯♥ ❤♦➷❝ t❤❛② ➤ỉ✐ ❦❤➠♥❣ ❣✐❛♥✳
❑❤➳✐ ♥✐Ư♠ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ➤➢ỵ❝ ❙✳ ❙✳ ❈❤❛♥❣ ✈➭ ❨✳ ❍✳ ▼❛ ❣✐í✐ t❤✐Ư✉
♥➝♠ ✶✾✾✶ ✈➭ s❛✉ ➤ã ➤➲ t❤✉ ❤ót sù q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉ ❝đ❛ ♥❤✐Ị✉ ♥❤➭ t♦➳♥
❤ä❝✳ ◆➝♠ ✷✵✵✻✱ ❚✳ ●✳ ❇❤❛s❦❛r ✈➭ ❱✳ ▲❛❦s❤♠✐❦❛♥t❤❛♠ ➤➲ t❤✐Õt ❧❐♣ ❝➳❝ ➤Þ♥❤
❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝
tù ❜é ♣❤❐♥ ✧≤✧ ❝❤♦ ❝➳❝ ➳♥❤ ①➵

F : X × X → X ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ ✈➭

t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦✿ ❚å♥ t➵✐ sè

x ≥ u, y ≤ v

(X, d) ❝ã tr❛♥❣ ❜Þ t❤ø

k ∈ (0, 1)

s❛♦ ❝❤♦ ✈í✐ ♠ä✐

x, y, u, v ∈ X

♠➭

t❛ ❝ã

d(T (x, y), T (u, v)) ≤

k
2

d(x, u) + d(y, v) .

◆➝♠ ✷✵✵✾✱ ❱✳ ▲❛❦s❤♠✐❦❛♥t❤❛♠ ✈➭ ▲✳ ❈✐r✐❝ ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ tÝ♥❤ ❝❤✃t

g ✲➤➡♥ ➤✐Ư✉ tré♥✱ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠
❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ♠➭ ❝❤ó♥❣ ❧➭ ♠ë ré♥❣ ❝đ❛
❝➳❝ ❦Õt q✉➯ ➤➲ t❤✉ ➤➢ỵ❝ ❝đ❛ ❚✳ ●✳ ❇❤❛s❦❛r ✈➭ ❱✳ ▲❛❦s❤♠✐❦❛♥t❤❛♠✱✳✳✳
➜Ĩ t❐♣ ❞➢ỵt ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝✱ ❝❤ó♥❣ t➠✐ t✐Õ♣ ❝❐♥ ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉
♥➭② ♥❤➺♠ t×♠ ❤✐Ĩ✉ ❝➳❝ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉②
ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✳ ❚r➟♥ ❝➡ së ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦✱ ❞➢í✐
sù ❤➢í♥❣ ❞➱♥ ❝đ❛ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✿
✷


✧ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣
❣✐❛♥ ♠➟tr✐❝ t❤ø tù✧
▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣

❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ❝➳❝ ➳♥❤ ①➵ ❣✐❛♦ ❤♦➳♥✱ tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉
tré♥✱ tÝ♥❤ ❝❤✃t

g ✲➤➡♥ ➤✐Ö✉ tré♥✱ ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é

➤➠✐✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐✱ ➤✐Ị✉ ❦✐Ư♥ ❝♦✱ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥
❤÷✉ tû✱ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵ ✈Ị ❝➳❝ ➳♥❤ ①➵ ➤ã✱ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣
❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ ❦❤➠♥❣ ➤ß✐ ❤á✐ tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥✱ ♠ét
sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❤÷✉ tû ✈➭ ♠ét
sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ❦Õt q✉➯ ➤ã✱✳✳✳
❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ♣❤✐
t✉②Õ♥ ❦❤➠♥❣ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø
tù✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ơ❝ ✶ ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ q✉❛ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠

❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝đ❛ ❧✉❐♥ ✈➝♥✳ ❈➳❝ ♥é✐ ❞✉♥❣ ❣å♠✿ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱
❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ❝➳❝ ➳♥❤ ①➵ ❣✐❛♦ ❤♦➳♥✱ tÝ♥❤ ❝❤✃t ➤➡♥
➤✐Ö✉ tré♥✱ tÝ♥❤ ❝❤✃t g ✲➤➡♥ ➤✐Ư✉ tré♥✱ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ➤✐Ó♠ ❜✃t ➤é♥❣
❜é ➤➠✐✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐✱ ➤✐Ị✉ ❦✐Ư♥ ❝♦✱ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥
❤÷✉ tû✱ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣
❦❤➠♥❣ ➤ß✐ ❤á✐ tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥✱ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝
➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❤÷✉ tû ✈➭ ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ết q
ó ụ trì ột số ị ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ♠➭ ❦❤➠♥❣ ➤ß✐ ❤á✐
tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥✱ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ✈Ị ❝➳❝ ➤Þ♥❤ ý ó r ò
trì ệ q ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳
❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛
♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✳ ❚r♦♥❣

❝❤➢➡♥❣ ♥➭②✱ ♠ơ❝ ✶ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é
➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø
tù✱ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ❦Õt q ợ trì ụ ú t trì

ột sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝
➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ✈➭

✸


❝➳❝ ❤Ư q✉➯ ❝đ❛ ❝❤ó♥❣✱ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ✈Ị ❝➳❝ ❦Õt q✉➯ ➤ã ✈➭ tr×♥❤ ❜➭② ♠ét
sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳
▲✉❐♥ ✈➝♥ ♥➭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✳ ◆❤➞♥ ❞Þ♣ ♥➭② t➳❝ ❣✐➯ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➳♠ ➡♥ ❇❛♥
❝❤đ ♥❤✐Ư♠ ❦❤♦❛ ❙➢ ♣❤➵♠ ❚♦➳♥ ❤ä❝✱ P❤ß♥❣ ➤➭♦ t➵♦ ❙❛✉ ➤➵✐ ❤ä❝✱ q✉ý t❤➬②✱ ❝➠
tr♦♥❣ tæ ●✐➯✐ ❚Ý❝❤ ❦❤♦❛ ❚♦➳♥ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ Pò ọ ợ
t qố tế rờ ọ ò ú ỡ tr q trì ọ t❐♣ ✈➭
❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ◆❤➞♥ ➤➞② t➳❝ ❣✐➯ ①✐♥ ❝➳♠ ➡♥ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ ❝❛♦ ❤ä❝
❦❤♦➳ ✷✶ ●✐➯✐ ❚Ý❝❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❙➭✐ ●ß♥✳ ❈✉è✐ ❝ï♥❣ ❝➳♠ ➡♥ ❣✐❛ ➤×♥❤
✈➭ ❇❛✱ ▼Đ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ợ ú t t ệ ụ tr
q trì ❤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❤➳♥❣ ✽ ♥➝♠ ✷✵✶✺

◆❣✉②Ơ♥ ❍♦➭♥❣ ❉ị

✹

❝❤➢➡♥❣ ✶

➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦
♣❤✐ t✉②Õ♥ ❦❤➠♥❣ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥
tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù
❈➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥

✶✳✶

P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ q✉❛ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ sở ệ
trì ủ

ị ĩ



ột tr tr

❈❤♦ t❐♣ ❤ỵ♣

X = φ✱ ➳♥❤ ①➵ d : X × 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

❦Ý ❤✐Ö✉ ❧➭

(X, d) ❤❛② ➤➡♥ ❣✐➯♥ ❧➭ X ✳ ❙è d (x, y) ❣ä✐ ❧➭

❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭

❦❤♦➯♥❣ ❝➳❝❤ tõ ➤✐Ĩ♠

x

➤Õ♥ ➤✐Ĩ♠ y ✳

✶✳✶✳✷

❱Ý ❞ơ✳

✶✮ ❳Ðt

X = R✱ d : R × R → R ❝❤♦ ❜ë✐ d (x, y) = |x − y|✱ ✈í✐ ♠ä✐

x, y ∈ R✳ ❑❤✐ ➤ã d ❧➭ ♠ét ♠➟tr✐❝ tr➟♥ R✳
✷✮ ❳Ðt

X = Rn ✳ ❱í✐ ❜✃t ❦ú x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ Rn t❛ ➤➷t
1
2

n

2

|xi − yi |

d1 (x, y) =

n

✈➭

i=1

i=1

tr➟♥

✶✳✶✳✸

X

|xi − yi |✳ ❑❤✐ ➤ã d1 , d2 ❧➭ ❝➳❝ ♠➟tr✐❝

d2 (x, y) =

Rn ✳

▼Ö♥❤ ➤Ị✳

✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

(X, d)✳ ❑❤✐ ➤ã ✈í✐ ♠ä✐ x, y, u, v ∈

✱ t❛ ❝ã

|d (x, y) − d (u, v)| ≤ d (x, u) + d (y, v) .
✺


➜Þ♥❤ ♥❣❤Ü❛✳

✶✳✶✳✹

✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

d(x, A) = inf d (x, y) ✈➭ ❣ä✐ d(x, A) ❧➭ ❦❤♦➯♥❣
y∈A

▼Ư♥❤ ➤Ị✳

✶✳✶✳✺

x, y ∈ X

(X, d)✱ A ⊂ X ✱ x ∈ X ✱ ❦Ý ❤✐Ư✉

❝➳❝❤ tõ ➤✐Ĩ♠

✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

x

➤Õ♥ t❐♣ ❤ỵ♣

(X, d)✱ A ⊂ X ✳

A✳

❑❤✐ ➤ã ✈í✐ ♠ä✐

t❛ ❝ã

|d (x, A) − d (y, A)| ≤ d (x, y) .

➜Þ♥❤ ♥❣❤Ü❛✳

✶✳✶✳✻

❧➭ ❤é✐ tơ ✈Ị ➤✐Ĩ♠

✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

(X, d)✱ ❞➲② {xn } ⊂ X ➤➢ỵ❝ ❣ä✐

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 xn → x ❦❤✐ n → ∞✳
n→∞

▼Ư♥❤ ➤Ị✳

✶✳✶✳✼

✶✮ ❚❐♣
✷✮

E

x∈E

✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

➤ã♥❣ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ✈í✐ ♠ä✐

♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ tå♥ t➵✐

➜Þ♥❤ ♥❣❤Ü❛✳

✶✳✶✳✽

{xn } ⊂ E

{xn } ⊂ E

✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

❧➭ ❞➲② ❈❛✉❝❤② ♥Õ✉ ✈í✐ ♠ä✐
❝ã

(X, d)✱ E ⊂ X ✱ x ∈ X ✳
♠➭

s❛♦ ❝❤♦

➜Þ♥❤ ♥❣❤Ü❛✳

❚❐♣ ❝♦♥
❣✐❛♥ ❝♦♥

✶✳✶✳✶✵

xn → x✳

ε > 0✱ tå♥ t➵✐ n0 ∈ N∗ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n, m ≥ n0 t❛

✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

♥Õ✉ ♠ä✐ ❞➲② ❈❛✉❝❤② tr♦♥❣

xn → x t❛ ❝ã x ∈ E ✳

(X, d)✳ ❉➲② {xn } ⊂ X ➤➢ỵ❝ ❣ä✐

d(xn , xm ) < ε✱ ❤❛② {xn } ❧➭ ❞➲② ❈❛✉❝❤② ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉

✶✳✶✳✾

❑❤✐ ➤ã

lim

n,m→+∞

d(xn , xm ) = 0✳

(X, d)✳ ❚❛ ♥ã✐ (X, d) ❧➭ ➤➬② ➤đ

X ➤Ị✉ ❤é✐ tơ✳

M ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ➤➢ỵ❝ ❣ä✐ ❧➭

➤➬② ➤đ ♥Õ✉ ❦❤➠♥❣

M ✈í✐ ♠➟tr✐❝ ❝➯♠ s✐♥❤ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ➤➬② ➤đ✳

❱Ý ❞ơ✳

✶✮ ❚❐♣ ❤ỵ♣ ❝➳❝ sè t❤ù❝

R ✈í✐ ♠➟tr✐❝ d (x, y) = |x − y| ❧➭ ❦❤➠♥❣

❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✳

✷✮ ❚❐♣ ❤ỵ♣

Rn ❣å♠ t✃t ❝➯ ❝➳❝ ❜é n sè t❤ù❝✱ ✈í✐ ♠➟tr✐❝ d1 (x, y)✱ d2 (x, y) ❧➭

❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✳

✻


✶✳✶✳✶✶

▼Ư♥❤ ➤Ị✳

✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

✶✮ ◆Õ✉

M

➤➬② ➤đ t❤×

✷✮ ◆Õ✉

M

❧➭ t❐♣ ➤ã♥❣ ✈➭

✶✳✶✳✶✷

➜Þ♥❤ ♥❣❤Ü❛✳

M

(X, d)✱ M ⊂ X ✳

❑❤✐ ➤ã

❧➭ t❐♣ ➤ã♥❣✳

X

➤➬② ➤đ t❤×

M

➤➬② ➤đ✳

✭❬✶❪✮ ❈❤♦ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

f : (X, d) → (Y, ρ) ➤➢ỵ❝ ❣ä✐ ❧➭

➳♥❤ ①➵ ❝♦ ♥Õ✉ tå♥ t➵✐

(X, d) ✈➭ (Y, ρ)✳

➳♥❤ ①➵

α ∈ [0, 1) s❛♦ ❝❤♦ ✈í✐ ♠ä✐

x, y ∈ X t❛ ❝ã

ρ[f (x) , f (y)] ≤ αd (x, y) .
❙è tự



[0, 1) ợ ọ ệ

ị ý

f :XX

x ∈ X

t➵✐ ❞✉② ♥❤✃t ➤✐Ó♠

①➵

f tr➟♥ X ✳

✭❬✶❪✮ ✭◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤✮ ●✐➯ sư

❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✱

➜✐Ĩ♠

sè ❝♦ ❝ñ❛

❧➭ ➳♥❤ ①➵ ❝♦ tõ

s❛♦ ❝❤♦

X

(X, d)

❧➭ ❦❤➠♥❣

✈➭♦ ❝❤Ý♥❤ ♥ã✳ ❑❤✐ ➤ã tå♥

f (x∗ ) = x∗ ✳

x∗ ∈ X ❝ã tÝ♥❤ ❝❤✃t f (x∗ ) = x∗ ➤➢ỵ❝ ❣ä✐ ❧➭

➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤

f✳

✶✳✶✳✶✹

➜Þ♥❤ ♥❣❤Ü❛✳

✭❬✷❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

(X, d) ✈➭ ❝➳❝ ➳♥❤ ①➵ F : X ×

X → X ✈➭ g : X → X ✳ ❍❛✐ ➳♥❤ ①➵ F ✈➭ g ➤➢ỵ❝ ❣ä✐ ❧➭ ❣✐❛♦

❤♦➳♥ ✈í✐ ♥❤❛✉ ♥Õ✉

F (gx, gy) = g(F (x, y)) ✈í✐ ♠ä✐ x, y ∈ X ✳

✶✳✶✳✶✺

➜Þ♥❤ ♥❣❤Ü❛✳

✭❬✷❪✮ ❈❤♦ t❐♣

X = φ✳ (X, d, ≤) ➤➢ỵ❝ ❣ä✐ ❧➭

❦❤➠♥❣ ❣✐❛♥

♠➟tr✐❝ t❤ø tù ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉
✭✶✮

(X, ≤) ❧➭ ♠ét t❐♣ s➽♣ t❤ø tù tõ♥❣ ♣❤➬♥✳

✭✷✮

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳

✶✳✶✳✶✻

➜Þ♥❤ ♥❣❤Ü❛✳

✭✶✮ ❈➳❝ ♣❤➬♥ tö

✭❬✷❪✮ ❈❤♦ t❐♣ s➽♣ t❤ø tù tõ♥❣ ♣❤➬♥

(X, ≤)✳

x, y ∈ X ➤➢ỵ❝ ❣ä✐ ❧➭ s♦ s➳♥❤ ➤➢ỵ❝ ✈í✐ ♥❤❛✉ ➤è✐ ✈í✐ t❤ø tù

≤✱ ♥Õ✉ ❤♦➷❝ x ≤ y ✱ ❤♦➷❝ y ≤ x✳

✼


✭✷✮

➳♥❤ ①➵ f : X → X ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐➯♠ ♥Õ✉ ✈í✐ x, y ∈ X ♠➭ x ≤ y t❛
❝ã

✭✸✮

f (x) ≤ f (y)✳

➳♥❤ ①➵ f : X → X ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ t➝♥❣ ♥Õ✉ ✈í✐ x, y ∈ X ♠➭ x ≤ y t❛
❝ã

f (y) ≤ f (x)✳
✭❬✷❪✮ ❈❤♦ t❐♣ ❤ỵ♣ ➤➢ỵ❝ s➽♣ t❤ø tù từ



ị ĩ



F : X ì X X ✈➭ g : X → X ✳ ➳♥❤ ①➵ F ➤➢ỵ❝ ❣ä✐ ❧➭ ❝ã

➤✐Ư✉ tré♥ ♥❣➷t ♥Õ✉

♥❣❤Ü❛ ❧➭ ✈í✐ ❜✃t ❦ú
♥Õ✉

(X, ≤) ✈➭ ✷

tÝ♥❤ ❝❤✃t

g ✲➤➡♥

F (x, y) t➝♥❣ ♥❣➷t t❤❡♦ ❜✐Õ♥ x✱ ✈➭ ❣✐➯♠ ♥❣➷t t❤❡♦ ❜✐Õ♥ y ✱

x, y ∈ X

x1 , x2 ∈ X, ♠➭ gx1 < gx2 , t❤× t❛ ❝ã F (x1 , y) < F (x2 , y),

✈➭
♥Õ✉ y1 , y2

∈ X, ♠➭ gy1 < gy2 , t❤× t❛ ❝ã F (x, y1 ) > F (x, y2 ).

❚r♦♥❣ ➤Þ♥❤ ♥❣❤Ü❛ tr➟♥✱ ♥Õ✉

g ❧➭ ồ t tì F ợ ọ

➤➡♥ ➤✐Ư✉ tré♥ ♥❣➷t✳

✶✳✶✳✶✽

❝đ❛ ❝➳❝ ➳♥❤ ①➵

✶✳✶✳✶✾

✭❬✷❪✮ P❤➬♥ tư

➜Þ♥❤ ♥❣❤Ü❛✳

(x, y) X ì X ợ ọ ể trù ❜é ➤➠✐

F : X × X → X ✈➭ g : X → X ♥Õ✉ F (x, y) = gx ✈➭ F (y, x) = gy ✳
✭❬✷❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

➜Þ♥❤ ♥❣❤Ü❛✳

➤➢➡❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ➳♥❤ ①➵
✈➭

(X, d)✳ P❤➬♥ tư (x, y) ∈ X × X

T : X × X → X ♥Õ✉ T (x, y) = x

T (y, x) = y ✳

✶✳✶✳✷✵

❱Ý ❞ô✳

❝➠♥❣ t❤ø❝

X = [0; +∞) ✈➭ ➳♥❤ ①➵ T : X × X X ợ ị ở

T (x; y) = x + y ✈í✐ ♠ä✐ x, y ∈ X ✳ ❉Ơ t❤✃② r➺♥❣ T ❝ã ♠ét ➤✐Ĩ♠ ❜✃t

➤é♥❣ ❜é ➤➠✐ ❧➭

✶✳✶✳✷✶

❈❤♦

❱Ý ❞ô✳

(0, 0)✳
❈❤♦

X = P([0; 1)) ❧➭ ❤ä t✃t ❝➯ ❝➳❝ t❐♣ ❝♦♥ ❝ñ❛ t❐♣ [0, 1) ✈➭ T :

X ì X X ợ ị ở T (A; B) = A − B ✈í✐ ♠ä✐ A, B ∈ X ✳ ❑❤✐ ➤ã✱ t❛
❝ã t❤Ó t❤✃② r➺♥❣ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛
❤❛✐ t❐♣ ❤ỵ♣ rê✐ ♥❤❛✉✳
✽

T ❧➭ ❝➷♣ (A, B)✱ tr♦♥❣ ➤ã A ✈➭ B ❧➭


t ợ ợ s tứ tự từ



ị ♥❣❤Ü❛✳

➳♥❤ ①➵

T : X × X → X✳

(X, ≤) ✈➭

➳♥❤ ①➵ T ➤➢ỵ❝ ❣ä✐ ❧➭ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ ♥Õ✉

T (x, y) ➤➡♥ ➤✐Ö✉ ❦❤➠♥❣ ❣✐➯♠ t❤❡♦ ❜✐Õ♥ x✱ ✈➭ ➤➡♥ ➤✐Ö✉ ❦❤➠♥❣ t➝♥❣ t❤❡♦ ❜✐Õ♥ y ✱
♥❣❤Ü❛ ❧➭ ✈í✐ ❜✃t ❦ú
♥Õ✉

x, y ∈ X

x1 , x2 ∈ X, ♠➭ x1 ≤ x2 , t❤× t❛ ❝ã T (x1 , y) ≤ T (x2 , y),

✈➭
♥Õ✉ y1 , y2

✶✳✶✳✷✸

✭❬✷❪✮ ❈❤♦ t❐♣ ❤ỵ♣ s➽♣ t❤ø tù tõ♥❣ ♣❤➬♥ (X, ≤) ✈➭ ❣✐➯ sư r➺♥❣

➜Þ♥❤ ❧ý✳

❝ã ♠ét ♠➟tr✐❝

∈ X, ♠➭ y1 ≤ y2 , t❤× t❛ ❝ã T (x, y1 ) ≥ T (x, y2 ).

d tr➟♥ X

F :X ×X →X

s❛♦ ❝❤♦

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

❧➭ ➳♥❤ ①➵ ❧✐➟♥ tơ❝ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ tr➟♥

k ∈ [0, 1)

r➺♥❣ tå♥ t➵✐ ♠ét sè

d(F (x, y), F (u, v)) ≤ k

X✳

❈❤♦

●✐➯ sö

s❛♦ ❝❤♦

d(x, u) + d(y, v)
2

✈í✐ ♠ä✐

x, y, u, v ∈ X

♠➭

x ≥ u, y ≤ v.
✭✶✳✶✮

◆Õ✉ tå♥ t➵✐

x, y ∈ X

✶✳✶✳✷✹

x0 , y0 ∈ X

x = F (x, y)

s❛♦ ❝❤♦

➜Þ♥❤ ❧ý✳

✐✮ ◆Õ✉

X

✈➭

x0 ≤ F (x0 , y0 )

✈➭

y0 ≥ F (y0 , x0 )✱

d tr➟♥ X

t❤× tå♥ t➵✐

y = F (y, x)✳

✭❬✷❪✮ ❈❤♦ t❐♣ ❤ỵ♣ ➤➢ỵ❝ t❤ø tù tõ♥❣ ♣❤➬♥

r➺♥❣ ❝ã ♠ét ♠➟tr✐❝
●✐➯ sư r➺♥❣

s❛♦ ❝❤♦

s❛♦ ❝❤♦

(X, ≤)

✈➭ ❣✐➯ sö

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

❝ã ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉ ➤➞②

{xn }

❧➭ ♠ét ❞➲② sè ❦❤➠♥❣ ❣✐➯♠ ✈í✐

xn → x✱

t❤×

xn ≤ x

✈í✐ ♠ä✐

❧➭ ♠ét ❞➲② sè ❦❤➠♥❣ t➝♥❣ ✈í✐

yn → y ✱

t❤×

yn ≥ y

✈í✐ ♠ä✐

n ≥ 1✳
✐✐✮ ◆Õ✉

{yn }

n ≥ 1✳
❈❤♦

F : X ×X → X

tå♥ t➵✐ ♠ét sè

❧➭ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ tr➟♥

k ∈ [0, 1)

d(F (x, y), F (u, v)) ≤ k

X ✳ ●✐➯ sö r➺♥❣

s❛♦ ❝❤♦

d(x, u) + d(y, v)
2

✈í✐ ♠ä✐

x, y, u, v ∈ X

♠➭

x ≥ u, y ≤ v.
✭✶✳✷✮

✾


◆Õ✉ tå♥ t➵✐

x, y ∈ X

✶✳✶✳✷✺

x0 , y 0 ∈ X

s❛♦ ❝❤♦

s❛♦ ❝❤♦

x = F (x, y)

✈➭

y0 ≥ F (y0 , x0 )

tì tồ t

y = F (y, x)





ị ♥❣❤Ü❛✳

x0 ≤ F (x0 , y0 )

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

X ×X → X ✳ ●✐➯ sư M ❧➭ t❐♣ ❤ỵ♣ ❝♦♥ ❦❤➳❝ rỗ ủ X 4 = X ìX ìX ìX ❚❛ ♥ã✐
r➺♥❣

M ❧➭ t❐♣

F ✲❜✃t

❝♦♥

❜✐Õ♥ q✉❛ ➳♥❤ ①➵

F ❝ñ❛ X 4 ♥Õ✉ ✈í✐ ♠ä✐ x, y, z, w ∈ X

t❛ ❝ã
✐✮

(x, y, z, w) ∈ M ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ (w, z, y, x) ∈ M ✳

✐✐✮ ◆Õ✉

✶✳✶✳✷✻

(x, y, z, w) ∈ M ✱ t❤× (F (x, y), F (y, x), F (z, w), F (w, z)) ∈ M ✳
✭❬✶✵❪✮

ị ĩ

t ợ ủ

(X, d) ột ♠➟tr✐❝ ✈➭ M ❧➭ ♠ét

X 4 ✳ ❚❛ ♥ã✐ M t❤á❛ ♠➲♥

tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉ ♥Õ✉ ✈í✐ ♠ä✐

x, y, z, w, a, b ∈ X ✱ ♠➭ (x, y, z, w) ∈ M ✈➭ (z, w, a, b) ∈ M ✱ t❤× t❛ ❝ã (x, y, a, b) ∈ M ✳

✶✳✶✳✷✼

◆❤❐♥ ①Ðt✳

❉Ơ ❞➭♥❣ t❤✃② r➺♥❣ t❐♣ ❤ỵ♣

M = X 4 ❧➭ t❐♣ F ✲❜✃t ❜✐Õ♥ t➬♠

t❤➢ê♥❣✱ ✈➭ ♥ã ❝ò♥❣ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉✳

✶✳✶✳✷✽

❱Ý ❞ô✳

❈❤♦

X = {0, 1, 2, 3} ✈í✐ ♠➟r✐❝ t❤➠♥❣ t❤➢ê♥❣ ✈➭ ➳♥❤ ①➵ F : X ì

X X ợ ị ở tứ
F (x, y) =
❉Ô t❤✃② r➺♥❣

1 ♥Õ✉ x, y = {1, 2},
3 tr trờ ợ ò .

M = {1, 2}4 ⊆ X 4 ❧➭ t❐♣ F ✲❜✃t ❜✐Õ♥✱ ✈➭ ♥ã ❝ị♥❣ t❤á❛ ♠➲♥

tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉✳

✶✳✶✳✷✾

❱Ý ❞ơ✳

❈❤♦

X = R ✈í✐ ♠➟tr✐❝ t❤➠♥❣ t❤➢ê♥❣ ✈➭ ➳♥❤ ①➵ F : X × X X

ợ ị ở tứ

F (x, y) =
❉Ô t❤✃② r➺♥❣

x

♥Õ✉

x, y ∈ (−∞, −1) ∪ (1, +∞),

cos(x + y) sin(x y) tr trờ ợ ò ❧➵✐.
M = [(−∞, −1) ∪ (1, ∞)]4 ⊆ X 4 ❧➭ t❐♣ F ✲❜✃t ❜✐Õ♥✱ ✈➭ ♥ã ❝ò♥❣

t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉✳
✶✵


✶✳✶✳✸✵

♣❤➬♥

❱Ý ❞ơ✳

❈❤♦

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➢ỵ❝ tr❛♥❣ ị tứ tự từ

F : X ì X → X ❧➭ ♠ét ➳♥❤ ①➵ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥✱

♥❣❤Ü❛ ❧➭ ✈í✐ ♠ä✐
♥Õ✉

x, y ∈ X t❛ ❝ã

x1 , x2 ∈ X, ♠➭ x1 ≤ x2 , t❤× t❛ ❝ã F (x1 , y) ≤ F (x2 , y),

✈➭
♥Õ✉ y1 , y2

∈ X, ♠➭ y1 ≤ y2 , t❤× t❛ ❝ã F (x, y1 ) ≥ F (x, y2 ).

ờ t ị ĩ t ợ

M X 4 ❝❤♦ ❜ë✐

M = {(a, b, c, d) ∈ X 4 : a ≥ c, b ≤ d}.
❑❤✐ ➤ã

M ❧➭ t❐♣ F ✲❜✃t ❜✐Õ♥✱ ✈➭ ♥ã ❝ò♥❣ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉✳

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

✶✳✷

❦❤➠♥❣ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥
♠➟tr✐❝ t❤ø tù

P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐
ò ỏ tí t ệ trộ



ị ý

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

t rỗ ủ
ớ

X 4

0 = (0) < ϕ(t) < t

F :X ×X →X

●✐➯ sư r➺♥❣ ❝ã ♠ét ❤➭♠ sè

✈➭

lim ϕ(r) < t

✈í✐ ♠ä✐

r→t+

ϕ : [0, +∞) → [0, +∞)

t > 0✱

❝ị♥❣ ❣✐➯ sư r➺♥❣

❧➭ ♠ét ➳♥❤ ①➵ s❛♦ ❝❤♦

d(F (x, y), F (u, v)) ≤ ϕ

d(x, u) + d(y, v)
2

,

✈í✐ ♠ä✐

(x, y, u, v) ∈ M.

✭✶✳✸✮

●✐➯ sư r➺♥❣ ❤♦➷❝
❛✮

F

❧➭ ➳♥❤ ①➵ ❧✐➟♥ tơ❝✱ ❤♦➷❝

❜✮ ◆Õ✉ ✈í✐ ❤❛✐ ❞➲② ❜✃t ❦ú

n≥1

s❛♦ ❝❤♦

xn → x

✈➭

{xn } , {yn }

yn → y ✱

t❤×

✶✶

♠➭

(xn+1 , yn+1 , xn , yn ) ∈ M ✱

(x, y, xn , yn ) ∈ M

✈í✐ ♠ä✐

✈í✐ ♠ä✐

n ≥ 1✳


◆Õ✉ tå♥ t➵✐
❧➭ t❐♣

F ✲❜✃t

x = F (x, y)

(x0 , y0 ) ∈ X × X

s❛♦ ❝❤♦

(F (x0 , y0 ), F (y, x0 ), x0 , y0 ) ∈ M

❜✐Õ♥ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉✱ t❤× tå♥ t➵✐

✈➭

y = F (y, x)✱

❈❤ø♥❣ ♠✐♥❤✳ ❱×

♥❣❤Ü❛ ❧➭

F

x, y ∈ X

✈➭

M

s❛♦ ❝❤♦

❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳

F (X × X) ⊆ X ❚❛ ❝ã t❤Ó ①➞② ❞ù♥❣ ❤❛✐ ❞➲② {xn } ✈➭

{yn }tr♦♥❣ X s❛♦ ❝❤♦
yn = F (yn−1 , xn−1 ), ✈í✐ ♠ä✐ n ∈ N.

xn = F (xn−1 , yn−1 ),
◆Õ✉ tå♥ t➵✐

n∗ ∈ N s❛♦ ❝❤♦ xn∗ −1 = xn∗ ✈➭ yn∗ −1 = yn∗ ✱ ♥❣❤Ü❛ ❧➭ t❛ ❝ã
xn∗ −1 = F (xn∗ −1 , yn∗ −1 ),

❱× t❤Õ✱

✭✶✳✹✮

yn∗ −1 = F (yn∗ −1 , xn∗ −1 ).

(xn∗ −1 , yn∗ −1 ) ❧➭ ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ F ✳

❱× ✈❐② t❛ ❝ã t❤Ĩ ❣✐➯ t❤✐Õt r➺♥❣

❱×

xn−1 = xn ❤♦➷❝ yn−1 = yn ✈í✐ ♠ä✐ n ∈ N✳

(F (x0 , y0 ), F (y0 , x0 ), x0 , y0 ) = (x1 , y1 , x0 , y0 ) ∈ M ✈➭ M ❧➭ ♠ét t❐♣ ❤ỵ♣ F ✲

❜✃t ❜✐Õ♥✱ t❛ s✉② r❛

(F (x1 , y1 ), F (y1 , x1 ), F (x0 , y0 ), F (y0 , x0 )) = (x2 , y2 , x1 , y1 ) ∈ M.
❚✐Õ♣ tơ❝ sư ❞ơ♥❣ ❣✐➯ t❤✐Õt

M ❧➭ ♠ét t❐♣ F ✲❜✃t ❜✐Õ♥✱ t❛ ❝ã

(F (x2 , y2 ), F (y2 , x2 ), F (x1 , y1 ), F (y1 , x1 )) = (x3 , y3 , x2 , y2 ) ∈ M.
❇ë✐ tÝ♥❤ t✉➬♥ ❤♦➭♥ ❝ñ❛ ❧❐♣ ❧✉❐♥ ♥➭②✱ t❛ ➤➢ỵ❝

(F (xn−1 , yn−1 ), F (yn−1 , xn−1 ), xn−1 , yn−1 ) = (xn , yn , xn−1 , yn−1 ) ∈ M, ✈í✐ ♠ä✐ n ∈ N.
❇➞② ❣✐ê ✈í✐ ♠ä✐

n ∈ N t❛ ➤➷t δn−1 := d(xn , xn−1 ) + d(yn , yn−1 ) > 0✳ ❚❛ sÏ

❝❤ø♥❣ ♠✐♥❤ r➺♥❣

δn ≤ 2ϕ
❚❤❐t ✈❐②✱ ✈×

δn−1
2

✈í✐ ♠ä✐

n ∈ N.

(xn , yn , xn−1 , yn−1 ) ∈ M ✈í✐ ♠ä✐ n ∈ N✱ tõ ✭✶✳✸✮ t❛ s✉② r❛
d(xn+1 , xn ) = d(F (xn , yn ), F (xn−1 , yn−1 ))
d(xn , xn−1 ) + d(yn , yn−1 )
≤ ϕ
2
δn−1
= ϕ
.
2
✶✷

✭✶✳✺✮


❱×

M ❧➭ ♠ét t❐♣ F ✲❜✃t ❜✐Õ♥ ✈➭ (xn , yn , xn−1 , yn−1 ) ∈ M ✈í✐ ♠ä✐ n ∈ N✱ t❛ ❝ã

(yn−1 , xn−1 , yn , xn ) ∈ M ✈í✐ ♠ä✐ n ∈ N✳ ❚õ ✭✶✳✸✮ ✈➭ (yn−1 , xn−1 , yn , xn ) ∈ M ✈í✐ ♠ä✐
n ∈ N✱ t❛ ➤➢ỵ❝
d(yn+1 , yn ) = d(F (yn , xn ), F (yn−1 , xn−1 ))
= d(F (yn−1 , xn−1 ), F (yn , xn ))
d(yn−1 , yn ) + d(xn−1 , xn )
≤ ϕ
2
δn−1
.

= ϕ
2

✭✶✳✻✮

❑Õt ❤ỵ♣ ✭✶✳✺✮ ✈➭ ✭✶✳✻✮✱ t❛ ➤➢ỵ❝

δn ≤ 2ϕ
❚õ ✭✶✳✺✮✱ sư ❞ơ♥❣ tÝ♥❤ ❝❤✃t

sè

✈í✐ ♠ä✐

n ∈ N.

✭✶✳✼✮

ϕ(t) < t ✈í✐ ♠ä✐ t > 0 t❛ ❝ã
δn−1
2

δn ≤ 2ϕ
❱× t❤Õ✱

δn−1
2

< δn−1 , ✈í✐ ♠ä✐ n ∈ N.

{δn } ❧➭ ♠ét ❞➲② ➤➡♥ ➤✐Ư✉ ❣✐➯♠✳ ❉♦ ➤ã✱ tå♥ t➵✐ ❣✐í✐ ❤➵♥ lim δn = δ ✈í✐
n→∞

δ ≥ 0 ♥➭♦ ➤ã✳
❇➞② ❣✐ê t❛ ❝❤Ø r❛ r➺♥❣

➤➻♥❣ t❤ø❝ ✭✶✳✼✮ ❝❤♦

δ = 0✳ ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐ r➺♥❣ δ > 0✳ ❑❤✐ ➤ã tõ ❜✃t

n → ∞ ✈➭ sư ❞ơ♥❣ ❣✐➯ t❤✐Õt lim+ ϕ(r) < t ✈í✐ ♠ä✐ t > 0✱ t❛
r→t

s✉② r❛

δ = lim δn ≤ 2 lim ϕ
n→∞

δn−1
2

=2

➜✐Ò✉ ♥➭② ❞➱♥ ➤Õ♥ ♠➞✉ t❤✉➱♥✳ ❱× ✈❐②

lim

δn−1 →δ

ϕ

+

δn−1
2

<2

δ
2

= δ.

δ = 0 ✈➭

lim δn = lim [d(xn+1 , xn ) + d(yn+1 , yn )] = 0.

n→∞

n→∞

❚✐Õ♣ t❤❡♦✱ t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣
♥❣➢ỵ❝ ❧➵✐ r➺♥❣ ♠ét tr♦♥❣ ❤❛✐ ❞➲②
❑❤✐ ➤ã tå♥ t➵✐ sè

✭✶✳✽✮

{xn } ✈➭ {yn } ❧➭ ❤❛✐ ❞➲② ❈❛✉❝❤②✳ ●✐➯ sö

{xn } ❤♦➷❝ {yn } ❦❤➠♥❣ ♣❤➯✐ ❧➭ ❞➲② ❈❛✉❝❤②✳

ε > 0 ✈➭ ❤❛✐ ❞➲② ❝♦♥ ❝➳❝ sè ♥❣✉②➟♥ nk ✈➭ mk ✈í✐ nk > mk ≥ k

s❛♦ ❝❤♦

rk := d(xmk , xnk ) + d(ymk , ynk ) ≥ ε, ✈í✐ ♠ä✐ k = 1, 2, 3, . . . .
❍➡♥ ♥÷❛✱ t➢➡♥❣ ø♥❣ ✈í✐

✭✶✳✾✮

mk ✱ t❛ ❝ã t❤Ĩ ❝❤ä♥ nk ❧➭ sè ♥❣✉②➟♥ ♥❤á ♥❤✃t ✈í✐

nk > mk ≥ k t❤á❛ ♠➲♥ ✭✶✳✾✮✳ ❑❤✐ ➤ã✱ t❛ ❝ã
d(xmk , xnk −1 ) + d(ymk , ynk −1 ) < ε.
✶✸

✭✶✳✶✵✮


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

ε ≤ rk
= d(xmk , xnk ) + d(ymk , ynk )
≤ d(xmk , xnk −1 ) + d(xnk −1 , xnk ) + d(ymk , ynk −1 ) + d(ynk −1 , ynk )

✭✶✳✶✶✮

= [d(xmk , xnk −1 ) + d(ymk , ynk −1 )] + [d(xnk , xnk −1 ) + d(ynk , ynk −1 )
< ε + δnk −1 .
❚r♦♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ ❝❤♦
❱×

k → ∞ ✈➭ sư ❞ơ♥❣ ✭✶✳✽✮ t❛ ❝ã lim rk = ε > 0✳
k→∞

nk > mk ✈➭ M t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉✱ t❛ ♥❤❐♥ ➤➢ỵ❝
(xnk , ynk , xmk , ymk ) ∈ M ✈➭ (ymk , xmk , ynk , xnk ) ∈ M.

✭✶✳✶✷✮

❚õ ✭✶✳✸✮ ✈➭ ✭✶✳✶✷✮✱ t❛ t❤✉ ➤➢ỵ❝

d(xmk +1 , xnk +1 ) = d(F (xmk , ymk ), F (xnk , ynk ))
= d(F (xnk , ynk ), F (xmk , ymk )
d(xnk , xmk ) + d(ynk , ymk )
2
rk
=ϕ
2

≤ϕ

✭✶✳✶✸✮

✈➭

d(ymk +1 , ynk +1 ) = d(F (ymk , xmk ), F (ynk , xnk ))
d(ymk , xnk ) + d(xmk , xnk )
≤ ϕ
2
rk

= ϕ
.
2

✭✶✳✶✹✮

❑Õt ❤ỵ♣ ✭✶✳✶✸✮ ✈➭ ✭✶✳✶✹✮✱ t❛ ➤➢ỵ❝

rk+1 ≤ 2ϕ

rk
, ✈í✐ ♠ä✐ k = 1, 2, 3, . . .
2

✭✶✳✶✺✮

❈❤♦

k → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✶✺✮ ✈➭ sö ❞ơ♥❣ ❣✐➯ t❤✐Õt lim+ ϕ(r) < t ✈í✐

♠ä✐

t > 0 t❛ s✉② r❛

r→t

ε = lim rk+1 ≤ 2 lim ϕ
k→∞

k→∞

rk
2

➜✐Ò✉ ♥➭② ❞➱♥ ➤Õ♥ ♠➞✉ t❤✉➱♥✳ ❱× t❤Õ✱
❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ tå♥ t➵✐

= 2 lim+ ϕ
rk →ε

rk
2

<2

ε
= ε.
2

{xn } ✈➭ {yn } ❧➭ ❤❛✐ ❞➲② ❈❛✉❝❤②✳ ❱× X ❧➭

x, y ∈ X s❛♦ ❝❤♦

lim xn = x,

lim yn = y.

n→∞

n→∞

✶✹

✭✶✳✶✻✮


❈✉è✐ ❝ï♥❣✱ t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣

x = F (x, y) ✈➭ y = F (y, x)✳

❚r➢ê♥❣ ❤ỵ♣ ✶✳ ◆Õ✉ ❣✐➯ tết ợ ĩ

F tụ tì t ❝ã

x = lim xn+1 = lim F (xn , yn ) = F ( lim xn , lim yn ) = F (x, y)

✭✶✳✶✼✮

y = lim yn+1 = lim F (yn , xn ) = F ( lim yn , lim xn ) = F (y, x).

✭✶✳✶✽✮

n→∞

n→∞

n→∞

n→∞

✈➭

n→∞

❱× t❤Õ✱

n→∞

n→∞

n→∞

x = F (x, y) ✈➭ y = F (y, x)✱ ♥❣❤Ü❛ ❧➭ F ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳

❚r➢ê♥❣ ❤ỵ♣ ✷✳ ◆Õ✉ tết ợ tì ì t ó

{xn } ❤é✐ tơ ✈Ị

x ✈➭ {yn } ❤é✐ tơ ✈Ị y ✈í✐ ❝➳❝ ♣❤➬♥ tư ♥➭♦ ➤ã x, y ∈ X ✈➭ (xn , yn , xn−1 , yn−1 ) ∈ M
✈í✐ ♠ä✐
♠ä✐

n ∈ N✱ ♥➟♥ t❛ ❝ã (x, y, xn , yn ) ∈ M ✈í✐ ♠ä✐ n ∈ N✳ ❱× (x, y, xn , yn ) ∈ M ✈í✐

n ∈ N✱ ♥❤ê ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝ ✈➭ ✭✶✳✸✮✱ t❛ ♥❤❐♥ ➤➢ỵ❝
d(F (x, y), x) ≤ d(F (x, y), xn+1 ) + d(xn+1 , x)
= d(F (x, y), F (xn , yn )) + d(xn+1 , x)
d(x, xn ) + d(y, yn )
+ d(xn+1 , x).
≤ ϕ
2

❈❤♦
r❛

✭✶✳✶✾✮

n → ∞ tr♦♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ ❝ã d(F (x, y), x) = 0✱ ✈➭ ✈× ✈❐② t❛ s✉②

x = F (x, y)✳ ❚➢➡♥❣ tù✱ t❛ t❤✉ ➤➢ỵ❝ y = F (y, x)✳ ❱× t❤Õ✱ F ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣

❜é ➤➠✐✳
❇➞② ❣✐ê✱ t❛ ➤➢❛ r❛ ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ị í



í ụ

ớ ọ

t ợ

X = R ớ ♠➟tr✐❝ t❤➠♥❣ t❤➢ê♥❣ d(x, y) = |x − y|

x, y ∈ X ✈➭ ✈í✐ q✉❛♥ ❤Ư t❤ø tù t❤➠♥❣ t❤➢ê♥❣ ợ ị ĩ ở

x y y x ∈ [0; ∞)✳ ❳Ðt ➳♥❤ ①➵ ❧✐➟♥ tô❝ F : X × X → X ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝
F (x, y) =
▲✃②

✈í✐ ♠ä✐

(x, y) ∈ X × X.

y1 = 2 ✈➭ y2 = 3✳ ❑❤✐ ➤ã t❛ ❝ã y1 ≤ y2 ✱ ♥❤➢♥❣ F (x, y1 ) ≤ F (x, y2 )✱ ✈➭ ✈× t❤Õ

➳♥❤ ①➵

F ❦❤➠♥❣ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥✳

❇➞② ❣✐ê✱ ❝❤♦
♠ä✐

x+y+2
,
3

ϕ : [0; ∞) → [0; ∞) ❧➭ ♠ét ❤➭♠ sè ➤➢ỵ❝ ❝❤♦ ❜ë✐ ϕ(t) = 32 t ✈í✐

t ∈ [0; ∞)✳ ❑❤✐ ➤ã t❛ ➤➢ỵ❝ 0 = ϕ(0) < ϕ(t) < t ✈➭ lim+ ϕ(r) < t ✈í✐ t > 0✳
r→t

✶✺


❇➺♥❣ tÝ♥❤ t♦➳♥ ➤➡♥ ❣✐➯♥✱ t❛ t❤✃② r➺♥❣ ✈í✐ ♠ä✐

x, y, u, v ∈ X t❛ ❝ã

x+y+2 u+v+2
−
3

3
1
≤ [d(x, u) + d(y, v)]
3
2 d(x, u) + d(y, v)
=
3
2
d(x, u) + d(y, v)
.
=ϕ
2

d(F (x, y), F (u, v)) =

❍➡♥ ♥÷❛✱ ♥Õ✉ t❛ ➳♣ ❞ơ♥❣ ➜Þ♥❤ ❧Ý ✶✳✷✳✶ ✈í✐
➤é♥❣ ❜é ➤➠✐ ❞✉② ♥❤✃t✳ ❉Ơ t❤✃② r➺♥❣ ➤✐Ĩ♠
♥❤✃t ❝đ❛

✶✳✷✳✸

M = X 4 ✱ t❤× F ❝ã ➤✐Ĩ♠ ❜✃t

(2, 2) ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❞✉②

F✳

◆❤❐♥ ①Ðt✳

▼➷❝ ❞ï tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ ❧➭ ❝➠♥❣ ❝ô ❝èt ②Õ✉ tr♦♥❣

❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ➤Ĩ ❝❤Ø r❛ sù tå♥ t➵✐ ❝đ❛ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ◆ã✐
❝❤✉♥❣✱ ❝➳❝ ➳♥❤ ①➵ ❝ã t❤Ó ❦❤➠♥❣ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥✱ ♥❤➢ tr♦♥❣ ✈Ý ❞ơ
tr➟♥✳ ❱× tế ị í ợ q t ó ♠ét ❝➠♥❣ ❝ơ ❜ỉ trỵ ♠í✐
tr♦♥❣ ✈✐Ư❝ ❝❤Ø r❛ sù tå♥ t➵✐ ❝đ❛ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳
◆Õ✉ t❛ ❧✃② ➳♥❤ ①➵

ϕ(t) = kt ✈í✐ k ∈ [0; 1) tr♦♥❣ ị í tì t t

ợ ết q s



ệ q



(X, d)

ột t ợ rỗ ủ
s ❝❤♦ tå♥ t➵✐

k ∈ [0, 1)

❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ ✈➭

X 4✳

●✐➯ sư r➺♥❣

✐✮

(x, y, u, v) ∈ M ✳

F

❧➭

❧➭ ♠ét ➳♥❤

t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥

d(F (x, y), F (u, v)) ≤ k
✈í✐ ♠ä✐

F :X ×X →X

M

d(x, u) + d(y, v)
2

✭✶✳✷✵✮

●✐➯ sư r➺♥❣ ❤♦➷❝

❧➭ ➳♥❤ ①➵ ❧✐➟♥ tơ❝✱ ❤♦➷❝

✐✐✮ ❱í✐ ❤❛✐ ❞➲② ❜✃t ❦ú
♥Õ✉

xn → x

✈➭

{xn } , {yn } ♠➭ (xn+1 , yn+1 , xn , yn ) ∈ M

yn → y

❦❤✐

n → ∞✱

✶✻

t❤×

(x, y, xn , yn ) ∈ M

✈í✐ ♠ä✐

✈í✐ ♠ä✐

n ∈ N✱

n ∈ N✳


◆Õ✉ tå♥ t➵✐
❤ỵ♣

F ✲❜✃t

x = F (x, y)

(x0 , y0 ) ∈ X × X

(F (x0 , y0 ), F (y0 , x0 ), x0 , y0 ) ∈ M

❜✐Õ♥ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉✱ t❤× tå♥ t➵✐
✈➭

y = F (y, x)✱

➜Þ♥❤ ❧ý✳

✶✳✷✳✺

s❛♦ ❝❤♦

F

x, y ∈ X

M

t❐♣

s❛♦ ❝❤♦

❝ã ➤✐Ĩ♠ t ộ ộ

ữ tết ủ ị ❧Ý ✶✳✷✳✶✱ t❛ ❣✐➯ t❤✐Õt

t❤➟♠ r➺♥❣ ✈í✐ ♠ä✐

(x, y, u, v) ∈ M

♥❣❤Ü❛ ❧➭

✈➭

✈➭

(x, y), (z, t) ∈ X × X ✱

(z, t, u, v) ∈ M ✳

❑❤✐ ➤ã

tå♥ t➵✐

F

(u, v) ∈ X × X

s❛♦ ❝❤♦

❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❞✉②

♥❤✃t✳
❈❤ø♥❣ ♠✐♥❤✳ ❱í✐ ❝➳❝ ❣✐➯ t❤✐Õt ❝đ❛ ➤Þ♥❤ ❧ý✱ tõ ➜Þ♥❤ ❧Ý ✶✳✷✳✶✱ t❛ ❜✐Õt r➺♥❣

F ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳
❇➞② ❣✐ê✱ ❣✐➯ sö r➺♥❣
♥❣❤Ü❛ ❧➭

(x, y) ✈➭ (z, t) ❧➭ ❤❛✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ F ✱

x = F (x, y), y = F (y, x), z = F (z, t), t = F (t, z)✳ ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣

x = z ✈➭ y = t✳ ❚❤❐t ✈❐②✱ ♥❤ê ❣✐➯ t❤✐Õt ✈í✐ (x, y) ✈➭ (z, t) tå♥ t➵✐ (u, v) ∈ X × X
s❛♦ ❝❤♦

(x, y, u, v) ∈ M ✈➭ (z, t, u, v) ∈ M ✳ ❚❛ ➤➷t u0 = u ✈➭ v0 = v ✈➭ ①➞② ❞ù♥❣

❤❛✐ ❞➲②

{un } ✈➭ {vn } ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝
un = F (un−1 , vn−1 ),

❱×

vn = F (vn−1 , un−1 ), ✈í✐ ♠ä✐ n ∈ N.

M ❧➭ F ✲❜✃t ❜✐Õ♥ ✈➭ (x, y, u0 , v0 ) = (x, y, u, v) ∈ M ✱ t❛ ❝ã
(F (x, y), F (y, x), F (u0 , v0 ), F (v0 , u0 )) ∈ M,

♥❣❤Ü❛ ❧➭

(x, y, u1 , v1 ) ∈ M.
❚õ

(x, y, u1 , v1 ) ∈ M ✱ ♥Õ✉ t❛ sö ụ ột ữ tí t F t ế tì t❛ ❝ã
(F (x, y), F (y, x), F (u1 , v1 ), F (v1 , u1 )) ∈ M,

✈➭ ✈× t❤Õ

(x, y, u2 , v2 ) ∈ M.
❇ë✐ tÝ♥❤ t✉➬♥ ❤♦➭♥ ❝đ❛ ❧❐♣ ❧✉❐♥ ♥➭②✱ t❛ ➤➢ỵ❝

(x, y, un , vn ) ∈ M, ✈í✐ ♠ä✐ n ∈ N.

✶✼

✭✶✳✷✶✮


❚õ ✭✶✳✸✮ ✈➭ ✭✶✳✷✶✮✱ t❛ ❝ã

d(x, un+1 ) = d(F (x, y), F (un , vn )) ≤ ϕ
❱×

d(x, un ) + d(y, vn )
2

✭✶✳✷✷✮

M ❧➭ F ✲❜✃t ❜✐Õ♥ ✈➭ (x, y, un , vn ) ∈ M ✈í✐ ♠ä✐ n ∈ N✱ t❛ ❝ã
(vn , un , y, x) ∈ M ✈í✐ ♠ä✐ n ∈ N.

✭✶✳✷✸✮

❚õ ✭✶✳✸✮ ✈➭ ✭✶✳✷✸✮✱ t❛ t❤✉ ➤➢ỵ❝

d(vn+1 , y) = d(F (vn , un ), F (y, x)) ≤ ϕ

d(vn , y) + d(un , x)
2

.

✭✶✳✷✹✮

❉♦ ➤ã✱ tõ ✭✶✳✷✷✮ ✈➭ ✭✶✳✷✸✮✱ t❛ ❝ã

d(x, un+1 ) + d(y, vn+1 )
≤ϕ
2

d(x, un ) + d(y, vn )
2

✈í✐ ♠ä✐

n ∈ N.

✭✶✳✷✺✮

✈í✐ ♠ä✐

n ∈ N.

✭✶✳✷✻✮

❇ë✐ tÝ♥❤ t✉➬♥ ❤♦➭♥ ❝đ❛ ❧❐♣ ❧✉❐♥ ♥➭②✱ t❛ ➤➢ỵ❝

d(x, un+1 ) + d(y, vn+1 )
≤ ϕn
2
❚õ ❣✐➯ t❤✐Õt

d(x, u1 ) + d(y, v1 )
2

ϕ(t) < t ✈➭ lim+ ϕ(r) < t✱ t❛ s✉② r❛ lim ϕn (t) = 0 ✈í✐ ♠ä✐ t > 0✳ ❱×
n→∞

r→t

t❤Õ✱ tõ ✭✶✳✷✻✮✱ t❛ ❝ã

lim [d(x, un+1 ) + d(y, vn+1 )] = 0.

✭✶✳✷✼✮

n→∞

❚➢➡♥❣ tù✱ t❛ ❝ã t❤Ó ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣

lim [d(z, un+1 ) + d(t, vn+1 )] = 0.

✭✶✳✷✽✮

n→∞

◆❤ê ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✱ ✈í✐ ♠ä✐

n ∈ N✱ t❛ ❝ã

d(x, z) + d(y, t) ≤ [d(x, un+1 ) + d(un+1 , z)] + [d(y, vn+1 ) + d(vn+1 , t)]
≤ [d(x, un+1 ) + d(y, vn+1 )] + [d(z, un+1 ) + d(t, vn+1 )].
❈❤♦

n → ∞ tr♦♥❣

✭✶✳✷✼✮

rå✐ sư ❞ơ♥❣ ✭✶✳✷✺✮ ✈➭ ✭✶✳✷✻✮✱ t❛ ❝ã

➜✐Ị✉ ♥➭② ①➮② r❛ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐
❱× t❤Õ✱

✭✶✳✷✾✮

d(x, z) + d(y, t) = 0✳

d(x, z) = 0 ✈➭ d(y, t) = 0✱ ♥❣❤Ü❛ ❧➭ x = z ✈➭ y = t✳

F ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❞✉② ♥❤✃t✳

✶✽


✶✳✷✳✻

❍Ö q✉➯✳

❝ã ♠ét ♠➟tr✐❝

✭❬✶✵❪✮ ❈❤♦

d

tr➟♥

r➺♥❣ ❝ã ♠ét ❤➭♠ sè

X

(X, ≤)

s❛♦ ❝❤♦

❧➭ ♠ét t❐♣ s➽♣ t❤ø tù tõ♥❣ ♣❤➬♥ ✈➭ ❣✐➯ sư

(X, d)

❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✳ ●✐➯ sư

ϕ : [0; ∞) → [0; ∞)

✈í✐

0 = ϕ(0) < ϕ(t) < t

t > 0 ✈➭ ❝ị♥❣ ❣✐➯ sư r➺♥❣ F : X × X → X

✈í✐ ♠ä✐

lim ϕ(r) < t

✈➭

r→t+

❧➭ ♠ét ➳♥❤ ①➵ s❛♦ ❝❤♦

F

❝ã

tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ ✈➭

d(x, u) + d(y, v)
2

d(F (x, y), F (u, v)) ≤ ϕ
x, y, u, v ∈ X

✈í✐ ♠ä✐

♠➭

x≥u

❛✮

F

❧✐➟♥ tơ❝✱ ❤♦➷❝

❜✮

X

❝ã ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉

✈➭

y ≤ v✳

●✐➯ sö r➺♥❣ ❤♦➷❝

✐✮

◆Õ✉

xn

❧➭ ❞➲② ❦❤➠♥❣ ❣✐➯♠ ✈í✐

✐✐✮

◆Õ✉

yn

❧➭ ❞➲② ❦❤➠♥❣ t➝♥❣ ✈í✐

◆Õ✉ tå♥ t➵✐

x0 , y 0 ∈ X

x, y ∈ X

xn → x✱

yn → y ✱

t❤×

t❤×

xn ≤ x

yn ≥ y

✈í✐ ♠ä✐

✈í✐ ♠ä✐

n ∈ N✳

n ∈ N✳

s❛♦ ❝❤♦

x0 ≤ F (x0 , y0 ),
t❤× tå♥ t➵✐

✭✶✳✸✵✮

s❛♦ ❝❤♦

y0 ≥ F (y0 , x0 ),

x = F (x, y)

✈➭

y = F (y, x)✱

♥❣❤Ü❛ ❧➭

F

❝ã ➤✐Ó♠ ❜✃t

➤é♥❣ ❜é ➤➠✐✳

❈❤ø♥❣ ♠✐♥❤✳ ➜➬✉ t✐➟♥✱ t❛ ①➳❝ ➤Þ♥❤ ♠ét t❐♣ ❝♦♥

M ⊆ X 4 ❜ë✐

M = (a, b, c, d) ∈ X 4 : a ≥ c, b ≤ d .
❉Ơ ❞➭♥❣ ❦✐Ĩ♠ tr❛ r➺♥❣

M ❧➭ t❐♣ ❤ỵ♣ F ✲❜✃t ❜✐Õ♥ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉✳

◆❤ê ✭✶✳✸✵✮ t❛ ❝ã

d(F (x, y), F (u, v)) ≤ ϕ
✈í✐ ♠ä✐

d(x, u) + d(y, v)
2

x, y, u, v ∈ X ♠➭ (x, y, u, v) ∈ M ✳ ❱× x0 , y0 ∈ X s❛♦ ❝❤♦
x0 ≤ F (x0 , y0 ),

y0 ≥ F (y0 , x0 ),

t❛ t❤✉ ➤➢ỵ❝

(F (x0 , y0 ), F (y0 , x0 ), x0 , y0 ) ∈ M.
✶✾

✭✶✳✸✶✮

ế tết ợ tỏ tì ớ ❞➲② ❜✃t ❦ú
❝❤♦

{xn } ✈➭ {yn } s❛♦

{xn } ❧➭ ❞➲② ❦❤➠♥❣ ❣✐➯♠ tr♦♥❣ X ♠➭ xn → x ✈➭ {yn } ❧➭ ❞➲② ❦❤➠♥❣ t➝♥❣

tr♦♥❣

X ♠➭ yn → y ✱ t❛ ❝ã
x1 ≤ x2 ≤ ... ≤ xn ≤ ... ≤ x

✈➭

y1 ≥ y2 ≥ ... ≥ yn ≥ ... ≥ y
✈í✐ ♠ä✐

n ∈ N✳ ❉♦ ➤ã✱ t❛ ❝ã (x, y, xn , yn ) ∈ M ✈í✐ ♠ä✐ n N ì tế tết

tr ị í ✶✳✷✳✶ ➤➢ỵ❝ t❤á❛ ♠➲♥✳
❚õ ❝➳❝ ❝❤ø♥❣ ♠✐♥❤ tr➟♥ t❛ s✉② r tt tết ủ ị í
ợ t❤á❛ ♠➲♥✱ ❞♦ ➤ã ➳♣ ❞ơ♥❣ ➤Þ♥❤ ❧ý ♥➭② t❛ s✉② r❛

F ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é

➤➠✐✳

✶✳✷✳✼

❍Ư q✉➯✳

✭❬✶✵❪✮ ◆❣♦➭✐ ♥❤÷♥❣ ❣✐➯ t❤✐Õt tr♦♥❣ ❍Ö q✉➯ ✶✳✷✳✻✱ t❛ ❣✐➯ t❤✐Õt

t❤➟♠ r➺♥❣ ✈í✐ ♠ä✐

u, y ≤ v

✈➭

(x, y), (z, t) ∈ X × X ✱

z ≥ u, t ≤ v ✳

❑❤✐ ➤ã

F

tå♥ t➵✐

(u, v) ∈ X × X

s❛♦ ❝❤♦

x≥

❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❞✉② ♥❤✃t✳

❈❤ø♥❣ ♠✐♥❤✳ ➜➬✉ t✐➟♥✱ t❛ ①➳❝ ➤Þ♥❤ ♠ét t❐♣ ❝♦♥

M ⊆ X 4 ❝❤♦ ❜ë✐

M = (a, b, c, d) ∈ X 4 : a ≥ c, b ≤ d .
❉Ơ ❞➭♥❣ ❦✐Ĩ♠ tr❛ r➺♥❣

M ❧➭ t❐♣ ợ F t ế tỏ tí t

ì ✈❐②✱ ➳♣ ❞ơ♥❣ ❍Ư q✉➯ ✶✳✷✳✻ t❛ s✉② r❛
❇➞② ❣✐ê ❣✐➯ sư

(x, y), (z, t) ∈ X × X ❧➭ ❝➳❝ ➤✐Ĩ♠ ❜✃t ❦ú tr♦♥❣ X × X ✱ ❦❤✐

➤ã t❤❡♦ ❣✐➯ t❤✐Õt tå♥ t➵✐
❝➳❝❤ ①➳❝ ➤Þ♥❤

F ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳

(u, v) ∈ X × X s❛♦ ❝❤♦ x ≥ u, y ≤ v ✈➭ z ≥ u, t ≤ v ✳ ❚õ

M t❛ s✉② r❛ (x, y, u, v) ∈ M ✈➭ (z, t, u, v) ∈ M

ì tế tt tết ủ ị í ợ tỏ ụ ị
ý t s✉② r❛

✶✳✷✳✽

❍Ư q✉➯✳

♠ét ♠➟tr✐❝

F ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❞✉② ♥❤✃t✳

✭❬✷❪✮ ❈❤♦

d tr➟♥ X

F : X×X → X

(X, ≤) ❧➭ t❐♣ ❤ỵ♣ s➽♣ t❤ø tù tõ♥❣ ♣❤➬♥ ✈➭ ❣✐➯ sö ❝ã

s❛♦ ❝❤♦

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✳

❧➭ ➳♥❤ ①➵ ❧✐➟♥ tơ❝ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ tr➟♥

✷✵

●✐➯ sö

X✳

●✐➯


t❤✐Õt r➺♥❣ tå♥ t➵✐

k ∈ [0, 1)

s❛♦ ❝❤♦

d(x, u) + d(y, v)
2

d(F (x, y), F (u, v)) ≤ k
✈í✐ ♠ä✐

x, y, u, v ∈ X

♠➭

x ≥ u, y ≤ v ✳

◆Õ✉ tå♥ t➵✐

x0 ≤ F (x0 , y0 ),
t❤× tå♥ t➵✐

x, y ∈ X

s❛♦ ❝❤♦

✭✶✳✸✷✮

x0 , y0 ∈ X

s❛♦ ❝❤♦

y0 ≥ F (y0 , x0 ),

x = F (x, y)

✈➭

y = F (y, x)✳

❈❤ø♥❣ ♠✐♥❤✳ ❚r♦♥❣ ❍Ö q✉➯ ✶✳✷✳✻✱ ♥Õ✉ t❛ ❧✃② ❤➭♠

ϕ(t) = kt ✈í✐ ♠ä✐

t ∈ [0, +∞)✱ tr♦♥❣ ➤ã k ∈ [0, 1) ❧➭ ❤➺♥❣ sè✱ t❤× ✈í✐ ❝➳❝ ❣✐➯ t❤✐Õt ➤➲ ❝❤♦ ❤➭♠ F
t❤á❛ ♠➲♥ ❣✐➯ t❤✐Õt ✭❛✮ ✈í✐ ❤➭♠

ϕ✳ ❉♦ ➤ã ➳♣ ❞ơ♥❣ ❤Ư q✉➯ ♥➭② t❛ t❤✉ ➤➢ỵ❝ ❦Õt

❧✉❐♥ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳

✶✳✷✳✾

❍Ư q✉➯✳

♠ét ♠➟tr✐❝
r➺♥❣

X

✭❬✷❪✮ ❈❤♦

d tr➟♥ X

(X, ≤) ❧➭ t❐♣ ❤ỵ♣ s➽♣ t❤ø tù tõ♥❣ ♣❤➬♥ ✈➭ ❣✐➯ sö ❝ã

s❛♦ ❝❤♦

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

t❤á❛ ♠➲♥ ❤❛✐ tÝ♥❤ ❝❤✃t s❛✉

✐✮

◆Õ✉

xn

❧➭ ❞➲② ❦❤➠♥❣ ❣✐➯♠ ✈➭

✐✐✮

◆Õ✉

yn

❧➭ ❞➲② ❦❤➠♥❣ t➝♥❣ ✈➭

●✐➯ sö

●✐➯ sö

F :X ×X →X

●✐➯ t❤✐Õt r➺♥❣ tå♥ t➵✐

xn → x✱

yn → y ✱

x, y, u, v ∈ X

k ∈ [0, 1)

x, y ∈ X

yn ≥ y

✈í✐ ♠ä✐

✈í✐ ♠ä✐

n ∈ N✳

n ∈ N✳

s❛♦ ❝❤♦

♠➭

x ≥ u, y ≤ v ✳

d(x, u) + d(y, v)
2

◆Õ✉ tå♥ t➵✐

x0 ≤ F (x0 , y0 ),
t❤× tå♥ t➵✐

t❤×

xn ≤ x

❧➭ ➳♥❤ ①➵ ❧✐➟♥ tơ❝ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ tr➟♥ ❳✳

d(F (x, y), F (u, v)) ≤ k
✈í✐ ♠ä✐

t❤×

s❛♦ ❝❤♦

x0 , y0 ∈ X

✭✶✳✸✸✮

s❛♦ ❝❤♦

y0 ≥ F (y0 , x0 ),

x = F (x, y)

✈➭

y = F (y, x)✳

❈❤ø♥❣ ♠✐♥❤✳ ❚r♦♥❣ ❍Ö q✉➯ ✶✳✷✳✻✱ ♥Õ✉ t❛ ❧✃② ❤➭♠

ϕ(t) = kt ✈í✐ ♠ä✐

t ∈ [0, +∞)✱ tr♦♥❣ ➤ã k ∈ [0, 1) ❧➭ ❤➺♥❣ sè✱ t❤× ✈í✐ ❝➳❝ ❣✐➯ t❤✐Õt ➤➲ ❝❤♦ t❛ s✉② r❛
❤➭♠

F t❤á❛ ♠➲♥ ❣✐➯ t❤✐Õt ✭❜✮ ✈í✐ ❤➭♠ ϕ✳ ❉♦ ➤ã ➳♣ ❞ơ♥❣ ❤Ư q✉➯ ♥➭② t❛ t❤✉

➤➢ỵ❝ ❦Õt ❧✉❐♥ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳
✷✶


❝❤➢➡♥❣ ✷

➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵
t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ❤÷✉ tû
tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù
➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛

✷✳✶

♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥
♠➟tr✐❝ tứ tự

P ú t trì ột số ị ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛
❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✳
❈❤♦

(X, ≤) ❧➭ ♠ét t❐♣ s➽♣ t❤ø tù tõ♥❣ ♣❤➬♥ ✈➭ d ❧➭ ♠ét ♠➟tr✐❝ tr➟♥ X s❛♦

❝❤♦

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✳ ❚❛ ũ tr ị

tí

X ì X ột q ❤Ö t❤ø tù tõ♥❣ ♣❤➬♥ ♥❤➢ s❛✉✿
(x, y), (u, v) ∈ X × X, (u, v) ≤ (x, y) ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x ≥ u, y ≤ v.

✷✳✶✳✶

➜Þ♥❤ ❧ý✳

✭❬✺❪✮ ❈❤♦

r➺♥❣ tå♥ t➵✐ ♠ét ♠➟tr✐❝

x≥u

✈➭

y ≤ v✱

α, β ∈ [0, 1)

y1 ✱

t❤×

T

(X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬②

♠➭

α+β < 1

✈➭ ✈í✐ ♠ä✐

x, y, u, v ∈ X ✱

t❛ ❝ã

d(T (x, y), T (u, v)) ≤ α
◆Õ✉ tå♥ t➵✐ ➤✐Ó♠

s❛♦ ❝❤♦

❧➭ ♠ét ➳♥❤ ①➵ ❧✐➟♥ tô❝ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥

s❛♦ ❝❤♦ ✈í✐ ❝➳❝ sè ♥➭♦ ➤ã
♠➭

❧➭ ♠ét t❐♣ s➽♣ t❤ø tù tõ♥❣ ♣❤➬♥ ✈➭ ❣✐➯ sö

d tr➟♥ X

T : X ×X → X

➤đ✳ ❈❤♦

(X, ≤)

d(x, T (x, y)).d(u, T (u, v)))
+ βd(x, u).
d(x, u)

(x0 , y0 ) ∈ X × X

s❛♦ ❝❤♦

x0 ≤ T (x0 , y0 ) = x1

✈➭

✭✷✳✶✮

y0 ≥ T (y0 , x0 ) =

❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳

❈❤ø♥❣ ♠✐♥❤✳ ▲✃②

(x0 , y0 ) ∈ X × X s❛♦ ❝❤♦ x0 ≤ T (x0 , y0 ) = x1 ✈➭ y0 ≥

T (y0 , x0 ) = y1 ✳ ❚❛ ➤➷t x1 = T (x0 , y0 ), y1 = T (y0 , x0 )✱ ✈➭ xn+1 = T (xn , yn ), yn+1 =
T (yn , xn ) ✈í✐ ♠ä✐ n ≥ 1✳ ❑❤✐ ➤ã t❛ ❝ã

T 2 (x0 , y0 ) = T (T (x0 , y0 ), T (y0 , x0 )) = T (x1 , y1 ) = x2 ,
✷✷


✈➭

T 2 (y0 , x0 ) = T (T (y0 , x0 ), T (x0 , y0 )) = T (y1 , x1 ) = y2 .
◆❤ê tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ ❝đ❛

T ✱ t❛ t❤✉ ➤➢ỵ❝

x2 = T 2 (x0 , y0 ) = T (x1 , y1 ) ≥ T (x0 , y0 ) = x1 , y2 = T 2 (y0 , x0 ) = T (y1 , x1 ) ≤ T (y0 , x0 ) = y1 .
❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ tỉ♥❣ q✉➳t✱ ✈í✐

n ∈ N✱ t❛ ❝ã

xn+1 = T n+1 (x0 , y0 ) = T (T n (x0 , y0 ), T n (y0 , x0 )), yn+1 = T n+1 (y0 , x0 ) = T (T n (y0 , x0 ), T n (x0 , y0 )).
❘â r➭♥❣✱ t❛ t❤✃② r➺♥❣

x0 ≤ T (x0 , y0 ) = x1 ≤ T 2 (x0 , y0 ) = x2 ≤ ... ≤ T n (x0 , y0 ) = xn ≤ ...,
✈➭

y0 ≥ T (y0 , x0 ) = y1 ≥ T 2 (y0 , x0 ) = y2 ≥ ... ≥ T n (y0 , x0 ) = yn ≥ ....
❉♦ ➤ã✱ ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✷✳✶✮ t❛ ❝ã

d(xn+1 , xn ) = d(T (xn , yn ), T (xn−1 , yn−1 ))
d(xn , T (xn , yn )).d(xn−1 , T (xn−1 , yn−1 ))
+ βd(xn , xn−1 )
d(xn , xn−1 )
d(xn , xn+1 ).d(xn−1 , xn )

=α
+ βd(xn , xn−1 )
d(xn , xn−1 )
≤α

= αd(xn , xn+1 ) + βd(xn , xn−1 ).
❚õ ➤✐Ị✉ ♥➭②✱ t❛ s✉② r❛ ➤➢ỵ❝

d(xn , xn+1 ) ≤

β
1−α

d(xn , xn−1 ).

✭✷✳✷✮

❚➢➡♥❣ tù✱ ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✷✳✶✮ t❛ ❧➵✐ ❝ã

d(yn+1 , yn ) = d(T (yn , xn ), T (yn−1 , xn−1 ))
≤α

d(yn , T (yn , xn )).d(yn−1 , T (yn−1 , xn−1 ))
+ βd(yn , yn−1 )
d(yn , yn−1 )

= αd(yn , yn+1 ) + βd(yn , yn−1 ),
✈➭ ♥❤ê ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❛ s✉② r❛

d(yn , yn+1 ) ≤

β
1−α
✷✸

d(yn , yn−1 ).

✭✷✳✸✮