❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❚❤Þ ❚❤ó② ❍✉ú♥❤ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ◆❣❤Ö ❆♥ ✲ ✷✵✶✻ ❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❚❤Þ ❚❤ó② ❍✉ú♥❤ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥ ◆❣❤Ư ❆♥ ✲ ✷✵✶✻ ▼ơ❝ ▲ơ❝ ❚r❛♥❣ ▼ơ❝ ❧ơ❝ ✶ ▲ê✐ ♥ã✐ ➤➬✉ ✐✐ ❈❤➢➡♥❣ ✶✳ ➜✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✶✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ➜✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❈❤➢➡♥❣ ✷✳ ❣✐❛♥ ✷✳✶ ✳ ✳ ✳ ✳ ✶ ✺ ➜✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ G✲♠➟tr✐❝ ✶✹ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ✷✳✷ G✲♠➟tr✐❝ ✶ G✲♠➟tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ➜✐Ó♠ trï♥❣ ♥❤❛✉ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ❑Õt ❧✉❐♥ ✸✸ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✸✺ ✐ ▲ê✐ ♥ã✐ ➤➬✉ ▲ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➲ ➤➢ỵ❝ ♥❣❤✐➟♥ ❝ø✉ ré♥❣ r➲✐ tr♦♥❣ ♥❤✐Ị✉ t❤❐♣ ❦û q✉❛✱ ✈× ❧ý t❤✉②Õt ♥➭② ➤ã♥❣ ♠ét ✈❛✐ trß q✉❛♥ trä♥❣ ❦❤➠♥❣ ❝❤Ø tr♦♥❣ t♦➳♥ ❤ä❝ ♠➭ ❝ß♥ tr♦♥❣ ❝➳❝ ♥❣➭♥❤ ❦❤♦❛ ❤ä❝ ø♥❣ ❞ơ♥❣ ❦❤➳❝✱ ❝❤➻♥❣ ❤➵♥ ♥❤➢ tè✐ ➢✉ ❤ã❛✱ ❝➳❝ ♠➠ ❤×♥❤ t♦➳♥ ❤ä❝ ✈➭ ❧ý t❤✉②Õt ❦✐♥❤ tÕ✳ ◆❤✐Ò✉ ♥❤➭ t♦➳♥ ❤ä❝ ➤➲ ❝è ❣➽♥❣ ♠ë ré♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ö♠ t❤➠♥❣ t❤➢ê♥❣ ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ♥❤➢✿ ❙✳ ●❛❤❧❡r ✈➭ ❇✳ ❈✳ ❉❤❛❣❡ ✈➭ ♠ë ré♥❣ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ó♠ ❜✃t ➤é♥❣ ➤➲ ❜✐Õt tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝❤♦ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ♥➭②✱ ♥❤➢♥❣ ❝➳❝ t➳❝ ũ ỉ r r ữ ỗ ự ♥➭② ❧➭ ❦❤➠♥❣ ➤➢ỵ❝ ➤➳♣ ø♥❣✳ ◆➝♠ ✶✾✾✷✱ ❇✳ ❈✳ ❉❤❛❣❡ ➤➲ ❣✐í✐ t❤✐Ư✉ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ❦❤➠♥❣ ❣✐❛♥ D✲♠➟tr✐❝✳ ◆➝♠ ✷✵✵✻✱ ❩✳ ▼✉st❛❢❛ ✈➭ ❇✳ ❙✐♠s ➤➲ ❝❤Ø r❛ r➺♥❣ ❤➬✉ ❤Õt ❝➳❝ ❦Õt q✉➯ ❧✐➟♥ q✉❛♥ ➤Õ♥ ❦❤➠♥❣ ❣✐❛♥ D✲♠➟tr✐❝ ❝đ❛ ❇✳ ❈✳ ❉❤❛❣❡ ❧➭ ❦❤➠♥❣ ❝ã ❤✐Ư✉ ❧ù❝✳ ❱× ✈❐②✱ ❤ä ➤➲ ❣✐í✐ t❤✐Ư✉ ♠ét ♣❤✐➟♥ ❜➯♥ ❝➯✐ t✐Õ♥ ❝đ❛ ❝✃✉ tró❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❣ä✐ ♥ã ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ ➤å♥❣ t❤ê✐ ❣✐í✐ t❤✐Ư✉ ♠ét ❧ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣ ♠í✐ ❝❤♦ ➳♥❤ ①➵ ❦❤➳❝ ♥❤❛✉ tr♦♥❣ ❝✃✉ tró❝ ♠í✐ ♥➭②✳ ◆➝♠ ✶✾✾✹✱ ❘✳ P✳ P❛♥t ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ➳♥❤ ①➵ R✲❣✐❛♦ ❤♦➳♥ ②Õ✉ t❤❡♦ ➤✐Ĩ♠ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣ tÝ♥❤ R✲❣✐❛♦ ❤♦➳♥ ②Õ✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣ ✈í✐ tÝ♥❤ ❣✐❛♦ ❤♦➳♥ t➵✐ ❝➳❝ ➤✐Ó♠ trï♥❣ ♥❤❛✉✳ ◆➝♠ ✶✾✾✻✱ ●✳ ❏✉♥❣❝❦ ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ✷ ➳♥❤ ①➵ r➺♥❣ S ✈➭ T S ✈➭ ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ S T ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ ✈➭ ❝❤Ø r❛ ✈➭ T ❧➭ R✲❣✐❛♦ ❤♦➳♥ ②Õ✉ t❤❡♦ ➤✐Ó♠✳ ❑❤➳✐ ♥✐Ư♠ ➤➷t ❝❤Ø♥❤ ❝đ❛ ❜➭✐ t♦➳♥ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤➲ ➤➢❛ r❛ ♠ét ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ♠í✐ t❤✉ ❤ót sù q✉❛♥ t➞♠ ❝đ❛ ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝✱ ♥❤➢✿ ❇✳ ❑✳ ▲❛❤✐r✐ ✱ P✳ ❉❛s✱ ❆✳ P❡tr✉s❡❧✱ ■✳ ❆✳ ❘✉s✱ ❏✳ ❈✳ ❨❛♦✳ ●➬♥ ➤➞②✱ ▼✳ ❆❦❦♦✉❝❤✐ ✈➭ ❱✳ P♦♣❛ ➤➲ ♥❣❤✐➟♥ ❝ø✉ ❜➭✐ t♦➳♥ ➤➷t ❝❤Ø♥❤ ❝❤♦ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ q✉❛♥ ❤Ö ➮♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ❚r➟♥ ❝➡ së ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ P●❙✳❚❙✳ ❚r➬♥ ✐✐ ❱➝♥ ➣♥ ❝❤ó♥❣ t➠✐ ➤➲ t✐Õ♣ ❝❐♥ ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ♥➭② ✈➭ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐ ✧➜✐Ĩ♠ ❜✃t ➤é♥❣ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✧✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ❧➭ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈➭ tÝ♥❤ ❝❤✃t ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ trì ột số ị ý ể t ộ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ t❤á❛ ♠➲♥ ❝➳❝ q✉❛♥ ❤Ö ➮♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị G✲♠➟tr✐❝✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ▼ơ❝ ✶ ♥❤➺♠ ❣✐í✐ t❤✐Ư✉ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝đ❛ ❧✉❐♥ ✈➝♥ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣ ❝➬♥ ❞ï♥❣ ❝❤♦ ❝➳❝ tr×♥❤ ❜➭② ✈Ị s❛✉✳ ▼ơ❝ ✷ ♥❤➺♠ tr×♥❤ ❜➭② ♠ét sè ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ G✲♠➟tr✐❝✳ ▼ơ❝ ✶ ♥❤➺♠ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ▼ơ❝ ✷ ♥❤➺♠ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ▲✉❐♥ ✈➝♥ ♥➭② ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ❝❤✉ ➤➳♦ ❝ñ❛ t❤➬② ❣✐➳♦ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ t➳❝ ❣✐➯ ①✐♥ ❜➭② tá sù ❜✐Õt ➡♥ s➞✉ s➽❝ tí✐ ❚❤➬②✳ ◆❤➞♥ ❞Þ♣ ♥➭② t➳❝ ❣✐➯ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❇❛♥ ❝❤đ ♥❤✐Ư♠ ❦❤♦❛ ❚♦➳♥✱ P❤ß♥❣ ➤➭♦ t➵♦ ❙❛✉ ➤➵✐ ❤ä❝✱ q✉ý ❚❤➬②✱ ❈➠ ❣✐➳♦ tr♦♥❣ tæ ●✐➯✐ ❚Ý❝❤ ❦❤♦❛ ❙➢ ♣❤➵♠ ❚♦➳♥ ❤ä❝✱ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ➤➲ t❐♥ t×♥❤ ❣✐ó♣ ➤ì t➳❝ ❣✐➯ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ❈✉è✐ ❝ï♥❣ t➳❝ ❣✐➯ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❇❛♥ ❣✐➳♠ ❤✐Ö✉ tr➢ê♥❣ ❚❍P❚ ❚r➢♥❣ ❱➢➡♥❣✱ t❤➭♥❤ ♣❤è ❍å ❈❤Ý ▼✐♥❤✱ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ ❝❛♦ ❤ä❝ ❦❤♦➳ ✷✷ ●✐➯✐ ❚Ý❝❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ✈➭ ❣✐❛ ➤×♥❤✱ ➤å♥❣ ♥❣❤✐Ư♣ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ♥❤✃t ➤Ĩ ❣✐ó♣ t➳❝ ❣✐➯ ❤♦➭♥ t❤➭♥❤ tèt ♥❤✐Ư♠ ✈ơ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣✳ ✐✐✐ ▼➷❝ ❞ï ➤➲ ❝ã ♥❤✐Ị✉ ❝è ❣➽♥❣ tr♦♥❣ ✈✐Ư❝ ♥❣❤✐➟♥ ❝ø✉✱ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✱ s♦♥❣ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ s sót ợ ữ ý ế ➤ã♥❣ ❣ã♣ ❝đ❛ q✉ý ❚❤➬②✱ ❈➠ ✈➭ ❜➵♥ ➤ä❝ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥✳ ❱✐♥❤✱ ♥❣➭② ✸✵ t❤➳♥❣ ✼ ♥➝♠ ✷✵✶✻ ◆❣✉②Ơ♥ ❚❤Þ ❚❤ó② ❍✉ú♥❤ ✐✈ ❝❤➢➡♥❣ ✶ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ✶✳✶ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❧✉❐♥ ✈➝♥✳ ◆é✐ ❞✉♥❣ ❣å♠✿ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ✈Ị ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ ❞➲② G✲❤é✐ tơ✱ G✲❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✱ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉✱ ➤✐Ó♠ ❞➲② ❜✃t ➤é♥❣ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉✱ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣ ❝➬♥ ❞ï♥❣ ❝❤♦ ❝➳❝ tr×♥❤ ❜➭② ✈Ị s❛✉✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✶ ❧➭ ♠ét ♠➟tr✐❝ tr➟♥ ✭❬✶❪✮ ❈❤♦ t❐♣ ❤ỵ♣ X X = φ✱ ➳♥❤ ①➵ d : X × X → R ➤➢ỵ❝ ❣ä✐ ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✶✮ d(x, y) ≥ ✈í✐ ♠ä✐ x, y ∈ X ✷✮ d(x, y) = d(y, x) ✈í✐ ♠ä✐ x, y ∈ X ✳ ✸✮ d(x, y) ≤ d(x, z) + d(z, y) ✈í✐ ♠ä✐ x, y, z ∈ X ✳ X ❚❐♣ ❤✐Ư✉ ❧➭ ✶✳✶✳✷ ❝ï♥❣ ✈í✐ ♠ét ♠➟tr✐❝ d(x, y) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x = y ✳ ✈➭ d tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ ❦Ý (X, d) ❤❛② ➤➡♥ ❣✐➯♥ ❧➭ X ✳ ❙è d (x, y) ❣ä✐ ữ x y ị ĩ X ột t rỗ G : X × X × X → R+ ❧➭ ♠ét ❤➭♠ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ò✉ s❛✉ ✭●✶✮ G(x, y, z) = ♥Õ✉ x = y = z ✱ ✭●✷✮ < G(x, x, y) ✈í✐ ♠ä✐ x, y ∈ X ✭●✸✮ G(x, x, y) ≤ G(x, y, z)✱ ✈í✐ ♠ä✐ x, y, z ∈ X ✶ ✈í✐ x = y✱ ✈í✐ z = y✱ ✭●✹✮ G(x, y, z) = G(x, z, y) = G(y, z, x) = ✭➤è✐ ①ø♥❣ ✈í✐ ❝➯ ✸ ❜✐Õ♥✮✱ ✭●✺✮ G(x, y, z) ≤ G(x, a, a) + G(a, y, z), ✈í✐ ♠ä✐ x, y, z, a ∈ X ✭❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✮✳ ❑❤✐ ➤ã✱ ❤➭♠ G ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ♠➟tr✐❝ s✉② ré♥❣✱ ❤❛② ❣ä♥ ❤➡♥ ❧➭ ♠ét G✲♠➟tr✐❝ X ✱ ✈➭ ❝➷♣ (X, G) ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ tr➟♥ ◆❤❐♥ ①Ðt✳ ✶✳✶✳✸ ◆Õ✉ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ t❤× ❝➳❝ ❤➭♠ Gs (d) : X × X × X → R+ ✈➭ Gm (d) : X × X × X → R+ ❝❤♦ ❜ë✐ ❝➳❝ ❝➠♥❣ t❤ø❝ (Es ) Gs (d)(x, y, z) = d(x, y) + d(y, z) + d(x, z)✱ (Em ) Gm (d)(x, y, z) = max{d(x, y), d(y, z), d(x, z)} ❧➭ ❝➳❝ G✲♠➟tr✐❝ tr➟♥ X ✳ ✶✳✶✳✹ ▼Ư♥❤ ➤Ị✳ x, y, z ✈➭ ✭❬✽❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐ a ∈ X ✱ t❛ ❝ã G(x, y, z) = 0✱ t❤× x = y = z ✱ ✭✶✮ ◆Õ✉ ✭✷✮ G(x, y, z) ≤ G(x, x, y) + G(x, x, z)✱ ✭✸✮ G(x, y, y) ≤ 2G(y, x, x)✱ ✭✹✮ G(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)✳ ✶✳✶✳✺ ▼Ư♥❤ ➤Ị✳ ✭❬✽❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ (X, G) ị dG : X ì X → R+ ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝ dG (x, y) = G(x, y, y) + G(y, x, x), ❑❤✐ ➤ã✱ dG ❧➭ ♠ét ♠➟tr✐❝ tr➟♥ X ✳ ✷ ✈í✐ ♠ä✐ x, y ∈ X ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✻ (X, G) ✭❬✽❪✮ ❈❤♦ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ x0 ∈ X ✈➭ r > 0✳ ❚❛ ❦ý ❤✐Ö✉ BG (x0 , r) = {y ∈ X : G(x0 , y, y) < r} ✈➭ ❣ä✐ BG (x0 , r) ❧➭ G✲❤×♥❤ ❝➬✉ ✈í✐ t➞♠ x0 ✈➭ ❜➳♥ ❦Ý♥❤ r✳ ✭❬✽❪✮ ▼Ư♥❤ ➤Ị✳ ✶✳✶✳✼ x0 ∈ X ✈➭ ❈❤♦ (X, G) ❧➭ ♠ét Gtr ó ỗ r > t❛ ❝ã ✭✶✮ ◆Õ✉ G(x0 , x, y) < r✱ t❤× x, y ∈ BG (x0 , r)✱ ✭✷✮ ◆Õ✉ y ∈ BG (x0 , r)✱ t❤× tå♥ t➵✐ ♠ét δ > ➤Ó BG (y, δ) ⊆ B(x0 , r)✳ ❈❤ø♥❣ ♠✐♥❤✳ ❑❤➻♥❣ ➤Þ♥❤ ✭✶✮ s✉② trù❝ t✐Õ♣ tõ ➤✐Ị✉ ❦✐Ư♥ ✭●✸✮ tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✷ ✈➭ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✻✳ ❑❤➻♥❣ ➤Þ♥❤ ✭✷✮ s✉② tõ ➤✐Ị✉ ❦✐Ư♥ ✭●✺✮ tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✷ ✈í✐ δ = r − G(x0 , y, y)✳ ◆❤❐♥ ①Ðt✳ ❝➬✉✱ ❚õ ❦❤➻♥❣ ➤Þ♥❤ ✭✷✮ ❝đ❛ ♠Ư♥❤ ➤Ị tr➟♥ t❛ s✉② r❛ ❤ä t✃t ❝➯ G✲❤×♥❤ B = {BG (x, r) : x ∈ X, r > 0}✱ ❧❐♣ t❤➭♥❤ ♠ét ❝➡ së ❝ñ❛ t➠♣➠ τ (G) tr➟♥ X ✱ ✈➭ ❣ä✐ ❧➭ t➠♣➠ G✲♠➟tr✐❝✳ ✶✳✶✳✽ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✽❪✮ ❈❤♦ ♠ét ❞➲② ❝➳❝ ➤✐Ĩ♠ ❝đ❛ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ●✐➯ sö {xn } ❧➭ X ✳ ➜✐Ó♠ x ∈ X lim G(x, xn , xm ) = 0✳ n,m→∞ ➤➢ỵ❝ ❣ä✐ ❧➭ ▲ó❝ ➤ã t❛ ♥ã✐ r➺♥❣ ❞➲② ❣✐í✐ ❤➵♥ ❝đ❛ ❞➲② {xn } ♥Õ✉ {xn } ❧➭ G✲❤é✐ tơ ✈Ị x✱ ➤➢ỵ❝ (G) ❦ý ❤✐Ö✉ xn → x✳ (G) ❉♦ ➤ã✱ ♥Õ✉ tå♥ t➵✐ ✶✳✶✳✾ ❧➭ xn → x✱ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ Gtr (X, G) ế ỗ > N ∈ N s❛♦ ❝❤♦ G(x, xn , xm ) < ε ✈í✐ ♠ä✐ n, m ≥ N ✳ ▼Ư♥❤ ➤Ị✳ ✭❬✽❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❉➲② {xn } ⊆ X G✲❤é✐ tơ ✈Ị x ♥Õ✉ ♥ã ❤é✐ tơ ✈Ị x t❤❡♦ t➠♣➠ G✲♠➟tr✐❝ τ (G) tr➟♥ X ✳ ✸ ✶✳✶✳✶✵ ▼Ư♥❤ ➤Ị✳ {xn } ⊆ X ✭❬✽❪✮ ✈➭ ➤✐Ó♠ ❈❤♦ (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❑❤✐ ➤ã✱ ✈í✐ ❞➲② x ∈ X ✱ ❝➳❝ ❦❤➻♥❣ ➤Þ♥❤ s❛✉ ➤➞② ❧➭ t➢➡♥❣ ➤➢➡♥❣ {xn } ❧➭ G✲❤é✐ tơ ✈Ị x❀ ✭✶✮ ❉➲② ✭✷✮ G(xn , xn , x) → 0✱ ❦❤✐ n → ∞❀ ✭✸✮ G(xn , x, x) → 0✱ ❦❤✐ n → ∞❀ ✭✹✮ G(xm , xn , x) → 0✱ ❦❤✐ m, n → ∞✳ ✭❬✽❪✮ ❈❤♦ (X, G) ✈➭ (X , G ) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ✶✳✶✳✶✶ ➜Þ♥❤ ♥❣❤Ü❛✳ ➳♥❤ ①➵ f : (X, G) → (X , G )✳ ❑❤✐ ➤ã✱ f ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ✈í✐ ♠ä✐ ➤➢ỵ❝ ❣ä✐ ❧➭ G✲❧✐➟♥ tơ❝ t➵✐ ➤✐Ĩ♠ a ∈ X ε > 0✱ tå♥ t➵✐ sè δ > s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X ♠➭ G(a, x, y) < δ t❛ ❝ã G(f (a), f (x), f (y)) < ε✳ ➳♥❤ ①➵ f : (X, G) → (X , G ) ➤➢ỵ❝ ❣ä✐ ❧➭ G✲❧✐➟♥ tơ❝ tr➟♥ X ♥Õ✉ ♥ã ❧➭ ✶✳✶✳✶✷ ♥Õ✉ ✈➭ ❝❤Ø G✲❧✐➟♥ tô❝ t➵✐ ♠ä✐ a ∈ X ✳ ▼Ư♥❤ ➤Ị✳ ➤ã✱ ➳♥❤ ①➵ ✭❬✽❪✮ (X, G), (X , G ) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❈❤♦ f :X →X ❧➭ G✲❧✐➟♥ tô❝ t➵✐ ➤✐Ĩ♠ x∈X G✲❧✐➟♥ tơ❝ ❞➲② t➵✐ x✱ ♥❣❤Ü❛ ❧➭ ✈í✐ ♠ä✐ ❞➲② {xn } ⊂ X ❝ã ❞➲② {f (xn )} ❧➭ G✲❤é✐ tơ ✈Ị f (x)✳ ✶✳✶✳✶✸ ▼Ư♥❤ ➤Ị✳ ✭❬✽❪✮ ❈❤♦ ❑❤✐ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ♥ã ❧➭ ❧➭ G✲❤é✐ tơ ✈Ị x✱ t❤× t❛ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❑❤✐ ➤ã✱ ❤➭♠ G(x, y, z) ❧✐➟♥ tơ❝ ➤å♥❣ t❤ê✐ t❤❡♦ t✃t ❝➯ ✸ ❜✐Õ♥ ❝đ❛ ♥ã✳ X ị ĩ ợ ọ G✲❈❛✉❝❤② (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❉➲② {xn } ⊆ ♥Õ✉ ✈í✐ ♠ä✐ ε > tå♥ t➵✐ sè tù ♥❤✐➟♥ N ∈ N s❛♦ G(xn , xm , xl ) < ε ✈í✐ ♠ä✐ n, m, l ≥ N ✱ ♥❣❤Ü❛ ❧➭ G(xn , xm , xl ) → ❦❤✐ n, m, l → ∞✳ ✹ ị ý ị ý ò ú ế tÝ♥❤ ❝❤✃t ✧t➢➡♥❣ t❤Ý❝❤ ✭❬✻❪✮ ②Õ✉✧ ➤➢ỵ❝ t❤❛② t❤Õ ❜ë✐ ♠ét tr♦♥❣ ♥❤÷♥❣ tÝ♥❤ ❝❤✃t s❛✉ ✭✈➭ ❣✐÷ ♥❣✉②➟♥ ❝➳❝ ❣✐➯ t❤✐Õt ❝ß♥ ❧➵✐✮ ✭✶✮ ❚Ý♥❤ ❝❤✃t ❘✲❣✐❛♦ ❤♦➳♥ ②Õ✉✱ ✭✷✮ ❚Ý♥❤ ❝❤✃t ❘✲❣✐❛♦ ❤♦➳♥ ②Õ✉ ❧♦➵✐ Af ✭✸✮ ❚Ý♥❤ ❝❤✃t ❘✲❣✐❛♦ ❤♦➳♥ ②Õ✉ ❧♦➵✐ (Ag )✱ ✭✹✮ ❚Ý♥❤ ❝❤✃t ❘✲❣✐❛♦ ❤♦➳♥ ②Õ✉ ❧♦➵✐ (P )✱ ✭✺✮ ❚Ý♥❤ ❝❤✃t ❣✐❛♦ ❤♦➳♥ ②Õ✉✳ ✱ ❈❤ø♥❣ ♠✐♥❤✳ ❱× t✃t ❝➯ ❣✐➯ t❤✐Õt ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✶✳✶✺ ➤Ị✉ t❤á❛ ♠➲♥✱ ♥➟♥ ❜➺♥❣ ❝➳❝❤ ❝❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù ♥❤➢ ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ✷✳✶✳✶✺ t❛ s✉② r❛ tå♥ t➵✐ ♠ét ➤✐Ó♠ trï♥❣ ♥❤❛✉ x∈X ♥❤❛✉ ❜✃t ❦ú ❝ñ❛ ❝➷♣ ❝ñ❛ ❝➷♣ ➳♥❤ ①➵ (f, g)✳ ❇➞② ❣✐ê ❣✐➯ sư x ❧➭ ➤✐Ĩ♠ trï♥❣ (f, g)✳ ✲ ❚r➢ê♥❣ ❤ỵ♣ ❝➷♣ (f, g) ❧➭ R✲❣✐❛♦ ❤♦➳♥ ②Õ✉✱ sư ❞ơ♥❣ tÝ♥❤ R✲❣✐❛♦ ❤♦➳♥ ②Õ✉✱ t❛ ➤➢ỵ❝ G(f gx, gf x, gf x) ≤ RG(f x, gx, gx) = 0, ✭✈× ❧ó❝ ➤ã t❛ ❝ã f x = gx✮✳ ❚õ ➤ã✱ t❛ s✉② r❛ ❉♦ ➤ã ❝➷♣ G(f gx, gf x, gf x) = 0✳ ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ f gx = gf x✳ (f, g) ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉✳ ❚õ ➜Þ♥❤ ❧ý ✷✳✶✳✶✺✱ t❛ ❦Õt ❧✉❐♥ ✲ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ❝➷♣ f ✈➭ g ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t✳ (f, g) ❧➭ R✲❣✐❛♦ ❤♦➳♥ ②Õ✉ ❧♦➵✐ Af ✱ t❤× t❛ ❝ã G(f gx, ggx, ggx) ≤ RG(f x, gx, gx) = ❚õ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❛ t❤✃② ✭●✺✮ ❝ñ❛ f gx = ggx✳ ❇➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ➤✐Ị✉ ❦✐Ư♥ G✲♠➟tr✐❝ t❛ ❝ã G(f gx, gf x, gf x) ≤ G(f gx, ggx, ggx) + G(ggx, gf x, gf x) = + G(gf x, gf x, gf x) = ✷✷ ❚õ ➤ã✱ ❧❐♣ ❧✉❐♥ t➢➡♥❣ tù ♥❤➢ tr➟♥ t❛ ❝ã ➤➢ỵ❝ ✲ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ❝➷♣ f gx = gf x✳ (f, g) ❧➭ R✲❣✐❛♦ ❤♦➳♥ ②Õ✉ ❧♦➵✐ (Ag ) t❤× t❛ ❝ã G(gf x, f f x, f f x) ≤ RG(f x, gx, gx) = 0✳ ❚õ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❛ ❝ã ➤➢ỵ❝ ✭●✺✮ ❝đ❛ gf x = f f x✳ ❇➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ➤✐Ị✉ ❦✐Ư♥ G✲♠➟tr✐❝ t❛ ❝ã G(f gx, gf x, gf x) ≤ G(f gx, f f x, f f x) + G(f f x, gf x, gf x) = G(f f x, f f x, f f x) + = ❚õ ➤ã✱ t➢➡♥❣ tù ♥❤➢ tr➟♥ t❛ ❝ã ➤➢ỵ❝ ❚➢➡♥❣ tù✱ ♥Õ✉ ❝➷♣ f gx = gf x✳ (f, g) ❧➭ R✲❣✐❛♦ ❤♦➳♥ ②Õ✉ ❧♦➵✐ (P ) ❤♦➷❝ ❣✐❛♦ ❤♦➳♥ ②Õ✉✱ (f, g) ❝ị♥❣ ❣✐❛♦ ❤♦➳♥ t➵✐ ♥❤÷♥❣ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ủ ú tì tt trờ ợ ị ý ✷✳✶✳✶✺✱ ❝➷♣ ❱× ✈❐②✱ tr♦♥❣ (f, g) ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t✳ ➜✐Ó♠ trï♥❣ ♥❤❛✉ ✈➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❝➳❝ ✷✳✷ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ✱ ❝➳❝ ❤Ư q✉➯ ❝đ❛ ❝❤ó♥❣ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ✷✳✷✳✶ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶✶❪✮ ❑ý ❤✐Ö✉ G ❧➭ t❐♣ t✃t ❝➯ ❝➳❝ ❤➭♠ ❧✐➟♥ tô❝ F : R6+ → R s❛♦ ❝❤♦ ✭❋✶✮ F ❧➭ ❤➭♠ ❦❤➠♥❣ t➝♥❣ t❤❡♦ ❜✐Õ♥ ✭❋✷✮ ❚å♥ t➵✐ h1 ∈ [0, 1) s❛♦ ❝❤♦ ✈í✐ ♠ä✐ u, v ≥ 0✱ F (u, v, v, u, u + v, 0) ≤ ❦Ð♦ t❤❡♦ ✭❋✸✮ ❚å♥ t➵✐ t❤❡♦ t5 ✳ u ≤ h1 v ✳ h2 ∈ [0, 1) s❛♦ ❝❤♦ ✈í✐ ♠ä✐ t, t > 0✱ F (t, t, 0, 0, t, t ) < ❦Ð♦ t ≤ h2 t ✳ ✷✸ ✷✳✷✳✷ ❱Ý ❞ô✳ ✭❬✶✶❪✮ ❳Ðt ❤➭♠ F : R6+ → R ❝❤♦ ❜ë✐ F (t1 , , t6 ) = t1 − at2 − bt3 − ct4 − dt5 − et6 , tr♦♥❣ ➤ã a, b, c, d, e ≥ ✈➭ < a + b + c + 2d + e < 1✳ ❑❤✐ ➤ã ✲ ➜✐Ị✉ ❦✐Ư♥ ✭❋✶✮ ❤✐Ĩ♥ ♥❤✐➟♥ t❤á❛ ♠➲♥✳ u, v ≥ F (u, v, v, u, u + v, 0) = a+b+d ✈➭ F (t, t, 0, 0, t, t ) = t − at − e dt − et < 0✱ t❤× ❜➺♥❣ ❝➳❝❤ ❝❤ä♥ ≤ h2 = < t❛ ❝ã ♥❣❛② t ≤ h2 t ✳ − (a + d) ✲ ➜✐Ị✉ ❦✐Ư♥ ✭❋✸✮ t❤á❛ ♠➲♥✱ ✈× ✈í✐ ✷✳✷✳✸ ❱Ý ❞ơ✳ ✭❬✶✶❪✮ ❳Ðt ❤➭♠ F : R6+ → R ❝❤♦ ❜ë✐ F (t1 , , t6 ) = t1 − k max{t2 , t3 , t4 , t5 , t6 }, tr♦♥❣ ➤ã k ∈ [0, 12 )✳ ❑❤✐ ➤ã ✲ ➜✐Ị✉ ❦✐Ư♥ ✭❋✶✮ ❤✐Ĩ♥ ♥❤✐➟♥ t❤á❛ ♠➲♥✳ ✲ ➜✐Ị✉ ❦✐Ư♥ ✭❋✷✮ t❤á❛ ♠➲♥✱ ✈× ✈í✐ u, v ≥ ✈➭ F (u, v, v, u, u + v, 0) = u − k max{u, v, u + v} ≤ 0✱ k t❤× ❜➺♥❣ ❝➳❝❤ ❝❤ä♥ ≤ h1 = < t❛ ❝ã ♥❣❛② u ≤ h1 v ✳ 1−k ✲ ➜✐Ị✉ ❦✐Ư♥ ✭❋✸✮ t❤á❛ ♠➲♥✱ ✈× ✈í✐ t, t > ✈➭ F (t, t, 0, 0, t, t ) = t − k max{t, t } < 0✳ ✷✳✷✳✹ ◆Õ✉ t > t ✱ t❤× t(1 − k) < 0✳ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ◆Õ✉ t ≤ t ✱ t❤× ❜➺♥❣ ❝➳❝❤ ❝❤ä♥ ≤ h2 = k < t❛ ❝ã ♥❣❛② t ≤ h2 t ✳ ❇ỉ ➤Ị✳ ✭❬✶✶❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ f, g (X, G) ❧➭ ❤❛✐ ➳♥❤ ①➵ s❛♦ ❝❤♦ ✷✹ : (X, G) → F (G(f x, f y, f y), G(gx, gy, gy), G(gx, f x, f x), G(gy, f y, f y), G(gx, f y, f y), G(gy, f x, f x)) ≤ 0, ✈í✐ ♠ä✐ f ❑❤✐ ➤ã✱ ✈➭ x, y ∈ X ✈➭ F ✭✷✳✶✮ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ✭❋✸✮ tr♦♥❣ ➤Þ♥❤ ♥❣❤Ü❛ ✷✳✷✳✶✳ g ❝ã ♥❤✐Ị✉ ♥❤✃t ♠ét ➤✐Ĩ♠ trï♥❣ ♥❤❛✉✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư u = f p = gp ✈➭ v = f q = gq ✱ ♥❣❤Ü❛ ❧➭ f ✈➭ g ❝ã ♥❤✐Ị✉ ❤➡♥ ♠ét ➤✐Ĩ♠ trï♥❣ ♥❤❛✉✳ ❑❤✐ ➤ã ❞ù❛ ✈➭♦ ✭✷✳✶✮ t❛ ❝ã F (G(f q, f p, f p), G(gq, gp, gp), G(gq, f q, f q), G(gp, f p, f p), G(gq, f p, f p), G(gp, f q, f q)) ≤ 0, F (G(gq, gp, gp), G(gq, gp, gp), 0, 0, G(gq, gp, gp), G(gq, gp, gp) ≤ ◆❤ê ✭❋✸✮✱ tõ ➤✐Ò✉ ♥➭② t❛ s✉② r❛ G(gq, gp, gp) ≤ h2 G(gp, gq, gq) ❚➢➡♥❣ tù✱ t❛ ❝ã G(gp, gq, gq) ≤ h2 G(gq, gp, gp) ➜✐Ò✉ ♥➭② s✉② r❛ r➺♥❣ ♥❣❤Ü❛ ❧➭ ✷✳✷✳✺ G(gq, gp, gp)(1 − h22 ) ≤ 0✳ ❉♦ ➤ã t❛ ❝ã G(gq, gp, gp) = 0, gq = gp✳ ❱× ✈❐② u = f p = gp = gq = f q = v ➜Þ♥❤ ❧ý✳ ✭❬✶✶❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ f, g : (X, G) → (X, G) ❧➭ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✶✮ ✈í✐ ♠ä✐ x, y ∈ X ✱ ➤ã F ∈ ➤ñ ❝ñ❛ f ✈➭ G ✳ ◆Õ✉ tr♦♥❣ f (X) ⊂ g(X) ✈➭ g(X) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ G✲♠➟tr✐❝ ➤➬② (X, G)✱ t❤× f ✈➭ g ó ột trị trù t ữ ♥Õ✉ g t➢➡♥❣ t❤Ý❝❤ ②Õ✉✱ t❤× f ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư ✈➭ g ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t✳ x0 ❧➭ ♠ét ➤✐Ó♠ ❜✃t ❦ú t❤✉é❝ X ✈➭ x1 ∈ X ➤Ĩ f x0 = gx1 ✳ ➜✐Ị✉ ó tể tự ệ ợ ì f (X) g(X)✳ ❚✐Õ♣ tơ❝ q✉➳ tr×♥❤ ♥➭②✱ ❦❤✐ t❛ ➤➲ ❝❤ä♥ ợ ể xn X tì t tì ợ ➤✐Ó♠ xn+1 s❛♦ ❝❤♦ f xn = gxn+1 ✳ ❑❤✐ ➤ã✱ ♥❤ê ✭✷✳✶✮ t❛ ❝ã ✷✺ F (G(f xn−1 , f xn , f xn ), G(gxn−1 , gxn , gxn ), G(gxn−1 , f xn−1 , f xn−1 ), G(gxn , f xn , f xn ), G(gxn−1 , f xn , f xn ), G(gxn , f xn−1 , f xn−1 )) ≤ 0, F (G(gxn , gxn+1 , gxn+1 ), G(gxn−1 , gxn , gxn ), G(gxn−1 , gxn , gxn ), G(gxn , gxn+1 , gxn+1 ), G(gxn−1 , gxn+1 , gxn+1 ), 0) ≤ ◆❤ê ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭❋✶✮ ✈➭ ✭●✺✮ t❛ ❝ã ➤➢ỵ❝ F (G(gxn , gxn+1 , gxn+1 ), G(gxn−1 , gxn , gxn ), G(gxn−1 , gxn , gxn ), G(gxn , gxn+1 , gxn+1 ), G(gxn−1 , gxn , gxn ) + G(gxn , gxn+1 , gxn+1 ), 0) ≤ ▲➵✐ ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ✭❋✷✮ t❛ ❝ã G(gxn , gxn+1 , gxn+1 ) ≤ h1 G(gxn−1 , gxn , gxn ) ✭✷✳✷✮ ❚✐Õ♣ tơ❝ q✉➳ tr×♥❤ tr➟♥ t❛ t❤✉ ➤➢ỵ❝ G(gxn , gxn+1 , gxn+1 ) ≤ hn1 G(gx0 , gx1 , gx1 ) ❑❤✐ ➤ã✱ ✈í✐ ✭✷✳✸✮ m > n t❛ ❝ã G(gxn , gxm , gxm ) ≤ G(gxn , gxn+1 , gxn+1 ) + G(gxn+1 , gxn+2 , gxn+2 )+ + · · · + G(gxm−1 , gxm , gxm ) + + hm−1 )G(gx0 , gx1 , gx1 ) ≤ (hn1 + hn+1 1 hn1 G(gx0 , gx1 , gx1 ) ≤ − h1 ❚õ ❜✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ❝ï♥❣ ♥➭② t❛ s✉② r❛ ∞✳ ❉♦ ➤ã {gxn } ❧➭ ♠ét ❞➲② G✲❈❛✉❝❤②✳ ➤✐Ó♠ q ➤✐Ó♠ p∈X t❤✉é❝ g(X) ➤Ó ➤Ó gxn → q ❦❤✐ G(gxn , gxm , gxm ) → ❦❤✐ n, m → ❱× g(X) ❧➭ G✲➤➬② ➤đ✱ ♥➟♥ tå♥ t➵✐ ♠ét n → ∞✳ ❉♦ ➤ã✱ t❛ ó tể tì ợ ột gp = q sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ f p = gp✳ ◆❤ê ➤✐Ò✉ ❦✐Ö♥ ✭✷✳✶✮ t❛ ❝ã F (G(f xn−1 , gp, gp), G(gxn−1 , gp, gp), G(gxn−1 , f xn−1 , f xn−1 ), G(gp, f p, f p), G(gxn−1 , f p, f p), G(gp, f xn−1 , f xn−1 )) ≤ 0, F (G(gxn , f p, f p), G(gxn−1 , gp, gp), G(gxn−1 , gxn , gxn ), G(gp, f p, f p), G(gxn−1 , f p, f p), G(gp, gxn , gxn ) ≤ ✷✻ ❤❛② ❈❤♦ n → ∞ t❛ t❤✉ ➤➢ỵ❝ F (G(gp, f p, f p), 0, 0, G(gp, f p, f p), G(gp, f p, f p), 0) ≤ ◆❤ê ➤✐Ị✉ ❦✐Ư♥ ✭❋✶✮ t❛ s✉② r❛ r➺♥❣ ♥➭② t❛ ❝ã G(gp, f p, f p) = 0✳ gp = f p✳ ❉♦ ➤ã✱ w = f p = gp ❧➭ ❣✐➳ trÞ trï♥❣ ♥❤❛✉ ❝đ❛ f ❇ỉ ➤Ị ✷✳✷✳✹✱ w ❧➭ ❣✐➳ trị trù t ủ t tí ế tì ♥❤ê ❇ỉ ➤Ị ✷✳✶✳✸✱ ✈➭ ❚õ ➤➻♥❣ t❤ø❝ ❝✉è✐ ❝ï♥❣ f ✈➭ ✈➭ g ✳ ◆❤ê g ✳ ❍➡♥ ♥÷❛✱ ♥Õ✉ f ✈➭ g w ❧➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t ❝đ❛ f g✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✷✳✷✳✻ ✭❬✶✶❪✮ ❈❤♦ (X, d) ❧➭ ➳♥❤ ①➵ tõ X (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ f : (X, d) → ✈➭♦ ❝❤Ý♥❤ ♥ã✳ ❇➭✐ t♦➳♥ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ f ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➷t ❝❤Ø♥❤ ♥Õ✉ ✭✶✮ f ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ✭✷✮ ❱í✐ ♠ét ❞➲② ❜✃t ❦ú x0 ∈ X ✱ {xn } ⊂ X ♠➭ lim d(xn , f xn ) = t❛ ❝ã lim d(xn , x0 ) = n→∞ n→∞ 0✳ ✷✳✷✳✼ ➜Þ♥❤ ♥❣❤Ü❛✳ ♥Õ✉ ✈í✐ ✭❬✶✶❪✮ ❍➭♠ u, v, w ≥ ✈➭ F : R6+ → R ➤➢ỵ❝ ❣ä✐ ❧➭ F (u, v, 0, w, u, v) ≤ 0, tå♥ t➵✐ ❝ã tÝ♥❤ ❝❤✃t p ∈ (0, 1) (F p) s❛♦ ❝❤♦ u ≤ p max {v, w} ❙❛✉ ➤➞② ❧➭ ❝➳❝ ✈Ý ❞ơ ✈Ị ❤➭♠ ❝ã tÝ♥❤ ❝❤✃t ✷✳✷✳✽ ❱Ý ❞ô✳ ✭❬✶✶❪✮ ❳Ðt ❤➭♠ (F p)✳ F : R6+ → R ❝❤♦ ❜ë✐ F (t1 , , t6 ) = t1 − at2 − bt3 − ct4 − dt5 − et6 , tr♦♥❣ ➤ã a, b, c, d, e ≥ ✈➭ < a + b + c + 2d + e < 1✳ u, v, w ≥ ✈➭ F (u, v, 0, w, u, v) = u−av−cw−du−ev ≤ 0✱ a+c+e < t❛ ❝ã ♥❣❛② u ≤ p max {u, v}✳ t❤× ❜➺♥❣ ❝➳❝❤ ❝❤ä♥ < p = 1−d ❑❤✐ ➤ã✱ ❣✐➯ sö ✷✼ ❱Ý ❞ô✳ ✷✳✷✳✾ ✭❬✶✶❪✮ ❳Ðt ❤➭♠ F : R6+ → R ❝❤♦ ❜ë✐ F (t1 , , t6 ) = t1 − k max{t2 , t3 , t4 , t5 , t6 }, tr♦♥❣ ➤ã k ∈ [0, 12 )✳ ❑❤✐ ➤ã✱ ❣✐➯ sö ◆Õ✉ t❛ ❝ã u, v, w ≥ ✈➭ F (u, v, 0, w, u, v) = u − k max {v, w} ≤ 0✳ u > max {v, w}✱ u ≤ max {v, w}✱ t❤× u(1 − k) ≤ 0✳ ❦❤✐ ➤ã ❜➺♥❣ ❝➳❝❤ ❝❤ä♥ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❱× t❤Õ✱ < p = k < t❛ ❝ã ♥❣❛② u ≤ p max {v, w} ✷✳✷✳✶✵ ❱Ý ❞ô✳ ✭❬✶✶❪✮ ❳Ðt ❤➭♠ F : R6+ → R ❝❤♦ ❜ë✐ F (t1 , , t6 ) = t1 − k max t2 , t3 , t4 , k ∈ [0, 1)✳ u+v k max v, w, tr♦♥❣ ➤ã ❑❤✐ ➤ã✱ ❣✐➯ sö u, v, w ≥ ✈➭ t5 + t6 , F (u, v, 0, w, u, v) = u − ✳ u+v ✳ ❚õ ➤ã s✉② r❛ u(1 − k) ≤ 0✳ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❱× t❤Õ✱ t❛ ❝ã u ≤ max {v, w}✱ ❦❤✐ ➤ã ❜➺♥❣ ❝➳❝❤ ❝❤ä♥ < p = k < ◆Õ✉ u > max {v, w}✱ t❤× u > t❛ ❝ã ♥❣❛② ✷✳✷✳✶✶ u ≤ p max {v, w} ❱Ý ❞ô✳ ✭❬✶✶❪✮ ❳Ðt ❤➭♠ F : R6+ → R ❝❤♦ ❜ë✐ F (t1 , , t6 ) = t21 − t2 (at2 + bt3 + ct4 ) − dt5 t6 , tr♦♥❣ ➤ã a, b, c, d ≥ ✈➭ ≤ a + b + c + d < 1✳ ❑❤✐ ➤ã✱ ❣✐➯ sö u, v, w ≥ ✈➭ F (u, v, 0, w, u, v) = u2 − u(av + cw) − duv ≤ 0✳ ◆Õ✉ u > 0✱ t❤× ❜➺♥❣ ❝➳❝❤ ❝❤ä♥ ≤ p = a+c+d < u ≤ p max {v, w}✳ ◆Õ✉ u = 0✱ t❤× u ≤ p max {v, w}✳ ✷✳✷✳✶✷ ❱Ý ❞ô✳ ✭❬✶✶❪✮ ❳Ðt ❤➭♠ F : R6+ → R ❝❤♦ ❜ë✐ F (t1 , , t6 ) = t1 − k.max t2 , ✷✽ t3 + t4 t5 + t6 , 2 , t❛ ❝ã ♥❣❛② k ∈ [0, 1) ❑❤✐ ➤ã✱ ❣✐➯ sö u, v, w ≥ ✈➭ F (u, v, 0, w, u, v) = u − w u+v w u+v k max v, , ✱ t❛ ❝ã u − k max v, , ≤ 0✳ 2 2 ◆Õ✉ u > max {v, w}✱ t❤× u(1 − k) ≤ 0✳ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❱× t❤Õ✱ tr♦♥❣ ➤ã t❛ ❝ã u ≤ max {v, w}✱ ❦❤✐ ➤ã ❜➺♥❣ ❝➳❝❤ ❝❤ä♥ < p = k < t❛ ❝ã ♥❣❛② u ≤ p max {v, w}✳ ✷✳✷✳✶✸ ❱Ý ❞ô✳ ✭❬✶✶❪✮ ❳Ðt ❤➭♠ F : R6+ → R ❝❤♦ ❜ë✐ F (t1 , , t6 ) = tr♦♥❣ ➤ã c ∈ [0, 1) ❑❤✐ ➤ã✱ t31 ❣✐➯ sö t23 t24 + t25 t26 , −c + t2 + t3 + t4 u, v, w ≥ ✈➭ F (u, v, 0, w, u, v) = u3 − u2 v ≤ 0✳ c 1+v+w ◆Õ✉ u > 0✱ t❤× ❜➺♥❣ ❝➳❝❤ ❝❤ä♥ ≤ p = c < t❛ ❝ã ♥❣❛② u ≤ cv v ≤ 1+v+w cv ≤ p max {v, w} ✳ ◆Õ✉ u = 0✱ t❤× u ≤ p max {v, w} ✷✳✷✳✶✹ ❱Ý ❞ô✳ ✭❬✶✶❪✮ ❳Ðt ❤➭♠ F : R6+ → R ❝❤♦ ❜ë✐ F (t1 , , t6 ) = t21 − at22 − c t5 t6 + t23 + t24 , a > ✈➭ a + c < ❑❤✐ ➤ã✱ ❣✐➯ sö u, v, w ≥ ✈➭ F (u, v, 0, w, u, v) = uv ≤ 0✱ t❛ ❝ã u2 − av − cuv ≤ 0✳ u2 − c + v2 u ◆Õ✉ v > 0✱ t❤× ❜➺♥❣ ❝➳❝❤ ❝❤ä♥ t = t❛ ❝ã ♥❣❛② f (t) = t − ct − a ✳ ❑❤✐ v ➤ã f (0) < ✈➭ f (1) > 0✱ tõ ➤ã tå♥ t➵✐ p ∈ (0, 1) s❛♦ ❝❤♦ f (t) ≤ ❦❤✐ t ≤ p✱ tr♦♥❣ ➤ã ❞♦ ➤ã ✷✳✷✳✶✺ u ≤ pv ≤ p max {v, w}✳ ◆Õ✉ v = 0✱ t❤× u = ✈➭ u ≤ p max {v, w}✳ ❱Ý ❞ô✳ ✭❬✶✶❪✮ ❳Ðt ❤➭♠ F : R6+ → R ❝❤♦ ❜ë✐ F (t1 , , t6 ) = t1 − at2 − c max {2t4 , t5 + t6 } , ✷✾ tr♦♥❣ ➤ã ≤ a + 2c < ❑❤✐ ➤ã✱ ❣✐➯ sö u, v, w ≥ ✈➭ F (u, v, 0, w, u, v) = u − av − c max {2w, u + v}✳ ◆Õ✉ t❤Õ✱ u > max {v, w}✱ u(1 − a − 2c) ≤ 0✳ t❤× ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❱× u ≤ max {v, w}✱ ❦❤✐ ➤ã ❜➺♥❣ ❝➳❝❤ ❝❤ä♥ < p = a + 2c < t❛ ❝ã ♥❣❛② u ≤ p max {v, w}✳ ✷✳✷✳✶✻ ❱Ý ❞ô✳ F : R6+ → R ❝❤♦ ❜ë✐ ✭❬✶✶❪✮ ❳Ðt ❤➭♠ √ √ F (t1 , , t6 ) = t1 − c max t2 , t3 , t4 t6 , t5 t6 , c ∈ [0, 1) ❑❤✐ ➤ã✱ √ √ c max {v, vw, uv} ≤ 0✳ tr♦♥❣ ➤ã ◆Õ✉ u > max {v, w}✱ u ≤ max {v, w}✱ ❣✐➯ sư t❤× u, v, w ≥ u(1 − c) ≤ 0✳ ✈➭ F (u, v, 0, w, u, v) = u − ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❱× t❤Õ < p = c < ❦❤✐ ➤ã ❜➺♥❣ ❝➳❝❤ ❝❤ä♥ t❛ ❝ã ♥❣❛② u ≤ p max {v, w}✳ ✷✳✷✳✶✼ ❱Ý ❞ô✳ ✭❬✶✶❪✮ ❳Ðt ❤➭♠ F : R6+ → R ❝❤♦ ❜ë✐ F (t1 , , t6 ) = t1 − k.max t2 , t3 , t4 , 2t4 + t6 2t4 + t5 t5 + t6 , , 3 k ∈ [0, 1)✳ ❑❤✐ ➤ã✱ ❣✐➯ sö u, v, w ≥ 2w + v 2w u + v k.max v, w, , , ≤ 0✳ 3 ◆Õ✉ u > max {v, w}✱ t❤× u(1 − k) ≤ 0✳ tr♦♥❣ ➤ã u ≤ max {v, w}✱ ❦❤✐ ➤ã ❜➺♥❣ ❝➳❝❤ ❝❤ä♥ ✈➭ , F (u, v, 0, w, u, v) = u − ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❱× t❤Õ < p = k < t❛ ❝ã ♥❣❛② u ≤ p max {v, w}✳ ✷✳✷✳✶✽ ➜Þ♥❤ ♥❣❤Ü❛✳ (X, G) → (X, G) ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ✭✶✮ f ✈➭ f ✭❬✶✶❪✮ ❈❤♦ (X, G) ❧➭ ❝➳❝ ➳♥❤ ①➵ tõ ✈➭ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ (X, G) ✈➭ f, g : ✈➭♦ ❝❤Ý♥❤ ♥ã✳ ❇➭✐ t♦➳♥ ➤✐Ĩ♠ ❜✃t g ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➷t ❝❤Ø♥❤ ♥Õ✉ g ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t✱ ✸✵ G✲♠➟tr✐❝ ✭✷✮ ❱í✐ ♠ét ❞➲② ❜✃t ❦ú {xn } ⊂ X ✈í✐ lim G(xn , f xn , f xn ) = 0, n→∞ ✈➭ lim G(xn , gxn , gxn ) = 0, n→∞ t❤× lim G(x, xn , xn ) = n→∞ ✷✳✷✳✶✾ ➜Þ♥❤ ❧ý✳ ❝➳❝ ➳♥❤ ①➵ ✈➭ ❤➭♠ F ✭❬✶✶❪✮ ●✐➯ sö (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤è✐ ①ø♥❣✳ ❱í✐ f, g : (X, G) → (X, G) t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✺ ❝ã tÝ♥❤ ❝❤✃t ✭❋♣✮✳ ❑❤✐ ➤ã✱ ❜➭✐ t♦➳♥ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ f ✈➭ g ợ t ỉ ứ ị ý t s✉② r❛ ❝❤✉♥❣ ❞✉② ♥❤✃t x ∈ X✳ ●✐➯ sö f ✈➭ g ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ {xn } ❧➭ ♠ét ❞➲② ❜✃t ❦ú tr♦♥❣ (X, G) s❛♦ ❝❤♦ lim G(xn , f xn , f xn ) = ✈➭ lim G(xn , gxn , gxn ) = 0✳ n→∞ n→∞ ◆❤ê ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✮ t❛ ❝ã F (G(f x, f xn , f xn ), G(gx, gxn , gxn ), G(gx, f x, f x), G(gxn , f xn , f xn ), G(gx, f xn , f xn ), G(gxn , f x, f x)) ≤ 0, ❤❛② F (G(x, f xn , f xn ), G(x, gxn , gxn ), 0, G(gxn , f xn , f xn ), G(x, f xn , f xn ), G(gxn , x, x)) ≤ ❱× G ❧➭ ♠ét G✲♠➟tr✐❝ ➤è✐ ①ø♥❣✱ ♥➟♥ t❛ ❝ã G(gxn , x, x) = G(x, gxn , gxn ), ✈➭ F (G(x, f xn , f xn ), G(x, gxn , gxn ), 0, G(gxn , f xn , f xn ), G(x, f xn , f xn ), G(x, gxn , gxn )) ≤ ✸✶ ◆❤ê tÝ♥❤ ❝❤✃t (F p) t❛ ❝ã G(x, f xn , f xn ) ≤ p max {G(x, gxn , gxn ), G(gxn , f xn , f xn )} ≤ p(G(x, gxn , gxn ) + G(gxn , f xn , f xn )) ❑❤✐ ➤ã✱ ♥❤ê ➤✐Ị✉ ❦✐Ư♥ (G5) ✈➭ ❣✐➯ t❤✐Õt (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤è✐ ①ø♥❣✱ t❛ ❝ã G(x, xn , xn ) ≤ G(x, f xn , f xn ) + G(f xn , xn , xn ) ≤ p(G(x, gxn , gxn ) + G(gxn , f xn , f xn )) + G(f xn , xn , xn ) ≤ p(G(x, xn , xn ) + G(xn , gxn , gxn )) + G(gxn , xn , xn )+ +G(xn , f xn , f xn )) + G(f xn , xn , xn ) = p(G(x, xn , xn ) + 2G(xn , gxn , gxn )+ +G(xn , f xn , f xn )) + G(f xn , xn , xn ) ❉♦ ➤ã G(x, xn , xn ) ≤ ❈❤♦ 2p p+1 G(xn , f xn , f xn ) + G(xn , gxn , gxn ) 1−p 1−p n → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ ❝ã ➤➢ỵ❝ lim G(x, xn , xn ) = n→∞ ❱× t❤Õ✱ ❜➭✐ t♦➳♥ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ✸✷ f ✈➭ g ❧➭ ❜➭✐ t♦➳♥ ➤➷t ❝❤Ø♥❤✳ ❑Õt ❧✉❐♥ ❙❛✉ t❤ê✐ ❣✐❛♥ ♥❣❤✐➟♥ ❝ø✉ ✈➭ t❤❛♠ ❦❤➯♦ ♥❤✐Ị✉ t➭✐ ❧✐Ư✉ ❦❤➳❝ ♥❤❛✉ ✈Ị ➤Ị t➭✐ ✧➜✐Ĩ♠ ❜✃t ➤é♥❣ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✧✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤✱ ❝❤✉ ➤➳♦ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤✉ ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ s❛✉✿ ✶✳ ❍Ư t❤è♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈Ị✿ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ G✲❤×♥❤ ❝➬✉✱ t➠♣➠ G✲♠➟tr✐❝✱ ❞➲② G✲❤é✐ tô✱ ➳♥❤ ①➵ G✲❧✐➟♥ tô❝✱ ❞➲② G✲❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ G✲➤è✐ ①ø♥❣✱ ❝➳❝ ➳♥❤ ①➵ ❣✐❛♦ ❤♦➳♥ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ (X, d) ✈➭ ❝➳❝ ➳♥❤ ①➵ ❣✐❛♦ ❤♦➳♥ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉✱ ➤✐Ó♠ trï♥❣ ♥❤❛✉✱ ❜➭✐ t♦➳♥ ➤✐Ó♠ ❜✃t ➤é♥❣ ✈➭ ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ❦Õt q✉➯ ➤ã✳ ✷✳ rì ó ệ tố ột số ị ý ề ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ✱ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✱ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ ✈➭ ❣✐í✐ t❤✐Ư✉ ♠ét sè ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✳ ✸✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ➤Þ♥❤ ❧ý✱ ❝➳❝ ❤Ư q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣✱ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉✱ ✈Ị ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ❝❤➻♥❣ ❤➵♥ ♥❤➢ ➜Þ♥❤ ❧ý ✶✳✷✳✶✱ ➜Þ♥❤ ❧ý ✶✳✷✳✺✱ ❍Ö q✉➯ ✶✳✷✳✻✱ ❍Ö q✉➯ ✶✳✷✳✼✱ ❍Ö q✉➯ ✶✳✷✳✽✱ ➜Þ♥❤ ❧ý ✷✳✶✳✶✹✱ ➜Þ♥❤ ❧ý ✷✳✶✳✶✻✱ ❇ỉ ➤Ị ✷✳✷✳✹✱ ➜Þ♥❤ ý ị ý rì tết ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛✿ ❱Ý ❞ô ✶✳✷✳✹ ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ❧ý ✶✳✷✳✶✱ ❝➳❝ ❱Ý ❞ơ ✷✳✶✳✶✶✱ ✷✳✶✳✶✷ ✈➭ ✷✳✶✳✶✸ ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✶✳✶✵✱ ✸✸ ❱Ý ❞ơ ✷✳✶✳✶✺ ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ❧ý ✷✳✶✳✶✹ ✈➭ ❝➳❝ ❱Ý ❞ơ ✷✳✷✳✷✱ ✷✳✷✳✸ ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✷✳✶✱ ❝➳❝ ❱Ý ❞ô ✷✳✷✳✽✱ ✷✳✷✳✾✱ ✷✳✷✳✶✵✱ ✷✳✷✳✶✶✱ ✷✳✷✳✶✷✱ ✷✳✷✳✶✸✱ ✷✳✷✳✶✹✱ ✷✳✷✳✶✺✱ ✷✳✷✳✶✻✱ ✷✳✷✳✶✼ ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✷✳✼✳ ✸✹ t ệ t ỗ ➤➵✐ ❝➢➡♥❣✱ ◆❤➭ ①✉✃t ❜➯♥ ❑❤♦❛ ❤ä❝ ✈➭ ❑ü t❤✉❐t✳ ❬✷❪ ▼✳ ❆❜❜❛s✱ ❇✳ ❊✳ ❘❤♦❛❞❡s ✭✷✵✵✾✮✱ ❈♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥t r❡s✉❧ts ❢♦r ♥♦♥❝♦♠✲ ♠✉t✐♥❣ ♠❛♣♣✐♥❣s ✇✐t❤♦✉t ❝♦♥t✐♥✉✐t② ✐♥ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ❆♣♣❧✳ ▼❛t❤✳ ❈♦♠♣✉t✳✱ ✷✶✺✱ ❬✸❪ ❉✳ ❙✳ ❏❛❣❣✐ ✭✷✵✶✶✮✱ ▼❛t❤✳✱ ✽✱ ✷✻✷✲✷✻✾✳ ❯♥✐q✉❡ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s✱ ■♥❞✐❛♥ ❏✳ P✉r❡ ❆♣♣❧✳ ✷✷✸✲✷✸✵✳ ❋✐①❡❞ ♣♦✐♥t ❢♦r s❡t ✈❛❧✉❡❞ ❢✉❝t✐♦♥s ✇✐t❤✲ ❬✹❪ ●✳ ❏✉♥❣❝❦✱ ❇✳ ❊✳ ❘❤♦❛❞❡s ✭✶✾✾✽✮✱ ♦✉t ❝♦♥t✐♥✉✐t②✱ ■♥❞✐❛♥ ❏✳ P✉r✳ ❆♣♣❧✳ ▼❛t❤✳✱ ❬✺❪ ❙✳ ❑✉♠❛r✱ ❙✳ ❑✳ ●❛r❣ ✭✷✵✵✾✮✱ ✷✷✼✲✷✸✽✳ ❊①♣❛♥s✐♦♥ ♠❛♣♣✐♥❣ t❤❡♦r❡♠s ✐♥ ♠❡tr✐❝ s♣❛❝❡s✱ ■♥t✳ ❏✳ ❈♦♥t❡♠♣✳ ▼❛t❤✳ ❙❝✐✳✱ ✹ ✭✸✻✮✱ ✶✼✹✾✲✶✼✺✽✳ ❬✻❪ ❙✳ ▼❛♥r♦✱ ❙✳ ❙✳ ❇❤❛t✐❛✱ ❙✳ ❑✉♠❛r ✭✷✵✶✵✮✱ ✐♥ ✷✾✱ ❊①♣❛♥s✐♦♥ ♠❛♣♣✐♥❣ t❤❡♦r❡♠s G✲♠❡tr✐❝ s♣❛❝❡s✱ ■♥t✳ ❏✳ ❈♦♥t❡♠♣✳ ▼❛t❤✳ ❙❝✐✳✱ ✺ ✭✺✶✮✱ ✷✺✷✾✲✷✺✸✺✳ ❬✼❪ ❩✳ ▼✉st❛❢❛ ❛♥❞ ❇✳❙✐♠s ✭✷✵✵✹✮✱ ❙♦♠❡ r❡♠❛r❦s ❝♦♥❝❡r♥✐♥❣ ❉✲ ♠❡tr✐❝ s♣❛❝❡s✱ ✐♥ Pr♦❝❡❡❡❞✐♥❣s ♦❢ t❤❡ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❋✐①❡❞ ♣♦✐♥t ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐t✐♦♥s✱ ❱❛❧❡♥❝✐❛✱ ❙♣❛✐♥✱ ❏✉❧② ✷✵✵✹✱ ✶✽✾✲✶✾✽✳ ❬✽❪ ❩✳ ▼✉st❛❢❛ ❛♥❞ ❇✳ ❙✐♠s ✭✷✵✵✻✮✱ ❆ ♥❡✇ ❛♣♣r♦❛❝❤ t♦ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ❏♦✉r♥❛❧ ♦❢ ◆♦♥❧✐♥❡❛r ❛♥❞ ❈♦♥✈❡① ❆♥❛❧②s✐s✱ ❬✾❪ ❩✳ ▼✉st❛❢❛ ❛♥❞ ❍✳ ❖❜✐❡❞❛t✱ ❋✳ ❆✇❛✇❞❡❤ ✭✷✵✵✽✮✱ r❡♠s ❢♦r ♠❛♣♣✐♥❣s ♦♥ ❝♦♠♣❧❡t❡ ✼ ✭✷✮✱ ✷✽✾✲✷✾✼✳ ❙♦♠❡ ❢✐①❡❞ ♣♦✐♥t t❤❡♦✲ G✲♠❡tr✐❝ s♣❛❝❡s✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❱♦❧✳ ✷✵✵✽✱ ❆rt✐❝❧❡ ■❉✹✵✶✻✽✹✱ ✶✷ ♣❛❣❡s✳ ❬✶✵❪ ❩✳ ▼✉st❛❢❛ ❛♥❞ ❲✳ ❙❤❛t❛♥❛✇✐✱ ▼✳ ❇❛t❛✐♥❡❤ ✭✷✵✵✾✮✱ ♣♦✐♥t r❡s✉❧ts ✐♥ G✲♠❡tr✐❝ ❊①✐st❡♥❝❡ ♦❢ ❢✐①❡❞ s♣❛❝❡s✱ ■♥t❡r✳ ❏✳ ▼❛t❤✳ ▼❛t❤✳ ❙❝✐✳✱ ❱♦❧✳ ✷✵✵✾✱ ❆rt✐❝❧❡ ■❉✹✷✸✽✵✷✽✱ ✶✵ ♣❛❣❡s✳ ✸✺ ❬✶✶❪ ❱✳ P♦♣❛✱ ❆✳ ▼✳ P❛tr✐❝✐✉ ✭✷✵✶✷✮✱ ❆ ❣❡♥❡r❛❧ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ❢♦r ♣❛✐r ♦❢ ✇❡❛❦❧② ❝♦♠♣❛t✐❜❧❡ ♠❛♣♣✐♥❣s ✐♥ G✲♠❡tr✐❝ s♣❛❝❡s✱ ❏✳ ◆♦♥❧✐♥❡❛r ❙❝✐✳ ❆♣♣❧✳✱ ✺✱ ✶✺✶✲✶✻✵✳ ✸✻ ... gp, gp), G( gq, f q, f q), G( gp, f p, f p), G( gq, f p, f p), G( gp, f q, f q)) ≤ 0, F (G( gq, gp, gp), G( gq, gp, gp), 0, 0, G( gq, gp, gp), G( gq, gp, gp) ≤ ◆❤ê ✭❋✸✮✱ tõ ➤✐Ò✉ ♥➭② t❛ s✉② r❛ G( gq, gp,... ), G( gxn , gxn+1 , gxn+1 ), G( gxn−1 , gxn+1 , gxn+1 ), 0) ≤ ◆❤ê ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭❋✶✮ ✈➭ ✭●✺✮ t❛ ❝ã ➤➢ỵ❝ F (G( gxn , gxn+1 , gxn+1 ), G( gxn−1 , gxn , gxn ), G( gxn−1 , gxn , gxn ), G( gxn , gxn+1 , gxn+1... gxn+1 ) ≤ hn1 G( gx0 , gx1 , gx1 ) ❑❤✐ ➤ã✱ ✈í✐ ✭✷✳✸✮ m > n t❛ ❝ã G( gxn , gxm , gxm ) ≤ G( gxn , gxn+1 , gxn+1 ) + G( gxn+1 , gxn+2 , gxn+2 )+ + · · · + G( gxm−1 , gxm , gxm ) + + hm−1 )G( gx0 , gx1