▼ô❝ ▲ô❝ ❚r❛♥❣ ▼ô❝ ❧ô❝ ✶ ▲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 ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❧➭ ♠ét tr♦♥❣ ♥❤÷♥❣ ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ q✉❛♥ trä♥❣✱ ❜ë✐ ♥ã ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ 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 ➤➲ ❝❤Ø r ữ ề ợ ý ề trú t➠♣➠ ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ 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 ❝ô❝ ❧✉❐♥ ✈➝♥ ❣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✐ tô✱ ❞➲② G✲❈➠s✐ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣✳ ❣✐❛♥ ❈❤➢➡♥❣ ✷✳ ▼ét sè ❦Õt q✉➯ ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲ ♠➟tr✐❝ ➤➬② ➤ñ✳ ❚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 ➤ñ✳ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✳ ❚➳❝ ❣✐➯ ①✐♥ ❜➭② tá ❧ß♥❣ ❜✐Õt ➡♥ s➞✉ s➽❝ ➤Õ♥ t❤➬②✳ ◆❤➞♥ ❞Þ♣ ♥➭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 ♥❤✐Ò✉ ❝è ❣➽♥❣✱ s♦♥❣ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ữ tế sót rt ợ ữ ý ❦✐Õ♥ ➤ã♥❣ ❣ã♣ ❝ñ❛ q✉ý ❚❤➬②✱ ❈➠ ❝ï♥❣ ❜➵♥ ➤ä❝ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥ ❤➡♥✳ ❱✐♥❤✱ t❤➳♥❣ ỗ ì ột ❧í♣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s rộ ị ĩ t ợ ❣ä✐ ❧➭ ✭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) = ♥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) = ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ x = y = z ❀ ✭D2 ✮ D(x, y, z) = D(k{x, y, z})✱ x, y, z 2✲♠➟tr✐❝ (X, d) ❤❛② X ✳ ➜Þ♥❤ ♥❣❤Ü❛ ✭❬✽❪✮✳ ❈❤♦ t❐♣ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➢ỵ❝ ❣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 ❞ô✳ (X, D) ❤❛② X ✳ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d)✳ ❳Ðt ➳♥❤ ①➵ D : X ×X × X → R+ ❝❤♦ ❜ë✐ D(x, y, z) = {d(x, y) + d(y, z) + d(z, x)}✱ ✈í✐ ♠ä✐ x, y, z ∈ X ✳ ✹ ❑❤✐ ➤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❝ ❦❤➠♥❣ {d(x, y) + d(y, z) + ➞♠ ✈í✐ ♠ä✐ x, y, z ∈ X ✳ ❉♦ ➤ã t❛ ❝ã G(x, y, z) = d(z, x)} ≥ ✈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 D(x, y, z) = {d(x, y) + d(y, z) + d(z, x)} ≤ {d(x, a) + d(y, a) + d(y, a) + d(a, z) + d(x, a) + d(z, a)} 1 ≤ {d(x, y)+d(y, a)+d(a, x)}+ {d(y, a) + d(a, z) + d(z, y)} 3 + {d(x, a) + d(a, z) + d(x, z)} = 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) ➤Ị✉ ❧➭ ❝➳❝ 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)} ≥ ✈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 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 = φ✳ ➳♥❤ ①➵ G : X × X × X → R+ G✲♠➟tr✐❝✱ ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿ ✭G1 ✮ G(x, y, z) = ♥Õ✉ x = y = z ❀ ✭G2 ✮ G(x, x, y) > ✈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❀ ✈í✐ ♠ä✐ x, y, z, a X t tứ ì ữ t X ❝ï♥❣ ✈í✐ G✲♠➟tr✐❝ G tr➟♥ X ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ❚❐♣ ✈➭ ❦ý ❤✐Ö✉ ❧➭ ✶✳✶✳✻ ❱Ý ❞ô✳ ➳♥❤ 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❛ ❦✐Ó♠ tr❛ G✲♠➟tr✐❝ ♥❤➢ s❛✉✿ ❧➭ ♠➟tr✐❝✱ ♥➟♥ ❦❤➠♥❣ ➞♠ ✈í✐ ♠ä✐ d(x, y), d(y, z), d(x, z)✱ ➤Ị✉ ❧➭ ❝➳❝ 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➵ Gs (x, y, z) = {d(x, y) + d(y, z) + d(x, z)} ✈í✐ ♠ä✐ x, y, z ∈ X ❑❤✐ ➤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 Gs (x, y, z) = {d(x, y) + d(y, z) + d(x, z)} ≤ [d(x, a) + d(a, y) + d(y, z) + d(x, a) + d(a, z) + d(a, a)] 1 = [d(x, a) + d(x, a) + d(a, a)] + [d(a, y) + d(y, z) + d(a, z)] 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✉ ✭✐✐✮ ●✐➯ sö (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❑❤✐ ➤ã t❛ ❝ã G(x, y, z) = 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)❀ ✭✈✮ G(x, y, z) ≤ {G(x, y, a) + G(x, a, z) + G(a, y, z)}❀ ✭✈✐✮ G(x, y, z) ≤ {G(x, a, a) + G(y, a, a) + G(z, a, a)}✳ ✭✐✈✮ ❈❤ø♥❣ ♠✐♥❤✳ ▲✃② ❜✃t ❦ú ✭✐✮ ❙✉② tõ ➤Þ♥❤ ♥❣❤Ü❛ ❝ñ❛ 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❝ G(x, y, z) ≤ {G(x, y, a) + G(x, a, z) + G(a, y, z)}✳ ✭✈✐✮ ❇➺♥❣ ❝➳❝❤ ➳♣ ❞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 ✭❬✼❪✮✳ ●✐➯ sö 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 ✳ + G(x, y, z) ❈❤ø♥❣ ♠✐♥❤✳ ❛✮ ❚❤❐t ✈❐②✱ t❛ ❦✐Ó♠ tr❛ Gi t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ❛✮ G✲♠➟tr✐❝ ♥❤➢ s❛✉✿ ✭G1 ✮ ❉Ô t❤✃② ❦❤✐ Gi (x, y, z) = ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ G(x, y, z) = 0✱ ❦❤✐ ✈➭ ❝❤Ø x = y = z✳ ❚➢➡♥❣ tù ❞Ô 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 ❞ô tr➟♥ t❛ s✉② r ế tr tì ó tể tr ị ✈➭ X ❝➳❝ (X, d) ❧➭ ❦❤➠♥❣ G✲♠➟tr✐❝ Gs ✈➭ Gm ➤Ó (X, Gs ) (X, Gm ) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ◆❣➢ỵ❝ ❧➵✐✱ ♥Õ✉ (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ t❤× ❝➠♥❣ 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), G(x, y, z) ≤ Gm (dG )(x, y, z) ≤ 2G(x, y, z), ✈➭ dGs (d) (x, y) = d(x, y), dGm (d) (x, y) = 2d(x, y) ❜✮ ◆Õ✉ (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ Gtr ố ứ tì tr ợ dG (x, y) = 2G(x, y, y) ✈í✐ ♠ä✐ x, y ∈ X ✳ (X, G) ❦❤➠♥❣ ➤è✐ ①ø♥❣✱ t❤× ♥❤ê ❝➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ G✲♠➟tr✐❝ t❛ ❝ã G(x, y, y) ≤ dG (x, y) ≤ 3G(x, y, y) ✈í✐ ♠ä✐ ❚✉② ♥❤✐➟♥ ♥Õ✉ ❦❤➠♥❣ ❣✐❛♥ x, y ∈ X ✳ ✶✳✶✳✶✷ (1.2) ▼Ư♥❤ ➤Ị ✭❬✼❪✮✳ ❈❤♦ (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♥❣ ♠✐♥❤✳ ✭✶✮⇒ ✭✷✮✳ ề ệ G3 tr ị ĩ ì (G, X) ❧➭ ➤è✐ ①ø♥❣ t❛ s✉② r❛ ✈í✐ ❜✃t ❦ú a = x t❛ G(x, y, y) ≤ G(x, y, a)✳ ✭✷✮⇒ ✭✸✮✳ ◆❤ê ▼Ư♥❤ ➤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) ❍♦➳♥ ổ trò ủ t Gtr ị ĩ ✳ X, r > 0✳ G✲❤×♥❤ ✭❬✼❪✮ ●✐➯ sư (X, G) a✱ ❝➬✉ ✈í✐ t➞♠ t➵✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❜➳♥ ❦Ý♥❤ r G✲♠➟tr✐❝✱ a ∈ ❦ý ❤✐Ö✉ ❧➭ 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 ✭❬✼❪✮✳ ●✐➯ sư (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 > s❛♦ ❝❤♦ BG (y, δ) ⊆ BG (x0 , r)✳ ❈❤ø♥❣ ♠✐♥❤✳ ✭✐✮ ●✐➯ sư ➤✐Ị✉ ❦✐Ư♥ (G3 ) ❝ñ❛ 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 > ➤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} ❝❤➢➡♥❣ ✷ ▼ét sè ❦Õt q✉➯ ➤✐Ó♠ ❜✃t ➤é♥❣ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ▼ét sè ❦❤➳✐ ♥✐Ö♠ ✈➭ ❦Õt q✉➯ ❝➡ ❜➯♥ ✷✳✶ ➜Þ♥❤ ♥❣❤Ü❛ ✭❬✼❪✮✳ ❈❤♦ (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❉➲② {xn } ✷✳✶✳✶ X G✲♠➟tr✐❝ ➤➬② ➤đ G✲❈➠s✐ ♥Õ✉ ✈í✐ sè ε > ❝❤♦ 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 ❝➳❝ ♠Ö♥❤ ➤Ò s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣ (i) ❉➲② {xn } ❧➭ G✲❈➠s✐❀ (ii) ❱í✐ sè ε > ❝❤♦ 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✳ ✭✐✐✮⇒✭✐✮✳ ●✐➯ sö sè N ∈ N s❛♦ ❝❤♦ G(xm , xn , xl ) < ε ✈í✐ ♠ä✐ n, m, l ≥ ▼➷t ❦❤➳❝ ♥❤ê ➤✐Ị✉ ❦✐Ư♥ G(xm , xn , xl ) {xn } ❧➭ ❞➲② G✲❈➠s✐ ♥➟♥ ✈í✐ sè ε > ❝❤♦ ε > ❧➭ 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♥❣ ♠✐♥❤✳ ●✐➯ sö ♠✐♥❤ {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) < ✳ N ∈ N ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐ G(xm , xm , xl ) < s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n, m, l > N ε ✳ n, m ∈ N✱ n, m > N t❛ ➤å♥❣ t❤ê✐ ❝ã G(xn , xm , xm ) < t❛ ❝ã ε ✈➭ (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 {xn } ❧➭ ❞➲② G✲❈➠s✐✳ ✷✳✶✳✹ ➜Þ♥❤ ♥❣❤Ü❛ ✭❬✼❪✮✳ ❑❤➠♥❣ ❣✐❛♥ ❣✐❛♥ G✲♠➟tr✐❝ tr♦♥❣ (X, G) ủ ế ỗ ị ý ✭❬✺❪✮✳ ❈❤♦ X→X G✲♠➟tr✐❝ (X, G) ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ (X, d) G✲❈➠s✐ tr♦♥❣ ❧➭ ❦❤➠♥❣ ❣✐❛♥ (X, G) d✲♠➟tr✐❝ ❧➭ G✲❤é✐ ➤➬② ➤đ ✈➭ tơ T : ❧➭ ♠ét ➳♥❤ ①➵ ✳ ●✐➯ sö tå♥ t➵✐ ❤➺♥❣ sè ❦❤➠♥❣ ➞♠✱ i = 1, , s❛♦ ❝❤♦ < 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 ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ❍Ư 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, , 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) tr♦♥❣ ➤ã ≤ a < ✈➭ ≤ b < ✳ ❑❤✐ ➤ã T ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣ u ∈ X ✈➭ T ❧➭ G✲❧✐➟♥ tô❝ t➵✐ u✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö 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 ≤ max{ dG (x, y)✱ b[ [dG (x, T (x)) + dG (y, T (y))]✱ 2 b [dG (x, T (y)) + dG (y, T (y)) + G(y, T (x))]}✳ (2.4) ❚✉② ♥❤✐➟♥ ♥Õ✉ (X, G) ❦❤➠♥❣ ố ứ ợ 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 ≤ max{ dG (x, y), b[ dG (x, T (x)) + dG (y, T (y))]✱ 3 2b {[dG (x, T (y)) + dG (y, T (y)) + G(y, T (x))]} 2a + max{ dG (x, y), b[ dG (x, T (x)) + dG (y, T (y))]✱ 3 2b [dG (x, T (y)) + dG (y, T (x)) + G(x, T (x))]}✳ (2.5) ❑❤✐ ➤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 ) ≤ − 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(xn1 , xn , xn ) ì ỗ tr➢ê♥❣ ❤ỵ♣ t❛ ❝ã G(xn , xn+1 , xn+1 ) ≤ qG(xn−1 , xn , xn ) (2.8) tr♦♥❣ ➤ã q ❂ b }✱ max{a, − 2b ❦❤✐ ≤ a < ✈➭ ≤ b < ✱ s✉② r❛ ≤ q < ✈➭ 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) ➳♣ ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ❤×♥❤ ❝❤÷ ♥❤❐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 ≤ q < ♥➟♥ 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 ❝ñ❛ ❝❤♦ u✳ ❚❤❐t ✈❐②✱ ❣✐➯ sö ❝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)✱ s✉② r❛ c < 1✱ ♥➟♥ tr♦♥❣ ➤ã c = max{a, 3b}✱ ❦❤✐ a < ✈➭ b < G(u, v, v) = 0✳ ❉♦ ➤ã u = v ✳ ➜Ó 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)✳ − 2b ❚r♦♥❣ ỗ trờ ợ n t❛ ➤➢ỵ❝ G(u, T (yn ), T (yn )) ❚r➢ê♥❣ ợ ì từ ệ ề t❛ ❝ã 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 = ➤ã t❛ ❝ã ✭✐✮ (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✳ xn = − ✱ t❛ ❝ã G(xn , xm , xl ) = n 1 1 1 max{|xn −xm |, |xm −xl |, |xl −xn |} = max{| − |, | − |, | − |} → n m m l l n ❦❤✐ n, m, l → ∞✱ ❞♦ ➤ã {xn } ❧➭ G✲❈➠s✐✳ 1 1 ▼➷t ❦❤➳❝ G(1, xn , xm ) = max{| |, | |, | − |} → ❦❤✐ n, m, l → n m n m ∞✱ ♥➟♥ ❞➲② {xn } ❧➭ G✲❤é✐ tơ ✈Ị ∈ [0, 1] = X ✱ ❞♦ ➤ã ❞➲② {xn } ❧➭ G✲❤é✐ ❚❤❐t ✈❐②✱ ①Ðt ❞➲② tô tr♦♥❣ ❱❐② ✭✐✐✮ {xn } ❝❤♦ ❜ë✐ X✳ (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✳ 3 ❈❤♦ a = ✈➭ b = t❤× T t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✮ ❝đ❛ ➜Þ♥❤ ❧ý 10 ✷✳✷✳✶✱ ♠➭ t❛ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ë tr➟♥✳ ✭✐✐✐✮ ❱× ✷✳✷✳✸ ✈➭ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ T (0) = = 0✳ ❍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) tr♦♥❣ ➤ã ≤ a < ✈➭ ≤ b < ✳ ❑❤✐ ➤ã T ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣ u ∈ X ✱ ✈➭ T m ❧➭ G✲❧✐➟♥ tơ❝ t➵✐ u✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚õ ➜Þ♥❤ ❧ý ✷✳✷✳✶ t❛ ❝ã ✈➭ Tm ❧➭ ♠ét G✲❧✐➟♥ tô❝ 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 ≤ a < ✈➭ ≤ 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♥❤ ❧➭ ➤✐Ò✉ ❦✐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 = ➤ã (i) (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✳ 32 (ii) ❈❤♦ a = ✈➭ b = t❤× T t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ (2.16) ❝đ❛ ➤Þ♥❤ 100 ❧ý ✷✳✷✳✹✱ ♠➭ t❛ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ë tr➟♥✳ ✷✹ (iii✮ T ❝ã ♠ét ➤✐Ó♠ ố ị x = 0 ì T (0) = = 0✳ ✷✳✷✳✻ ❍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) tr♦♥❣ ➤ã ≤ a < ✈➭ ≤ b < ✳ ❑❤✐ ➤ã T ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣ u✱ ✈➭ T ❧➭ G✲❧✐➟♥ tô❝ t➵✐ u✳ ❈❤ø♥❣ ♠✐♥❤✳ ❱í✐ ♠ä✐ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤×Ị✉ ❦✐Ư♥ ✭✷✳✶✼✮✱ sÏ tỏ ìề ệ ủ ị ý ì ✈❐② ❝❤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) tr♦♥❣ ➤ã k ∈ [0, )✳ ❑❤✐ ➤ã T ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣ u ∈ X ✱ ✈➭ T ❧➭ 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))]}✳ ●✐➯ sö (X, G) ❧➭ ➤è✐ ①ø♥❣✱ tõ ➜Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ♠➟tr✐❝ ✭✷✳✶✾✮ t❛ ♥❤❐♥ ➤➢ỵ❝ ✷✺ (X, dG )✱ (2.20) ✭✶✳✶✮ ✈➭ dG (T x, T y) = 2G(T x, T y, T y) ≤ 2k max{[ dG (x, T x) + dG (y, T y)]✱ [dG (x, T y) + dG (y, T x) + dG (y, T y)]}✳ (2.21) ❚✉② ♥❤✐➟♥✱ ♥Õ✉ (X, G) ❧➭ ❦❤➠♥❣ ➤è✐ ①ø♥❣ tõ ➤ã ❞♦ ➜Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ♠➟tr✐❝ (X, dG )✱ ✭✶✳✷✮✱ ✭✷✳✶✾✮ ✈➭ ✭✷✳✷✵✮ t❛ ♥❤❐♥ ➤➢ỵ❝ dG (T x, T y) = G(T x, T y, T y)+G(T y, T x, T x) ≤ k max{[ dG (x, T x)+ + dG (y, T y)]✱ [dG (x, T y) + dG (y, T x) + dG (y, T y)]} 3 2 +k max{[ dG (y, T y)+ dG (x, T x)]✱ [dG (x, T y)+dG (y, T x)+dG (x, T x)]}✳ 3 (2.22) ➜Ĩ tr×♥❤ ❜➭② T ❧➭ G✲❧✐➟♥ tơ❝ t➵✐ u t❛ ❝❤♦ ❞➲② {yn } ⊆ X ✈➭ ❝❤Ø r❛ r➺♥❣ lim yn = u✱ ❦❤✐ ➤ã tõ ✭✷✳✶✽✮ t❛ s✉② r❛ G(T (yn ), T (u), T (yn )) ≤ k max{[2G(yn , T (yn ), T (yn )) + G(u, T (u), T (u))]✱ [G(yn , T (u), T (u))+G(u, T (yn ), T (yn ))+G(yn , T (yn ), T (yn ))]✱ [G(yn , yn , T (u)) + G(yn , yn , T (yn )) + G(u, u, T (yn ))]}✳ (2.23) ◆❤➢♥❣ ❞♦ (G5 ) tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺ t❛ ❝ã G(yn , T (yn ), T (yn )) ≤ G(yn , u, u) + G(u, T (yn ), T (yn ))✱ (2.24) ❞♦ ➤ã tõ ✭✷✳✷✹✮ t❛ ♥❤❐♥ ➤➢ỵ❝ 2G(yn , T (yn ), T (yn )) ≤ G(yn , u, u) + G(u, T (yn ), T (yn )) + G(yn , T (yn ), T (yn ))✳ ❑❤✐ ➤ã✱ (2.23) trë t❤➭♥❤ G(T (yn ), u, T (yn )) ≤ k max{[G(yn , u, u)+G(u, T (yn ), T (yn ))+G(yn , T (yn ), T (yn ))]✱ [G(yn , yn , u) + G(yn , yn , T (yn )) + G(u, u, T (yn ))]}✳ (2.25) ❑❤✐ ➤ã tõ ✭✷✳✷✺✮ t❛ ❝ã ❤❛✐ tr➢ê♥❣ ❤ỵ♣ ❚r➢ê♥❣ ❤ỵ♣ ✶✿ G(T (yn ), u, T (yn )) ≤ k{G(yn , u, u)+G(u, T (yn ), T (yn ))+G(yn , T (yn ), T (yn ))}✱ (2.26) trë ❧➵✐ ✭✷✳✷✹✮ t❛ ❝ã G(u, T (yn ), T (yn )) + G(yn , T (u).T (u)) + G(yn , T (yn ), T (yn )) ≤ 2G(yn , u, u) + 2G(u, T (yn ), T (yn ))✳ 2k ❉♦ ➤ã ✭✷✳✷✻✮ trë t❤➭♥❤ G(u, T (yn ), T (yn )) ≤ G(yn , u, u) − 2k ✷✻ (2.27) (2.28) ❚r➢ê♥❣ ❤ỵ♣ ✷✿ G(T (yn ), u, T (yn )) ≤ k{G(yn , yn , u) + G(yn , yn , T (yn )) + G(u, u, T (yn ))}✱ (2.29) ♥❤➢♥❣ ❞♦ (G5 ) tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺ t❛ ❝ã G(T yn , yn , yn ) ≤ G(T (yn ), u, u) + G(u, yn , yn ) (2.30) ✈× ✈❐② ✭✷✳✷✾✮ trë t❤➭♥❤ G(T (yn ), u, T (yn )) ≤ k{2G(u, yn , yn ) + 2G(u, u, T (yn ))}✱ ❝ò♥❣ tõ (2.31) (G5 ) tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺ t❛ ❝ã G(u, yn , yn ) ≤ 2G(yn , u, u) ✈➭ (2.32) G(u, u, T (yn )) ≤ 2G(u, T (yn ), T (yn ))✳ (2.33) ❉♦ ➤ã✱ tõ ✭✷✳✷✷✮ ✈➭ ✭✷✳✸✸✮ t❤× ✭✷✳✸✶✮ trë t❤➭♥❤ G(T (yn ), u, T (yn )) ≤ k{4G(yn , u, u) + 4G(u, T (yn ), T (yn ))} (2.34) ♥❣❤Ü❛ ❧➭ t❛ ❝ã 4k G(yn , u, u)✳ − 4k ❚õ ✭✷✳✷✽✮ ✈➭ ✭✷✳✸✺✮ ❧✃② ❣✐í✐ ❤➵♥ ❦❤✐ n → ∞ t❛ ➤➢ỵ❝ G(T (yn ), u, T (yn )) ≤ (2.35) G(T (yn ), u, T (yn )) → ✈➭ ✈× ✈❐②✱ tõ ▼Ư♥❤ ➤Ị ✶✳✷✳✾ t❛ ❝ã T (yn ) → u = T (u)✳ ❉♦ ➤ã ✷✳✷✳✽ T G✲❧✐➟♥ tô❝ t➵✐ u✳ ❍Ö q✉➯ ✭❬✾❪✮✳ T :X →X ♠ä✐ ❧➭ ❈❤♦ (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ❧➭ ♠ét ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ✈í✐ ➤➬② ➤đ ✈➭ m∈N ✈➭ ✈í✐ x, y ∈ X ✳ G(T m (x), T m (y), T m (y)) ≤ k max{[G(x, T m (x), T m (x)) + 2G(y, T m (y), T m (y))]✱ [G(x, T m (y), T m (y))+G(y, T m (y), T m (y))+G(y, T m (x), T m (x))]✱ [G(y, y, T m (x)) + G(y, y, T m (y)) + G(x, x, T m (y))]} (2.36) tr♦♥❣ ➤ã k ∈ [0, )✳ ❑❤✐ ➤ã ➳♥❤ ①➵ T ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣ u ∈ X m ✈➭ T ❧➭ G✲❧✐➟♥ tô❝ t➵✐ u✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư T t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✻✮✱ ✈í✐ ♠ä✐ t❛ ❝ã G(T m (x), T m (y), T m (y)) ≤ k max{[G(x, T m (x), T m (x)) + 2G(y, T m (y), T m (y))]✱ ✷✼ x, y ∈ X ✱ [G(x, T m (y), T m (y))+G(y, T m (y), T m (y))+G(y, T m (x), T m (x))]✱ [G(y, y, T m (x)) + G(y, y, T m (y)) + G(x, x, T m (y))]}✱ ✈➭ (2.37) G(T m (y), T m (x), T m (x)) ≤ k max{[G(y, T m (y), T m (y)) + 2G(x, T m (x), T m (x))]✱ [G(y, T m (x), T m (x))+G(x, T m (x), T m (x))+G(x, T m (y), T m (y))]✱ [G(x, x, T m (y)) + G(x, x, T m (x)) + G(y, y, T m (x))]}✳ (2.38) ●✐➯ sö (X, G) ố ứ ợ ị ĩ ủ ♠➟tr✐❝ (X, dG ) ✈➭ ✭✶✳✶✮ ✈➭ ✭✷✳✸✼✮ t❛ ♥❤❐♥ ➤➢ỵ❝ ❚✉② ♥❤✐➟♥✱ ♠➟tr✐❝ dG (T m x, T m y) = 2G(T m x, T m y, T m y) ≤ 2k max{[ dG (x, T m x) + dG (y, T m y)]✱ [dG (x, T m y) + dG (y, T m x) + dG (y, T m y)]}✳ ♥Õ✉ (X, G) ❧➭ ❦❤➠♥❣ ➤è✐ ①ø♥❣ tõ ➤ã ❞♦ ➜Þ♥❤ (2.39) ♥❣❤Ü❛ ❝đ❛ (X, dG )✱ ✭✶✳✷✮✱ ✭✷✳✸✼✮ ✈➭ ✭✷✳✸✽✮ t❛ ♥❤❐♥ ➤➢ỵ❝ dG (T m x, T m y) = G(T m x, T m y, T m y) + G(T m y, T m x, T m x) ≤ k max{[ dG (x, T m x) + dG (y, T m y)]✱ 3 [dG (x, T m y) + dG (y, T m x) + dG (y, T m y)]} +k max{[ dG (y, T m y) + dG (x, T m x)]✱ 3 [dG (x, T m y) + dG (y, T m x) + dG (x, T m x)]}✳ ➜Ĩ tr×♥❤ ❜➭② r➺♥❣ Tm ❧➭ (G)✲❧✐➟♥ tơ❝ t➵✐ u t❛ ❝❤♦ ❞➲② {yn } ⊆ X (2.40) ✈➭ ❝❤Ø r❛ lim yn = u✳ ❑❤✐ ➤ã tõ ✭✷✳✸✻✮ t❛ s✉② r❛ G(T m (yn ), T m (u), T m (yn )) ≤ k max{[2G(yn , T m (yn ), T m (yn )) + G(u, T m (u), T m (u))]✱ [G(yn , T m (u), T m (u)) + G(u, T m (yn ), T m (yn )) + G(yn , T m (yn ), T m (yn ))]✱ [G(yn , yn , T m (u)) + G(yn , yn , T m (yn )) + G(u, u, T m (yn ))]}✳ (2.41) ◆❤➢♥❣ ❞♦ (G5 ) tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺ t❛ ❝ã G(yn , T m (yn ), T m (yn )) ≤ G(yn , u, u) + G(u, T m (yn ), T m (yn ))✱ (2.42) ❞♦ ➤ã✱ tõ ✭✷✳✹✷✮ t❛ ♥❤❐♥ ➤➢ỵ❝ 2G(yn , T m (yn ), T m (yn )) ≤ G(yn , u, u) + G(u, T m (yn ), T m (yn )) + G(yn , T m (yn ), T m (yn ))✳ ❑❤✐ ➤ã✱ ✭✷✳✹✶✮ trë t❤➭♥❤ ✷✽ G(T m (yn ), u, T m (yn )) ≤ k max{[G(yn , u, u) + G(u, T m (yn ), T m (yn )) + G(yn , T m (yn ), T m (yn ))]✱ [G(yn , yn , u) + G(yn , yn , T m (yn )) + G(u, u, T m (yn ))]}✳ (2.43) ❑❤✐ ➤ã tõ ✭✷✳✹✸✮ t❛ ❝ã ❤❛✐ tr➢ê♥❣ ❤ỵ♣ ❚r➢ê♥❣ ❤ỵ♣ ✶✿ G(T m (yn ), u, T m (yn )) ≤ k{G(yn , u, u) + G(u, T m (yn ), T m (yn )) + G(yn , T m (yn ), T m (yn ))}✱ (2.44) trë ❧➵✐ ✭✷✳✹✷✮ t❛ ❝ã G(u, T m (yn ), T m (yn )) + G(yn , T m (u), T m (u)) + G(yn , T m (yn ), T m (yn )) ≤ 2G(yn , u, u) + 2G(u, T m (yn ), T m (yn ))✱ (2.45) ❞♦ ➤ã ✭✷✳✹✹✮ trë t❤➭♥❤ G(u, T m (yn ), T m (yn )) ≤ ❚r➢ê♥❣ ❤ỵ♣ ✷✿ G(T m (yn ), u, T m (yn )) ≤ 2k G(yn , u, u) − 2k (2.46) k{G(yn , yn , u) + G(yn , yn , T m (yn )) + G(u, u, T m (yn ))}✱ (2.47) ♥❤➢♥❣ ❞♦ (G5 ) tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺ t❛ ❝ã G(T m yn , yn , yn ) ≤ G(T m (yn ), u, u) + G(u, yn , yn )✳ (2.48) ❱× ✈❐② ✭✷✳✹✼✮ trë t❤➭♥❤ G(T (yn ), u, T (yn )) ≤ k{2G(u, yn , yn ) + 2G(u, u, T (yn ))}✱ (2.49) ❝ị♥❣ tõ (G5 ) tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺ t❛ ❝ã ✈➭ G(u, yn , yn ) ≤ 2G(yn , u, u) (2.50) G(u, u, T m (yn )) ≤ 2G(u, T m (yn ), T m (yn ))✳ (2.51) ❉♦ ➤ã✱ tõ ✭✷✳✺✵✮ ✈➭ ✭✷✳✺✶✮ t❤× ✭✷✳✹✾✮ trë t❤➭♥❤ G(T m (yn ), u, T m (yn )) ≤ k{4G(yn , u, u) + 4G(u, T m (yn ), T m (yn ))} (2.52) ♥❣❤Ü❛ ❧➭ t❛ ❝ã 4k G(yn , u, u)✳ − 4k ❚õ ✭✷✳✹✻✮ ✈➭ ✭✷✳✺✸✮ ❧✃② ❣✐í✐ ❤➵♥ ❦❤✐ n → ∞✱ t❛ ➤➢ỵ❝ G(T m (yn ), u, T m (yn )) ≤ G(T m (yn ), u, T m (yn )) → 0✳ ❱× ✈❐②✱ t❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✷✳✾ t❛ ❝ã T m (yn ) → u = T (u)✳ ❱➞② T m ❧➭ G✲❧✐➟♥ tô❝ t➵✐ u✳ ✷✾ (2.53) ✷✳✷✳✾ G✲♠➟tr✐❝ ➤➬② ➤ñ ✈➭ ❧➭ ♠ét ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ✈í✐ ♠ä✐ x, y, z ∈ X ✳ ➜Þ♥❤ ❧ý ✭❬✾❪✮✳ T :X→X ❈❤♦ (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G(T (x), T (y), T (z)) ≤ k max{[G(x, T (x), T (x))+G(y, T (y), T (y))+G(z, T (z), T (z))]✱ [G(x, T (y), T (y))+G(y, T (z), T (z))+G(z, T (x), T (x))]✱ [G(y, y, T (x)) + G(z, z, T (y)) + G(x, x, T (z))]} (2.54) tr♦♥❣ ➤ã k ∈ [0, )✳ ❑❤✐ ➤ã ➳♥❤ ①➵ T ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣ u ∈ X ✈➭ T ❧➭ G✲❧✐➟♥ tô❝ t➵✐ u✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚❤❛② z = y ✈➭♦ ✭✷✳✺✹✮ t❛ ❝ã G(T (x), T (y), T (y)) ≤ k max{[G(x, T (x), T (x))+G(y, T (y), T (y))+G(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))]} = 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 ❝❤Ý♥❤ ❧➭ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✽✮ tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✷✳✼ ♠➭ t❛ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ë tr➟♥✳ ✷✳✷✳✶✵ ❍Ö q✉➯ ✭❬✾❪✮✳ T :X→X ❈❤♦ (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ❧➭ ♠ét ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ✈í✐ ♠ä✐ ➤➬② ➤ñ ✈➭ x, y ∈ X ✳ G(T (x), T (y), T (y)) ≤ k max{[G(x, T (x), T (x)) + G(y, T (y), T (y))]✱ [G(x, T (y), T (y)+G(y, T (x), T (x))], [G(y, y, T (x))+G(x, x, T (y))]}✳ (2.55) tr♦♥❣ ➤ã k ∈ [0, )✳ ❑❤✐ ➤ã ➳♥❤ ①➵ T ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣ u ∈ X ✈➭ T ❧➭ G✲❧✐➟♥ tơ❝ t➵✐ u✳ ❈❤ø♥❣ ♠✐♥❤✳ ❱í✐ ♠ä✐ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤×Ị✉ ❦✐Ư♥ ✭✷✳✺✺✮✱ sÏ t❤á❛ ♠➲♥ ➤×Ị✉ ❦✐Ư♥ ✭✷✳✶✽✮ ủ ị ý ì ứ tõ ➜Þ♥❤ ❧ý ✷✳✷✳✼✳ ✸✵ ❑Õt ❧✉❐♥ ▲✉❐♥ ✈➝♥ ➤➲ t ợ ột số ết q s rì ❜➭② ♠ét ❝➳❝❤ ❤Ư t❤è♥❣ ❦❤➳✐ ♥✐Ư♠ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ G✲♠➟tr✐❝ ➤➬② ➤ñ ✈➭ ♠ét sè ❦❤➳✐ ♥✐Ư♠ ❧✐➟♥ q✉❛♥✳ ✷✳ ❚r×♥❤ ❜➭② ❝❤✐ t✐Õt ♠ét sè ❦Õt q✉➯ ➤✐Ó♠ ❜✃t ➤é♥❣ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ ➤ã ❧➭ ➜Þ♥❤ ❧ý ✷✳✷✳✹✱ ➜Þ♥❤ ❧ý ✷✳✷✳✾✱ ❍Ư q✉➯ ✷✳✷✳✽✱ ❱Ý ❞ơ ✷✳✷✳✷✱ ❱Ý ❞ơ ✷✳✷✳✺✳ ✸✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ♠ét sè ▼Ư♥❤ ➤Ị ✶✳✶✳✽✱ ▼Ư♥❤ ➤Ị ✶✳✶✳✾✱ ▼Ư♥❤ ➤Ị ✶✳✷✳✶✸✱ ▼Ư♥❤ ➤Ị ✷✳✶✳✸✱ tr♦♥❣ ❧✉❐♥ ✈➝♥ ♠➭ tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ö✉ ❝❤➢❛ ❝❤ø♥❣ ♠✐♥❤ ❤♦➷❝ ❝❤ø♥❣ ♠✐♥❤ ✈➽♥ t➽t✳ ✸✶ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ❇✳ ❈✳ ❉❤❛❣❡ ✭✶✾✾✷✮✱ ●❡♥❡r❛❧✐③❡❞ ▼❡tr✐❝ ❙♣❛❝❡ ❛♥❞ ▼❛♣♣✐♥❣s ✇✐t❤ ❢✐①❡❞ ♣♦✐♥t✱ ❇✉❧❧✳ ❈❛❧✳ ▼❛t❤✳ ❙♦❝✱ ✽✹ ✱ ✸✷✾✲✸✸✻✳ ❬✷❪ ❇✳ ❈✳ ❉❤❛❣❡ ✭✷✵✵✵✮✱ ●❡♥❡r❛❧✐③❡❞ ▼❡tr✐❝ ❙♣❛❝❡ ❛♥❞ ❚♦♣♦❧♦❣②❝❛❧ ❙tr✉❝t✉r❡ ■✱ ❆♥✳ st✐✐♥t✳ ❯♥✐✈✳ ❆❧✳ ■✳ ❈✉③❛ ■❛s✐✳ ▼❛t❤✳ ✭◆✳❙✮✱ ✹✻✱ ✸✲✷✹✳ ❬✸❪ ❙✳ ●❛❤❧❡r ✭✶✾✻✸✮✱ ✷✲▼❡tr✐❝❤❡ r❛✉♠❡ ✉♥❞✐❤r❡ t♦♣♦❧♦❣✐s❝❤❡ str✉❝✲ t✉r❡✱ ▼❛t❤✳◆❛❝❤r✳✱ ✷✻✱ ✶✶✺✲✶✹✽✳ ❬✹❪ ❙✳ ●❛❤❧❡r ✭✶✾✻✻✮✱ ❩✉r ❣❡♦♠❡tr✐❝ ✷✲♠❡tr✐❝❤❡ r❛✉♠❡✱ ❘❡❡✈✉❡ ❘♦✉♠❛✐♥❡ ❞❡ ▼❛t❤✳ P✉r❡s ❡t ❆♣♣❧✳✱ ✶✶✱ ✻✻✹✲✻✻✾✳ ❬✺❪ ●✳ ❊✳ ❍❛r❞②✱ ❚✳ ❉✳ ❘♦❣❡rs ✭✶✾✼✸✮✱ ❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❛ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ♦❢ ❘❡✐❝❤✱ ❈❛♥❛❞✳ ▼❛t❤✳ ❇✉❧❧✳✱ ✶✻✱ ✷✵✶✲✷✵✻✳ ❬✻❪ ❩✳ ▼✉st❛❢❛ ✭✷✵✵✺✮✱ ❆ ◆❡✇ ❙tr✉❝t✉r❡ ❋♦r ●❡♥❡r❛❧✐③❡❞ ▼❡tr✐❝ ❙♣❛❝❡s ❲✐t❤ ❆♣♣❧✐❝❛t✐♦♥s ❚♦ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r②✱ P❤❉ ❚❤❡s✐s ❚❤❡ ❯♥✐✈❡rs✐t② ♦❢ ◆❡✇❝❛st❧❡✱ ❆✉str❛❧✐❛✳ ❬✼❪ ❩✳ ▼✉st❛❢❛✱ ❇✳ ❙✐♠s ✭✷✵✵✻✮✱ ❆ ◆❡✇ ❆♣r♦❛❝❤ t♦ ●❡♥❡r❛❧✐③❡❞ ▼❡t✲ r✐❝ ❙♣❛❝❡s✱ ❏♦✉r♥❛❧ ♦❢ ◆♦♥❧✐♥❡❛r ❛♥❞ ❈♦♥✈❡① ❆♥❛❧②③s✐s✱ ✼✭✷✮✱ ✷✽✾✲✷✾✼✳ ❬✽❪ ❩✳ ▼✉st❛❢❛✱ ❇✳ ❙✐♠s ✭✷✵✵✹✮✱ ❙♦♠❡ ❘❡♠❛r❦s ❈♦♥❝❡r♥✐♥❣ ❉✲▼❡tr✐❝ ❙♣❛❝❡s✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡s ♦♥ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r❡② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❱❛❧❡♥❝✐❛ ✭❙♣❛✐♥✮✱ ❏✉❧② ✷✵✵✸✱ ✶✽✾✲✶✾✽✱ ❨♦❦♦❤❛♠❛ P✉❜❧✳✱ ❨♦❦♦❤❛♠❛✱ ✷✵✵✹ ✳ ❬✾❪ ❩✳ ▼✉st❛❢❛✱ ▼✳ ❑❤❛♥❞❛❣❥✐✱ ❲✳ ❙❤❛t❛♥❛✇✐ ✭✷✵✶✶✮✱ ❋✐①❡❞ ♣♦✐♥t r❡s✉❧ts ♦♥ ❝♦♠♣❧❡t❡ G✲♠❡tr✐❝ s♣❛❝❡s✱ ❙t✉❞✐❛ ❙❝✐❡♥t✐❛r✉♠ ▼❛t❤✳ ❍✉♥❣❛r②✱ ✹✽ ✭✸✮✱ ✸✵✹✲✸✶✾✳ ✸✷ ... ❝❤ó♥❣ t➠✐ G✲♠➟tr✐❝ ✈➭ ❣✐❛♥ G✲♠➟tr✐❝ sÏ tr×♥❤ ❜➭② ♠ét sè ❦Õt q✉➯ ✈Ị ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ tì ể ột số ị ý ề ể ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❝❤♦ ❝➳❝ ➳♥❤ ①➵ ❞➢í✐ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❦❤➳❝ ♥❤❛✉✳ ❇è ❝ơ❝ ❧✉❐♥ ✈➝♥ ❣å♠... ❧➭ ❝➳❝ ❤ä❝ ✈✐➟♥ ❝❛♦ ❤ä❝ ❦❤ã❛ ✶✽ ❚♦➳♥✲●✐➯✐ tÝ❝❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ➤➲ t➵♦ ➤✐Ò✉ ệ t ợ ú t tr sốt q trì ❤ä❝ t❐♣✳ ▼➷❝ ❞ï ❝ã ♥❤✐Ò✉ ❝è ❣➽♥❣✱ s♦♥❣ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ t❤✐Õ✉ sãt✳ ❚➳❝