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▼ơ❝ ▲ơ❝
❚r❛♥❣
▼ơ❝ ❧ơ❝

✶

▲ê✐ ♥ã✐ ➤➬✉

✷

❈❤➢➡♥❣ ✶✳

▼ét ❧í♣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣

✹

✶✳✶

❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✷

❚➠♣➠

G✲♠➟tr✐❝

✹

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

❈❤➢➡♥❣ ✷✳ ▼ét sè ❦Õt q✉➯ ➤✐Ó♠ ❜✃t ➤é♥❣ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥

G✲

♠➟tr✐❝ ➤➬② ➤đ

✶✽

✷✳✶

▼ét sè ❦❤➳✐ ♥✐Ư♠ ✈➭ ❦Õt q✉➯ ❝➡ ❜➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✷✳✷

▼ét sè ❦Õt q✉➯ ➤✐Ó♠ ❜✃t ➤é♥❣ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥
➤➬② ➤ñ

G✲♠➟tr✐❝

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

❑Õt ❧✉❐♥

✸✶

❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦

✸✷

✶

❧ê✐ ♥ã✐ ➤➬✉
▲ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ ♠ét tr♦♥❣ ♥❤÷♥❣ ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ q✉❛♥
trä♥❣✱ ❜ë✐ ♥ã ❝ã ♥❤✐Ị✉ ø♥❣ ❞ô♥❣ tr♦♥❣ ●✐➯✐ tÝ❝❤ ✈➭ ♠ét sè ♥❣➭♥❤ q✉❛♥
trä♥❣ ❦❤➳❝✳ ❉♦ ➤ã ♥ã ➤➢ỵ❝ ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝ q✉❛♥ t➞♠ ứ
t ợ ề ết q ữ ♥➝♠ ✻✵ ❝đ❛ t❤Õ ❦û tr➢í❝✱
❦❤➳✐ ♥✐Ư♠ ❦❤➠♥❣ ❣✐❛♥ ✷✲♠➟tr✐❝ ➤➲ ➤➢ỵ❝ ❣✐í✐ t❤✐Ư✉ ❜ë✐ ❙✳ ●❛❤❧❡r ✭①❡♠
❬✸✱✹❪✮✳ ◆➝♠ ✶✾✽✹✱ tr♦♥❣ ❧✉❐♥ ➳♥ t✐Õ♥ sü ❝đ❛ ♠×♥❤ ❇✳ ❉❤❛❣❡ ➤➲ ➤Ị ①✉✃t
♠ét ❤➢í♥❣ tỉ♥❣ q✉➳t ❝❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ➤ã ❧➭ ➤➢❛ r❛ ❦❤➠♥❣ ❣✐❛♥

D✲

♠➟tr✐❝ ✭①❡♠ ❬✶✱✷❪✮ ✈➭ ➠♥❣ ➤➲ ♣❤➳t tr✐Ĩ♥ ❝✃✉ tró❝ t➠♣➠ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥
➤ã ❜➺♥❣ ❝➳❝❤ tr×♥❤ ❜➭② ❦❤➳✐ ♥✐Ư♠ ❤×♥❤ ❝➬✉ ♠ë tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

D✲

♠➟tr✐❝ ✭①❡♠ ❬✷❪✮✳ ❱➭♦ ♥➝♠ ✷✵✵✸✱ ❩✳ ▼✉st❛❢❛ ✈➭ ❇✳ ❙✐♠s ỉ r ữ
ề ợ ý ề tró❝ t➠♣➠ ❝đ❛ ❦❤➠♥❣ ❣✐❛♥

D✲♠➟tr✐❝ ✭①❡♠ ❬✽❪✮✱

❤ä ➤➲ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ö♠ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝✳ ❙❛✉ ➤ã ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝ ♥➭② ➤➲ ➤➢❛ r❛ ✈➭ ♥❣❤✐➟♥ ❝ø✉ ♠ét sè
➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✭①❡♠
❬✼❪✮✳ ❍✐Ư♥ ♥❛② ❧ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ tr➟♥ ❦❤➠♥❣
❣✐❛♥

G✲♠➟tr✐❝

➤❛♥❣ t❤✉ ❤ót ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝ tr♦♥❣ ✈➭ ♥❣♦➭✐ ♥➢í❝

♥❣❤✐➟♥ ❝ø✉ ✈➭ ➤➲ ❝ã ♥❤÷♥❣ ❦Õt q✉➯ ♥❤✃t ➤Þ♥❤✳
❚r➟♥ ❝➡ së ❝➳❝ ❜➭✐ ❜➳♦ ❆ ♥❡✇ ❛♣♣r♦❛❝❤ t♦ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s
❝ñ❛ ❝➳❝ t➳❝ ❣✐➯ ❩✳ ▼✉st❛❢❛ ✈➭ ❇✳ ❙✐♠s ✈➭ ❋✐①❡❞ ♣♦✐♥t r❡s✉❧ts ♦♥ ❝♦♠✲
♣❧❡t❡

G✲♠❡tr✐❝

s♣❛❝❡s ❝ñ❛ ❝➳❝ t➳❝ ❣✐➯ ❩✳ ▼✉st❛❢❛✱ ▼✳ ❑❤❛♥❞❛❣❥✐ ✈➭ ❲✳

❙❤❛t❛♥❛✇✐✱ ❝ï♥❣ ✈í✐ ♠ét sè t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❦❤➳❝✱ ❞➢í✐ sù ❤➢í♥❣
❞➱♥ t❐♥ t×♥❤ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t✐Õ♣ ❝❐♥
❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ♥➭② ✈➭ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✿ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉②
ré♥❣ ✈➭ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ❱í✐ ➤Ị t➭✐ ♥➭②✱ ❝❤ó♥❣ t➠✐

G✲♠➟tr✐❝ ✈➭
❣✐❛♥ G✲♠➟tr✐❝

sÏ tr×♥❤ ❜➭② ♠ét sè ❦Õt q✉➯ ✈Ị ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ❦❤➠♥❣ ❣✐❛♥
t×♠ ❤✐Ĩ✉ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣

❝❤♦ ❝➳❝ ➳♥❤ ①➵ ❞➢í✐ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❦❤➳❝ ♥❤❛✉✳ ❇è ❝ơ❝ ❧✉❐♥ ✈➝♥ ❣å♠ ❤❛✐
❝❤➢➡♥❣✳
❈❤➢➡♥❣ ✶✳ ▼ét ❧í♣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳
❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ơ❝ ✶ ❞➭♥❤ ❝❤♦ ✈✐Ư❝ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ✈Ị
❦❤➠♥❣ ❣✐❛♥

2✲♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ D✲♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ ❦❤➠♥❣

✷


G✲♠➟tr✐❝ ➤è✐ ①ø♥❣ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣✳ ▼ơ❝
✷ tr×♥❤ ❜➭② ❝➳❝ ❦Õt q✉➯ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ✱ ❞➲② G✲❧✐➟♥ tơ❝✱ ❞➲②
G✲❤é✐ tơ✱ ❞➲② G✲❈➠s✐ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣✳
❣✐❛♥

❈❤➢➡♥❣ ✷✳ ▼ét sè ❦Õt q✉➯ ➤✐Ó♠ ❜✃t ➤é♥❣ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥

G✲

♠➟tr✐❝ ➤➬② ➤đ✳

❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ơ❝ ✶ tr×♥❤ ❜➭② ♠ét sè ❦❤➳✐ ♥✐Ư♠ ✈➭ ❦Õt q✉➯ ❝➡
❜➯♥ ❝đ❛ ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝

➤➬② ➤đ✳ ▼ơ❝ ✷

tr×♥❤ ❜➭② ♠ét sè ❦Õt q✉➯ ✈Ị ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝ ➤➬② ➤đ✳
▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❞➢í✐ sù ❤➢í♥❣
❞➱♥ t❐♥ t×♥❤ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✳ ❚➳❝ ❣✐➯ ①✐♥ tỏ
ò ết s s ế t ị ♥➭② t➳❝ ❣✐➯ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠
➡♥ ❇❛♥ ❝❤đ ♥❤✐Ư♠ ❦❤♦❛ ❚♦➳♥✱ P❤ß♥❣ ➤➭♦ t➵♦ ❙❛✉ ➤➵✐ ❤ä❝✱ q✉ý ❚❤➬②✱ ❈➠
tr♦♥❣ tæ ●✐➯✐ tÝ❝❤ ❦❤♦❛ ❚♦➳♥ tr➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ ➤➲ ❣✐ó♣ ➤ì tr♦♥❣ s✉èt
q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ❈✉è✐ ❝ï♥❣ ①✐♥ ❝➯♠ ➡♥✱ ❣✐❛

➤×♥❤✱ ❝➡ q✉❛♥✱ ❝➳❝ ➤å♥❣ ♥❣❤✐Ö♣✱ ❜➵♥ ❜❒ ➤➷❝ ❜✐Öt ❧➭ ❝➳❝ ❤ä❝ ✈✐➟♥ ❝❛♦ ❤ä❝
❦❤ã❛ ✶✽ ❚♦➳♥✲●✐➯✐ tÝ❝❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥
❧ỵ✐ ❣✐ó♣ t➳❝ ❣✐➯ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤ä❝ t❐♣✳
▼➷❝ ❞ï ❝ã ♥❤✐Ị✉ ❝è ❣➽♥❣✱ s♦♥❣ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣
t❤✐Õ✉ sãt✳ ❚➳❝ rt ợ ữ ý ế ó ó ❝đ❛ q✉ý
❚❤➬②✱ ❈➠ ❝ï♥❣ ❜➵♥ ➤ä❝ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ tệ

t

ỗ ì




❝❤➢➡♥❣ ✶
♠ét ❧í♣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣

❦❤➠♥❣ ❣✐❛♥ tr s rộ





ị ĩ t

ợ ọ
A1 ❱í✐
✭A2 ✮

X = φ✳

➳♥❤ ①➵ d : X × X × X → R+

2✲♠➟tr✐❝ tr➟♥ X ✱ ♥Õ✉ d t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿
x, y ∈ X

♠➭

x = y tå♥ t➵✐ z ∈ X

s❛♦ ❝❤♦

d(x, y, z) = 0❀

d(x, y, z) = 0 ♥Õ✉ ❝ã Ýt ♥❤✃t ❤❛✐ tr♦♥❣ ❝➳❝ ♣❤➬♥ tö x, y, z ∈ X

❧➭

❜➺♥❣ ♥❤❛✉❀
✭A3 ✮

d(x, y, z) = d(k{x, y, z})

x, y, z ✱ ✈í✐ ♠ä✐ x, y, z ∈ X

tr♦♥❣ ➤ã

k{x, y, z}

❧➭ ❝➳❝ ❤♦➳♥ ✈Þ ❝đ❛

✭ tÝ♥❤ ❝❤✃t ➤è✐ ①ø♥❣✮❀

✭A4 ✮

d(x, y, z) ≤ d(x, y, a) + d(x, a, z) + d(a, y, z)✱ ✈í✐ ♠ä✐ x, y, z, a ∈

❚❐♣

X

X✳
❝ï♥❣ ✈í✐

✈➭ ❦ý ❤✐Ö✉ ❧➭

✶✳✶✳✷

2✲♠➟tr✐❝ d tr➟♥ X

X = φ✳ ➳♥❤ ①➵ D : X × X × X → R+

D✲♠➟tr✐❝✱ ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿

✭D1 ✮

D(x, y, z) = 0 ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ x = y = z ❀

✭D2 ✮

D(x, y, z) = D(k{x, y, z})✱

x, y, z

2✲♠➟tr✐❝

(X, d) X

ị ĩ t

ợ ọ

ợ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥

✈í✐ ♠ä✐

x, y, z ∈ X

✭D3 ✮ D(x, y, z)

tr♦♥❣ ➤ã

k{x, y, z}

❧➭ ❤♦➳♥ ✈Þ ❝đ❛

✭tÝ♥❤ ❝❤✃t ➤è✐ ①ø♥❣✮❀

≤ D(x, y, a) + D(x, a, z) + D(a, y, z)✱ ✈í✐ ♠ä✐ x, y, z, a ∈

X✳
❚❐♣

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

✈➭ ❦ý ❤✐Ư✉ ❧➭

✶✳✶✳✸

❱Ý ❞ơ✳

(X, D) ❤❛② X ✳
❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

(X, d)✳

❳Ðt ➳♥❤ ①➵

D : X ×X ×

X → R+ ❝❤♦ ❜ë✐
1
D(x, y, z) = {d(x, y) + d(y, z) + d(z, x)}✱ ✈í✐ ♠ä✐ x, y, z ∈ X ✳
3
✹


❑❤✐ ➤ã

(X, D) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ D✲♠➟tr✐❝✳ ❚❤❐t ✈❐②✱ t❛ ❦✐Ĩ♠ tr❛ ❝➳❝

➤✐Ị✉ ❦✐Ư♥ ❝đ❛

D✲♠➟tr✐❝ ♥❤➢ s❛✉✿

(D1 ) ❱× d ❧➭ ♠➟tr✐❝ ♥➟♥ d(x, y), d(y, z), d(z, x) ➤Ò✉ ❧➭ ❝➳❝ sè t❤ù❝ ❦❤➠♥❣
1
{d(x, y) + d(y, z) +
➞♠ ✈í✐ ♠ä✐ x, y, z ∈ X ✳ ❉♦ ➤ã t❛ ❝ã G(x, y, z) =
3
d(z, x)} ≥ 0 ✈í✐ ♠ä✐ x, y, z ∈ X ✳ ➜➻♥❣ t❤ø❝ ①➯② r❛ ❦❤✐ x = y = z ✳
(D2 ) ❚Ý♥❤ ❝❤✃t ➤è✐ ①ø♥❣ ❧➭ ❤✐Ĩ♥ ♥❤✐➟♥✳
(D3 ) ❱í✐ ♠ä✐ x, y, z, a ∈ X ✱ t❛ ❝ã
1
D(x, y, z) = {d(x, y) + d(y, z) + d(z, x)}
3
1
≤ {d(x, a) + d(y, a) + d(y, a) + d(a, z) + d(x, a) + d(z, a)}
3
1
1
≤ {d(x, y)+d(y, a)+d(a, x)}+ {d(y, a) + d(a, z) + d(z, y)}
3
3
1
+ {d(x, a) + d(a, z) + d(x, z)}
3
= D(x, y, a) + D(x, a, z) + D(a, y, z)✳
❱❐②

✶✳✶✳✹

(X, D) ❧➭ ❦❤➠♥❣ ❣✐❛♥ D✲♠➟tr✐❝✳

❱Ý ❞ô✳

❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

(X, d)✳

❳Ðt ➳♥❤ ①➵

Dm (d) : X ×

X × X → R+ ❝❤♦ ❜ë✐
Dm (d)(x, y, z) = max{d(x, y), d(y, z), d(z, x)}✳
D✲♠➟tr✐❝ tr➟♥ X ✳

❑❤✐ ➤ã

❚❤❐t ✈❐②✱ t❛ ❦✐Ĩ♠ tr❛ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛

Dm (d)

❧➭ ♠ét

D✲♠➟tr✐❝ ♥❤➢

s❛✉✿

(D1 ) ❱× d ❧➭ ♠➟tr✐❝ ♥➟♥ d(x, y), d(y, z), d(z, x) ➤Ò✉ ❧➭ ❝➳❝ sè t❤ù❝ ❦❤➠♥❣
➞♠ ✈í✐ ♠ä✐

x, y, z ∈ X ✳ ❉♦ ➤ã t❛ ❝ã

Dm (d)(x, y, z) = max{d(x, y), d(y, z), d(z, x)} ≥ 0

✈í✐ ♠ä✐

x, y, z ∈

X ✳ ➜➻♥❣ t❤ø❝ ①➯② r❛ ❦❤✐ x = y = z ✳
(D2 ) ❚Ý♥❤ ❝❤✃t ➤è✐ ①ø♥❣ ❧➭ ❤✐Ó♥ ♥❤✐➟♥✳
(D3 ) ❱í✐ ♠ä✐ x, y, z, a ∈ X ✱ t❛ ❝ã
Dm (d)(x, y, z) = max{d(x, y), d(y, z), d(z, x)}
≤ max{d(x, y), d(y, a), d(a, x)}+max{d(y, a), d(a, z), d(z, y)}
+ max{d(x, a), d(a, z), d(x, z)}
= Dm (d)(x, y, a) + Dm (d)(x, a, z) + Dm (d)(a, y, z)✳
❱❐②

(X, Dm (d)) ❧➭ ❦❤➠♥❣ ❣✐❛♥ D✲♠➟tr✐❝✳

✺




ị ĩ

ợ ọ

t

X = G : X × X × X → R+

G✲♠➟tr✐❝✱ ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿

✭G1 ✮

G(x, y, z) = 0 ♥Õ✉ x = y = z ❀

✭G2 ✮

G(x, x, y) > 0 ✈í✐ ♠ä✐ x, y ∈ X

✭G3 ✮

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

✭G4 ✮

G(x, y, z) = G(k{x, y, z})

x, y, z

✈í✐ ♠ä✐

x, y, z ∈ X

♠➭

tr♦♥❣ ➤ã

♠➭

z = y❀

k{x, y, z}

❧➭ ❤♦➳♥ ✈Þ ❝đ❛

✭tÝ♥❤ ❝❤✃t ➤è✐ ①ø♥❣✮❀

G(x, y, z) ≤ G(x, a, a) + G(a, y, z)✱

✭G5 ✮

x = y

ớ ọ

x, y, z, a X

t

tứ ì ữ ♥❤❐t✮✳

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

❚❐♣

✈➭ ❦ý ❤✐Ư✉ ❧➭

✶✳✶✳✻

❱Ý ❞ơ✳

➳♥❤ ①➵

(X, G) ❤❛② X ✳
(X, d)

❈❤♦

❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤➠♥❣ t❤➢ê♥❣ ①Ðt

Gm : X × X × X → R+ ❝❤♦ ❜ë✐

Gm (x, y, z) = max{d(x, y), d(y, z), d(x, z)} ✈í✐ ♠ä✐ x, y, z ∈ X ✳
❑❤✐ ➤ã

(X, Gm )

❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛

(G1 )

❱×

d

❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝✳

❚❤❐t ✈❐②✱ t❛ ❦✐Ĩ♠ tr❛

G✲♠➟tr✐❝ ♥❤➢ s❛✉✿

❧➭ ♠➟tr✐❝✱ ♥➟♥

❦❤➠♥❣ ➞♠ ✈í✐ ♠ä✐

d(x, y), d(y, z), d(x, z)✱

➤Ò✉ ❧➭ ❝➳❝ sè t❤ù❝

x, y, z ∈ X ✳ ❱× t❤Õ t❛ ❝ã

Gm (x, y, z) = max{d(x, y), d(y, z), d(x, z)} ≥ 0,
✈í✐ ♠ä✐

x, y, z ∈ X ✳ ➜➻♥❣ t❤ø❝ ①➯② r❛ ❦❤✐ x = y = z ✳

(G2 ) ❱× d ❧➭ ♠➟tr✐❝✱ ♥➟♥ t❛ ❝ã
Gm (x, x, y) = max{d(x, x), d(x, y), d(x, y)} = d(x, y) > 0,
✈í✐ ♠ä✐

x, y ∈ X

♠➭

x = y✳

(G3 ) Gm (x, x, y) = max{d(x, x), d(x, y), d(x, y)} = d(x, y)
≤ max{d(x, y), d(y, z), d(x, z)} = G(x, y, z)✱

✈í✐ ♠ä✐

z = y✳
✭G4 ) ❚Ý♥❤ ❝❤✃t ➤è✐ ①ø♥❣ ❧➭ ❤✐Ó♥ ♥❤✐➟♥✳
✭G5 )

Gm (x, y, z) = max{d(x, y), d(y, z), d(x, z)}
✻

x, y, z ∈ X

♠➭


≤ max{d(x, a) + d(a, y), d(y, z), d(x, a) + d(a, z) + d(a, a)}
≤ max{d(x, a), d(x, a), d(a, a)}+max{d(a, y), d(a, z), d(y, z)}
= Gm (x, a, a) + Gm (a, y, z)✱ ✈í✐ ♠ä✐ x, y, z, a ∈ X ✳
✶✳✶✳✼

❱Ý ❞ô✳

❈❤♦

(X, d)

❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤➠♥❣ t❤➢ê♥❣✳ ❳Ðt

Gs : X × X × X → R+ ❝❤♦ ❜ë✐

➳♥❤ ①➵

1
Gs (x, y, z) = {d(x, y) + d(y, z) + d(x, z)} ✈í✐ ♠ä✐ x, y, z ∈ X
3
❑❤✐ ➤ã

(X, Gs ) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❚❤❐t ✈❐②✱ t❛ ❦✐Ĩ♠ tr❛ ❝➳❝

➤✐Ị✉ ❦✐Ư♥ ❝đ❛

G✲♠➟tr✐❝ ♥❤➢ s❛✉✿

❉Ơ t❤✃② ➳♥❤ ①➵

Gs

t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭G1 ✮✱✭G2 ✮✱✭G3 ✮✱✭G4 ✮ tr♦♥❣

➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺✳ ❚❛ ❦✐Ĩ♠ tr❛ ➤✐Ị✉ ❦✐Ư♥ ✭G5 ✮✳ ●✐➯ sư ✈í✐ ♠ä✐

x, y, z, a ∈

X ✱ ❦❤✐ ➤ã t❛ ❝ã

1
Gs (x, y, z) = {d(x, y) + d(y, z) + d(x, z)}
3
1
≤ [d(x, a) + d(a, y) + d(y, z) + d(x, a) + d(a, z) + d(a, a)]
3
1
1
= [d(x, a) + d(x, a) + d(a, a)] + [d(a, y) + d(y, z) + d(a, z)]
3
3
= Gs (x, a, a) + Gs (a, y, z)✳

❱❐②

✶✳✶✳✽

(X, Gs ) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳

▼Ư♥❤ ➤Ị ✭❬✼❪✮✳

✈í✐ ♠ä✐

x, y, z, a ∈ X

✭✐✮ ◆Õ✉
✭✐✐✮

●✐➯ sư

(X, G)

❧➭ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝✳

❑❤✐ ➤ã

t❛ ❝ã

G(x, y, z) = 0 t❤× x = y = z ❀

G(x, y, z) ≤ G(x, x, y) + G(x, x, z)❀

✭✐✐✐✮

G(x, y, y) ≤ 2G(y, x, x)❀

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)}✳
✭✐✈✮

❈❤ø♥❣ ♠✐♥❤✳ ▲✃② ❜✃t ❦ú

✭✐✮ ❙✉② tõ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛

x, y, z, a ∈ X ✳ ❑❤✐ ➤ã t❛ ❝ã

G✲♠➟tr✐❝✳

✭✐✐✮ ◆❤ê ❝➳❝ tÝ♥❤ ❝❤✃t ✭G4 ✮ ✈➭ ✭G5 ✮✱ ❧✃②

a = x t❛ ❝ã

G(x, y, z) = G(y, x, z) ≤ G(y, x, x) + G(x, x, z) = G(x, x, y) + G(x, x, z).
✼


✭✐✐✐✮ ❙✉② tõ ✭✐✐✮ ❜➺♥❣ ❝➳❝❤ ❧✃②

y = z✳

✭✐✈✮ ◆❤ê ❝➳❝ tÝ♥❤ ❝❤✃t ✭G3 ✮ ✈➭ ✭G5 ✮ t❛ ❝ã

G(x, y, z) ≤ G(x, a, a) + G(a, y, z) ≤ G(x, a, z) + G(a, y, z).
✭✈✮ ◆❤ê tÝ♥❤ ❝❤✃t ✭✐✈✮ ❜➺♥❣ ❝➳❝❤ ❤♦➳♥ ✈Þ

x, y, z

✈➭ sư ❞ơ♥❣ tÝ♥❤ ❝❤✃t

✭G4 ✮ t❛ ❝ã

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

❈é♥❣ ✈Õ ✈í✐ ✈Õ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② ✈➭ sư ❞ơ♥❣ tÝ♥❤ ❝❤✃t ✭G4 ✮ t❛
t❤✉ ➤➢ỵ❝

2
G(x, y, z) ≤ {G(x, y, a) + G(x, a, z) + G(a, y, z)}✳
3
✭✈✐✮ ❇➺♥❣ ❝➳❝❤ ➳♣ ❞ô♥❣ tÝ♥❤ ❝❤✃t ✭G5 ✮ ❧✐➟♥ t✐Õ♣ t❛ t❤✉ ➤➢ỵ❝
G(x, y, z) ≤ G(x, a, a)+G(a, y, z) ≤ G(x, a, a)+G(y, a, a)+G(z, a, a).

✶✳✶✳✾

▼Ư♥❤ ➤Ị ✭❬✼❪✮✳ ●✐➯ sư

k > 0✳

❑❤✐ ➤ã ❝➳❝ ❤➭♠

♠➟tr✐❝ tr➟♥

(X, G)

Gi ✱Gj

G✲♠➟tr✐❝

✈➭

❝❤♦ ❜ë✐ ❝➳❝ ❝➠♥❣ t❤ø❝ s❛✉ ❝ò♥❣ ❧➭

G✲

❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥

X✳

Gi (x, y, z) = min{k, G(x, y, z)} ✈í✐ ♠ä✐ x, y, z ∈ X ✱
G(x, y, z)
❜✮ Gj (x, y, z) =
✈í✐ ♠ä✐ x, y, z ∈ X ✳
1 + G(x, y, z)
❈❤ø♥❣ ♠✐♥❤✳ ❛✮ ❚❤❐t ✈❐②✱ t❛ ❦✐Ó♠ tr❛ Gi t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛
❛✮

G✲♠➟tr✐❝ ♥❤➢ s❛✉✿
✭G1 ✮ ❉Ơ t❤✃②
❦❤✐

Gi (x, y, z) = 0 ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ G(x, y, z) = 0✱ ❦❤✐ ✈➭ ❝❤Ø

x = y = z✳
❚➢➡♥❣ tù ❞Ơ t❤✃② ➳♥❤ ①➵

tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺✳

Gi

t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭✭G2 ✮✱✭G3 ✮✱✭G4 ✮

❚❛ ❦✐Ĩ♠ tr❛ ➤✐Ị✉ ❦✐Ư♥ ✭G5 ✮✳

●✐➯ sư ✈í✐ ♠ä✐

x, y, z, a ∈ X ✱ ❦❤✐ ➤ã t❛ ❝ã
Gi (x, y, z) = min{k, G(x, y, z)} ≤ min{k, G(x, a, a) + G(a, y, z)}
≤ min{k, G(x, a, a)} + min{k, G(a, y, z)}
≤ Gi (x, y, z) + Gi (x, y, z)✳
✽


Gi ❧➭ ♠ét G✲♠➟tr✐❝✳

❱❐②

❜✮ ❈❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù✳

✶✳✶✳✶✵

➜Þ♥❤ ♥❣❤Ü❛ ✭❬✼❪✮✳ ❑❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝ (X, G) ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝ ➤è✐ ①ø♥❣ ♥Õ✉ G(x, y, y) = G(y, x, x) ✈í✐ ♠ä✐ x, y ∈ X ✳
✶✳✶✳✶✶

◆❤❐♥ ①Ðt ✭❬✼❪✮✳ ❛✮ ❚õ ❝➳❝ ✈Ý ❞ơ tr➟♥ t❛ s✉② r❛ ♥Õ✉

❣✐❛♥ ♠➟tr✐❝✱ t❤× ❝ã t❤Ĩ tr❛♥❣ ❜Þ ❝❤♦
✈➭

X

❝➳❝

(X, d) ❧➭ ❦❤➠♥❣

G✲♠➟tr✐❝ Gs ✈➭ Gm ➤Ó (X, Gs )

(X, Gm ) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ Gtr ợ ế (X, G)



Gtr tì t❤ø❝ dG (x, y) = G(x, x, y) + G(x, y, y)

✈í✐ ♠ä✐

x, y ∈ X

✈➭ ✈í✐ ♠➟tr✐❝

①➳❝ ➤Þ♥❤ ♠ét ♠➟tr✐❝ tr➟♥

d ❝❤♦ tr➢í❝ tr➟♥ X

(1.1)

X ✳ ❍➡♥ ♥÷❛ ✈í✐ ♠ä✐ x, y ∈ X

t❛ ❝ã

G(x, y, z) ≤ Gs (dG )(x, y, z) ≤ 2G(x, y, z),

1
G(x, y, z) ≤ Gm (dG )(x, y, z) ≤ 2G(x, y, z),
2
4
✈➭ dGs (d) (x, y) = d(x, y), dGm (d) (x, y) = 2d(x, y).
3
❜✮ ◆Õ✉ (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ Gtr ố ứ tì tr ợ
ở

dG (x, y) = 2G(x, y, y) ✈í✐ ♠ä✐ x, y ∈ X ✳
(X, G)

❦❤➠♥❣ ➤è✐ ①ø♥❣✱ t❤× ♥❤ê ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛

G✲♠➟tr✐❝ t❛ ❝ã

3
G(x, y, y) ≤ dG (x, y) ≤ 3G(x, y, y) ✈í✐ ♠ä✐
2

❚✉② ♥❤✐➟♥ ♥Õ✉
❦❤➠♥❣ ❣✐❛♥

x, y ∈ X ✳
✶✳✶✳✶✷

(1.2)

▼Ư♥❤ ➤Ò ✭❬✼❪✮✳ ❈❤♦

(X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❑❤✐ ➤ã ❝➳❝

➤✐Ị✉ ❦✐Ư♥ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣

(X, G) ❧➭ ➤è✐ ①ø♥❣❀

✭✶✮
✭✷✮

G(x, y, y) ≤ G(x, y, a) ✈í✐ ♠ä✐ x, y, a ∈ X ❀

✭✸✮

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

❈❤ø♥❣ ♠✐♥❤✳ ✭✶✮⇒ ✭✷✮✳ ❚õ ➤✐Ị✉ ❦✐Ư♥

G3 tr ị ĩ ì

(G, X) ố ứ t❛ s✉② r❛ ✈í✐ ❜✃t ❦ú a = x t❛ G(x, y, y) ≤ G(x, y, a)✳
✭✷✮⇒ ✭✸✮✳ ◆❤ê ▼Ö♥❤ ➤Ị ✶✳✶✳✽ ✈➭ ❣✐➯ t❤✐Õt ❝ã ✭✷✮✱ ♥➟♥ ✈í✐ ♠ä✐ x, y, z, a, b

X

t❛ ❝ã

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

∈

❉♦ ➤ã t❛ ❝ã ✭✷✮⇒ ✭✸✮✳
✭✸✮⇒ ✭✶✮✳ ❚❤❛②

z = y ✈➭♦ ✭✸✮ t❛ ❝ã

G(x, y, y) ≤ G(x, y, a) + G(y, y, b)
❚❤❛②

(∗)

a = x✱ b = y ✈➭♦ (∗) t❛ ❝ã
G(x, y, y) ≤ G(x, y, x) ❤❛② G(x, y, y) ≤ G(y, x, x)

❍♦➳♥ ➤ỉ✐ ✈❛✐ trß ủ





t

Gtr

ị ĩ

X, r > 0 Gì

sử

(X, G)
a

✈í✐ t➞♠ t➵✐

❧➭ ❦❤➠♥❣ ❣✐❛♥

❜➳♥ ❦Ý♥❤

r

G✲♠➟tr✐❝✱ a ∈

❦ý ❤✐Ư✉ ❧➭

BG (a, r)

BG (a, r) = {x ∈ X : G(a, x, x) < r}✳

❧➭ t❐♣ ❤ỵ♣

✶✳✷✳✷

x ✈➭ y t❛ ❝ã G(y, x, x) ≤ G(x, y, y)

▼Ư♥❤ ➤Ị ✭❬✼❪✮✳ ●✐➯ sö

(X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ x0 ∈ X

✈➭

r > 0✳ ❑❤✐ ➤ã t❛ ❝ã✿
✭✐✮ ♥Õ✉

G(x0 , x, y) < r✱ t❤× x, y ∈ BG (x0 , r)✳

✭✐✐✮ ♥Õ✉

y ∈ BG (x0 , r)✱

t❤× tå♥ t➵✐ sè

δ > 0

s❛♦ ❝❤♦

BG (y, δ) ⊆

BG (x0 , r)✳
❈❤ø♥❣ ♠✐♥❤✳ ✭✐✮ ●✐➯ sư

➤✐Ị✉ ❦✐Ư♥

(G3 )

❝đ❛

x, y ∈ X

G✲♠➟tr✐❝

t❛ ❝ã

s❛♦ ❝❤♦

G(x0 , x, y) < r✳

◆❤ê

G(x0 , x, x) ≤ G(x0 , x, y) < r

✈➭

G(x0 , y, y) ≤ G(x0 , x, y) < r✳ ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ x, y ∈ BG (x0 , r).
✭✐✐✮ ❱×
➤ã t❛ ❝ã

y ∈ BG (x0 , r)✱ ♥➟♥ G(x0 , y, y) < r✳ ➜➷t δ = r − G(x0 , y, y)✳ ❑❤✐

δ > 0✳

❧✃② ❜✃t ❦ú

❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣

z ∈ BG (y, δ)✱

t❛ ❝ã

G(y, z, z) + G(x0 , y, y) < r✳
t❛ ❝ã

❚❤❐t ✈❐②✱

G(y, z, z) < δ = r − G(x0 , y, y)✱

▼➷t ❦❤➳❝ ♥❤ê ➤✐Ị✉ ❦✐Ư♥

G(x0 , z, z) ≤ G(x0 , y, y) + G(y, z, z) < r✳

✈❐② tå♥ t➵✐

✶✳✷✳✸

BG (y, δ) ⊆ BG (x0 , r)✳

s✉② r❛

(G5 ) ❝ñ❛ G✲♠➟tr✐❝

❙✉② r❛

z ∈ BG (x0 , r)✳

❱×

δ > 0 ➤Ĩ BG (y, δ) ⊆ BG (x0 , r).

◆❤❐♥ ①Ðt✳

❈❤♦

(X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳

➤Þ♥❤ ✭✐✐✮ ❝đ❛ ▼Ư♥❤ ➤Ị ✶✳✷✳✷ t❛ s✉② r❛ ❤ä

✶✵

❑❤✐ ➤ã tõ ❦❤➻♥❣

B = {BG (x, r) : x ∈ X, r > 0}


❧❐♣ t❤➭♥❤ ♠ét ❝➡ së ❝ñ❛ ♠ét t➠♣➠
❧➭ t➠♣➠

✶✳✷✳✹
✈➭

τ (G) ♥➭♦ ➤ã tr➟♥ X ✳ ❚❛ ❣ä✐ t➠♣➠ τ (G)

G✲♠➟tr✐❝✳
(X, G)

▼Ư♥❤ ➤Ị✳ ✭❬✼❪✮ ●✐➯ sư

r > 0✳

❑ý ❤✐Ư✉

❦❤➠♥❣ ❣✐❛♥

BG (x0 , r), BdG (x0 , r)

G✲♠➟tr✐❝ (X, G)

❝ã

❧➭ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝✱ x0 ∈ X

t❤ø tù ❧➭ ❝➳❝ ❤×♥❤ ❝➬✉ tr♦♥❣

✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝

(X, dG )✳

❑❤✐ ➤ã t❛

1
BG (x0 , r) ⊂ BdG (x0 , r) ⊂ BG (x0 , r).
3
❈❤ø♥❣ ♠✐♥❤✳ ❙✉② trù❝ t✐Õ♣ tõ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✷✳✶ ✈➭ ◆❤❐♥ ①Ðt



X

ị ĩ

ợ ọ

Gộ

tụ ế

tr

(G) tr X ✳

✶✳✷✳✻

▼Ư♥❤ ➤Ị ✭❬✼❪✮✳ ❈❤♦

(X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❉➲② (xn ) ⊆
x ∈ X

(X, G)

♥Õ✉ ♥ã ❤é✐ tô ➤Õ♥

❧➭ ❦❤➠♥❣ ❣✐❛♥

x

t❤❡♦ t➠♣➠

G✲♠➟tr✐❝✳

G✲

❑❤✐ ➤ã ❝➳❝

♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣

(1) {xn } ❧➭ G✲❤é✐ tơ tí✐ x❀
(2) dG (xn , x) → 0✱
♠➟tr✐❝

❦❤✐

n → ∞✱

♥❣❤Ü❛ ❧➭ ❞➲②

{xn }

❤é✐ tơ tí✐

x

t❤❡♦

dG ❀

(3) G(xn , xn , x) → 0✱ ❦❤✐ n → ∞❀
(4) G(xn , x, x) → 0✱ ❦❤✐ n → ∞❀
(5) G(xm , xn , x) → 0✱ ❦❤✐ n, m → ∞✳

❈❤ø♥❣ ♠✐♥❤✳ ❚õ ✭✶✮⇔✭✷✮✳ ❙✉② tõ trù❝ t✐Õ♣ ▼Ư♥❤ ➤Ị ✶✳✷✳✹✳

❚õ ✭✷✮⇒✭✸✮ s✉② trù❝ t✐Õ♣ tõ ➤➻♥❣ t❤ø❝ dG (x, y)
✈í✐ ♠ä✐

= G(x, x, y)+G(x, y, y)

x, y ∈ X ✳

❚õ ✭✸✮⇒✭✹✮ ❧➭ ❤Ö q✉➯ trù❝ t✐Õ♣ ❝đ❛ ❦❤➻♥❣ ➤Þ♥❤ ✭✐✐✐✮ tr♦♥❣ ▼Ư♥❤ ➤Ị
✶✳✶✳✽✳
❚õ ✭✹✮⇒✭✺✮ s✉② tõ ❦❤➻♥❣ ➤Þ♥❤ ✭✐✐✮ tr♦♥❣ ▼Ư♥❤ ➤Ị ✶✳✶✳✽✳
❚õ ✭✺✮⇒✭✷✮ s✉② tõ ➤➻♥❣ t❤ø❝

dG (x, y) = G(x, x, y) + G(x, y, y)

❦❤➻♥❣ ➤Þ♥❤ ✭✐✐✐✮ tr♦♥❣ ▼Ư♥❤ ➤Ị ✶✳✶✳✽✳

✶✶

✈➭


✶✳✷✳✼

➜Þ♥❤ ♥❣❤Ü❛ ✭❬✾❪✮✳ ❈❤♦

f : X → X

♠➟tr✐❝ ✈➭ ➳♥❤ ①➵

➤✐Ĩ♠

♠ä✐

a∈X

♥Õ✉ ✈í✐ sè

x, y ∈ X

❍➭♠

f

♠➭

(X, G)

✈➭

(X , G )

❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥

✳ ❚❛ ♥ã✐ r➺♥❣ ➳♥❤ ①➵

f

❧➭

G✲❧✐➟♥

G✲

tô❝ t➵✐

ε > 0 ❝❤♦ tr➢í❝ tå♥ t➵✐ ♠ét sè δ > 0✱ s❛♦ ❝❤♦ ✈í✐

G(a, x, y) < δ t❛ ❝ã G(f (a), f (x), f (y)) < ε✳

➤➢ỵ❝ ❣ä✐ ❧➭

G✲❧✐➟♥ tơ❝ tr➟♥ X ♥Õ✉ f

❧➭

G✲❧✐➟♥ tơ❝ t➵✐ ♠ä✐ ➤✐Ĩ♠

a ∈ X✳
◆❤❐♥ ①Ðt✳ ●✐➯ sö (X, G) ✈➭ (X

✶✳✷✳✽

f :X→X
f

❧➭

❧➭ ➳♥❤ ①➵ tõ

X ✈➭♦ X

G✲❧✐➟♥ tơ❝ t➵✐ ➤✐Ĩ♠ x0 ∈ X

✈í✐ ♠ä✐

❦❤✐ ✈➭ ❝❤Ø ❦❤✐

f −1 (BG (f (x0 ), r)) ∈ τ (G)✱

r > 0✳

❑❤✐ ➤ã ➳♥❤ ①➵
♥Õ✉ ♥ã ❧➭

f :X →X

❧➭

G✲❧✐➟♥

, G ) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳
tơ❝ t➵✐ ➤✐Ĩ♠

x∈X

♥Õ✉ ✈➭ ❝❤Ø

G✲❧✐➟♥ tơ❝ ❞➲② t➵✐ x❀ ♥❣❤Ü❛ ❧➭✱ ✈í✐ ❜✃t ❦ú ❞➲② {xn } ❧➭ G✲❤é✐

x t❤× ❞➲② {f (xn )} ❧➭ G✲❤é✐ tơ tí✐ f (x)✳

❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư

❞➲② ❜✃t ❦ú
tí✐

✳ ❑❤✐ ➤ã ♥❤ê ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✷✳✶ t❛ s✉② r❛

▼Ư♥❤ ➤Ị ✭❬✼❪✮✳ ❈❤♦ (X, G) ✈➭ (X

✶✳✷✳✾

tơ tí✐

, G ) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭

f (x)✳

f

❧➭

G✲❧✐➟♥ tô❝ t➵✐ ➤✐Ĩ♠ x ∈ X

✈➭ ❞➲②

{xn } ❧➭

G✲❤é✐ tơ tí✐ x✱ t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ {f (xn )} ❧➭ ❞➲② G✲❤é✐ tơ

❚❤❐t ✈❐②✱

tr➢í❝ tå♥ t➵✐ sè

f

❧➭

G✲❧✐➟♥ tơ❝ t➵✐ ➤✐Ĩ♠ x ∈ X

δ > 0 s❛♦ ❝❤♦ ✈í✐ ♠ä✐ y, z ∈ X

♥➟♥ ✈í✐ sè

♠➭

ε > 0 ❝❤♦

G(x, y, z) < δ

t❛ ❝ã

G(f (x), f (y), f (z)) < ε✳ ▲➵✐ ✈× {xn } ❧➭ G✲❤é✐ tơ tí✐ x ♥➟♥ ✈í✐ sè δ > 0 ➤ã
tå♥ t➵✐ sè
s✉② r❛

N ∈ N s❛♦ ❝❤♦ G(x, xn , xm ) < δ

✈í✐ ♠ä✐

n, m ≥ N ✳

❚õ ➤ã t❛

G(f (x), f (xn ), f (xm )) < ε ✈í✐ ♠ä✐ n, m ≥ N ✳ ❱❐② {f (xn )} ❧➭ ❞➲②

G✲❤é✐ tơ tí✐ f (x)✳
◆❣➢ỵ❝ ❧➵✐✱ ❣✐➯ sư r➺♥❣ ♥Õ✉
❝ã

{xn } ❧➭ ❞➲② ❜✃t ❦ú G✲❤é✐ tơ tí✐ x ∈ X

{f (xn )} ❧➭ ❞➲② G✲❤é✐ tơ tí✐ f (x)✱ ♥❤➢♥❣ f

❧➭ ❦❤➠♥❣ ❧➭

t❛

G✲❧✐➟♥ tụ t

0 > 0 s ớ ỗ n N tå♥ t➵✐ xn , yn ∈
1
X s❛♦ ❝❤♦ G(x, xn , yn ) < ✱ ♥❤➢♥❣ G(f (x), f (xn ), f (yn )) ≥ ε0 ✳ ◆❤ê
n
1
tÝ♥❤ ❝❤✃t (G3 ) t❛ ❝ã G(x, xn , xn ) ≤ G(x, xn , yn ) <
✈➭ G(x, yn , yn ) ≤
n
1

G(x, xn , yn ) < ✳ ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ G(x, xn , xn ) → 0 ✈➭ G(x, yn , yn ) →
n
0 ❦❤✐ n → ∞✳ ◆❤ê ❣✐➯ t❤✐Õt t❛ s✉② r❛ G(f (x), f (xn ), f (xn )) → 0 ✈➭
➤✐Ó♠

x✳

❑❤✐ ➤ã tå♥ t➵✐ sè

✶✷


G(f (x), f (yn ), f (yn )) → 0 ❦❤✐ n → ∞✳

◆❤ê ▼Ư♥❤ ➤Ị ✶✳✶✳✽✭✐✐✐✮ t❛ s✉② r❛

G(f (x), f (x), f (xn )) → 0 ✈➭ G(f (x), f (x), f (yn )) → 0 ❦❤✐ n → ∞✳ ❱× t❤Õ
❧➵✐ ♥❤ê ▼Ư♥❤ ➤Ị ✶✳✶✳✽✭✐✐✮ t❛ s✉② r❛
➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❱❐②

✶✳✷✳✶✵

f

▼Ư♥❤ ➤Ị ✭❬✼❪✮✳ ❈❤♦

❧➭

G(f (x), f (xn ), f (yn )) → 0 ❦❤✐ n → ∞✳

G✲❧✐➟♥ tơ❝ t➵✐ ➤✐Ĩ♠ x✳

(X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❑❤✐ ➤ã ❤➭♠

G(x, y, z) ❧✐➟♥ tô❝ ➤å♥❣ t❤ê✐ t❤❡♦ ❝➯ ❜❛ ❜✐Õ♥✳
❈❤ø♥❣ ♠✐♥❤✳

tí✐ ❝➳❝ ➤✐Ĩ♠

●✐➯ sư

x, y, z ∈ X ✳

{xn }✱ {yn }✱ {zn }

❧➭ ❝➳❝ ❞➲②

G✲❤é✐

tô ❧➬♥ ợt

ờ ề ệ G5 ) tr ị ĩ t

ó

G(x, y, z) ≤ G(y, ym , ym ) + G(ym , x, z)✱
G(z, x, ym ) ≤ G(x, xk , xk ) + G(xk , ym , z)

✈➭

G(z, xk , ym ) ≤ G(z, zn , zn ) + G(zn , ym , xk )✳
❚õ ➤ã t❛ ❝ã

G(x, y, z) ≤ G(y, ym , ym ) + G(x, xk , xk ) + G(z, zn , zn ) + G(xk , ym , zn )✳
❙✉② r❛

G(x, y, z) − G(xk , ym , zn ) ≤ G(y, ym , ym ) + G(x, xk , xk ) + G(z, zn , zn )✳
▲❐♣ ❧✉❐♥ t➢➡♥❣ tù t❛ ❝ã

G(xk , ym , zn ) − G(x, y, z) ≤ G(x, x, xk ) + G(y, y, ym ) + G(z, z, zn )✳
▼➷t ❦❤➳❝ tõ ➤✐Ị✉ ❦✐Ư♥ ✭G5 ) tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺ t❛ ❝ã

G(x, x, xk ) ≤ G(x, xk , xk ) + G(xk , x, xk ) = 2G(x, xk , xk )✳ ❚➢➡♥❣ tù
G(y, y, ym ) ≤ 2G(y, ym , ym )✱ G(z, z, zn ) ≤ 2G(z, zn , zn )✳ ❚õ ➤ã s✉② r❛
G(xk , ym , zn ) − G(x, y, z) ≤ G(x, x, xk ) + G(y, y, ym ) + G(z, z, zn )
≤ 2{G(x, xk , xk )+G(y, ym , ym )+G(z, zn , zn )}✳
▲❐♣ ❧✉❐♥ tr➟♥ ❞➱♥ ➤Õ♥

|G(xk , ym , zn )−G(x, y, z)| ≤ 2{G(x, xk , xk )+G(y, ym , ym )+G(z, zn , zn )}✳
❱× ✈❐② G(xk , ym , zn )

→ G(x, y, z) ❦❤✐ k, m, n → ∞✳ ❱× t❤Õ ❤➭♠ (x, y, z) →

G(x, y, z) ❧✐➟♥ tô❝ ➤å♥❣ t❤ê✐ t❤❡♦ t❐♣ ❝➳❝ ❜✐Õ♥ x, y, z ✳
✶✳✷✳✶✶

➜Þ♥❤ ♥❣❤Ü❛ ✭❬✼❪✮✳ ●✐➯ sư

❧➭ sè ❞➢➡♥❣ ❝❤♦ tr➢í❝✳ ❚❐♣
♥Õ✉ ✈í✐ ❜✃t ❦ú

x∈X

(X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ε > 0

A ⊆ X

➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét

❝ã Ýt ♥❤✃t ♠ét ➤✐Ĩ♠

✶✸

ε✲❧➢í✐

❝đ❛

(X, G)

a ∈ A s❛♦ ❝❤♦ x ∈ BG (a, ε)✳


A

ế

t ữ tì

A

ợ ọ ột

ớ

ữ ủ

(X, G)✳
❉Ơ t❤✃② r➺♥❣ ♥Õ✉

A ❧➭ ♠ét ε✲❧➢í✐ t❤× X =

BG (a, ε)✳
a∈A

✶✳✷✳✶✷

➜Þ♥❤ ♥❣❤Ü❛ ✭❬✼❪✮✳ ❑❤➠♥❣ ❣✐❛♥

t♦➭♥ ❜Þ ❝❤➷♥ ♥Õ✉ ✈í✐ ♠ä✐

ε>0

G✲♠➟tr✐❝ (X, G) ➤➢ỵ❝ ❣ä✐ ❧➭ ❤♦➭♥

tå♥ t➵✐ ♠ét t❐♣

Aε ⊂ X

s❛♦ ❝❤♦

A

❧➭

ε✲❧➢í✐ ❤÷✉ ❤➵♥ ❝đ❛ (X, G)✳
✶✳✷✳✶✸

▼Ư♥❤ ➤Ị ✭❬✼❪✮✳

❈❤♦ ♠ét ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝ (X, G)✳

❑❤✐ ➤ã

❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣
✭✶✮

(X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ❤♦➭♥ t♦➭♥ ❜Þ ❝❤➷♥❀

✭✷✮

(X, dG ) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❤♦➭♥ t♦➭♥ ❜Þ ❝❤➷♥✳

❈❤ø♥❣ ♠✐♥❤✳ ✭✶✮⇒ ✭✷✮✳ ●✐➯ sö

(X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ Gtr
1
t ị ó ớ ỗ > 0 t❛ ❧✃② ε = ε > 0✳ ❱× (X, G) ❧➭ ❦❤➠♥❣
3

❣✐❛♥ G✲♠➟tr✐❝ ❤♦➭♥ t♦➭♥ ❜Þ ❝❤➷♥✱ ♥➟♥ tå♥ t➵✐ t❐♣ ❤÷✉ ❤➵♥ A ⊂ X s❛♦ ❝❤♦
X =

BG (a, ε )✳

◆❤ê ▼Ư♥❤ ➤Ị ✶✳✷✳✹ t❛ s✉② r❛

X =

BdG (a, ε)✳

a∈A

❱❐②

a∈A

(X, dG ) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❤♦➭♥ t♦➭♥ ❜Þ ❝❤➷♥✳
✭✷✮⇒ ✭✶✮✳ ❙✉② trù❝ t✐Õ♣ tõ ▼Ư♥❤ ➤Ị ✶✳✷✳✹ ✈➭ ✭✶✮✳

✶✳✷✳✶✹

◆❤❐♥ ①Ðt✳

●✐➯ sư

n

1, 2..., n✱

❦Ý ❤✐Ư✉

X =

(Xi , Gi )

Xi ✳

❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝

✈í✐

i =

❑❤✐ ➤ã ❝➳❝ ➤Þ♥❤ ♥❣❤Ü❛ ♠ét ❝➳❝❤ tù ♥❤✐➟♥

i=1
➤è✐ ✈í✐ ❝➳❝

G✲♠➟tr✐❝ tr➟♥ X

❝ã t❤Ĩ ❧➭

n

Gm (x, y, z) = max Gi (xi , yi , zi )

✈➭

1≤i≤n

✈í✐ ♠ä✐

Gs (x, y, z) =

Gi (xi , yi , zi )
i=1

x = (x1 , . . . , xn ), y = (y1 , . . . , yn ), z = (z1 , . . . , zn ) ∈ X ✳

♥❤✐➟♥ trõ ❦❤✐ t✃t ❝➯ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥

(Xi , Gi ) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤è✐

①ø♥❣✱ ò trờ ợ ò tì
í ụ sử



Gm ✈➭ Gs ❝ã t❤Ó ❦❤➠♥❣ ❧➭ G✲♠➟tr✐❝✳

X1 = {a, b} ✈➭ X2 = {1, 2}✳ ❳Ðt ➳♥❤ ①➵ G1 : X1 × X1 ×

X1 → R ❝❤♦ ❜ë✐ G1 (a, a, a) = G1 (b, b, b) = 0✱ G1 (a, a, b) = 1✱ G1 (a, b, b) =
2 ✈➭ G1 ❜➺♥❣ ♥❤❛✉ t➵✐ t✃t ❝➯ ❝➳❝ ❤♦➳♥ ✈Þ✱ ①Ðt ➳♥❤ ①➵ G2 : X2 ×X2 ×X2 → R
✶✹

❝❤♦ ❜ë✐

G2 (x, y, z) = max{|x − y|, |y − z|, |z − x|} ✈í✐ ♠ä✐ x, y, z ∈ X2 ✳
Gm (x, y, z) = max{G1 (x1 , y1 , z1 ), G2 (x2 , y2 , z2 )} ❦❤➠♥❣

❑❤✐ ➤ã ❝➠♥❣ t❤ø❝
❧➭

G✲♠➟tr✐❝ tr➟♥ X = X1 × X2 ✳
❚❤❐t ✈❐②✱ ❞Ơ ❞➭♥❣ t❤ư t❤✃② r➺♥❣

(G1 ), (G2 ), (G4 ), (G5 )✳
✈× ♥Õ✉ t❛ ❧✃②
♥❤➢♥❣

❚✉② ♥❤✐➟♥

Gm

Gm

t❤á❛ ♠➲♥ t✃t ❝➯ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥

❦❤➠♥❣ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥

x = (a, 1), y = (b, 1), z = (a, 2)✱

t❤× t❛ ❝ã

(G3 )✱

Gm (x, y, y) = 2✱

Gm (x, y, z) = 1✳

✶✳✷✳✶✺

➜Þ♥❤ ❧ý ✭❬✼❪✮✳

❈❤♦

n

i = 1, 2, ..., n

✈➭

X =

(Xi , Gi )

Xi ✳

G✲♠➟tr✐❝✱

❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥

❳Ðt ➳♥❤ ①➵

G

✈í✐

❝❤♦ ❜ë✐ ♠ét tr♦♥❣ ✷ ❝➠♥❣

i=1
t❤ø❝ s❛✉✳

n

G(x, y, z) = max Gi (xi , yi , zi )
1≤i≤n

❑❤✐ ➤ã✱

Gi (xi , yi , zi ).
i=1

i = 1, 2, ..., n✳

❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư r➺♥❣

i = 1, 2, ..., n✳

❦✐Ư♥

G(x, y, z) =

(X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤è✐ ①ø♥❣ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ (Xi , Gi )

❧➭ ➤è✐ ①ø♥❣✱ ✈í✐

✈í✐

❤♦➷❝

(Xi , Gi ) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤è✐ ①ø♥❣

❑❤✐ ➤ã ❞Ô ❞➭♥❣ ❦✐Ĩ♠ tr❛ r➺♥❣

G

t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉

(G1 ), (G2 ), (G4 )✱ (G5 ) ❝đ❛ G✲♠➟tr✐❝✳ ❈ß♥ G t❤á❛ ♠➲♥ ề ệ (G3 )

ợ s r từ ị ủ ệ ề
ợ ớ ỗ
ề ệ

j = 1, 2, ..., n ❞Ơ ❞➭♥❣ ❦✐Ĩ♠ tr❛ r➺♥❣ Gj

(G1 ), (G2 ), (G4 )✱ (G5 )

♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣

i = 1, 2, ..., n

Gj

❝đ❛

G✲♠➟tr✐❝✳

t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥

t❛ ❧✃② ♣❤➬♥ tư

pi ∈ X i ✳

t❤á❛ ♠➲♥

❱✃♥ ➤Ị ❝ß♥ ❧➵✐ ❧➭ t

(G3 )

rớ ết ớ ỗ

ờ ớ ỳ

xj , yj ∈ Xj

➤➷t

x = (p1 , . . . , pj−1 , xj , pj+1 , . . . , pn ) ∈ X

✈➭

y = (p1 , . . . , pj−1 , yj , pj+1 , . . . , pn ) ∈ X.
❑❤✐ ➤ã t❛ ❝ã

❱❐②

Gj

Gj (xj , yj , yj ) = G(x, y, y) = G(y, x, x) = Gj (yj , xj , xj )✳

❧➭

G✲♠➟tr✐❝✳

✶✺

t❛


✶✳✷✳✶✻

➜Þ♥❤ ❧ý ✭❬✼❪✮✳

❈❤♦

n

i = 1, 2, ..., n

X =

✈➭

(Xi , Gi )

Xi ✳
i=1

❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝✱

✈í✐

❑❤✐ ➤ã ❝➳❝ ❝➠♥❣ t❤ø❝ s❛✉ ①➳❝ ➤Þ♥❤ ❝➳❝

n

G✲♠➟tr✐❝ ➤è✐ ①ø♥❣ tr➟♥ X =

Xi ✳
i=1

(1) Gm
s (x, y, z) = max{Gs (dGi )(xi , yi , zi ) : i = 1, . . . , n}✱
(2)

Gss (x, y, z)

n

=

Gs (dGi )(xi , yi , zi )✱

i=1

(3) Gm
m (x, y, z) = max{Gm (dGi )(xi , yi , zi ) : i = 1, . . . , n}✱
n

(4) Gsm (x, y, z) =

Gm (dGi )(xi , yi , zi )✳
i=1

❈❤ø♥❣ ♠✐♥❤✳ ✭✶✮ ❉Ô ❞➭♥❣ ❦✐Ĩ♠ tr❛ r➺♥❣

Gm
s t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥

(G1 ), (G2 ), (G4 )✳ ❚❛ ❝❤Ø ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ ♥ã t❤á❛ ♠➲♥ (G3 ) ✈➭ (G5 ) tr♦♥❣
➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺✳ ❚❤❐② ✈❐②✱ ❧✃② ❜✃t ❦ú x

(z1 , z2 , ..., zn ) ∈ X

= (x1 , x2 , ..., xn ), y = (y1 , y2 , ..., yn ), z =

a = (a1 , a2 , ..., an ) ∈ X

✈➭

❚r➢í❝ ❤Õt t❛ ❦✐Ĩ♠ tr❛ ➤✐Ị✉ ❦✐Ư♥

(G3 )✳

❚õ ❱Ý ❞ơ s r ớ ỗ

i = 1, 2, ..., n t❛ ❝ã

2
Gs (dGi )(xi , yi , yi ) = dGi (xi , yi )
3
1
≤ (dGi (xi , yi ) + dGi (xi , zi ) + dGi (zi , yi ))
3
= Gs (dGi )(xi , yi , zi )✳
m
Gm
s (x, y, y) ≤ Gs (x, y, z)✳

➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦

◆❤ê ▼Ư♥❤ ➤Ị ✶✳✶✳✶✷ t❛ s✉② r❛

Gm
s ❧➭ ➤è✐ ①ø♥❣✳

(G5 ) ớ ỗ i = 1, 2, ..., n t ❝ã
1
Gs (dGi )(xi , yi , zi ) = (dGi (xi , yi ) + dGi (xi , zi ) + dGi (zi , yi ))
3
1
≤ (dGi (xi , ai ) + dGi (ai , yi )
3

+dGi (yi , zi ) + dGi (xi , ai ) + dGi (ai , zi ))

❇➞② ❣✐ê t❛ ❦✐Ĩ♠ tr❛ ➤✐Ị✉ ❦✐Ư♥

= Gs (dGi )(xi , ai , ai ) + Gs (dGi )(ai , yi , zi )✳
❉♦ ➤ã t❛ ❝ã

✶✳✷✳✶✼

m
m
Gm
s (x, y, z) ≤ Gs (x, a, a) + Gs (a, y, z)✳

➜Þ♥❤ ❧ý ✭❬✼❪✮✳

❈❤♦

n

i = 1, 2, ..., n ✈➭ X =

(Xi , Gi )

❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝✱

✈í✐

s
m
s
Xi ✳ ❑❤✐ ➤ã t✃t ❝➯ ❝➳❝ G✲♠➟tr✐❝ Gm
s , Gs , Gm , Gm

i=1
t t s ở ỗ
t➠♣➠

τ (Gi )✱ i = 1, 2, ..., n✳
✶✻

G✲♠➟tr✐❝ ➤ã ❧➭ t➠♣➠ tÝ❝❤ ❝ñ❛


❈❤ø♥❣ ♠✐♥❤✳ ❚õ ❝➳❝❤ ①➳❝ ➤Þ♥❤ ❝➳❝

s
m
s
G✲♠➟tr✐❝ Gm
s , Gs , Gm , Gm tr♦♥❣

➜Þ♥❤ ❧ý ✱ ❜➺♥❣ tÝ♥❤ t♦➳♥ trù❝ t✐Õ♣ t❛ ♥❤❐♥ ➤➢ỵ❝

m
s
s
m

Gm
s (x, y, z) ≤ Gm (x, y, z) ≤ Gm (x, y, z) ≤ 3Gs (x, y, z) ≤ 3nGs (x, y, z)
✈í✐ ♠ä✐

x, y, z ∈ X ✳

❚õ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❛ s✉② r❛ ❝➳❝ t➠♣➠ ❝➯♠ s✐♥❤ ❜ë✐ ❝➳❝
tr✐❝

G✲♠➟

s
m
s
Gm
s , Gs , Gm , Gm ❧➭ trï♥❣ ♥❤❛✉ ✈➭ trï♥❣ ✈í✐ t➠♣➠ tÝ❝❤ ❝ñ❛ ❝➳❝ t➠♣➠

τ (Gi )✱ i = 1, 2, ..., n✳

✶✼


❝❤➢➡♥❣ ✷
▼ét sè ❦Õt q✉➯ ➤✐Ó♠ ❜✃t ➤é♥❣
tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥

▼ét sè ❦❤➳✐ ♥✐Ư♠ ✈➭ ❦Õt q✉➯ ❝➡ ❜➯♥

✷✳✶

➜Þ♥❤ ♥❣❤Ü❛ ✭❬✼❪✮✳ ❈❤♦ (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❉➲② {xn }

✷✳✶✳✶

X

G✲♠➟tr✐❝ ➤➬② ➤đ

G✲❈➠s✐ ♥Õ✉ ✈í✐ sè ε > 0 ❝❤♦ tr➢í❝ tå♥ t➵✐ ♠ét sè N ∈ N✱

➤➢ỵ❝ ❣ä✐ ❧➭

s❛♦ ❝❤♦

⊂

G(xm , xn , xl ) < ε ✈í✐ ♠ä✐ n, m, l ≥ N ❀ ♥❣❤Ü❛ ❧➭ G(xm , xn , xl ) →

0✱ ❦❤✐ n, m, l → ∞✳
✷✳✶✳✷

▼Ư♥❤ ➤Ị ✭❬✼❪✮✳ ❈❤♦

(X, G)

❧➭ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝✳

❑❤✐ ➤ã ❝➳❝

♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣

(i)

❉➲②

{xn } ❧➭ G✲❈➠s✐❀

(ii) ❱í✐ sè ε > 0 ❝❤♦ tr➢í❝✱ tå♥ t➵✐ ♠ét sè N ∈ N s❛♦ ❝❤♦ G(xn , xm , xm )
< ε ✈í✐ ♠ä✐ n, m ≥ N✳
❈❤ø♥❣ ♠✐♥❤✳ ✭✐✮⇒✭✐✐✮✳ ❱×

tr➢í❝ tå♥ t➵✐ ♠ét sè

N✳

♠ä✐

✈í✐ ♠ä✐

m = l✳

(G3 )

❝đ❛

G✲♠➟tr✐❝

➜✐Ị✉ ♥➭② ❦Ð♦ t❤❡♦

t❛ ❝ã

G(xn , xm , xm ) <

G(xn , xm , xm ) < ε

✈í✐

n, m ≥ N✳

✭✐✐✮⇒✭✐✮✳ ●✐➯ sư
sè

N ∈ N s❛♦ ❝❤♦ G(xm , xn , xl ) < ε ✈í✐ ♠ä✐ n, m, l ≥

▼➷t ❦❤➳❝ ♥❤ê ➤✐Ị✉ ❦✐Ö♥

G(xm , xn , xl )

{xn } ❧➭ ❞➲② G✲❈➠s✐ ♥➟♥ ✈í✐ sè ε > 0 ❝❤♦

ε > 0 ❧➭ sè ❞➢➡♥❣ ❜Ð tï② ý ❝❤♦ tr➢í❝✳ ❑❤✐ ➤ã tå♥ t➵✐

N ∈ N s❛♦ ❝❤♦ G(xm , xn , xn ) < ε ✈í✐ ♠ä✐ n, m ≥ N✳

(G5 )

tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛

G✲♠➟tr✐❝

t❛ ❝ã

G(xm , xn , xl ) ≤ G(xm , xn , xn ) +

G(xn , xn , xl ) = G(xm , xn , xn ) + G(xl , xn , xn )
♥➭② ❦Ð♦ t❤❡♦

G(xm , xn , xl ) ≤ 2ε

◆❤ê ➤✐Ị✉ ❦✐Ư♥

✈í✐ ♠ä✐

✈í✐ ♠ä✐

n, m, l ≥ N✳

n, m, l ≥ N✳

❱× t❤Õ

➜✐Ị✉

{xn }

❧➭

G✲❈➠s✐✳

✷✳✶✳✸

▼Ư♥❤ ➤Ị ✭❬✼❪✮✳ ▼ä✐ ❞➲②

G✲❤é✐

tô tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝

(X, G) ❧➭ ❞➲② G✲❈➠s✐✳
❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư

♠✐♥❤

{xn }

❧➭ ❞➲②

G✲❤é✐

tơ tí✐

x ∈ X

t❛ ❝➬♥ ❝❤ø♥❣

{xn } ❧➭ ❞➲② G✲❈➠s✐✳ ❚❤❐t ✈❐② ✈× {xn } ❧➭ ❞➲② G✲❤é✐ tơ tí✐ x ♥➟♥ ✈í✐
✶✽

♠ä✐

ε > 0✱

tå♥ t➵✐

ε
G(xn , xm , x) < ✳
2

N ∈ N

❑❤✐ ➤ã✱ ✈í✐ ♠ä✐

G(xm , xm , xl ) <

s❛♦ ❝❤♦ ✈í✐ ♠ä✐

n, m, l > N

ε
✳
2

n, m ∈ N✱ n, m > N

t❛ ➤å♥❣ t❤ê✐ ❝ã

G(xn , xm , xm ) <

t❛ ❝ã

ε
2

✈➭

(G5 ) tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺✱ t❛ ❝ã

▼➷t ❦❤➳❝ ♥❤ê ➤✐Ị✉ ❦✐Ư♥

G(xn , xm , xl ) ≤ G(xn , xm , xm ) + G(xm , xm , xl )✳
ε ε
❱× t❤Õ t❛ ❝ã G(xn , xm , xl ) <
+ = ε✳
2 2
❱❐② {xn } ❧➭ ❞➲② G✲❈➠s✐✳
✷✳✶✳✹

➜Þ♥❤ ♥❣❤Ü❛ ✭❬✼❪✮✳ ❑❤➠♥❣ ❣✐❛♥

❣✐❛♥

G✲♠➟tr✐❝

tr♦♥❣

(X, G)✳

✷✳✶✳✺

➤➬② ➤đ ế ỗ

ị ý

XX

Gtr (X, G) ợ ọ ❧➭ ❦❤➠♥❣

(X, d)

G✲❈➠s✐

tr♦♥❣

❧➭ ❦❤➠♥❣ ❣✐❛♥

(X, G)

d✲♠➟tr✐❝

❧➭

G✲❤é✐

➤➬② ➤đ ✈➭

tơ

T :

❧➭ ♠ét ➳♥❤ ①➵ ✳
5

●✐➯ sö tå♥ t➵✐ ❤➺♥❣ sè

ai

❦❤➠♥❣ ➞♠✱

i = 1, . . . , 5 s❛♦ ❝❤♦

ai < 1
i=1

✈➭ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ s❛✉

d(T (x), T (y)) ≤ a1 d(x, y) + a2 d(x, T (x)) + a3 d(y, T (y))
+ a4 d(x, T (y))+a5 d(y, T (x))✱ ✈í✐ ♠ä✐ x, y ∈ X ✳ (1.4)
❑❤✐ ➤ã✱

✷✳✶✳✻

T

❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t

❍Ö q✉➯ ✭❬✼❪✮✳ ◆Õ✉ ♠ét ❞➲②

u ∈ X ❀ T u = u✳

G✲❈➠s✐

tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝

(X, G) ❝❤ø❛ ♠ét ❞➲② ❝♦♥ G✲❤é✐ tơ✱ t❤× ♥ã ❧➭ ♠ét ❞➲② G✲❤é✐ tơ✳
✷✳✶✳✼

▼Ư♥❤ ➤Ị ✭❬✼❪✮✳ ❑❤➠♥❣ ❣✐❛♥

❝❤Ø ❦❤✐

G✲♠➟tr✐❝ (X, G) ❧➭ G✲➤➬② ➤ñ ❦❤✐ ✈➭

(X, dG ) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ➤➬② ➤đ✳

➜Þ♥❤ ❧ý ✭❬✼❪✮✳ ●✐➯ sư (Xi , Gi ) ✈í✐ i = 1, 2..., n ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥
n
G✲♠➟tr✐❝ ✈➭ X =
Xi ✳ ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐ ❝➳❝❤ ❧✃② j, k ∈ {s, m} t❛ ❝ã
i=1
(X, Gjk ) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ (Xi , Gi ) ❧➭ ❦❤➠♥❣

✷✳✶✳✽

❣✐❛♥

G✲♠➟tr✐❝ ➤➬② ➤đ✱ ✈í✐ ♠ä✐ i = 1, 2, ..., n✳
✶✾


▼ét sè ❦Õt q✉➯ ➤✐Ó♠ ❜✃t ➤é♥❣ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥

✷✳✷

G✲♠➟tr✐❝ ➤➬② ➤đ
✷✳✷✳✶

➜Þ♥❤ ❧ý ✭❬✾❪✮✳ ❈❤♦

X→X

(X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ ✈➭ T :

❧➭ ♠ét ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ✈í✐ ♠ä✐

x, y ∈ X ✳

G(T (x), T (y), T (y))
≤ max{aG(x, y, y), b[G(x, T (x), T (y)) + 2G(y, T (y), T (y))]✱
b[G(x, T (y), T (y)) + G(y, T (y), T (y)) + G(y, T (x), T (x))]} (2.1)
1
tr♦♥❣ ➤ã 0 ≤ a < 1 ✈➭ 0 ≤ b < ✳ ❑❤✐ ➤ã T ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣
3
u ∈ X ✈➭ T ❧➭ G✲❧✐➟♥ tơ❝ t➵✐ u✳
❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư

T

t❤á❛ ♠➲♥ ✭✷✳✶✮✳ ❑❤✐ ➤ã ♠ä✐

x, y ∈ X ✱ t❛ ❝ã

G(T (x), T (y), T (y))
≤ max{aG(x, y, y), b[G(x, T (x), T (x)) + 2G(y, T (y), T (y))]✱
b[G(x, T (y), T (y)) + G(y, T (y), T (y)) + G(y, T (x), T (x))]} (2.2)
✈➭

G(T (y), T (x), T (x))
≤ max{aG(y, x, x), b[2G(x, T (x), T (x)) + G(y, T (y), T (y))]✱
b[G(x, T (y), T (y))+G(y, T (x), T (x))+G(x, T (x), T (x))]}✳ (2.3)
●✐➯ sö

(X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ➤è✐ ①ø♥❣✱ tõ ➜Þ♥❤ ♥❣❤Ü❛ ♠➟tr✐❝ (X, dG )✱

✭✶✳✶✮ ✈➭ ✭✷✳✷✮ t❛ ♥❤❐♥ ➤➢ỵ❝

dG (T (x), T (y)) = 2G(T (x), T (y), T (y))
a
1
≤ 2 max{ dG (x, y)✱ b[ [dG (x, T (x)) + dG (y, T (y))]✱
2
2
b
[dG (x, T (y)) + dG (y, T (y)) + G(y, T (x))]}✳
(2.4)
2

❚✉② ♥❤✐➟♥ ♥Õ✉ (X, G) ❦❤➠♥❣ ố ứ ợ ở ị ĩ
tr

(X, dG ) ✭✶✳✷✮✱ ✭✷✳✷✮ ✈➭ ✭✷✳✸✮ t❛ ❝ã

dG (T (x), T (y)) = G(T (x), T (y), T (y)) + G(T (y), T (x), T (x))
2a
2
4
≤ max{ dG (x, y), b[ dG (x, T (x)) + dG (y, T (y))]✱
3
3
3
2b
{[dG (x, T (y)) + dG (y, T (y)) + G(y, T (x))]}
3
2a
4
2
+ max{ dG (x, y), b[ dG (x, T (x)) + dG (y, T (y))]✱
3
3
3
2b
[dG (x, T (y)) + dG (y, T (x)) + G(x, T (x))]}✳ (2.5)
3
❑❤✐ ➤ã sư ❞ơ♥❣ ✭✷✳✶✮ t❛ ❝ã

✷✵

G(xn , xn+1 , xn+1 )
≤ max{aG(xn−1 , xn , xn )✱ b[G(xn−1 , xn , xn ) + 2G(xn , xn+1 , xn+1 )]✱
b[G(xn−1 , xn+1 , xn+1 ) + G(xn , xn+1 , xn+1 )]}✳
◆❤➢♥❣ ❞♦

(2.6)

(G5 ) tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺ t❛ ❝ã

G(xn−1 , xn+1 , xn+1 ) ≤ G(xn−1 , xn , xn ) + G(xn , xn+1 , xn+1 )✱ ✈❐②
b[G(xn−1 , xn+1 , xn+1 ) + G(xn , xn+1 , xn+1 )] ≤ b[G(xn−1 , xn , xn )
+2G(xn , xn+1 , xn+1 )]✳
❑❤✐ ➤ã ✭✷✳✻✮ trë t❤➭♥❤

G(xn , xn+1 , xn+1 ) ≤ max{aG(xn−1 , xn , xn ), b[G(xn−1 , xn , xn )
+2G(xn , xn+1 , xn+1 )]}✳

(2.7)

❱× ✈❐②✱ t❛ ❝ã ❤❛✐ tr➢ê♥❣ ❤ỵ♣
❚r➢ê♥❣ ❤ỵ♣ ✶✿ ◆Õ✉

max{aG(xn−1 , xn , xn ), b[G(xn−1 , xn , xn )+

+2G(xn , xn+1 , xn+1 )]} = b[G(xn−1 , xn , xn ) + 2G(xn , xn+1 , xn+1 )]✱ t❤× ✭✷✳✼✮
b
G(xn−1 , xn , xn )✳
trë t❤➭♥❤ G(xn , xn+1 , xn+1 ) ≤
1 − 2b

❚r➢ê♥❣ ❤ỵ♣ ✷✿ ◆Õ✉ max{aG(xn−1 , xn , xn ), b[G(xn−1 , xn , xn )+
+2G(xn , xn+1 , xn+1 )]} = aG(xn−1 , xn , xn )✱ t❤× ✭✷✳✼✮ trë t❤➭♥❤
G(xn , xn+1 , xn+1 ) ≤ aG(xn−1 , xn , xn )
ì ớ ỗ trờ ợ t❛ ❝ã G(xn , xn+1 , xn+1 )

≤ qG(xn−1 , xn , xn )
(2.8)

tr♦♥❣ ➤ã q ❂

b
}✱
max{a,
1 − 2b

❦❤✐

0 ≤ a < 1

✈➭

1
0 ≤ b < ✱
3

s✉② r❛

0 ≤ q < 1 ✈➭ sư ❞ơ♥❣ ♥❤✐Ị✉ ❧➬♥ ✭✷✳✽✮ t❛ ❝ã
G(xn , xn+1 , xn+1 ) ≤ q n G(x0 , x1 , x1 )✳
❱× t❤Õ✱ ✈í✐ ♠ä✐

n, m ∈ N❀ n < m

(2.9)

ụ t tứ ì ữ

t ✭✷✳✾✮ t❛ ❝ã

G(xn , xm , xm ) ≤ G(xn , xn+1 , xn+1 ) + G(xn+1 , xn+2 , xn+2 )+
+G(xn+2 , xn+3 , xn+3 )+ ✳✳✳ +G(xm−1 , xm , xm )

❉♦

≤ (q n + q n+1 + ... + q m−1 )G(x0 , x1 , x1 )
qn
≤
G(x0 , x1 , x1 )✳
1−q
0 ≤ q < 1 ♥➟♥ lim G(xn , xm , xm ) → 0✳ ❱× t❤Õ {xn }
n,m→∞

❈➠s✐✳ ◆❤ê tÝ♥❤ ➤➬② ➤ñ ❝ñ❛
➤Õ♥

(X, G) tå♥ t➵✐ u ∈ X

u tr♦♥❣ (X, G)✳ ●✐➯ sö T (u) = u✱ ❦❤✐ ➤ã
✷✶

s❛♦ ❝❤♦

❧➭ ♠ét

G✲

{xn } G✲❤é✐ tô


G(xn , T (u), T (u))
≤ max{aG(xn−1 , u, u), b[G(xn−1 , xn , xn )+ 2G(u, T (u), T (u))]✱
b[G(xn−1 , T (u), T (u)) + G(u, T (u), T (u)) + G(u, xn , xn )]}✳
▲✃② ❣✐í✐ ❤➵♥ ❦❤✐

n → ∞✱

✈➭ sư ❞ơ♥❣ tÝ♥❤ ❝❤✃t

G✲❧✐➟♥

tơ❝ t❤❡♦ ❝➳❝

❜✐Õ♥ ❝đ❛ ♥ã t❛ ❝ã

G(u, T (u), T (u)) ≤ 2bG(u, T (u), T (u))✱ t❛ ❣➷♣ ♠➞✉ t❤✉➱♥ ✈× 2b < 1✳
❉♦ ➤ã

u = T (u)✳

❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❞✉② ♥❤✃t ❝đ❛

❝❤♦

u✳ ❚❤❐t ✈❐②✱ ❣✐➯ sư ❝ã v ∈ X

s❛♦

T (v) = v ✳ ❑❤✐ ➤ã t❛ ❝ã
G(u, v, v) ≤ max{aG(u, v, v), b[G(u, u, u) + 2G(v, v, v)]✱
b[G(u, v, v)+G(v, u, u)]}✳ (2.10)

◆❤➢♥❣ ❞♦
✈× ✈❐②

(G5 ) tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺ t❛ ❝ã G(v, u, u) ≤ 2G(u, v, v)✱

bG(u, v, v) + bG(v, u, u) ≤ 3bG(u, v, v)✳ ❑❤✐ ➤ã ✭✷✳✶✵✮ trë t❤➭♥❤

G(u, v, v) ≤ max{aG(u, v, v), b[G(u, v, v) + G(v, u, u)]} ≤ cG(u, v, v)✱
1
s✉② r❛ c < 1✱ ♥➟♥
tr♦♥❣ ➤ã c = max{a, 3b}✱ ❦❤✐ a < 1 ✈➭ b <
3
G(u, v, v) = 0✳ ❉♦ ➤ã u = v ✳
➜Ĩ t❤✃② r➺♥❣

X

s❛♦ ❝❤♦

T

❧➭

G✲❧✐➟♥ tơ❝ t➵✐ u✱ t❛ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ ✈í✐ ♠ä✐ {yn } ⊆

lim yn = u✳ ❚❛ ❝ã

G(T (yn ), T (u), T (u)) ≤ max{aG(yn , u, yn )✱
b[G(u, T (u), T (u)) + 2G(yn , T (yn ), T (yn ))]✱
b[G(yn , T (u), T (u))+G(u, T (yn ), T (yn ))+G(yn , T (yn ), T (yn ))]}✳ (2.11)
◆❤➢♥❣ ❞♦

(G5 ) tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺ t❛ ❝ã

G(yn , T (yn ), T (yn )) ≤ G(yn , u, u) + G(u, T (yn ), T (yn ))✳
❱× ✈❐②

(2.12)

2G(yn , T (yn ), T (yn ))
≤ {G(yn , u, u) + G(u, T (yn ), T (yn )) + G(yn , T (yn ), T (yn ))}✳

❙✉② r❛ ✭✷✳✶✶✮ trë t❤➭♥❤

G(T (yn ), u, T (yn )) ≤ max{aG(yn , u, yn )✱

b[G(yn , u, u) + G(u, T (yn ), T (yn )) + G(yn , T (yn ), T (yn ))]}✳ (2.13)
▼ét ❧➬♥ ♥÷❛ tõ ✭✷✳✶✷✮ t❛ ❝ã

b[G(yn , u, u) + G(u, T (yn ), T (yn )) + G(yn , T (yn ), T (yn ))]

≤ 2bG(yn , u, u) + 2bG(u, T (yn ), T (yn ))✳
❱❐② ✭✷✳✶✸✮ trë t❤➭♥❤

G(T (yn ), u, T (yn ))
✷✷


≤ max{aG(yn , u, yn ), 2bG(yn , u, u)+2bG(u, T (yn ), T (yn ))}✳ (2.14)
❚õ ✭✷✳✶✹✮ t❛ ❝ã ❤❛✐ tr➢ê♥❣ ❤ỵ♣

G(u, T (yn ), T (yn )) ≤ aG(yn , u, yn )✱ ❤♦➷❝
2b
❚r➢ê♥❣ ❤ỵ♣ ✷✿ G(u, T (yn ), T (yn )) ≤
G(yn , u, u)✳
1 − 2b
r ỗ trờ ợ ớ n ∞✱ t❛ ➤➢ỵ❝ G(u, T (yn ), T (yn ))
❚r➢ê♥❣ ợ

0

ì từ ệ ề t ó

T (yn ) → u = T (u)✱

♥❣❤Ü❛ ❧➭

T

❧➭

G✲❧✐➟♥ tô❝ t➵✐ u ✳
✷✳✷✳✷

❱Ý ❞ô ✭❬✾❪✮✳ ❈❤♦

X = [0, 1] ✈➭ G✲♠➟tr✐❝ ①➳❝ ➤Þ♥❤ ❜ë✐

G(x, y, z) = max{|x − y|, |y − z|, |z − x|}✱ ✈í✐ ♠ä✐ x, y, z ∈ X
x
✱ ✈í✐ ♠ä✐ x ∈ X ✳ ❑❤✐
t❛ ①➳❝ ➤Þ♥❤ ➳♥❤ ①➵ T : X → X ❝❤♦ ❜ë✐ T x =
5
➤ã t❛ ❝ã
✭✐✮

(X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✳

1
xn = 1 − ✱ t❛ ❝ã G(xn , xm , xl ) =
n
1 1
1 1 1 1
max{|xn −xm |, |xm −xl |, |xl −xn |} = max{| − |, | − |, | − |} → 0
n m m l l n
❦❤✐ n, m, l → ∞✱ ❞♦ ➤ã {xn } ❧➭ G✲❈➠s✐✳
1 1
1
1
▼➷t ❦❤➳❝ G(1, xn , xm ) = max{| |, |
|, | − |} → 0 ❦❤✐ n, m, l →

n m n m
∞✱ ♥➟♥ ❞➲② {xn } ❧➭ G✲❤é✐ tơ ✈Ị 1 ∈ [0, 1] = X ✱ ❞♦ ➤ã ❞➲② {xn } ❧➭ G✲❤é✐
❚❤❐t ✈❐②✱ ①Ðt ❞➲②

tô tr♦♥❣
❱❐②
✭✐✐✮

{xn }

❝❤♦ ❜ë✐

X✳

(X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ✳
3
3
❈❤♦ a =
✈➭ b =
t❤× T t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✮ ❝đ❛ ➜Þ♥❤ ❧ý
4
10

✷✳✷✳✶✱ ♠➭ t❛ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ë tr➟♥✳
✭✐✐✐✮
❱×

✷✳✷✳✸
✈➭

T

❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣

T (0) =

0
= 0✳
5

❍Ö q✉➯ ✭❬✾❪✮✳ ❈❤♦

T :X→X

✈➭ ♠ä✐

x = 0✳

(X, G)

❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥

G✲

♠➟tr✐❝ ➤➬② ➤đ

❧➭ ♠ét ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ✈í✐ ♠ét sè

m∈N

x, y ∈ X ✳

G(T m (x), T m (y), T m (z))
≤ max{aG(x, y, y), b[G(x, T m (x), T m (x)) + 2G(y, T m (y), T m (y))]✱

✷✸


b[G(x, T m (y), T m (y))+G(y, T m (y), T m (y))+G(y, T m (x), T m (x))]} (2.15)
1
tr♦♥❣ ➤ã 0 ≤ a < 1 ✈➭ 0 ≤ b < ✳ ❑❤✐ ➤ã T ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣
3
u ∈ X ✱ ✈➭ T m ❧➭ G✲❧✐➟♥ tô❝ t➵✐ u✳
❈❤ø♥❣ ♠✐♥❤✳ ❚õ ➜Þ♥❤ ❧ý ✷✳✷✳✶ t❛ ❝ã

✈➭

Tm

❧➭ ♠ét

G✲❧✐➟♥

tơ❝ t➵✐

u✳

❉♦

T m ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ u ∈ X ✱

T m (u) = u

♥➟♥

T (u) = T (T m (u)) =

T m+1 (u) = T m (T (u))✱ ✈× ✈❐② T (u) ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦❤➳❝ ❝đ❛ T m ✈➭
❞♦ ➤ã

T (u) = u ✳

✷✳✷✳✹

➜Þ♥❤ ❧ý ✭❬✾❪✮✳

T : X → X

❈❤♦

(X, G)

❧➭ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝

➤➬② ➤ñ ✈➭

❧➭ ♠ét ➳♥❤ ①➵ t❤á❛ ♠➲♥ ♥❤÷♥❣ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ✈í✐ ♠ä✐

x, y, z ∈ X ✳
G(T (x), T (y), T (z)) ≤ max{aG(x, y, z), b[G(x, T (x), T (x))+
+G(y, T (y), T (y)) + G(z, T (z), T (z))]✱
b[G(x, T (y), T (y)) + G(y, T (z), T (z)) + G(z, T (x), T (x))]}
tr♦♥❣ ➤ã

0 ≤ a < 1 ✈➭ 0 ≤ b < 1✳ ❑❤✐ ➤ã T

u ∈ X ✱ ✈➭ T

❧➭

(2.16)

❝ã ❞✉② ♥❤✃t ➤✐Ĩ♠ ❜✃t ➤é♥❣

G✲❧✐➟♥ tơ❝ t➵✐ u✳

❈❤ø♥❣ ♠✐♥❤✳ ❚❤❛②

z = y ✈➭♦ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✻✮ t❛ ➤➢ỵ❝

G(T (x), T (y), T (y)) ≤ max{aG(x, y, y), b[G(x, T (x), T (y))+
+G(y, T (y), T (y)) + G(y, T (y), T (y))]✱
b[G(x, T (y), T (y)) + G(y, T (y), T (y)) + G(y, T (x), T (x))]}
= max{aG(x, y, y), b[G(x, T (x), T (y)) + 2G(y, T (y), T (y))]✱
b[G(x, T (y), T (y)) + G(y, T (y), T (y)) + G(y, T (x), T (x))]}✳
➜➞② ❝❤Ý♥❤ ❧➭ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✮ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✶ ♠➭ t❛ ➤➲ ❝❤ø♥❣ ♠✐♥❤
ë tr➟♥✳

✷✳✷✳✺

❱Ý ❞ô ✭❬✾❪✮✳ ❈❤♦

X = [0, 1] ✈➭ G✲♠➟tr✐❝ ①➳❝ ➤Þ♥❤ ❜ë✐

G(x, y, z) = |x − y| + |y − z| + |z − x|✱ ✈í✐ ♠ä✐ x, y, z ∈ X ✳
x
✱ ✈í✐ ♠ä✐ x ∈ X ✳ ❑❤✐
❚❛ ①➳❝ ➤Þ♥❤ ➳♥❤ ①➵ T : X → X ❝❤♦ ❜ë✐ T x =
7
➤ã

(i) (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✳
4
32
(ii) ❈❤♦ a = ✈➭ b =
t❤× T t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ (2.16) ❝đ❛ ➤Þ♥❤
5
100
❧ý ✷✳✷✳✹✱ ♠➭ t❛ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ë tr➟♥✳

✷✹


(iii T

ó ột ể ố ị

x = 0

0
ì T (0) =
= 0✳
7
✷✳✷✳✻

❍Ö q✉➯ ✭❬✾❪✮✳

T :X→X

❈❤♦

(X, G)

❧➭ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝

❧➭ ♠ét ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ✈í✐ ♠ä✐

➤➬② ➤đ ✈➭

x, y ∈ X ✳

G(T (x), T (y), T (y))
≤ max{aG(x, y, y), b[G(x, T (x), T (x))+G(y, T (y), T (y))]✱
b[G(x, T (y), T (y)) + G(y, T (x), T (x))]}
(2.17)
1

tr♦♥❣ ➤ã 0 ≤ a < 1 ✈➭ 0 ≤ b < ✳ ❑❤✐ ➤ã T ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣
3
u✱ ✈➭ T ❧➭ G✲❧✐➟♥ tô❝ t➵✐ u✳
❈❤ø♥❣ ♠✐♥❤✳ ❱í✐ ♠ä✐ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤×Ị✉ ❦✐Ư♥ ✭✷✳✶✼✮✱ sÏ tỏ

ìề ệ ủ ị ý ì ❝❤ø♥❣ ♠✐♥❤ ➤➲ ❝❤♦ tõ ➜Þ♥❤ ❧ý
✷✳✷✳✶✳

✷✳✷✳✼

➜Þ♥❤ ❧ý ✭❬✾❪✮✳

T : X → X

❈❤♦

(X, G)

❧➭ ❦❤➠♥❣ ❣✐❛♥

G✲♠➟tr✐❝

➤➬② ➤ñ ✈➭

❧➭ ♠ét ➳♥❤ ①➵ t❤á❛ ♠➲♥ ♥❤÷♥❣ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ✈í✐ ♠ä✐

x, y ∈ X ✳
G(T (x), T (y), T (y)) ≤ k max{[G(x, T (x), T (x)) + 2G(y, T (y), T (y))]✱
[G(x, T (y), T (y)) + G(y, T (y), T (y)) + G(y, T (x), T (x))]✱
[G(y, y, T (x)) + G(y, y, T (y)) + G(x, x, T (y))]}

(2.18)
1
tr♦♥❣ ➤ã k ∈ [0, )✳ ❑❤✐ ➤ã T ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣ u ∈ X ✱ ✈➭ T
4
❧➭ G✲❧✐➟♥ tơ❝ t➵✐ u✳
❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư

T

t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✽✮✱ ✈í✐ ♠ä✐

x, y ∈ X ✱

t❛ ❝ã

G(T (x), T (y), T (y)) ≤ k max{[G(x, T (x), T (x)) + 2G(y, T (y), T (y))]✱
[G(x, T (y), T (y)) + G(y, T (y), T (y)) + G(y, T (x), T (x))]✱
[G(y, y, T (x)) + G(y, y, T (y)) + G(x, x, T (y))]}
✈➭

(2.19)

G(T (y), T (x), T (x)) ≤ k max{[G(y, T (y), T (y)) + 2G(x, T (x), T (x))]✱
[G(y, T (x), T (x)) + G(x, T (x), T (x)) + G(x, T (y), T (y))]✱
[G(x, x, T (y)) + G(x, x, T (x)) + G(y, y, T (x))]}✳

●✐➯ sư

(X, G)

❧➭ ➤è✐ ①ø♥❣✱ tõ ➜Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ♠➟tr✐❝

✭✷✳✶✾✮ t❛ ♥❤❐♥ ➤➢ỵ❝

✷✺

(X, dG )✱

(2.20)
✭✶✳✶✮ ✈➭